Cohomological descent for logarithmic differential forms in the log etale topology
Dissertation zur Erlangung des Doktorgrades
vorgelegt von
Vetere Elmiro
an der Fakult¨atf¨urMathematik und Physik der Albert-Ludwigs-Universit¨atFreiburg
Oktober 2018 Dekan: Prof. Dr. Gregor Herten Referenten: Prof. Dr. Annette Huber-Klawitter Prof. Dr. Stefan Kebekus Prof. Dr. Wieslawa Nizio l
Datum der Promotion: 28.11.2018 Abstract (in German)
In dieser Dissertation beweisen wir kohomologischen Abstieg zwischen dem (kleinen) ´etalen Situs und dem logarithmischen Situs f¨urdie Garbe der logarithmischen Differentialformen. Die existierenden Methoden in der Literatur erlauben uns, kohomologischen Abstieg f¨ureine gr¨oßereKlasse von Garben zu beweisen. Ins- besondere vergleichen wir die Kummer log ´etaleKohomologie mit der ´etalenKo- homologie f¨urquasi-koh¨arente Garben auf dem Kummer log ´etalenSitus. Damit beweisen wir, dass Morphismen von log Schemata, f¨urdie der zugrun- deliegende Morphismus von Schemata affin ist, triviale h¨oheredirekte Bilder f¨ur ket lokal klassische quasi-koh¨arente Garben auf dem Kummer log ´etalenSitus haben. In derselben Weise vergleichen wir die log ´etaleKohomologie mit der ´etalen Kohomologie f¨urklassische Vektorb¨undel auf dem log ´etalenSitus. Wir geben außerdem einen Beweis f¨urdie als bekannt angesehenen Tatsache, dass log regul¨arelog Schemata rational singul¨ar sind. Der Fall von Zariski log Schemata wurde bereits von K. Kato in [Kat94] behandelt. Wir geben einen Beweis f¨ur den Fall von ´etalenlog Schemata. Als letzte Anwendung definieren wir algebraische log de Rham Kohomologie f¨urfs log Schemata, mithilfe des log etalen Situs. Das liefert einen neuen Beweis f¨ur den Satz von A. Ogus, [Ogu18, Theorem V.4.2.5(1)], ¨uber die Verallgemeinerung algebraischer de Rham Kohomologie f¨urSchemata mit torischen Singularit¨aten.
Abstract (in English)
In this thesis we prove cohomological descent between the (small) etale site and the logarithmic sites for the sheaf of logarithmic differential forms. The existing techniques in the literature allow us to prove cohomological descent for a broader class of sheaves. Specifically, we compare the Kummer log etale cohomology with the etale coho- mology for quasi-coherent sheaves defined on the Kummer log etale site. By using this, we prove that morphisms of log schemes where the underlying morphism of schemes is affine have trivial higher direct images for ket locally classical quasi- coherent sheaves on the Kummer log etale site. In the same spirit, we compare the log etale cohomology with the etale cohomology for classical vector bundles on the log etale site. We also give a proof for the fact that log regular log schemes are rationally singular, something which is considered known. The case of Zariski log schemes has been already treated by K. Kato in [Kat94]. We provide a proof for the case of etale log schemes. As a final application we define algebraic log de Rham cohomology for fs log schemes using the log etale site. Along these lines, we prove with a different method the theorem of A. Ogus, [Ogu18, Theorem V.4.2.5(1)], regarding the gen- eralization of algebraic de Rham cohomology to schemes with toric singularities.
QwrÐc mtaiec antarsÐec na deic kai na deqteÐc ta sÔnora tou anjr¸pinou nou, kai mèsa st' austhr toÔta sÔnora adiamartÔrhta, akatpauta na douleÔeic. Na poio eÐnai to pr¸to sou qrèoc. Me antreÐa, me sklhrìthta sterèwse pnw sto saleuìmeno qoc to katastrìg- gulo, to kataf¸tisto al¸ni tou nou, n' alwnÐseic, na liqnÐseic, sa noikokÔrhc, ta sÔmpanta. Kajar na xeqwrÐseic ki hrwik na deqteÐc tic pikrèc gìnimec toÔtec, anjr¸pinec, srka apì th srka mac, al jeiec:
a) O nouc tou anjr¸pou fainìmena monqa mporeÐ na sullbei, potè thn ousÐa.
b) Ki ìqi ìla ta fainìmena, par monqa ta fainìmena thc Ôlhc.
g) Ki akìma sten¸tera: ìqi kan ta fainìmena toÔta thc Ôlhc, par monqa touc metaxÔ touc suneirmoÔc.
d) Ki oi suneirmoÐ toÔtoi den eÐnai pragmatikoÐ, anexrthtoi apì ton njrwpo. EÐnai ki autoÐ genn mata tou anjr¸pou.
e) Kai den eÐnai oi mìnoi dunatoÐ anjr¸pinoi, par monqa oi pio bolikoÐ gia tic praqtikèc kai nohtikèc tou angkec.
Mèsa sta sÔnora toÔta, o nouc eÐnai o nìmimoc apìlutoc monrqhc. Kami llh exousÐa sto basÐleio tou den uprqei. AnagnwrÐzw ta sÔnora toÔta, ta dèqoumai m' egkartèrhsh, gennaiìthta ki agph, ki agwnÐzoumai mèsa sthn perioq touc neta sa na 'moun eleÔteroc. Upotzw thn Ôlh, thn anagkzw na gÐnei kalìc agwgìc tou mualoÔ mou. Qa- Ðroumai ta fut, ta z¸a, touc anjr¸pouc, touc jeoÔc san paidi mou. 'Olo to SÔmpanto to ni¸jw na sofilizei apnw mou kai na me akoloujei sa s¸ma. Se xafnec foberèc stigmèc astrftei mèsa mou: - 'Ola toÔta eÐnai paiqnÐdi sklhrì kai mtaio, dÐqwc arq , dÐqwc tèloc, dÐqwc nìhma . Ma xanazeÔoumai, pli, gorg ston troqì thc angkhc ki ìlo to SÔmpanto xanarqinei gÔra trogÔra mou thn peristrof tou. PeijarqÐa, na h an¸tath aret . 'Etsi monqa sozugizetai h dÔnamh me thn epijumÐa kai karpÐzei h prospjeia tou anjr¸pou. Na pwc me saf neia kai me sklhrìthta na kajorÐzeic thn pantodunamÐa tou nou mèsa sta fainìmena kai thn anikanìthta tou nou pèra apì ta fainìmena, prin na kin seic gia th lÔtrwsh. Alli¸c den mporeÐc na lutrwjeÐc.
Askhtik , NÐkoc Kazantzkhc
To see and accept the boundaries of the human mind without vain rebellion, and in these severe limitations to work ceaselessly without protest - this is where man’s first duty lies. Build over the unsteady abyss, with manliness and austerity, the fully round and luminous arena of the mind where you may thresh and winnow the universe like a lord of the land. Distinguish clearly these bitter yet fertile human truths, flesh of our flesh, and admit them heroically: a) The mind of man can perceive appearances only, and never the essence of things.
b) And not all appearances but only the appearances of matter.
c) And more narrowly still: not even these appearances of matter, but only relationships between them.
d) And these relationships are not real and independent of man, for even these are his creations.
e) And they are not the only ones humanly possible, but simply the most convenient for his practical and perceptive needs. Within these limitations the mind is the legal and absolute monarch. No other power reigns within its kingdom. I recognize these limitations, I accept them with resignation, bravery, and love, and I struggle at ease in their closure, as though I were free. I subdue matter and force it to become my mind’s good medium. I rejoice in plants, in animals, in man and in gods, as though they were my children. I feel all the universe nestling about me and following me as though it were my own body. In sudden dreadful moments a thought flashes through me: -”This is all a cruel and futile game, without beginning, without end, without meaning”. But again I yoke myself swiftly to the wheels of necessity, and all the universe begins to revolve around me once more. Discipline is the highest of all virtues. Only so may strength and desire be counterbalanced and the endeavors of man bear fruit. This is how, with clarity and austerity, you may determine the omnipotence of the mind amid appearances and the incapacity of the mind beyond appearances - before you set out for salvation. You may not otherwise be saved.
The Saviors of God (Salvatores dei), Nikos Kazantzakis
Contents
Introduction
Acknowledgments
1 Monoids and Log schemes1 1.1 Monoids...... 1 1.2 Kummer and exact homomorphisms of monoids...... 6 1.3 Pushouts in the category of fs monoids...... 8 1.4 Monoid algebras and Toric varieties...... 10 1.5 Log structures and log schemes ...... 16 1.6 Fs log schemes and charts ...... 20 1.7 Strict, Kummer morphisms of log schemes...... 24 1.8 Fiber products in the category of coherent log schemes ...... 25
2 Differential forms on Log schemes 28 2.1 Logarithmic differential forms...... 28 2.2 Log deformation theory...... 34
3 Log regularity and Log blow-ups 38 3.1 Log regularity...... 38 3.2 Divisorial log structures ...... 43 3.3 Log blow-ups ...... 45
4 Log sites 51 4.1 The Kummer log etale topology...... 52 4.2 The log etale topology ...... 57
5 Sheaves on Log sites 60 5.1 Intoduction ...... 60 5.2 ket descent for the extension presheaf...... 62 5.3 let descent for the extension presheaf...... 69 5.4 Classical and locally classical sheaves...... 73
6 Cohomological descent for Log sites 77 6.1 Comparison of ket cohomology and etale cohomology ...... 77 6.2 Comparison of let cohomology and etale cohomology ...... 80
7 Applications and concluding remarks 84 7.1 Applications: Algebraic log de Rham cohomology for fs log schemes 84 7.2 Concluding remarks...... 90
8 Appendix 92 Notation (X, triv) The log scheme with underlying space X and the trivial log structure. Xtriv The trivial locus of the log structure of X. X The underlying scheme of a log scheme X. f The underlying morphism of schemes of a log morphism f. X´et The etale site for the underlying scheme of a log scheme (X,M). Xket The Kummer log etale site for a log scheme (X,M). Xlet The log etale site for a log scheme (X,M). Fτ The extension presheaf of a module F for the topology τ.
OXτ The sheaf of rings on the site X equiped with the τ topology. Sh(Xτ ) The category of sheaves for the topology τ over the scheme/log scheme (X,M). Mod The category of modules over a scheme/log scheme. Qcoh The category of quasicoherent modules over a scheme/log scheme. Vect The category of locally free sheaves of finite rank over a scheme/log scheme. fs × The product in the category of fs log schemes. ηX The inclusion of sites ηX : XZar → X´et. X The inclusion of sites X : X´et → Xket. κX The inclusion of sites κX : Xket → Xlet. ρX The inclusion of sites κX : X´et → Xlet. f∗ The pushforward on the level of sheaves for a functor of sites f : C → D. f −1 The pullback on the level of abelian groups for a functor of sites f : C → D. f ∗ The pullback on the level of modules for a functor of ringed sites f : C → D. log f∗ The pushforward of log structures for a continuous map f : X → Y. ∗ flog The pullback of log structures for a continuous map f : X → Y. P The constant sheaf of the monoid P on X. M∗ The sheaf of invertible sections of M . Introduction
Logarithmic geometry is a part of algebraic geometry and serves as a natural habitat for the notion of differential forms with logarithmic poles. The theory was started by J. Fontaine and L. Illusie and has been continued by K. Kato, C. Nakayama and others, having its main applications in arithmetic geometry. Nowadays logarithmic geometry is expanding and is used in various domains of mathematics, such as in the theory of Berkovich spaces, Mirror symmetry, etc. Its foundation lies in what people nowadays call geometry over F1, which is an algebro-geometrical theory emerging from monoids. In this fashion, logarithmic schemes can be seen as a generalization of toric varieties, and they provide a framework to define base-free analogues of the theory of toroidal embeddings. A great feature of the theory is that it provides the notion of log smoothness, which detects hidden smoothness on singular schemes. Our connection to logarithmic geometry started as follows: In [HJ14], A. Hu- ber and C. J¨orderdefined an algebraic de Rham cohomology using the h-topology of A. Voevodsky. In particular, they defined de Rham cohomology by using hy- percoverings and the h-sheafification of the de Rham complex. It is then natural to ask what are the possible generalizations for log de Rham cohomology in the h-topology. This led us naturally to the theory of logarithmic schemes. In partic- ular the log etale topology can be seen as a resolutions of singularities topology, in the sense of [GK15], analogous to the h-topology. Having this in mind, the log etale topology can be used, at least morally, as an analogue for the h-topology but the picture is still not clear on how to mathematically formulate this. The main aim of this thesis is to prove cohomological descent for the sheaf of logarithmic differential forms in the logarithmic sites namely, the Kummer log etale site Xket and the log etale site Xlet. This means: Theorem 0.1. Let X be an fs log scheme. 1. The presheaf: Ωlog : U 7→ Ωlog (U) U´et
is a quasi-coherent sheaf on Xket and we have an isomorphism: Hi(X , Ωlog) ∼= Hi(X , Ωlog ). ket ´et X´et
log 2. In the case where X is log regular, Ω is a vector bundle on Xlet and we have an isomorphism:
Hi(X , Ωlog) ∼= Hi(X , Ωlog ). let ´et X´et
The methods that we use allows us to prove cohomological descent for a larger class of sheafs. For the Kummer log etale site, we compare the cohomology groups i i H (Xket,F ) and H (X´et, ∗F ) for any quasi-coherent sheaf F on Xket in Propo- sition 6.4. The case for the log etale site is more restrictive. We compare in i ∗ i Theorem 6.14, the cohomology groups H (Xlet, ρ E) and H (X´et,E) where X is a log regular log scheme and E is a vector bundle coming from the (small) etale site. To do this we prove cohomological descent between the logarithmic sites in Theorem 6.13. By using Theorem 0.1 and Hodge theoretic techniques, were able to prove the following:
Theorem 0.2. Let X be a log regular, fs log scheme over (C, triv) with proper underlying scheme, and let Xtriv be the trivial locus of the log structure. Then, we have an isomorphism:
Hi(X , Ωlog,•) ∼= Hi (X , ) ´et X´et sing triv C This generalizes algebraic log de Rham cohomology to the case of proper toroidal embeddings with self-intersections. It is also the proper case of [Ogu18, Theorem V.4.2.5(1)]. Since the existing literature is not always written for etale log schemes, we tried to fill in the gaps by making our exposition for etale log schemes. For this, we also provide a small survey-like section from the existing literature for connections between Zariski log structures and etale log structures, and how divisorial log structures and toroidal embeddings fit into this picture. We also give details for the arguments, in the case where the statements appear in the literature but the details of the statements are not given. In all of our exposition we use fs log schemes except from small parts where for reasons of completeness we use fine or coherent log schemes. Specifically in Section1 we give a basic introduction to the theory of monoids and log schemes that we will use in the following sections. It is based mainly on [Ogu18, Sections I and III]. Section2 is dedicated to logarithmic deformation theory. Since we are inter- ested in coherent log schemes, we use the exposition of [Kat96] in combination with Section IV of [Ogu18]. Also, we use some elements from [Niz08], since we will need them in later sections for Kummer log etale descent. Section3 has a more geometric nature. It starts with the theory of log regu- lar log schemes and the connections with toroidal embeddings and divisorial log structures. In Remark 3.5, we slightly elaborate on [Ols03] and [Ogu18] about the differences that occur when we define log structures in Zariski topology and etale topology. We also write the details for the equivalence of toroidal embeddings and log regular schemes in Proposition 3.8.Section 3.3 continues with log blow-ups and it is based on [Nak17] and [Niz06]. Also, we prove in Theorem 3.28 that toroidal embeddings have rational singularities, something which is considered known. For this we exploit the fact that log regular log schemes have rational singularities, as well as the fact that log etale morphisms behave with respect to log differential forms similar to etale morphisms with respect to differential forms. This can be understood as an instance of Itaka philosophy [Mat02, Chapter 2]. In Section4, we give the basic definitions of the Kummer log etale site and the log etale site following C. Nakayama from [Nak97] and [Nak17]. We fill in the details of the descent arguments for the logarithmic topologies, for example the choice of good coverings, see Proposition 4.17 and Remark 4.27. In Section5 we prove the sheaf condition for specific modules on the logarith- mic sites. Namely, for the Kummer log etale topology we extend quasi-coherent sheaves from the (small) etale site in Theorem 5.11, and for log regular log schemes we extend locally free sheaves of finite rank from the (small) etale site to the log etale site in Theorem 5.21. In contrast with [Niz08] and [Hag03], the main differ- ence in our exposition is that we use the (small) etale site to transfer the modules of our interest instead of the (small) Zariski site. This does not bring up any issue since the categories of quasi-coherent sheaves on the (small) Zariski site and the (small) etale site are equivalent [Sta18, Sect. 03DX]. We also note the same holds for vector bundles from Hilbert’s 90. In Section6 we prove the comparison theorems. The comparison between the Kummer log etale cohomology and the etale cohomology is given in Subsection 6.1 in Proposition 6.4. The comparison of the full log etale cohomology and the Kummer log etale cohomology is given in Subsection 6.2, Theorem 6.13. Finally in Section7, we define algebraic log de Rham cohomology for the log etale site associated to an fs log scheme in Definition 7.13. In Theorem 7.8, we give a different proof for the proper case of [Ogu18, Theorem V.4.2.5(1)] which is a generalization of Deligne’s theorem [Del71, Paragraph 3.2.2] for log regular log schemes where the underlying scheme is proper. We note that this also serves as a generalization of the work of T.Oda in [Oda93] from the case of toric varieties to the case of toroidal embeddings with self intersections. Our proof uses Hodge theoretic methods and the fact that log regular log schemes have rational singular- ities. Also, we give a second proof based on the fact that the full log etale toposes of a log regular log scheme and its resolution of singularities are equivalent.
Acknowledgments
First and foremost I would like to thank my supervisors Annette Huber-Klawitter and Stefan Kebekus, for exposing me to the mathematical ideas of arithmetic and birational geometry. I am particularly grateful for their time and patience in this endeavour. Truly, their guidance and support has been indispensable. I would also like to thank, deeply from my heart, Wies lawa Nizio lfor her help with Proposition 6.4 and for her wonderful hospitality during my visit to ENS in Lyon back in the fall of 2016. The conversations that we had about logarithmic geometry shaped the form of this thesis the most. I also thank L. Illusie for providing a copy of [Kat91] and A. Ogus for granting me access to his manuscript [Ogu18]. In the past three years I have benefited greatly from the many mathemati- cal discussions I have had with others. So I would like to thank Brad Drew and Shane Kelly for the talks we had about algebraic geometry and cohomology; Giulia Battiston, Salvatore Floccari, Pietro Gatti, Lars Halvard Halle, Enrica Mazzon, Emanuel Scheidegger, Arne Smeets for our conversations for logarithmic geometry and toric varieties; Veronica Erlt and Helene Sigloh for the conversations and their hospitality during my visit in Regensbourg in the Spring of 2017; George Dalezios for the conversations we had about homological algebra; Panagis Karazeris - my mathematical godfather- and Pavlos Jeremias for their hospitality in the Summer meeting that we had in the university of Patras the past 3 years. I would also like to thank the staff at the university of Freiburg, all the members of GK and in particular Hannah Bergner, Oliver Brauling, Yohan Brunebarbe, my mentor Sebastian Goette, Fritz H¨ormann, Behrouz Taji, Matthias Wend; my cohomo- logical brothers and sisters Florian Beck, Valeria Bertini, Jens Eberhardt, Nelvis Fornasin, Yuhang Hou, Fabian Kartels, Jørgen Olsen Lye, Ben MacDonnell, Rene Recktenwald, Maximillian Schmidtke, Martin Schwald, Konrad Voelkel, Yi-Sheng Wang, Anja Wittmann, and Giovanni Zaccanelli. Outside the university, there are some people whom I owe a great debt of gratitude. My parents Vasiliki and Silvio who believe in my dream and supported me to come this far; my sister Elena who was always there to shed some light in the darkness; my friends in Friburg Chris F., Iliana I., Irina K. and Chris M. for providing me with homemade food and shelter; my friends from Corinth and Athens Elena I., Vasilis K., Runa S., Chris V., for keeping me sane, everyone with his/hers own special way; and my friends from Copenhagen Alvin Sipragaˇ and Andrien Vakil, whose existence reminds me that there is still hope in this world. Last but not least, I would like to thank my teacher and brother Panagioths Iliadhs for introducing me to mathematics. A big part of who I am, mathematically and non-mathematically, is due to the ideas that we explored the past 13 years. This thesis is dedicated to my parents and to him. This research was financially supported by the Graduiertenkolleg 1821 ”Coho- mological Methods in Geometry”.
Freiburg, October 2018 Elmiro Vetere
1 Monoids and Log schemes
The theory of logarithmic geometry is based in the spirit of toric geometry. The foundations of the theory rely heavily on the language of monoid geometry or more specific algebraic geometry over the field with one element F1. This kind of algebraic geometry considers as the fundamental building block not rings but monoids. For a monoid M the notion of ideal and prime ideals makes sense hence we can define the spectrum of a monoid as the set of all prime ideals of M, see Paragraph 1.9. This gives us a set that we can topologise with a Zariski topology and the resulting topological space is called a monoscheme [Ogu18]. Many notions of logarithmic geometry are based in this language. For example, logarithmic blow-ups were first approximated by Kato [Kat94] in the language of monoschemes (or monoidal spaces). In this chapter, we give the basic material of the theory of monoids and log schemes that we will use in the rest of the manuscript. The basic reference that we use in this chapter is [Ogu18, Sections I,II,III].
1.1 Monoids 1.1. (The category of commutative monoids, [Ogu18, Section I.1]). In what fol- lows, by monoid we mean a commutative monoid which is a triple (P, +, eP ) where P is a set equipped with a binary symmetric operation + and eP is the identity element. A homomorphism of monoids f : P → Q need to preserve the identity elements of the monoids i.e., f(eP ) = eQ. For reasons of simplicity we use 0 for denoting the identity element. There are two ways of writing down a monoid, we can use additive notation (M, +, 0), or multiplicative notation (M, ·, 1). We will note which kind of notation we are using every time. For example, we denote with (N, +, 0) the monoid of natural numbers including zero. Another basic example is the monoid F := ({0, ∞}, +, 0) or in multiplicative notation ({1, 0}, ·, 1). We denote the category of monoids by Mon which has as objects monoids and as morphisms homomorphisms of monoids. We have the following adjunction from [Ogu18, Section I.1.1]:
For Mon Sets Free with Free a For, where For is just the forgetful functor and Free is the functor that takes a set S and maps it to the free monoid generated by S. This is the S S monoid (N , +, 0), where N consist of all elements f ∈ HomSets(S, N) such that
1 f(s) 6= 0 only for a finite number of s ∈ S, equipped with point wise addition and identity element the function which is equal to zero. For a monoid (P, +, 0), and subsets A, B of P we define:
A + B := {a + b | a ∈ A, b ∈ B} ⊂ P
n−times z }| { We write x + B in the case that A = {x} and we denote by nB = B + ... + B where n is a positive natural number.
1.2. (Groupification and the subgroup of invertible elements) There is an adjunc- tion between the category on monoids and the category of abelian groups, [Ogu18, Section I.1.3]: ∗ Inc Ab Mon Gr with Gr a Inc a ∗. We explain these functors. Since every abelian group is a monoid, the functor Inc : Ab → Mon takes an abelian group G and maps it to the underlying monoid. The functor ∗ : Mon → Ab takes a monoid (P, +, 0) and maps it to the group of units of P [Ogu18, Section I.1.3] defined by:
P ∗ := {x ∈ P | there exists y ∈ P such that x + y = 0}
We note that P ∗ is an abelian group since P is commutative as a monoid and for every x ∈ P ∗ there exists a y ∈ P ∗ with x + y = 0. The functor Gr is the Grothendieck group associating to a monoid M its group completion M gp. This is defined as follows: for a monoid (M, +, 0) we consider the monoid M × M with the usual point-wise addition and define an equivalence relation:
(x, y) ∼ (z, w) ⇔ there exists k ∈ M such that x + w + k = z + y + k
Then we define M gp := M × M/ ∼ where the equivalence class [(x, y)] ∈ M gp represents the element ”x − y”. For example, in the examples of the previous r gp ∼ r gp paragraph (N ) = Z but F = {0}. This functor comes together with a gp morphism λM : M → M which sends an element x ∈ M to the class [(x, 0)]. We are interested in specific classes of monoids. We give the following defini- tion.
Definition 1.3. ([Ogu18, Section I.1.3]). A monoid (P, +, 0) is said to be:
1. integral if the cancellation law x + y = x + z ⇒ y = z holds for all x, y, z ∈ P , or equivalently, the natural map of groupification P → P gp is injective;
2 2. saturated if it is integral and for all x ∈ P gp such that there exists n ∈ N with n · x ∈ P , we have that x ∈ P ;
3. sharp if the subgroup of invertible elements P ∗ is trivial;
4. coherent if it is finitely generated over N i.e., there exist a surjective ho- momorphism of monoids NS → P for some finite set S; 5. fine if it is integral and coherent;
6. fs if it is fine and saturated;
7. toric if it is fs and without torsion. 1.4. We define the categories Monint, Monsat, Moncoh, Monfin, Monfs which are the full subcategories of Mon having as objects integral, saturated, coherent, fine, fs monoids respectively. We also have the following adjunctions:
Inc Inc
Monfs Monfin Moncoh
sat int Inc Inc Inc Inc Inc
Monsat Monint Mon
sat int where in all cases Inc is the inclusion functor and the functors sat, int are left adjoint to their inclusion functors [Ogu18, Section I.1.3 and Proposition I.1.3.5 (3)]. We give their definitions. 1. The functor int is defined by associating to each monoid M the monoid:
int gp M := im(λM : M → M )
which we call the integralization of M;
2. and sat, is the functor which associates to every monoid (M, +, 0) the monoid: sat gp M := {x ∈ M | n · x ∈ M for some n ∈ N} which we will call the saturation of M. Our main interest lies in the top row of the diagram. In particular, we are inter- ested in the category of fs monoids.
3 Remark 1.5. (Sharpification). Consider a monoid P where the subgroup of invertible elements P ∗ is not trivial. Then, we can always sharpify P by taking the orbits in P of the natural action of P ∗ on P defined by:
P ∗ × P → P (x, y) 7→ x + y
We denote the monoid of orbits with P := P/P ∗ and we call it the sharpification of P since the corresponding group of invertible elements (P )∗ := ((P/P ∗)∗) is trivial. For the theory of actions on monoids we refer the reader to [Ogu18, Section I.1.2]. We continue with the theory of integral monoids. Remark 1.6. If M be a fine monoid, then M gp is a finitely generated abelian gp ∼ r gp group hence M = Z ⊕G where G is the torsion subgroup of M . First note that since M is integral we can embed M in M gp hence, fine monoids are submonoids of finitely generated abelian groups. A second observation is that if M gp has torsion, then the torsion is already contained in M. In fact, every finite integral monoid is already an abelian group. This comes from Dirichlet’s box principle. Lemma 1.7. Let P be an integral monoid. We have the following: 1. The natural map P gp/P ∗ → (P )gp is an isomorphism. Moreover, if P is saturated, then P gp/P ∗ is a free abelian group.
2. Assume that P is toric, then the natural homomorphism π : P → P has a section i.e., there exists a homomorphism of monoids s : P → P such that
π ◦ s = idP . In particular, both statements hold when P is fs. Proof. The first assertion is [Ogu18, Proposition I.1.3.5]. For the second assertion, let P be an integral monoid such that P is toric. We argue that there exists a homomorphism of monoids s : P → P such that π ◦ s : P → P → P is the identity on P i.e. the natural map π has a section. Note that the morphism P gp → (P )gp is a surjective homomorphism of abelian groups mapping to a free group of finite rank. In particular, (P )gp is a projective module over Z and by the lifting property of projective modules we obtain a sections ˜ :(P )gp → P gp. Now consider the following commutative diagram: π P P
λP λP πgp s˜ P gp (P )gp P gp
4 where the vertical arrows are the groupification homomorphisms. Since π is sur- jective and the diagram is commutative, the image ofs ˜ ◦ λP ◦ π is contained in the image of λP . Hence, the homomorphism of monoids s : P → P given by −1 s := λP ◦ s˜ ◦ λP is well defined. We prove that π ◦ s = idP . For that note that:
−1 gp −1 λP ◦ (π ◦ s) = (λP ◦ π) ◦ λP ◦ s˜ ◦ λP = π ◦ λP ◦ λP ◦ s˜ ◦ λP = λP
Since λP is a monomorphism we conclude that:
−1 π ◦ λP ◦ s˜ ◦ λP = idP ⇒ π ◦ s = idP Thus, π admits a section. By using the above lemma we can describe the structure of fs monoids.
Theorem 1.8. Let P be an fs monoid. Then, we have the following decomposition:
P gp = P ∗ ⊕ (P )gp
Moreover, the above decomposition induces a decomposition of monoids:
P = P ∗ ⊕ P
Proof. First note that since P is fs, the monoid P is a toric monoid. Hence, we have a section π : P → P by Lemma 1.7. This means that the following sequence is exact: φ 0 → P ∗ → P gp → (P )gp → 0 and we get the first decomposition. For the second decomposition consider an element in a ∈ P with image [a] in P . We need to show that b := a − π([a]) is in P and it is invertible. For this gp consider the image λP (b) via the groupification homomorphism λP : P → P . Then:
φ(λP (b)) = φ(λP (a − π([a]))) = φ(λP (a)) − φ(λP (π([a]))) = 0
∗ gp which means that λP (b) lies in im((λP )|P ∗ : P → P ) and a − π([a]) is invertible in P . See also [Uli15, Lemma I.2.1]
1.9. (Ideals and faces of monoids, [Ogu18, Section I.1.4]). Let (P, +, 0) be a monoid. We can define the notion of an ideal in P to be a subset I of P such that: for every x ∈ P and y ∈ I we have that x + y ∈ I. Note that we also consider the empty set as an ideal. Similarly as in the case of rings, a prime ideal is an ideal p of P with p 6= P such that for every x, y ∈ P with x + y ∈ p we have that x ∈ p or y ∈ p. Then, we define the prime spectrum Spec(P ) of a monoid P to be the set of prime ideals in P and we topologize the set Spec(P ) with the a Zariski topology to make it a topological space. Note that when P
5 is finitely generated over (N, +, 0) the topological space Spec(P ) is finite [Kat94, Proposition 5.5]. A face of a monoid P is a submonoid F of P such that for every x, y ∈ P such that x + y ∈ F we have that x ∈ F and y ∈ F . Note that the faces are complements of prime ideals meaning that the correspondence p 7→ P − p is an inclusion reversing correspondence between the poset of prime ideals and the poset of faces of P . In the case that P is finitely generated since the set of prime ideal is finite we can use the above correspondence and get that the set of faces of P is finite.
1.2 Kummer and exact homomorphisms of monoids Definition 1.10. ([Ogu18, Definition I.4.3.1 (2)]). A homomorphism θ : P → Q of fs monoids is said to be of Kummer type, or simply Kummer, if it is injective and Q-surjective i.e., for every y ∈ Q there exists a natural number n ∈ N with n ≥ 1 such that n · y ∈ im θ.
Remark 1.11. In the above definition, note that Q-surjectivity is equivalent to the existence of a natural number n ≥ 1 such that n · Q ⊂ im θ. To see this, since Q is finitely generated. we can consider the generating set {y1, ..., yk} ⊂ Q and by Q-surjectivity we can find natural numbers n1, ..., nk such that ni · yi ∈ im θ. Qk By taking n = i=1 ni, we have that n · y ∈ im θ for all y ∈ Q, hence n · Q ⊂ im θ. The other implication is trivial.
Example 1.12. The multiplication by n homomorphism:
θn :(N, +) → (N, +) x 7→ n · x is a Kummer homomorphism of monoids. Note that we can define such homomor- phism for any monoid P , since every monoid is an ”N-module”, but it might not be of Kummer type. For example, consider the multiplication by 2 homomorphism θ2 : Z/2Z → Z/2Z, which is not Kummer since θ2 is not injective. Let P be a toric monoid, and consider the multiplication by n homomorphism 1 θn : P → P . Since we can view θn as the inclusion P,→ · P , we will denote this ∼ n homomorphism by θn : P → P1/n even though P1/n = P . We claim that θn is of Kummer type. To see this, first note that P is fine gp ∼ r and without torsion, we have from Remark 1.6 that P = Z for some positive gp r r number r. Now, since θn : Z → Z is injective we conclude that θn : P → P1/n is also injective. On the other hand, for every a ∈ P1/n, we have n · a ∈ im θn and θn is Q−surjective. This means that θn is of Kummer type. Remark 1.13. Let P,Q be toric monoids and consider u : P → Q a homomor- phism of Kummer type. Then, there exists a natural number n ∈ N≥1 such that
6 the homomorphism θn : P → P1/n factors as:
u P → Q → P1/n
To see this, let us use the Kummer condition for u, namely, since P,Q are fs, they are finitely generated and there exists a natural number n0 ∈ N≥1 such that 1 ∼ 1 n0 · Q ⊂ im u. In other words Q ⊂ · im u = · P := P . By taking n = n0 n0 n0 1/n0 we have the factorization.
Lemma 1.14. Let u : P → Q be a Kummer homomorphism of fs monoids. Then there exists a natural number n ≥ 1 such that coker ugp is annihilated by n.
Proof. By using the Kummer condition we for u, there exists a natural number n ≥ 1 such that n · Q ⊂ im u. Take a ∈ coker ugp then, a = b + P gp for some b ∈ Q, which implies that na = nb + P gp = P gp and n annihilates coker ugp.
Definition 1.15. ([Ogu18, Definition I.2.1.14]). Let θ : P → Q be a homomor- gp phism of monoids. We say that θ is exact if the homomorphism P → P ×Qgp Q is an isomorphism.
Exact morphism play an important role in logarithmic geometry. In what follows, we will not treat the theory of exact morphism in great detail except but their connections with Kummer homomorphisms.
Lemma 1.16. ([Ogu18, Proposition I.4.2.1 (1,2)]). Let θ : P → Q, φ : Q → W be homomorphisms of integral monoids.
1. The homomorphism P → P is exact.
2. If φ ◦ θ is exact the same is true for θ.
Lemma 1.17. (Kummer vs. Exact, [Ogu18, Proposition I.4.3.5 (3)]). Every Kummer homomorphism in the category of saturated monoids is exact. A homo- morphism of fs monoids θ : P → Q is of Kummer type if and only if the following conditions hold:
1. θ is exact,
2. θ is injective
3. θgp is Q−surjective. We note that homomorphisms of monoids that satisfy condition 3 are called small.
7 1.3 Pushouts in the category of fs monoids The category of monoids is complete and cocomplete [Ogu18, Section I.1]. Since the forgetful functor For : Mon → Sets is a right adjoint, the formation of limits in the category of monoids commutes with the formation of limits in the category of sets. So, in some extent, limits in the category of monoids are easy to describe. On the other hand the groupification functor Gr : Mon → Ab is a left adjoint. This means that the formation of colimits in the category of monoids commutes with the formation of colimits in the category of abelian groups, a fact that we will heavily use to compute colimits of monoids. Having said that, we continue with some terminology. Definition 1.18. (Amalgamated sum, [Ogu18, Section I.1]). We define the amal- gamated sum of two homomorphism of monoids φ1 : P → Q1, φ2 : P → Q2 to be the pushout of the following diagram:
φ1 P Q1
φ2
Q2 Q1 ⊕P Q2
in the category of monoids and we denote it by Q1 ⊕P Q2. We denote with superscripts ⊕int, ⊕sat to specify the category in which the pushout is formed. Since we are interested in inductive limits and amalgamated sums in the cate- gory of integral or saturated monoids, we give the next remark which is a collection of statements regarding the formation of colimits in Monint, Monsat. Remark 1.19. We state some things about colimits in various categories of monoids.
int 1. (Amalgamated sum in Mon ,[Ogu18, Proposition I.1.3.4]). Let P → Q1, P → Q2 be a pair of homomorphisms of integral monoids. The pushout of the two morphisms in Monint is given by:
int gp gp Q1 ⊕P Q2 = im(Q1 ⊕ Q2 → Q1 ⊕P gp Q2 ) This is possible since the groupification functor Gr : Mon → Ab is a left adjoint and hence it commutes with colimits. This means that (Q1 ⊕P gp ∼ gp gp int Q2) = Q1 ⊕P gp Q2 and since Q1 ⊕P Q2 = im(Q1 ⊕ Q2 → Q1 ⊕P Q2 → gp (Q1 ⊕P Q2) ) we get the above description.
Note that in general Q1 ⊕P Q2 might not be integral (the same is true for the category of saturated monoids see, Example 1.21). If one of the integral monoids P,Q1,Q2 is a group then Q1 ⊕P Q2 is integral .
8 2. (Quotients of integral monoids, [Ogu18, Proposition I.1.3.3]). If P is a submonoid of an integral monoid Q, then the quotient Q/P in the category of monoids is already integral and given by:
Q/P = im(Q → Qgp/P gp)
3. (Colimits of directed systems, [Ogu18, Proposition I.1.3.6]). If {Qi}i∈I be an direct system of integral (saturated) monoids then colimi∈I Qi is integral (saturated).
The following lemma describes some specific pushouts in the category of fs monoids. We will need it later in Section5 for Kummer log etale descent. Lemma 1.20. ([Ill02, Lemma 3.3]). Let u : P → Q be a homomorphism of fs monoids such that Γ := coker(ugp) in annihilated by an integer n ≥ 1. Consider the amalgamated sum: u P Q
u
fs Q Q ⊕P Q in the category of fs monoids. Then the natural map:
fs Q ⊕P Q → Γ ⊕ Q (a, b) 7→ ([b], a + b) is an isomorphism. Example 1.21. ([Sam10, Example 1.1.16]). We give an example of an amal- gamated sum of saturated monoids in the category of integral monoids which is not saturated. Let θ2 : N → N be the multiplication by 2 homomorphism, and consider the following amalgamated sum in the category of monoids:
θ2 N N
θ2 v u N M
We want to compute M int so we use Remark 1.19 (1) to get a description of the int gp integralization M := im(λM : M → M ) where λM is groupification homomor- phism of Paragraph 1.2. Since the groupification functor Gr : Mon → Ab commutes with arbitrary colimits, we can identify M gp as the following push out in the category of abelian groups:
9 gp θ2 Z Z
gp gp θ2 v ugp Z M gp
Now we can identify M gp as (Z ⊕ Z)/Λ where: gp gp Λ := {(θ2 (x), −θ2 (x)) ∈ Z ⊕ Z | x ∈ Z} = 2 · h(1, −1)i ⊂ Z ⊕ Z and the morphisms ugp, vgp are given by ugp(a) = [(a, 0)] and vgp(a) = [(0, a)]. Note also that the abelian group (Z ⊕ Z)/Λ is isomorphic to Z ⊕ (Z/2Z) via the homomorphism ψ defined by (a, b) 7→ (a + b, [b]). int gp gp Hence M is the image of the homomorphism of monoids (u ◦θ2)⊕(v ◦θ2): N ⊕ N → M gp which is identified via ψ as: int M = {(a, i) ∈ Z ⊕ (Z/2Z) | 0 ≤ i ≤ 1, a ≥ i} Note that the element (0, 1) is not in M int but the element 2·(0, 1) is in M int which means that M int is not saturated. We saturate the monoid M int by adjoining the element (0, 1) and we get that M sat = N ⊕ (Z/2Z).
1.4 Monoid algebras and Toric varieties 1.22. (Monoid algebras, [Ogu18, Section I.3]). Let R be a commutative ring with identity, then for a monoid (M, +, 0) we can form the monoid algebra of M over R to be the R-module defined by: ( ) X m R[M] := rmx | m ∈ M, rm ∈ R with rm 6= 0 only for finite amount of m m∈M with multiplication defined by xm · xn = xm+n. The association M 7→ R[M] is a functor which forms an adjunction: For
R -Alg Mon
R[−] where R[−] a For and For maps a ring to its underlying multiplicative monoid. Definition 1.23. Let M be a monoid and R a ring, we define the scheme as- sociated to the monoid M, or the R-realization of M, to be the scheme M AR := Spec(R[M]) and we call such schemes monoid schemes. Note that we will omit the subscript when R = C.
10 Remark 1.24. Let R be a ring, S an R−algebra and let θ : P → (S, ·, 1) be a monoid homomorphism where P is a monoid and S is considered as a monoid with multiplication. Then, we can define a morphism of R-algebras θ˜ : R[P ] → S which is defined by: ! ˜ X m X θ amχ = amθ(m) m∈P m∈P Using the above adjunction: ∼ HomMon(P,S) = HomR−Alg(R[P ],S) we see that giving θ is equivalent to give θ˜ and in this way we will use θ to denote both θ, θ˜ as well as the induced morphism of schemes θ] : Spec(S) → Spec(R[P ]). A standard example of monoid schemes are affine toric varieties which we will now give the construction.
1.25. (Affine Toric Varieties, [Oda88, Chapter 1]). We fix a lattice N := Zr and the dual M := HomZ(N, Z) together with the Z-bilinear pairing:
h , iZ : M × N → Z
We extend all the above to R by tensoring with the real numbers to get the R−vector spaces NR := N ⊗Z R,MR := M ⊗Z R and the R−bilinear pairing:
h , iR : MR × NR → R
Now we consider a convex cone (or simply a cone) σ in NR. This is a convex subset of NR which is a ”R≥0−module” i.e.: i) σ is closed under addition,
ii) for all x ∈ R≥0 we have that x · σ ⊂ σ.
A subset σ ⊂ NR is called a strongly convex rational polyhedral cone (with apex at the origin 0 ∈ NR), if there exists a finite number of elements u1, ..., un ∈ N such that: σ = R≥0 · u1 + ... + R≥0 · un and σ ∩ (−σ) = {0}. For a cone σ we can associate to it the dual cone which is the cone defined by: ∨ σ := {y ∈ MR | hx, yi ≥ 0, ∀x ∈ σ} Having said that, a face of a cone σ is a subset τ ⊂ σ such that τ = σ ∩ {m}⊥ for some m ∈ σ∨ and by doing some of the, see [Oda88, Corollary A.7], we get a description for the dual of a face:
∨ ∨ τ = σ + R≥0{−m}
11 If σ is strongly convex rational polyhedral cone then, every face τ of σ is strongly convex rational polyhedral and the same is true for the dual cones. Note that by strongly convexity the zero cone {0} is always a face of σ. For a strongly rational polyhedral cone σ, we can consider the integral points ∨ of the dual cone Sσ := σ ∩ M which is a monoid. By Gordan’s lemma [CLS11, Proposition 1.2.17], the monoid Sσ is finitely generated and it is also saturated by [Oda88, Proposition 1.1 (iv)]. In other words Sσ is a toric monoid. Then, we ∨ consider the monoid scheme Uσ := Spec(C[σ ∩ M]) and we define the affine toric variety associated to σ to be Uσ. The associated torus is Tσ := Spec(C[S{0}]) := U{0} and the torus embedding is given by the map induced by the natural map Sσ → S{0}. See [CLS11] and [Oda88] for more on toric varieties. ∼ r Pr Example 1.26. (The affine n-space). Let N = Z and σ := i=1 ei · R≥0 where ∨ Pr r ei is the standard basis of N. Then σ = i=1 mi · R≥0 where {mi}i=1 is the dual r basis, i.e., hmi, eji = δi,j. Then Sσ = N and the corresponding toric variety is: r r Ak := Spec(k[N ]) The torus embedding is induced by the morphism S → S ∼= r: σ MR Z r r TN = Spec(k[Z ]) ,→ Ak 1.27. (The fan construction, [Oda88, Theorem 1.4]). We can construct more general toric varieties by the fan construction. Let the setup be as the above paragraph. A fan ∆ in N is a finite, nonempty collection of strongly convex rational polyhedral cones in NR that satisfies the following: 1. For every σ ∈ ∆, every face τ of σ is in ∆, 2. For every σ, ω ∈ ∆ the intersection σ ∩ ω is in ∆. From a fan ∆ we can produce a toric variety in the following way. For every σ ∈ ∆ we construct Uσ. Since every cone is strongly convex, the zero cone {0} is a face of all members of ∆ hence for every cone σ ∈ ∆ we have the torus inclusion ∨ ∨ U{0} ,→ Uσ. In general, if τ is a face of σ then the description τ = σ +R≥0{−m} ∨ implies that Sτ = Sσ ⊕ N · {−m} for some m ∈ σ ∩ M. With this we can realize Uτ as a localization of Uσ and hence the induced morphism Uτ ,→ Uσ is a Zariski inclusion. Having said that, we can glue the family {Uσ}σ∈∆ over U{0} and we have the resulting toric variety:
U∆ :=−−−−−→ colim Uσ σ∈∆ In general, we can define morphisms of fans which induce morphisms in the cor- responding toric varieties [CLS11, Section 3.3]. Another key concept associated with fans is that of subdivisions [CLS11, Section 3, Refinements of Fans and Blowups]. The idea is that subdivisions of fans correspond to blow-ups. We give the following example.
12 ω τ → γ σ
Figure 1: On the right, we see the fan Σ and a subdivision Σ0 on the left
Example 1.28. (Blow-up of the affine plane at the origin). Let N := Z2 with basis e1, e2 and consider the cone γ := R≥0e1 ⊕ R≥0e2. We denote with Σ the fan associated to γ that is, Σ contains all the faces of γ. Consider the subdivision Σ0 of Σ as in Figure1. where the maximal cones in Σ 0 are given by:
ω := R≥0e2 ⊕ R≥0(e1 + e2), σ := R≥0e1 ⊕ R≥0(e1 + e2), τ := ω ∩ σ By taking the integral points of the dual cones, we can form the associated monoids:
S := ∗ ⊕ ∗ ∗ S := ∗ ⊕ ∗ ∗ S := ⊕ ∗ ∗ = ∗ ⊕ ∗ ∗ ω Ne1 Ne2−e1 σ Ne2 Ne1−e2 τ Ne1 Ze2−e1 Ne2 Ze1−e2
∗ ∗ where e1, e2 is the dual basis of MR. Note that since τ is a face ω we can write S as S ⊕ ∗ ∗ and in the same way we can write S as S ⊕ ∗ ∗ . τ ω N−(e2−e1) τ σ N−(e1−e2) This should be seen as a localization but for monoids. We identify the associated k-algebras:
φ −1 −1 −1 ±1 ∼ −1 ±1 k[Sω] = k[x, yx ], k[Sσ] = k[y, xy ], k[Sτ ] = k[x, (yx ) ] = k[y, (xy ) ] where the isomorphism φ is defined by sending x 7→ y · (xy−1) and yx−1 7→ (xy−1)−1. So we have the corresponding diagram on toric varieties:
φ Uτ Uτ
inc inc
Uω Uσ
Where we glue Uσ,Uω along Uτ via the map:
−1 φ : Uω|Uτ → Uσ|Uτ , (a, b) 7→ (ab, b )
2 The resulting scheme is the blow-up of Ak at the origin equipped with the blow-up
13 ω ω∨
τ ∨ τ Taking duals
σ σ∨
Figure 2: The dual cones of the fan Σ’. Note the the trivial cone {0} has as dual cone MR.
∼ 2 map to π : UΣ0 → UΣ = Ak:
φ Uτ Uσ
inc
Uω UΣ0 πω π πσ
UΣ
Note that in order to describe π we only need to describe πσ and πω. Looking at the fans, the identity map id : NR → NR induces a morphism sending σ, ω to γ. The induced map on k-algebras are:
inc −1 inc −1 πω : k[x, y] → k[x, yx ], πσ : k[x, y] → k[y, xy ] and the induced morphism on the toric varieties are:
πω :(a, b) 7→ (a, ab), πσ :(a, b) 7→ (ab, a)
This determines π.
We close this subsection by giving the definition of a log ring. This should be considered as an introductory definition of the notion of log schemes.
14 Definition 1.29. ([Ogu18, Definition 1.2.3]). Let R be a ring and (P, +, 0) a monoid. A log ring is a homomorphism of monoids α :(P, +, 0) → (R, ·, 1) where R is considered as a monoid using its multiplicative structure. We denote a log ring by (R, P, α).
Definition 1.30. A morphism of log rings ϕ :(R, P, α) → (S, Q, β) is a pair (φ, φ[) where φ : R → S is a homomorphism of rings, φ[ : P → Q is a morphism of monoids such that the following diagram:
φ[ P Q
α β
R S φ is commutative.
15 1.5 Log structures and log schemes The notion of log structures has its roots in the works of Deligne, Falting, Fontaine and Illusie. In the early days, Deligne and Faltings used the notion of DF- structures which is a particular class of log structures, see [Ogu18, Introduction]. Loosely speaking, the idea of a log scheme is that of a scheme (X, OX ) equipped with a sheaf of monoids M over OX , specifying for which functions f ∈ OX we want to consider logarithms log(f) ∈ M [Ogu18, Introduction of Section II]. In general, log structures can be defined for ringed spaces as in [Ogu18] or for ringed sites as in [GR04]. Here we define log structures for the etale topology and every sheaf will be considered in the etale topology.
1.31. (Log structures, [Ogu18, Section III.1]). Let X be a scheme, with structure ∗ sheaf OX , and OX the sheaf of invertible elements of OX .A pre-log structure on a scheme X is a tuple (M, α) where M is a sheaf of monoids over X´et and α : M → (OX , 1, ·) is a homomorphism of sheaves of monoids. A log structure is a pre-log structure (M, α) such that the morphism α restricts to an isomorphism:
−1 ∗ ∗ α : α (OX ) → OX
A morphism of (pre-)log structures f :(M, α) → (N , β) is a homomorphism of sheaves of monoids f : M → N such that α = β ◦ f. We denote the category of log structures (resp. pre-log structures) over X by Log(X) (resp. Plog(X)).
Remark 1.32. Let X be a scheme and consider a sheaf of monoids M over X´et. We define the sheaf of invertible elements M∗ of M to be the functor:
∗ M := ∗ ◦ M : X´et → Mon → Ab where ∗ is the functor defined in Subsection 1.1. Since ∗ is a right adjoint functor it follows that it preserves limits, hence M∗ satisfies the sheaf condition. We define the sharpification M of M to be the sheafification of the presheaf:
U 7→ M(U)/M(U)∗ where the quotient should be interpreted as the orbits of the free action of M(U)∗ on M(U)[Ogu18, Section III, Remark 1.1.7].
Definition 1.33. ([Ogu18, Section II, Definition 1.1.1 and 1.1.4]) Let X be a scheme and let θ : M → N be a homomorphism of sheaves of monoids. We say that θ is:
1. local if for every geometric pointx ¯ of X the induced morphism on the stalk −1 ∗ ∗ θx¯ : Mx¯ → N x¯ satisfies θx¯ (N x¯) = Mx¯. 2. logarithmic if it induces an isomorphism θ−1(N ∗) ∼= N ∗.
16 Remark 1.34. Let X be a scheme equipped with a log structure (M, α). Then, the structure morphism of the log structure α : M → OX is a local homomorphism −1 ∗ ∼ ∗ of sheaves of monoids. To prove that we use the isomorphism α (OX ) = OX to −1 ∗ ∗ conclude that α (OX ) is a sheaf of subgroups of M . In particular is a subsheaf. ∗ −1 ∗ Hence we need to prove that M is a subsheaf of α (OX ). Since α maps invertible ∗ sections of M to invertible sections of OX we conclude that M is a subsheaf of −1 ∗ ∗ −1 ∗ ∼ ∗ ∗ ∗ α (OX ). Hence M = α (OX ) = OX . In practice, we identify M and OX via this isomorphism. Example 1.35. There are two basic examples of log structures on a scheme X:
∗ ∗ 1. The trivial log structure, which is given by (OX , inc) where inc : OX → OX is the natural inclusion. This is an initial element in the category of log structures over X and we denote it by triv.
2. The full log structure which is given by (OX , idOX ). This is the final element in the category of log structures over X and we never use it. Notation 1.36. Let X be a scheme a let (α, M) be log structure over X. We define the triviality locus of the log structure (α, M) to be:
∗ (X, α, M)triv := {x ∈ X | Mx¯ = OX,x¯} wherex ¯ varies over all geometric points above x. 1.37. (Logarithmification, [Ogu18, Proposition III.1.1.3]). For a pre-log structure
(M, α) on X we define the induced log structure to be the pushout in MonX´et of the following diagram:
−1 ∗ α ∗ α (OX ) OX
inc αlog
M Mlog
We denoted by (Mlog, αlog) the resulting log structure. This construction induces an adjunction between the category of pre-log structures and the category of log structures over X: Log
PLog(X) Log(X)
Inc with Log a Inc where Inc is just the inclusion functor and Log is the logarithmi- cation functor described above taking a pre-log structure (M, α) and mapping it to (Mlog, αlog).
17 Remark 1.38. A minor remark about the terminology. The logarithmification process in the literature appears under the name ”logification” which refers to ”making something log” and sounds like ”making something logical” in contrast with the term used in this text which sound more like ”making something loga- rithmic”.
1.39. (Pushforward and pull-back of log structures, [Ogu18, Definition III.1.1.5]). Let f : X → Y be a morphism of schemes, then we can pushforward log structures from X to Y in the following way: Let (α, M) be a log structure over X then log log we define the pushforward of M along f to be the log structure (f∗ α, f∗ M) over Y defined by the pullback diagram:
log f∗ M f∗ M
log f∗ α f∗α f ] OY f∗ OX
We can also pullback log structures from Y to X is the following way: Let (β, N ) be a log structure over Y , then we can define the pullback of N along f to be the log structure over X defined in the following way. We consider the composition: ] −1 −1 −1 f ◦ f β : f N → f OY → OX which is a pre-log structure on X and then we logarithmify to obtain a log struc- ∗ ∗ ture on X which we denote with (flog N , flogβ). These processes form an adjunc- tion: log f∗
Log(X) Log(Y )
∗ flog
∗ log with flog a f∗ . 1.40. (The category of log schemes, [Ogu18, Definition III.1.2.1]). A log scheme is a triple (X, M, α) where X is a scheme and (M, α) is a log structure over X. We call M the structure monoid of X and α the log structure morphism of the log structure. A morphism of log schemes is given by a morphism of schemes [ f : X → Y and a homomorphism of sheaves of monoids f : MY → f∗ MX such
18 that the following diagram is commutative:
f [ −1 f MY MX
−1 f αY αX f ] −1 f OY OX where f −1 is the inverse image functor in the level of sheaves of monoids. Notation 1.41. We will use two kind of notation for log schemes. We use the notation (X, M) to specify a log scheme, where X is a scheme and M is the log structure on X. When we say ”...Let X be a log scheme...” we will denote with MX and αX the sheaf of monoids and the structure morphism respectively. In this case, we will write X for the underlying scheme of the log scheme X. For simplicity, we sometimes denote the underlying scheme of X also by X when confusion does not arise. In accordance with Notation 1.36, we write Xtriv for the triviality locus of the log structure of the log scheme X. Example 1.42. (Divisorial log schemes, [GR04, Example 12.1.12]). Let X be a scheme and let D be a codimension 1 closed subset of X and denote with j : U := X − D,→ X the inclusion of the complement. Then we can define the log structure (in the etale topology):
∗ V 7→ MD(V ) := {s ∈ OX (V ) | s|U×X V ∈ OX (U ×X V )} which we will call the divisorial log structure associated to D. A log scheme equipped with a divisorial log structure is said to be divisorial. See Subsection 3.2 for more about divisorial log structures. Example 1.43. (Log scheme associated to log ring, [Kat89b, Examples 1.5 (3)]). A log ring, Definition 1.29, induces a log scheme in a natural way. Let α : P → R be a log ring, let X := Spec(R) and P the constant sheaf associated to P . The homomorphism α : P → R induces a homomorphism of sheaves of monoids:
α : P → (OX , ·, 1) which is a pre-log structure over X and we logarithmify to get a log structure:
log log α :(P ) → OX over X. We call the log scheme (X, (P )log, αlog) the log scheme associated to the log ring α : P → R. M In general, we can consider any monoid scheme AR , see Definition 1.23, as a log scheme by considering the log scheme associated to the log ring α : M → R[M].
19 M We refer to the induced log structure as the canonical log structure on AR and M we denote the resulting log scheme also with AR . Having said that, note that by Paragraph 1.25 we can consider affine toric varieties as log schemes by considering the log scheme associated to the log ring α : P → R[P ] where P = σ∨ ∩ M, and
σ is a strongly rational polyhedral cone in NR.
1.6 Fs log schemes and charts We are interested in a specific class of log schemes called fs log schemes which have a more geometric nature. In order to define what an fs log scheme is we need the notion of coherence which is due to Kato [Kat89b, Section 2]. This is a notion dependent on the topology in which we define our log structure [Ogu18, Remark II.1.6.4]. We start with the following definition.
Definition 1.44. Let X be a scheme and (M, α) a log structure. We say that M is a log structure of integral (resp. saturated) monoids, if for every x ∈ X the stalk Mx¯ is an integral (resp. saturated) monoid. Remark 1.45. By [Ogu18, Proposition II.1.1.3], a log structure (M, α) over X is integral (resp. saturated) if for every etale morphism U → X the monoid M(U) is integral (resp. saturated).
A key concept for dealing with log schemes is the notion of a chart for a log structure introduced by K. Kato. We give the definition.
Definition 1.46. Let (X, M) be a log scheme, a (global) chart for M is tu- ple (P, θ) where P is a monoid and θ : P → Γ(X, M) such that the induced homomorphism of sheaves of monoids (P )log → M is an isomorphism.
Remark 1.47. ([Nak97, Definition 1.5]). There are some variations about the notion of a chart. A local chart for a log structure M over X is a triple (V, P, θ) such that V → X is an etale morphism of schemes and (P, θ) is a chart for the log structure M |V over V . In the same spirit, a local chart covering for M is a family (Vi,Pi, θi) such that (Vi → X)i∈I is an etale covering of X and for each i ∈ I the tuple (Pi, θi) is a chart for M |Vi . Definition 1.48. ([Nak97, Definition 1.7]). A log scheme (X, M) is said to be coherent (resp. fine, fs) if there exists a local chart covering (Vi,Pi, θi)i∈I where the monoids Pi are finitely generated (resp. and M is integral, saturated). In other words a log structure is coherent if it is locally given by finitely gen- erated monoids, which is similar to the notion of coherence of sheaves of modules. The next lemma deals with the pullback of log structures and how this preserves the various categories of log structures.
20 Lemma 1.49. ([Kat89a, Lemma 1.2.8]). Let f : Y → X be a morphism of schemes and consider a log structure (M, α) over X. If (M, α) is integral (resp., ∗ ∗ saturated, coherent, fine) then (flog M, flogα) is also integral (resp., saturated, coherent, fine). Remark 1.50. Note, that in comparison with Remark 1.45 we cannot define the notion of coherence locally, meaning that, if M is coherent then the monoid of local sections of M might not be finitely generated as a monoid. The reason is ∗ that the monoid of local sections of M contains local sections from OX where the latter is not coherent as a sheaf of monoids. The correct way to characterize coherence is after modding out the invertible elements i.e.: M is coherent if and only if the monoid of local sections of M is finitely generated. Remark 1.51. Let (X, M, α) be a log scheme equipped with a (global) chart (P, θ). Note that giving a a morphism θ : P → Γ(X, M) induces a morphism P → Γ(X, M) → Γ(X, OX ) and by Remark 1.24 we have an induced morphism of schemes X → Spec(Z[P ]). In this context, we can rephrase that (P, θ) is a chart, by saying that the log structure on X is isomorphic to the pullback of the canonical log structure of Spec(Z[P ]). Hence a chart for (M, α) can be given as a morphism of schemes X → Spec(Z[P ]). This is [Ogu18, Proposition III 1.2.4], and an analogues statement holds for local chart coverings. Remark 1.52. (Integralization and saturation, [Kat89b, Proposition 2.7]). We are interested in the category of coherent log schemes. By [Ogu18, Proposition I.2.1.5], we have the following adjunctions:
Int Sat
Logcoh Logfin Logfs
Inc Inc where Int is the integralization functor and Sat is the saturation functor which we will now describe. We note that both of these functors are defined in an analogues way and so we will only give the description of Int. We start with the case of a coherent log scheme X equipped with a chart θ : X → Spec(Z[P ]) where P is a finitely generated monoid. Then, we can consider the integralization homomorphism for P → P int which induces a morphism of schemes Spec(Z[P int]) → Spec(Z[P ]). Consider the fiber product in the category of log schemes: int int X := X ×Spec(Z[P ]) Spec(Z[P ]) We equip Xint with the inverse log structure induced by φ and we define the integralization functor Int(X) := Xint. For the general case, if X admits a local chart covering of finitely generated int monoids (Ui,Pi, θi)i∈I then, as above, we form the log schemes Ui and then we
21 glue them up to get a fine log scheme Xint. The Sat functor is defined analogously, where we replace the functor int by the functor sat. In the end, we get a fs log scheme which we denote by Xsat. We also denote with Xfs the fs log scheme (Xint)sat.
Proposition 1.53. [Ogu18, Proposition II.1.2.5]. Let X be a coherent log scheme then:
1. Xint → X is a closed immersion,
2. Xsat → Xint is finite and surjective.
Let (X, M) be an fs log scheme and let us pick a geometric pointx ¯ of X. log ∼ Assume we have a homomorphism of monoids θ : P → Mx¯ such that (P )x¯ = Mx¯. Then we can ask ourselves if we can extend θ to a chart on X. The next lemma of Kato, says that we can do this if we restrict to an etale neighborhood ofx ¯.
Lemma 1.54. ([Kat89b, Lemma 2.10]). Let X be a scheme with a fine log struc- gp ture M, let x ∈ X, G a finitely generated abelian group, and let h : G → Mx¯ be gp ∗ gp −1 a homomorphism such that G → Mx¯ / OX,x¯ is surjective. Let P = (h ) (Mx¯). Then, P → Mx¯ can be extended to a chart PU → M |U for an etale neighborhood U of x¯.
Consider a log scheme (X, M) equipped with a global chart (P, θ). The rel- ∗ evant part of the log structure lies in M since OX is always present for any log structure. Having said that, a monoid P can be used as a chart, if for every x ∈ X and for every geometric pointx ¯ over x the monoid P already contains the ∼ generators of Mx¯. Note that this does not mean that P = Mx¯. Let us see an example. Consider the log scheme associated to the log ring N → C[x], n 7→ xn. Then is a chart for 1 . If we pick a geometric pointx ¯ of 1 with M ∼ O∗ N AC AC x¯ = X,x¯ then P → Mx¯ is the zero map. Having said that we give the following definition. Definition 1.55. ([Ogu18, Definition II.2.3.1]). Let (X, M, α) be a log scheme equipped with a chart θ : P → Γ(X, M) andx ¯ a geometric point of X. We say that the chart (P, θ) is:
1. exact atx ¯ if θx¯ := αx¯ ◦ θ : P → Mx¯ is a local homomorphism of monoids −1 ∗ ∗ i.e., (θx¯) (Mx¯) = P ,
2. neat atx ¯ if the homomorphism P → MX,x¯ → Mx¯ is an isomorphism, or equivalently if the chart is exact atx ¯ and P is sharp.
Remark 1.56. We use the above definition also for Zariski points in the following way. We say that a chart (P, θ) is exact (resp. neat) at x ∈ X if the chart (P, θ) is exact (resp. neat) atx ¯ for every geometric pointx ¯ over x.
22 The next proposition states that etale locally we can always find a chart neat at a geometric pointx ¯ of X. This is also the main ingredient of Proposition 2.1 in [Hag03]. Proposition 1.57. ([Ogu18, Etale analog of Proposition II.2.3.7]). Let (X, M) be a fine log scheme and x¯ a geometric point of X. The log structure M admits a local chart which is neat at x¯ (after X is replaced by an etale neighborhood of x¯) if and only if the sequence:
∗ gp gp 0 → Mx¯ → Mx¯ → (Mx¯) → 0 splits. This holds in particular when M is saturated Proof. Assume that the splitting exists, and consider the following diagram:
0 gp s π M × gp (M ) M M x¯ Mx¯ x¯ x¯ x¯ ρ κ
gp gp s gp π gp (Mx¯) Mx¯ (Mx¯) where the left square is cartesian. By Lemma 1.16 the natural homomorphism Mx¯ → Mx¯ is exact hence the right square is cartesian. By composing the left gp and the right cartesian squares we get that M ∼ M × gp (M ) and that x¯ = x¯ Mx¯ x¯ 0 gp π ◦ s = id(Mx¯) |Mx¯ = idMx¯ . Now by Lemma 1.54 we can extend the tuple 0 (Mx¯, s ) to a chart for an etale neighborhood ofx ¯ and by what we proved already the chart is neat atx ¯. Now, assume that there exist a chart P → M in an etale neighborhood U → X of fs monoids neat atx ¯. This implies that the composition P → Mx¯ → Mx¯ is an isomorphism from which we can induce a splitting of the exact sequence in Proposition 1.57. If P is saturated then (P )gp is torsion free by Lemma 1.7 (1). Hence in the category of fs log schemes we can always find charts neat at a point after etale localization. Definition 1.58. Let f : X → Y be a morphism of log schemes, and consider the 5-tuple Ψ := (P, Q, ρ, κ, θ) where θ : Q → P is a homomorphism of monoids, (P, ρ) is a chart for X and (Q, κ) is a chart for Y . We say that Ψ is a chart for f if the following diagram is commutative: θ Q P
κ ρ f [ −1 f MY MX
23 Conventions 1.59. When we refer to a chart of a morphism we omit the 5−tuple and we say ”...Let (P, Q, θ) be a chart for f...” and we use similar phrases as in Remark 1.47.
1.7 Strict, Kummer morphisms of log schemes There are some special classes of morphism of log schemes. A morphism of log ∗ ∼ schemes f :(Y, N ) → (X, M) is said to be strict if flog M = N . Strict morphisms are morphisms induced by the underlining morphism of schemes. In the following example we describe how to construct strict morphisms. Example 1.60. (Construction of strict morphisms). Let (X, M, α) be an fs log scheme and consider a morphism of schemes f : Y → X. Then we equip Y with ∗ ∗ the inverse log structure from M via f and define the log scheme (Y, flog M, flogα) which comes together with a morphism of log schemes:
f [ −1 ∗ f M flog M
∗ α flogα f ] −1 f OX OY
[ ] log −1 −1 log ∗ where f := (f ◦ α) : f M → (f M) := flog M is the homomorphism coming from the logarithmification process of the pre-log structure (f −1 M, f ] ◦α) on Y . By construction, the morphism of log schemes f˜ := (f, f ], (f ] ◦ α)log) is a strict morphism of log schemes. We note that, from Lemma 1.49, if M is integral (coherent, saturated, fine, ∗ fs) then flog M is also integral (coherent, saturated, fine, fs).
Remark 1.61. Consider an fs log scheme X with a local chart covering (Ui,Pi, φi)i∈I of fs monoids. We make the following remarks: log 1. The covering ((Ui, (φi)∗ MX ) → X)i∈I consists of strict morphisms such log that (Ui → X)i∈I is an etale covering and every log scheme (Ui, (φi)∗ MX ) is globally charted.
2. We can further refine the covering in such a way that all Ui are affine. log For the first assertion note that each morphism (Ui, (φi)∗ MX ) → X is con- structed as in Example 1.60. Hence is strict by construction, and for every i ∈ I log the log structure (φi)∗ MX over Ui has as chart Pi, which means that Ui is globally charted. For the second remark, note that for every scheme Ui we can find an open covering of schemes (ψi,j : Vi,j → Ui)j∈J such that Vi,j is affine. Observe that if we equip the schemes Vi,j with the inverse log structure from Ui and they are still globally charted by Pi.
24 Another really important class of morphisms of log schemes are morphisms of Kummer type. These morphisms are of main importance when we need to prove arguments analogous to ”etale descent” but in logarithmic geometry. Definition 1.62. ([Niz08, Definition 2.11 (2)]). A morphism of fs log schemes f :(X, M) → (Y, N ) is of Kummer type, or simply Kummer, if for every geo- ∗ ∗ metric pointx ¯ the induced homomorphism of monoids (N /OY )f(x) → (M /OX )x¯ is of Kummer type. See Definition 1.10 for Kummer homomorphism of monoids. Lemma 1.63. Strict morphisms of fs log schemes are of Kummer type. Proof. If f : X → Y is a strict morphism of fs log schemes we have by defi- ∗ ∼ ∗ ∼ nition that flog MY = MX and by sharpening we get flog MY = MX . Using −1 ∗ [Ogu18, Remark III.1.1.6], we get that the morphism f (MY ) → flog MY is −1 an isomorphism, hence the morphism f (MY ) → MX is an isomorphism. For every geometric pointx ¯, localizing atx ¯ gives an isomorphism on the stalks:
(MY )f(x) → (MX )x¯ Since isomorphisms are of Kummer type, f is of Kummer type. Definition 1.64. (Standard Kummer Galois covers, [Ill02, Definition 3.5]). Let X be an fs log scheme endowed with a global chart X → Spec(Z[P ]), where P is an fs monoid and let u : P → Q be a Kummer map of fs monoids. By a standard Kummer Galois cover of X we mean a morphism of fs log schemes f : XQ → X where XQ := X ×Spec(Z[P ]) Spec(Z[Q]) and the log structure on XQ is given by the inverse image of the canonical log structure on Spec(Z[Q]). Proposition 1.65. ([Ill02, Proposition 3.2]). Let X be an fs log scheme endowed with a global chart and consider a standard Kummer Galois cover f : Y → X. Then f is open, finite and surjective on the underlying schemes, and remains so after any fs base change.
1.8 Fiber products in the category of coherent log schemes We start with the definition of the fiber product in the category of log schemes. 1.66. (Fiber products in Log,[Ogu18, Proposition 2.1.2]). Consider two mor- phisms of log schemes f : Y → X, g : Z → X. Then the fiber product, in the category of log schemes, has as underlying scheme the fiber product of the underlying schemes Y ×X Z.
pZ Y ×X Z Z
pX pY g
Y X f
25 and the log structure is given as the logarithmification of the pre-log structure given by the pushout diagram in MonX´et :
p−1(g[) −1 Z −1 pX MX pZ MZ
p−1(f [) Y ] pZ ◦ αZ −1 −1 −1 p MY p MY ⊕ −1 p MZ Y Y pX MX Z ∃!γ
OY ×X Z ] pY ◦ αY
log We denote the resulting log structure on Y ×X Z by (MY ×X X , γ ). Lemma 1.67. ([Ogu18, Proposition II.2.1.1 and Corollary II.2.1.6]). The category of coherent log structures is closed under finite colimits. Moreover, the category of coherent log schemes is stable under finite products.
In general, the fiber product in the categories Log fin, Log fs behaves strangely. As we saw in Remark 1.52, the underlying scheme of the fiber product changes and it is not the same as the fiber product of the underlying schemes [Ogu18, Section II.2.1]).
Proposition 1.68. ([Kat96, Lemma 3.4]). Let f : X → Y be a strict morphism of fs log schemes. Then, for any morphism g : Z → Y of fs log schemes the fs fs ∼ projection Z ×Y X → X is also strict. Moreover, Z ×Y X = Z ×Y X. Remark 1.69. By Nakayama, [Nak97, Section 2.1.2], Kummer morphisms of fs log schemes are stable under fs-base change.
Lemma 1.70. Let X be an fs log scheme equipped with a global chart (P, θ) and consider a Kummer homomorphism of monoids ρ : P → Q. Consider the standard Kummer Galois cover associated to ρ : P → Q:
XQ := X ×Spec(Z[P ]) Spec(Z[Q])
fs Then, the underlying scheme of the fiber product XQ ×X XQ is:
fs gp gp XQ ×X XQ = X ×Spec(Z[P ]) Spec(Z[Q ⊕ (Q /P )])
26 fs Proof. We are going to compute the underlying scheme of XQ ×X XQ. Note that since XQ → X is not necessarily strict we need to fs-ize. First we do some calculations:
fs fs fs fs XQ ×X XQ = (X ×Spec(Z[P ]) Spec(Z[Q])) ×X (X ×Spec(Z[P ]) Spec(Z[Q])) fs fs = X ×Spec(Z[P ]) (Spec(Z[Q]) ×Spec(Z[P ]) Spec(Z[Q])) fs = X ×Spec(Z[P ]) Spec(Z[Q ⊕P Q]) where in the last line we do not need to fs-ize by the strictness of the morphism of log schemes X → Spec(Z[P ]). By Lemma 1.20 we have the isomorphism: fs ∼ gp gp Q ⊕P Q = Q ⊕ (Q /P )
fs ∼ gp gp and therefore Spec(Z[Q ⊕P Q]) = Spec(Z[Q ⊕ (Q /P )]). Putting everything together, we conclude that:
fs gp gp XQ ×X XQ = X ×Spec(Z[P ]) Spec(Z[Q ⊕ (Q /P )])
Remark 1.71. Let X be a coherent log scheme. By Proposition 1.53, the mor- phism Xfs → X is given as the composition:
Xfs →π Xint →i X see Remark 1.52 for notation. Since π is finite and surjective, and i is a closed immersion, they are both affine and the same is true for the composition i ◦ π. Since forgetting the log structure is functorial i ◦ π = i ◦ π. Hence, the underlying morphism of schemes i ◦ π : Xfs → X is affine.
27 2 Differential forms on Log schemes
Caution 2.1. In what follows we give the definition of log deformation theory from [Ogu18] and [Kat96]. The main difference between these two references is that the exposition of [Kat96] is for fine log schemes and the exposition of [Ogu18] is for general log schemes. Conventions 2.2. We note that in this section the sheaves of logarithmic differ- ential forms are given for the Zariski topology. For example, in Propositions 2.13, ∗ 2.14 the pullback f refers to pullback of OX -modules in the Zariski topology.
2.1 Logarithmic differential forms
Let us start with a motivating example. Consider the ring homomorphism C → C[x], then, we can define the module of K¨ahler differential forms: 1 ∼ ΩC[x]/C = dx · C[x] Assume that we want to consider a differential form for log(x). Then we can take dx 1 d log(x) = x but this would be problematic because x ∈/ C[x]. So, we can enhance our module of K¨ahlerdifferentials by adjoining a symbol δ(x) which it should be treated as the differential of the logarithm of x. Thinking formally, we can take:
dx · C[x] ⊕ δ(x) · C[x]/hdx − x · δ(x)iC[x] Then δ(x) behaves as d log(x) with a small advantage, namely, we bypassed the 1 problem of the need of x in C[x]. This is the idea behind the definition of log K¨ahlerdifferential forms [Ogu18, Section IV] which can be formalized in the setup of logarithmic geometry. Before starting exposing the theory logarithmic deformation theory, we give a small review of the classical case [Gro67, Section 20]. In algebraic geometry, for a morphism of rings f : R → S we can define the module of K¨ahlerdifferentials in the following way. First we define the notion of an S/R−derivation with values in an S−module M. The set of all such derivations is denoted with DerS/R(E) and it is easy to see that it is an S−module as well. This defines an endofunctor DerS/R(−) of S −mod and we can ask the question if this functor is representable. The answer is affirmative and the representing object is the module of K¨ahler 1 differential forms ΩS/R which comes together with a universal derivation d : S → 1 ΩS/R. Note that, for the global case (i.e., for a morphism of schemes f : X → Y ), there are two equivalent ways to construct differential forms. We can either work 1 locally and then glue up to get the global object ΩX/Y together with a universal 1 1 derivation d : OX → ΩX/Y or we can identify the sheaf of differentials forms ΩX/Y with the conormal sheaf of the diagonal embedding X → X ×Y X. This second approach is a bit more technical in the category of log schemes as we can see in [Ogu18, Remark IV.1.1.8], so we will use the first approach to define logarithmic differentials.
28 Definition 2.3. (Log derivations, [Kat96, Definition 5.1], [Ogu18, Definition IV.1.2.1]). Let f : X → Y be a morphism of fine log schemes and E be a sheaf of OX −modules. A log derivation of f with values on E is a pair (∂, δ) such that ∂ : OX → E is a homomorphism of abelian sheaves, δ :(MX , +) → (E, +) is a homomorphism of sheaves of monoids such that:
1. ∂(xy) = x∂(y) + y∂(x) for every local section x, y of OX . (Leibniz rule)
] −1 2. ∂(f (z)) = 0 for every local section z of f (OY ). (derivation relative to Y )
3. ∂(αX (m)) = αX (m)δ(m) for every local section m in MX . (”exponential” derivation)
[ −1 4. δ(f (n)) = 0 for every local section n in f MY . (derivation relative to MY )
log We denote the OX -module of log derivations with DerX/Y (E). Proposition 2.4. ([Kat96, Proposition 5.3]). Let f : X → Y be a morphism log of fine log schemes. Then, the functor DerX/Y (−) : Mod(OX ) → Mod(OX ) is log representable by the sheaf of logarithmic differential forms ΩX/Y i.e.:
log ∼ log DerX/Y (E) = HomOX (ΩX/Y ,E) for every E ∈ Mod(OX ) and the isomorphism is natural in E. There are many constructions for the sheaf of logarithmic differential forms. We give the following construction.
Definition 2.5. ([Ogu18, Proposition IV.1.1.2]). We define the sheaf of loga- rithmic differential forms to be:
log gp ΩX/Y := ΩX/Y ⊕ [OX ⊗Z MX ]/ ∼ where ∼ is generated locally by:
• h(dαX (m), −αX (m) ⊗ m) | m ∈ MX iOX
[ −1 • h(0, 1 ⊗ f (n)) | n ∈ f MY iOX
log The universal derivation (∂, δ):(OX , MX ) → ΩX/Y is defined locally by:
∂ : f 7→ [(df, 0)] and δ : m 7→ [(0, 1 ⊗ m)]
log We write ΩX for the sheaf of logarithmic differential forms associated to morphism of log schemes f : X → (Spec(k), triv) where k is a fixed base field.
29 2.6. (Log differentials and log rings, [Ogu18, Proposition IV 1.1.2]). We give the definition of log differential forms for log rings. Let φ :(R, P, α) → (S, Q, β) be a morphism of log rings, see Definition 1.30. We define the module of logarithmic differential forms associated to φ to be the S-module:
log 1 gp gp Ω(S,Q)/(R,P ) := ΩS/R ⊕ (S ⊗Z Q /P )/ ∼ where Qgp/P gp is the cokernel of group homomorphism P gp → Qgp. The equiva- lence relation ∼ is generated by elements of the form:
(d(β(n)), −β(n) ⊗ π(n)) for n ∈ Q where π : Q → Qgp/P gp is the composition of the map Q → Qgp followed by the quotient map Qgp → Qgp/P gp. The universal derivation is given log log by the morphisms d : S → ΩS/R by sending s to the class (ds, 0) and δ : Q → ΩS/R sends n to (0, 1 ⊗ π(n)). We note that by [Ogu18, Corollary IV.1.2.6] if a morphism of log schemes f : Y → X comes from a morphism of log rings ϕ :(R, P, α) → (S, Q, β) then the log sheaf of logarithmic differential forms ΩY/X is the sheaf associated to the module log of logarithmic differential forms Ω(S,Q)/(R,P ). So, in order to compute logarithmic differential forms we can use log rings instead. Note, that in the definition of a log ring we do not require the monoidal structure over the ring to be a log structure. Hence, computing log differentials for log rings is easier since we can skip logarithmification.
Example 2.7. (Differential forms of the affine plane). Let X := Spec(C[x, y]) be the affine plane. Note that in Example 1.26 we realized X as a toric variety and by Paragraph 1.25 it has a natural log structure given by the log ring α : N2 → C[x, y] where (n, m) 7→ xnym. We consider X as a log scheme over (Spec(C), triv) and we log calculate the module of log differentials ΩX by using log rings. The log scheme (Spec(C), triv) is induced by the log ring β : {1} ,→ C hence, the structure morphism X → (Spec(C), triv) is given by the morphism of log rings:
2 ϕ :(C, {1}, β) → (C[x, y], N , α) where φ] in the canonical inclusion C ,→ C[x, y] and φ[ maps {1} to (0, 0) ∈ N2. By the definition of the module of logarithmic differentials:
log 1 2 ΩX := ΩX ⊕ [C[x, y] ⊗Z Z ]/ ∼
2 ∼ 2 where ∼ is given as above. Note that C[x, y] ⊗Z Z = C[x, y] and we denote the C[x, y]-basis with d log x, d log y. The equivalence relation dictates that: dx = x · d log x, dy = y · d log y
30 log ∼ dx dy We claim that ΩX = x C[x, y] ⊕ y C[x, y]. To see this consider the map defined on the basis: dx dx 7→ d log x, 7→ d log y x y An easy check shows that this homomorphism is an isomorphism. In the following example we compute the global sections of the sheaf loga- rithmic differential forms related to the blow-up of the affine plane at the origin relative to (C, triv). The log structure that we use on the blow-up is the log structure of the log blow-up, see Paragraph 3.19. Example 2.8. (Log differentials on the blow-up). Let X := Spec(C[x, y]). Since X is a toric variety we can consider naturally as a log scheme, see Remark 1.43. Consider the blow up π : X˜ → X of the affine plane at the origin and the Zariski covering of the blow-up: −1 −1 U1 := Spec(C[x, yx ]),U2 := Spec(C[y, xy ]) as in of Example 1.28. We equip the blow up with the log structure MX˜ which is defined as the log structure induced by the pre-log structures on the covering: 2 −1 n −1 m α1 : N → C[x, yx ], (n, m) 7→ x (yx ) 2 −1 n −1 m α2 : N → C[y, xy ], (n, m) 7→ y (xy ) Note that the above log structures are the divisorial log structures associated to the union of the coordinate axis of U1,U2 respectively and we have already calculated the module logarithmic differential forms for such log structures in Example 2.7 ˜ Now consider the log scheme (X, MX˜ ) → (Spec(C), triv). Let us denote at the log log log moment with Ω ˜ the sheaf of differential forms Ω ˜ and by ΩX the sheaf of X (X,MX˜ ) logarithmic differential forms of the log scheme X. We are going to compute the global sections of the sheaf Ωlog For this we use the fact that Ωlog is a sheaf on X˜ X˜ the Zariski topology. Consider the following commutative diagram of modules:
0
π∗ Ωlog(X) Ωlog(X˜) X X˜
id res
Ωlog(X) Ωlog(U ) ⊕ Ωlog(U ) X ∗ ∗ X˜ 1 X˜ 2 p1 ⊕ p2
res1 − res2
Ωlog(U ∩ U ) X˜ 1 2
31 ∗ where pi := π|Ui is restriction of the blow up map on Ui. In order to show that π ∗ ∗ is injective we will show that p1 ⊕p2 is injective and then by the commutativity of the diagram π∗ will be forced to be injective. Let us pick an element α in Ωlog(X). This element will be of the form: dx dy α := f(x, y) + g(x, y) , f, g ∈ [x, y] x y C
The logarithmic differentials on U1 corresponding to the union of the two axis are identified as: da db Ωlog(U ) = [a, b] ⊕ [a, b] (relabeling x with a and yx−1 with b) 1 a C b C ∗ Then, p1 is the coordinate change by the map x 7→ a, y 7→ ab. Hence dx 7→ da and dy 7→ adb + bda, so we have the following: dx dy da da db p∗ f(x, y) + g(x, y) =f(a, ab) + g(a, ab) + 1 x y a a b da db =(f + g) + g a b ∗ We do the same for p2, we write: dc de Ωlog(U ) = [c, e] ⊕ [c, e] (relabeling y with c and xy−1 with e) X˜ 2 c C e C ∗ and p2 will be the coordinate change by the map x 7→ ce, y 7→ c. Hence dx 7→ cde + edc and dy 7→ dc, so we have the following: dx dy dc de dc p∗ f(x, y) + g(x, y) =f(ce, c) + + g(ce, c) 2 x y c e c dc de =(f + g) + f c e ∗ ∗ Now if we assume that p1 ⊕ p2(α) = 0 we have that: f(a, ab) = −g(a, ab), g(a, ab) = 0 and g(ce, e) = 0, f(ce, e) = −g(ce, e) which means that g = 0 and hence f = 0 (because that map sending g(x, y) 7→ g(a, ab) is injective). So the map π∗ is injective. ∗ ∗ ∗ In order to show that π is surjective we show that p1 ⊕ p2 is surjective on the kernel of res1 − res2. Calculating the logarithmic forms of the intersection: da db Ωlog(U ∩U ) = [a, b±1]⊕ [a, b±1] (relabeling x with a and yx−1 with b) X˜ 1 2 a C b C and the map res :Ωlog(U ) ,→ Ωlog(U ∩ U ) is just the inclusion. For the map 1 X˜ 1 X˜ 1 2 res :Ωlog(U ) → Ωlog(U ∩ U ) we will have to change coordinates by the map 2 X˜ 2 X˜ 1 2
32 −1 db c 7→ ab, e 7→ b , hence dc 7→ adb + bda and de 7→ − b2 . So if we take an element dc de f(c, e) c + g(c, e) e this will be mapped to: da db db da db f(ab, b−1) + − g(ab, b−1)b = f + (f − g) a b b2 a b
So if we pick (h, z, f, g) ∈ Ωlog(U )⊕Ωlog(U ) such that (res −res )(h, z, f, g) = 0 X˜ 1 X˜ 2 1 2 then we get the equations: da db da db h(a, b) + z(a, b) − f(ab, b−1) + f(ab, b−1) − g(ab, b−1) = 0 a b a b ⇒ h(a, b) = f(ab, b−1) and z(a, b) = f(ab, b−1) − g(ab, b−1) Because the following diagram is a pullback diagram:
] p2 C[x, y] C[c, e]
] c 7→ ab p1 e 7→ b−1 C[a, b] C[a, b±1] inc there exist a unique element f˜ such that:
] ˜ ˜ ] ˜ ˜ p1(f(x, y)) = f(a, ab) = h(a, b) and p2(f(x, y)) = f(ce, c) = f(c, e) and a uniqueg ˜ such that:
] ] p1(˜g(x, y)) =g ˜(a, ab) = z(a, b) and p2(˜g(x, y)) =g ˜(ce, c) = (f − g)(c, e)
We define the element: dx dy β := (f˜(x, y) − g˜(x, y)) +g ˜(x, y) x y and we have that: da da db p∗(β) = (f˜(a, ab) − g˜(a, ab)) +g ˜(a, ab) + 1 a a b da db = f˜(a, ab) +g ˜(a, ab) a b da db = h(a, b) + z(a, b) a b
33 and dc de dc p∗(β) = (f˜(ce, c) − g˜(ce, c)) + +g ˜(ce, c) 2 c e c dc de = f˜(ce, c) + (f˜− g˜)(ce, c) c e dc de = f(c, e) + g(c, d) c e Hence every element of Ωlog(U )⊕Ωlog(U ) which goes to zero by res − res comes X˜ 1 X˜ 2 1 2 log ∗ from a unique element in ΩX (X). This means that π is surjective and hence an isomorphism.
2.2 Log deformation theory Smooth, etale, unramified morphisms find an analogue in logarithmic geometry. In this way, some non smooth morphisms behave as ”smooth” in the logarithmic category. The theory is defined similarly as the approach of Grothendieck [Gro67]. Namely, for a morphism of schemes f : X → Y we define an infinitesimal thick- ening of order n to be a square:
S X
i f
T Y where T,S are schemes, i is a closed immersion and the ideal of S in T is a n+1 nil ideal of order n i.e., IS = 0. Then we characterize f as formally smooth, (formally etale, formally unramified) if for every infinitesimal thickening as the above, we have at least one, (exactly one, at most one) lifts j : T → X locally for T . Following [Kat96] and [Ogu18], we give the analogous definitions in the category of fine log schemes. 2.9. (Log deformation theory, [Kat96, Section 3]). A log thickening of order n is a strict morphism of fine log schemes i : S → T such that i is a closed immersion and the ideal I of S in T satisfies In+1 = 0. A log thickening over X/Y is a commutative diagram of fine log schemes: g S X
i f
T Y
34 where i is a log thickening. A deformation of g to T is a lifting t : T → X such that the two triangles that come up are commutative. Having said that, a morphism of fine log schemes f : X → Y is said to be formally log smooth (resp. formally log etale, formally log unramified) if for any diagram as the above, with i : S → Y a log thickening over X/Y , there exists at least one (resp. exactly one, at most one) deformation of g to T , locally on T . We say that f is log smooth, (resp. log etale) if f is formally log smooth (resp. formally log etale), MX , MY are coherent and f is locally of finite presen- tation. We say that f is log unramified if f is formally log unramified and f is locally of finite presentation.
Proposition 2.10. ([Ogu18, Remark IV.3.1.2]). The family of formally log smooth, formally log etale, formally log unramified morphisms is stable under composition in the category of fine log schemes. Consider the morphisms of fine log schemes f : X → Y and g : Y → Z.
1. If g is formally log etale then g ◦ f is formally log smooth if and only if f is formally log smooth.
2. If g ◦ f, g are formally log etale then f is formally log etale.
3. If g ◦ f is formally log unramified the f is as well.
Lemma 2.11. ([Niz08, Lemma 2.7]). Log etale morphisms are stable under fs-base changes.
Proposition 2.12. ([Ogu18, Proposition IV.3.1.6]). Let f : X → Y be a strict morphism of fine log schemes such that f is locally of finite presentation. Then f is formally log smooth (resp. log etale, log unramified) if and only if the underlying morphism of schemes f is formally smooth (resp, etale, unramified).
f g Proposition 2.13. ([Ogu18, Theorem IV.3.2.3]). Let X → Y → Z be morphisms of fine log schemes. We have the following exact sequence of OX -modules:
∗ log log log f ΩY/Z → ΩX/Z → ΩX/Y → 0 If f is log smooth then the above sequence is also left exact and locally splits.
Proposition 2.14. ([Ogu18, Corollary IV.3.2.4]). If f : X → Y is a log etale morphism over a log scheme Z, then:
∗ log ∼ log f ΩY/Z = ΩX/Z Remark 2.15. We note that the above proposition in an instance of how log schemes realize Itaka’s philosophy [Mat02, Chapter 2].
35 Proposition 2.16. Let f : X → Y be a morphisms of coherent log schemes. log Then ΩX/Y is a quasi-coherent sheaf on XZar . In particular, if f is log smooth log and f is of finite type, then the OX -module ΩX/Y is locally free of finite type. Proof. The first part is [Ogu18, Corollary IV.1.2.8] and the second part is [Ogu18, Proposition IV.3.2.1]. Theorem 2.17. ([Ogu18, Theorem IV.3.1.8, Corollary VI.3.1.10]). Let R be a ring and θ : P → Q be a homomorphism of monoids and consider the induced mor- phism of log rings θ : Spec(R[Q]) → Spec(R[P ]). Then, the following conditions are equivalent: Q P 1. The induced morphism of log schemes θ : AR → AR is log smooth (resp. log etale) over (Spec(R), triv). 2. The kernel and the torsion part of the cokernel (resp. the kernel and the cokernel) of θgp are finite groups whose order invertible in R.
∗ Q∗ P ∗ 3. The morphism of group schemes f : AR → AR is smooth (resp. etale) over R. See Definition 1.23 for notation. The following example is an immediate consequence of the above theorem. Example 2.18. (Monoid algebras are log smooth). Let k be a field of character- istic zero and let X be the spectrum of the monoid algebra k[P ] where P is finitely generated. By Example 1.43, we can consider Spec(k[P ]) as a log scheme over (Spec(k), triv) where the log structure is induced by the log ring αP : P → k[P ]. By Theorem 2.17, to verify log smoothness we just need to check if the kernel and the torsion part of the cokernel of the map 0 → P gp are finite groups with order invertible over k. Since k is of characteristic zero, every finite group has order which is invertible over k. The kernel is a finite group, namely the trivial group, and since P is a finitely generated abelian group, the torsion subgroup is finite. This means that X is log smooth. Note that if we ask for P to be toric the corresponding variety will be an affine toric variety. Hence, toric varieties are log smooth considered as log schemes. We state two important characterizations of K.Kato [Kat89b] which gives nice conditions about when a morphism of log schemes is log smooth (resp. log etale). First we introduce some terminology in the following remark. Remark 2.19. ([Niz08, Remark 2.3]). Let f : Y → X be a morphism of fs log schemes. When we say that f satisfies classically etale locally, or (cl) etale locally for short, a property P , we mean that there exists an etale covering (φi : Ui → X)i∈I and for every i ∈ I etale coverings (ψj : Vi,j → Y ×X Ui)j∈J , such that the induced morphisms of log schemes Vi,j → Ui satisfies property P , where each Ui,Vi,j is equipped with the inverse image log structure from X and Y respectively, see Example 1.60.
36 Theorem 2.20. ([Kat89b, Theorem 3.5]). Let f : Y → X be a morphisms of fine log schemes. Then, f is log smooth (resp. log etale) if and only if (cl) etale locally on X there exists a chart (P, Q, θ) for f satisfying the following conditions:
1. The induced morphism of schemes Y → X ×Spec(Z[Q]) Spec(Z[P ]) is etale. 2. The kernel and the torsion part of the cokernel (resp. the kernel and the cokernel) of θgp are finite groups with orders invertible on Y . Theorem 2.21. ([Niz08, Definition 2.1]). Let f : Y → X be a morphism of fs log schemes. Then f is log etale if and only if (cl) etale locally on X and Y , there exists a chart (P, Q, θ) for f such that the induced morphisms of schemes:
1. Y → X ×Spec(Z[P ]) Spec(Z[Q]) gp gp 2. Spec(OY [Q ]) → Spec(OY [P ]) are etale. Remark 2.22. For what regards Section5 we are going to introduce the following terminology. We say that a morphism of fs log schemes f : Y → X satisfies the log etale condition if there exists a chart (P, Q, θ) for f (and not a local chart covering) such that the morphisms of schemes:
1. Y → X ×Spec(Z[P ]) Spec(Z[Q]) gp gp 2. Spec(OY [Q ]) → Spec(OY [P ]) are etale. We use this terminology because morphisms that satisfy the log etale condition will enable us to factor any log etale morphism which is of Kummer type, as compositions of strict log etale covers and standard Kummer Galois covers, see Proposition 4.14. Note that for a log etale morphism of fs log schemes f : X → Y the above two theorems are equivalent. The first conditions are the same and for the equivalence of the second conditions we state the relevant part of [Niz08, Lemma 2.5]. Lemma 2.23. Let Y be a nonempty scheme and let φ : G → H be a homomor- phism of finitely generated abelian groups. Then the morphism OY [G] → OY [H] is etale if and only if the kernel and the cokernel of h are finite groups whose orders are invertible on Y . Remark 2.24. Let X be an fs log scheme over (C, triv). Assume that X is log smooth (resp. log etale), then by Theorem 2.20, there exists a chart (P, Q, θ) for f such that the morphism of schemes:
X → Spec(C) ×Spec(Z[Q]) Spec(Z[P ]) is etale. In the case that Q is the trivial monoid, the morphism of schemes X → ∼ Spec(C) ×Spec(Z) Spec(Z[P ]) = Spec(C[P ]) is etale hence the structure morphism of the chart of X, X → Spec(C[P ]) is an etale morphism of schemes.
37 3 Log regularity and Log blow-ups
Logarithmic regularity is a central notion in logarithmic geometry introduced by Kato in [Kat94]. Log regular schemes have toric singularities and essentially are a generalization of toroidal embeddings. In general, we can resolve toric singularities in a similar fashion as in the case of toric varieties using subdivisions of fans [Kat94]. In the context of logarithmic geometry we can also use log blow-ups, which is a class of morphisms similar to toric blow-ups. We refer the reader to [Niz06] for resolving toric singularities via log blow-ups. Surprisingly, log blow-ups are log etale, despite the fact that the underlying morphism of schemes is not even flat, and we will use them in Section4 to define the log etale topology. In general, log blow-ups have many non-expected properties (see for example Lemma 3.23 and Remark 6.15).
Conventions 3.1. In the rest of this section all schemes are defined over a field of characteristic zero.
3.1 Log regularity Definition 3.2. ([Niz08, Definition 2.26]). Let X be scheme equipped with an fs log structure M. We say that (X, M) is log regular at x ∈ X if:
1. OX,x¯ /(Ix¯ ·OX,x¯) is a regular local ring,
gp ∗ 2. dim(OX,x¯) = dim(OX,x¯ /(Ix¯ ·OX,x¯)) + rankZ(MX / OX )x¯ wherex ¯ is any geometric point over x, dim is the Krull dimension of rings [Mat89, ∗ Section 5] and Ix¯ is the ideal generated by MX,x¯ − OX,x¯ in OX,x¯. We say that X is log regular if X is log regular at point x ∈ X.
Remark 3.3. (Singularities and log regularity). We note if a log scheme (X,M) is log regular by Definition 3.2,(1) the singularity locus Sing(X) of X is contained in supp(M).
Remark 3.4. For the notion of log regularity on fine log schemes we refer to [IT14, Section 3.5].
K. Kato in [Kat94] defined log regularity in a slightly different way, since he is using log structures in the Zariski topology. Basically, he asks the same definition but for points in Zariski topology. When we want to refer in the definition of Kato we will say ”...log regular in the sense of Kato...”. In the next remark we say some things about log structures in the Zariski topology and how the notion of log regularity in the sense of Kato compares to the above given one.
Remark 3.5. (Zariski vs etale and log regularity). Let X be a scheme and fin fin consider the categories of fs log structures over X, LogZar(X), Log´et (X), where
38 the subscript indicates in which topology we are considering the log structure. fin For simplicity we will call elements of LogZar(X) Zariski log structures and fin the elements of Log´et (X) etale log structures. Consider the natural functor of sites η : XZar → X´et. From [Ols03, Appendix], we have an induced adjunction in the same spirit as Paragraph 1.39:
log η∗
fin fin Log´et (X) LogZar(X)
∗ ηlog
log ∗ where η∗ restricts an etale log structure to the Zariski site. The functor ηlog is log the left adjoint of η∗ and can been seen as giving the inverse image log structure but for the morphism of sites η : X´et → XZar. In particular, we can see in [Ols03, log ∗ Corollary A.7] that the canonical morphism id → η∗ ◦ ηlog is an isomorphism. fin So we can abuse the language a bit and call an element in Log´et (X) Zariski if it ∗ is in the essential image of ηlog. As we can see in [GR04, Proposition 12.2.3] and [Ols03, Theorem A.1], we have the following equivalence:
η∗ Fine log structures log Fine log structures M on X´et ∗ log ∼ on XZar such that ηlog ◦ η∗ M → M Considering log regularity in this setup, we note that by [GR04, Lemma 12.5.37] an fs log scheme (X, M) with Zariski log structure is log regular at x ∈ X, in the sense of Kato, if and only if for every geometric point ξ¯ over x the log scheme ∗ ¯ (X, ηlog M) is log regular at ξ, in the sense of Definition 3.2. Remark 3.6. ([Kat94, Proposition 8.3]). Log regularity and log smoothness have similar connection as in classical algebraic geometry. Let X be an fs log scheme over (Spec(k), triv) where k is a field and X is locally of finite type over k. Assuming that X is log smooth over (Spec(k), triv) implies that X is log regular. The converse also holds when k is a perfect field. Hence, when we work over fields of characteristic zero the notion of log regularity and log smoothness coincide. The theory of log regular fs log schemes can also be used as a base free analogue of the theory of toroidal embeddings [Kat89b, Example 3.7 (1)]. In this setup, log regular log schemes with Zariski log structures (i.e. Zariski log regular log schemes) correspond to toroidal embeddings without self-intersections i.e., the irreducible components of the divisor associated with the toroidal embedding are normal, and log regular log schemes with etale log structures (i.e. etale log regular log schemes) correspond to toroidal embeddings with self-intersections. The standard source for the theory of toroidal embeddings is [KKMSD73]. For our purposes we will use the definition from [ACM+16].
39 Definition 3.7. A toroidal embedding is a pair (X,U) where X is a normal variety and U is an open dense subset such that for any point p ∈ X there exists an etale neighborhood φp : Vp → X of p, an affine toric variety Xσp with open −1 −1 torus Tp and an etale morphism ψp : Vp → Xσp such that φp (U) = ψ (Tp). We say that the torus embedding is without self-intersections in the case φp is a Zariski neighborhood. Otherwise we say that the torus embedding is with self-intersections. We call D = X − U the divisor associated to the toroidal embedding. The next proposition identifies toroidal embeddings with self-intersections with log smooth fs log schemes. We refer the reader to [Uli15, Sections 4.2 and 4.4] for more details. Proposition 3.8. Let (X,U) be a toroidal embedding over a field k of character- istic zero, and consider the associated log scheme (X, MD), where D := X − U. Then (X, MD) is log smooth over (k, triv). Moreover, every log smooth, fs log scheme over (k, triv) has the structure of a toroidal embedding. Proof. Assume that (X,U) is a toroidal embedding defined over Spec(k) and consider the associated log scheme (X, MD). We want to prove that (X, MD) is log smooth over (Spec(k), triv) so we are using Theorem 2.20. It suffices to find a local chart covering (ψi : Vi → Xσi )i∈I for (X, MD), where Xσi := Spec(k[Pi]) are toric varieties, such that the morphisms of schemes ψi : Vi → Xσi are etale. By the definition of toroidal embeddings, for every p ∈ X we have the following diagram: −1 U φp (U) Tσp λp
inc jp ip
X Vp Xσp φp ψp
where Xσp = Spec(k[Pp]) are toric varieties associated to some toric monoids Pp and φi, ψi are etale morphisms. We will show that the family (ψp : Vp → Xσp )p∈X is the wanted local chart covering for (X, MD). Note that the restriction of the log structure of X on Vp is the divisorial log −1 structure induced by Ep := Vp −φp (V ), so we only need to prove that the induced morphism of log schemes:
ψp :(Vp, MEp ) → (Xσp , MDp) ∗ ∼ is strict i.e., (ψp)log MDp = MEp . ∼ First we claim that MDp |Vp = MEp . Consider an etale morphism f : Y → Vp. Then:
f ψp◦f [MDp |Vp ](Y → Vp) := MDp (Y → Xσp ) :={s ∈ O (Y ) | s| ∈ O∗ (Y × T )} Xσp Y ×Xσp Tσp X Xσp σp
40 ∼ −1 Note that OXσp (Y ) = OVp (Y ) = OY (Y ) and that Y ×Xσp Tσp = Y ×Vp φp (Tσp ) over Y , and so we get that:
f [M | ](Y → V ) :={s ∈ O (Y ) | s| ∈ O∗ (Y × T )} Dp Vp p Xσp Y ×Xσp Tσp X Xσp σp ∗ −1 ={s ∈ O (Y ) | s| −1 ∈ O (Y × φ (T ))} Vp Y ×Vp φ (Tσp ) X Vp σp f = MEp (Y → Vp)
which proves our first claim. Now by [Kat89b, Item 1.4.1], we have that:
∼ ∗ (MDp )|Vp = (ψp)log MDp
∗ which, in combination with our first claim proves that (ψp)log MDp , (MEp ) are isomorphic. Hence, we get that:
∗ ∼ (ψp)log MDp = MEp Now, by Lemma 1.54 we can extend the charts to some etale neighborhoods and by quasi-compactness we can do it in such way that the index set I is finite. The resulting family of morphisms (φ : V → X ) is a chart for (X, M ) satisfying i i σpi i∈I D Theorem 2.20. This makes (X, MD) log smooth over (k, triv). Now starting from a log smooth log scheme (X, M) and using the existence of local chart coverings of Theorem 2.20, we can easily reconstruct the above diagram proving that log smooth schemes are toroidal embeddings.
Remark 3.9. Note that we can always consider toroidal embeddings without self intersections as toroidal embeddings with (trivial) self intersections (as every Zariski inclusion is etale). In the language of log schemes this translates as: Zariski log schemes which are log regular in the sense of Kato are log regular in the above sense. So, the toroidal interpretation of log regular fs log schemes is compatible with Remark 3.5.
Remark 3.10. Let us mention some things about the connection of log regularity, divisorial log structures and toroidal embeddings.
1.([Kat94, Example 1.7]). In Proposition 3.8 we proved that toroidal embed- dings are log smooth log schemes over (Spec(k), triv) and visa-versa. Note that by Remark 3.6 log smoothness implies log regularity and are equivalent when k is perfect, hence:
(X,U) toroidal ⇔ (X, MD) log regular
where D := X − U.
2.([Kat94, Theorem 11.6]). Let ( X, M) be a Zariski log regular log scheme such that X is locally noetherian and let D = X − U be the complement of
41 Fs log schemes Divisorial Log regular
Figure 3: Diagram representation of fs log schemes log, regular log schemes and divisorial log schemes. Note that this is independent of the topology that we use.
the trivial locus of (X, M). Then we can reconstruct the log structure of X ∼ from D, meaning that M = MD, and the log structure on X is divisorial. This means that: (X, M) log regular ⇒ (X, M) divisorial The analogue for etale log regular schemes can be found [Niz06, Proposition 2.6]. Keep in mind that in general:
(X, M) divisorial ; (X, M) log regular since a log scheme (X, M) such that, the support of the log structure does not contain the singular locus of X, can never be log regular: (X, M) log regular ⇒ Sing(X) ⊂ supp(M) see Remark 3.3. Figure3 depicts what we said above. Note that we will show examples of Divisorial log schemes which are not coherent in the next subsection.
Example 3.11. (An etale log regular log scheme which is not log regular in the sense of Kato). We will give an example of a scheme equipped with a log structure defined in the etale topology (X, M) which is log regular such that log (X, η∗ M) is not log regular in the sense of Kato. Consider X := Spec(C[x, y]) and M to be the divisorial log structure associated to the normal crossings divisor 2 2 3 log D := Z(y − (x − x )). Note that the log scheme (X, η∗ MD) has the log structure given by the pre-log structure n 7→ (y2 − (x2 − x3))n. Since the variety 2 2 3 ∼ D has a singularity at p := (1, 0) the local ring C[x, y]p/((y −(x −x ))·C[x, y]p) = 2 2 3 log C[x, y]/(y −(x −x ) is not regular hence and the Zariski log scheme (X, η∗ MD) is not log regular since the first condition of the definition is not satisfied. On the other hand the etale log scheme (X, MD) is log regular in our sense, since D is a NCD [GR04, Example 12.2.11] and X is smooth. This holds in general meaning that if X is smooth and D is a NCD in X then the etale log scheme (X, MD) is log smooth [Uli15, Example I.4.22].
42 2 Example 3.12. (The nodal curve is NCD). Let X = Ak where k is algebraically closed field and consider the nodal curve D = Z(y2−(x2−x3)) in X. Note that D is a normal crossings divisor since we have the etale morphism Spec(C[y, t±1]) → X defined by the map in the level of rings y 7→ y and x 7→ (1 − t2), pulling back D to a simple normal crossings divisor.
3.2 Divisorial log structures In this subsection we say some things about divisorial log structures and their formation in the Zariski and in the etale topology. In general, many things might go wrong when forming divisorial log structures. First of all, they might not be coherent or their coherence might be dependent on the topology that we use as we can see in Example 3.16. The picture is much nicer if the divisorial log structure induces a log regular scheme as we can see in Lemma 3.15. Remark 3.13. (Divisorial log structures Zariski vs Etale, [Ogu18, Proposition III.1.6.5]). Let X be a scheme and consider i : U → X an open immersion such that D := X − U is of pure codimension one. Then we can form the divisorial log structure MD,Zar, in the Zariski topology, and the divisorial log structure MD,´et, ∗ in the etale topology. In general, the natural morphism ηlog MD,Zar → MD,´et is not an isomorphism, as we will see in Example 3.16, where η : XZar → X´et is the natural functor from the (small) Zariski site to the (small) etale site. Sufficient conditions for the above morphism to be an isomorphism are 1. X is locally noetherian and normal, 2. U is containing all of its associoated points, 3. every irreducible component of D is geometrically unibranched [Gro65], ∗ ∼ When the above conditions are satisfied we get ηlog MD,Zar = MD,´et. In other words the divisorial log structure MD,´et is Zariski (accessible). Remark 3.14. (Coherence of divisorial log structures, [Ogu18, Proposition III.1.9.1]). We give sufficient conditions for a divisorial log structure to be coherent. Suppose that: 1. X is locally noetherian, normal and locally factorial; 2. U is dense in X, then MD,Zar is fine. Note that, in the etale topology, we need an extra condi- tion, namely, that there exists an etale cover f : Y → X such that f −1(D) is unibranched. We also note by [GR04, Example 12.2.11] that, if X is noetherian and D is a normal crossings divisor regularly embedded in X then, (X, MD,´et) is fine. As we can see in [Ogu18, Proposition III.1.2.8], the same is true for a simple normal crossings and MD,Zar assuming that X is regular.
43 We give the following Proposition which is similar to Remark 3.13. It states that the formation of a divisorial log structure is independent of the topology that we use as soon as the resulting log scheme is log regular.
Lemma 3.15. (Log regularity in etale and Zariski topology vs Divisorial). Let (X, M) be a Zariski fs log scheme equipped with a divisorial log structure M := MD,Zar such that X is log regular in the sense of Kato. Then, we have that ∗ ∼ ηlog MD,Zar = MD,´et. Proof. We make the following observations:
1. We can define the following morphism of ringed sites equipped with log struc- tures: 0 ∗ η :(X´et, ηlog MD,Zar) → (XZar, MD,Zar)
which is strict since X´et is equipped with the inverse log structure induced by ∗ MD,Zar. Note that the log ringed site (X´et, ηlog MD,Zar) is equivalent, as log ∗ ringed sites, to the log site induced by the etale log scheme (X, ηlog MD,Zar).
2. Since (X, MD,Zar) is log regular in the sense of Kato we have by Remark 3.5 ∗ that (X, ηlog MD,Zar) is log regular in the sense of Definition 3.2.
∗ 3. By [Ols03, Proposition A.10] we have that ηlog MD,Zar is a fine log structure on X and arguing similarly as [GR04, Lemma 12.1.14] the fine log scheme ∗ (X, ηlog MD,Zar) is also saturated.
∗ By the second assertion, (X, ηlog MD,Zar) is log regular hence by Remark 3.10 ∗ (2) the log structure ηlog MD,Zar is divisorial. So, we just need to show that the ∗ triviality locus (X´et, ηlog MD,Zar)triv is U := X − D = (XZar, MD,Zar)triv. 0 ∗ Since η :(X´et, ηlog MD,Zar) → (XZar, MD,Zar) is a strict morphism of log sites, it is local and we have the analogous statement as in [GR04, Remark 12.2.8] for ∗ log sites hence we get the inclusion (X´et, ηlog MD,Zar)triv ⊂ (XZar, MD,Zar)triv. For the other implication consider the logarithmification processes in the for- ∗ mation of ηlog MD,Zar, in particular this is the sheafification of the following pushout: (η−1 O∗ → η−1 M , η−1 O∗ → O∗ ) XZar D,Zar XZar X´et ∼ Note that for any geometric point ξ lying above a Zariski point p with MD,Zar,p = O∗ , the log structure η∗ M is trivial since by localizing at p the first XZar,p log D,Zar component of the above pushout is the identity. This gives us the other inclusion. Having said that, if the Zariski fs log scheme (X, MD,Zar) is log regular in the sense ∗ ∼ of Kato, then ηlog MD,Zar = MD,´et. Again this is compatible with the toroidal interpretation of fs log regular log schemes.
Example 3.16. We note counterexample from [GR04, Example 12.2.12] of a divisorial log structure which is not fine in the etale topology, but it is fine in the Zariski topology.
44 2 Let k be an algebraically closed field, and we set C ⊂ Ak to be the nodal cubic. 3 Consider X := Ak and let D ⊂ X be the reduced, affine cone over C with vertex 3 p ∈ Ak. Then, away from p the divisor D is a normal crossings divisor and so the log structure is fine on X − {p} by [GR04, Example 12.2.11]. A closer inspection on the origin p shows that the log structure cannot be fine over p. Since the full argument is extensive, we refer the reader to [GR04]. We note that this serves as a counterexample of a divisorial log structure with ∗ ηlog MD,Zar MD,´et. To see this, since MD,Zar is fine then by [Ols03, Proposition ∗ A.10] the log structure ηlog MD,Zar is fine as well, but since MD,´et is not fine we ∗ conclude that ηlog MD,Zar MD,´et and by the Lemma 3.15 we conclude that (X, MD,Zar) is also not log regular in the sense of Kato.
3.3 Log blow-ups Log blow-ups are analogues of blow-ups in the logarithmic category. In the log- arithmic setup we restrict ourselves to blow-up of ideals coming from the log structure and then, we equip the resulting scheme with a log structure. Again we have similar universal properties as in the classical case but there are huge differences how log blow-ups behave under fs fiber products and what phenom- ena they induce. There are two ways to define log blow-ups, both defined by K. Kato, mainly with Kato fans as it is done in [Kat94] and hands on, as it is done in [Niz06], [Kat99]. Here we are going to use the hands on approach. For connections between the two approaches we refer to [Niz06].
Notation 3.17. In the rest of this subsection, for a fs log scheme X we denote with OXτ for τ ∈ {Zar, ´et} the structure sheaf of X in the τ-topology and any reference on sites and cohomology is for the small τ-site of X.
Definition 3.18. Let X be an fs log scheme. A subsheaf I of MX is called a log ideal if the stalk Ix¯ is an ideal of MX,x¯ for all x ∈ X. 3.19. (Definition of log blow-ups, [Nak17, Section II]). We we will use multiplica- tive notation for the monoids in this paragraph. We describe the construction of log blow-ups and we start by blowing up log schemes which come from log rings. Consider the log scheme X := Spec(Z[P ]) where P is an fs monoid and X is equipped with the natural log structure induced by P . Take I an ideal in P and denote with I the ideal generated by I in Z[P ]. We denote with XI the scheme theoretic blow-up along I. The scheme BlowI (X) comes naturally with a fine log structure which we will describe now. n The underlying scheme of the log blow-up BlowI (X) := Proj(⊕ I ) has an affine open covering of the form:
−1 {Spec(Z[P ha Ii])}a∈I
45 then we equip Spec(Z[P ha−1Ii]) with the log structure induced by P ha−1Ii → Z[P ha−1Ii] where P ha−1Ii is the saturation of monoid generated by i ∈ P gp, i ∈ I a
gp inside P . So we glue up the log structure to obtain a fine log structure on XI . Since the log structure might not be fs we need to saturate. Now we do the general case. Let X be an fs log scheme equipped with a local chart covering (uj : Uj → X,Pj, θj)j∈J . A morphism of fs log schemes f : Y → X is a log blow-up if for every j ∈ J there exists ideals Ij in Pj such that the base change of the log blow-up πj : XIj → Spec(Z[Pj]) along Uj → Spec(Z[Pj]) fs coincides with the base changed morphism Uj ×X Y → Uj. Note that for every j ∈ J the log scheme Uj is equipped with the pullback log structure induced by uj, see Example 1.60. Hence, by Proposition 1.68 we can omit the superscript fs in the fiber products. We also note that log blow-ups are log etale [Nak17, Proposition 2.6 (1)].
Remark 3.20. There is an equivalent way to define log blow-ups by introducing the notion of coherent ideals [Kat99, Definitions 3.6 and 3.8]. Loosely speaking, coherent log ideals are log ideals which are accessible by the local chart covering of the log scheme. Since we are not going deep to the theory of log blow-ups we use the definition given above.
We give the following example of a log blow-up.
Example 3.21. (Log blow-up of the affine plane). Consider the affine plane equipped with the divisorial log structure induced by the union of the two co- ordinate axis from Example 1.26 and denote it by X. We remind that this is the log scheme associated to the log ring α : N2 → k[x, y]. Consider the ideal I := N2 − {(0, 0)} of N2 which is generated by x := (1, 0) and y := (0, 1) i.e., I := (x+N2)+(y+N2). We denote with m the ideal generated by I in k[x, y]. The n 2 log blow-up Xm has as underlying scheme Proj(⊕m ) which is the blow-up of Ak at along m. Note that this is the toric blow-up constructed in Example 1.28. Let us explain the log structure on the affine pieces Uω,Uσ. The associated monoids, in additive notation, are:
2 2 N h−x + mi = Nx ⊕ Ny−x ⊂ Z
2 2 N h−y + mi = Ny ⊕ Nx−y ⊂ Z and we can write the associated log rings by:
−1 Nx ⊕ Ny−x → k[x, yx ] = k[Uω]
−1 Ny ⊕ Nx−y → k[y, xy ] = k[Uσ]
46 • • • ◦ ◦ • • • [ πω • • • ,→ ◦ • • • • • • • • •
Figure 4: Local picture of the log blow-up π : Xm → X.
By restricting to one of the pieces of the cover we have the following picture in the level of log rings:
[ πω Nx ⊕ Ny Nx ⊕ Ny−x
α αω
k[x, y] k[x, yx−1] πω
[ where πω(n, m) = (n + m, m), see Figure4. We note that that the log structure of the log blow-up is different from the ∗ pullback log structure πlog MX . Otherwise, this would mean that the morphism of log schemes π : Xm → X is strict and since log blow-ups are log etale the underlying morphism of schemes would have been etale, which is definitely not the case.
We have the following properties.
Lemma 3.22. Composition and fs base change of log blow ups are log blow ups.
Proof. For composition of log blow ups see [Niz06, Corollary 4.11] or [Nak17, Proposition 2.5]. For fs base change see [Niz06, Corollary 4.8].
Lemma 3.23. ([Nak17, Proposition 2.7]). Let X be an fs log scheme and let π : X˜ → X be a log blow up of X. Then the following commutative diagram:
idX˜ X˜ X˜
idX˜ π
˜ X π X
˜ fs ˜ ∼ ˜ is cartesian in the category of fs log schemes i.e. X ×X X = X.
47 Remark 3.24. Let π : X˜ → X be a log blow up and consider a morphism of fs log schemes f : Y → X˜. We argue that the following diagram is fs cartesian:
f Y X˜
idY π
Y X f ◦ π
To see this, note that both of the inner squares, of the following commutative diagram, are fs cartesian:
f idX˜ Y X˜ X˜
idY idX˜ π f π Y X˜ X and the same is true for the outer square. In the view of Remark 3.5 we can ask if performing log blow ups in the category of fs log schemes for a Zariski log scheme over a Zariski log ideal, will give us a Zariski log scheme. The answer is affirmative as the following proposition states.
Proposition 3.25. ([Niz06, Proposition 4.5]). Let (X, MX ) be a Zariski fs log scheme and let J ⊂ MX be a Zariski coherent ideal. Let (Y, MY ) be the log blow- up of the coherent ideal J ·MX . Then the etale log scheme (Y, MY ) is Zariski log and the corresponding Zariski log scheme (Y, η∗ MY ) is uniquely isomorphic over log (X, η∗ MX ) to the log blow-up of J. The following theorem is just a reformulation of a theorem of Kato in the language of log blow-ups. Theorem 3.26. ([Kat94, Theorem 11.3]). Let X be a Zariski log regular fs log scheme, then for every Zariski log blow up π : Y → X:
OX´et → Rπ´et∗ OY´et is a quasi-isomorphism. We note that in [Kat94] the above theorem is stated for the Zariski topol- ogy. Using the etale topology makes no difference, since the derived categories of quasi-coherent sheaves on the etale and the Zariski site are equivalent [Sta18, Sect. 071Q]. We now prove the analogous statement for etale log regular log schemes.
48 Theorem 3.27. (Analogue of [Kat94, Theorem 11.3] for etale log schemes). Let X be an etale log regular log scheme, then for any etale log blow-up π : Y → X:
OX´et → Rπ´et∗ OY´et is a quasi-isomorphism.
Proof. By [Uli15, Example I.4.22], we can find an etale cover f : U → X of X such ∗ that f pullbacks the etale log structure MX to a Zariski log structure flog MX on U. Now, consider the fs-pullback:
g fs U ×X Y Y
s π
U X f where U is equipped with the inverse log structure via f. Then since f is strict we use Proposition 1.68 to omit the superscript fs from the fiber product. The projec- tion s is a log blow up, because log blow ups are preserved under fs-pullbacks, and by Proposition 3.25 the log scheme U ×X Y is also Zariski. Note also that U ×X Y is log regular because U ×X Y → Y is strict log etale and Y → (Spec(k), triv) is log smooth. Since U is a Zariski log scheme, we can use Theorem 3.26 and to get that the homomorphism:
OU´et → Rs´et∗ O(U×X Y )´et is a quasi-isomorphism. Now we can use etale descent and conclude that the homomorphism:
OX´et → Rπ´et∗ OY´et is also a quasi-isomorphism. In order to see that, note that f is surjective etale, in particular faithfully flat, hence it reflects isomorphisms in the derived category + D (U´et). Having said that, we have the following:
! ∼ ∗ ∗ ∼ ∗ ∼ OU´et = f´et OX´et → f´etRπ´et∗ OY = Rs´et∗(g OY´et ) = Rs´et∗ O(U×X Y )´et where ! comes from flat base change [Sta18, Sect. 02KH], since f is flat and the above commutative diagram is a pullback diagram in the category of schemes.
Since OU´et → Rs´et∗ O(U×X Y )´et is a quasi-isomorphism the same is true for OX´et →
Rπ´et∗ OY´et and the proof is complete. Theorem 3.28. Log regular fs log schemes have rational singularities.
49 Proof. Since for every log regular fs log scheme X we can resolve singularities via log blow-ups [Niz06, Theorem 5.10], and since the underlying scheme of a log regular fs log scheme is normal [GR04, Corollary 12.5.29], we can use Theorem 3.27 to conclude that toroidal embeddings with self-intersections have rational singularities i.e., X is normal and there exist a proper birational map f : Y → X where Y is a regular scheme such that OXZar → Rf´et ∗ OYZar is a quasi-isomorphism in the derived category. For the birationality see [GR04, Corollary 12.6.53].
50 4 Log sites
In this section we give the basic definitions regarding the theory of logarithmic sites. The main goal is to prepare the setup for what comes in Sections5,6. We will define the Kummer log etale topology as in [Nak97] and the log etale topology as in [Nak17]. We also define the various functors of sites between logarithmic sites. In general we have in mind the following: In Section5 we will prove various descent arguments for sheaves defined over logarithmic sites. This will require to transfer sheaves from the (cl) etale site to the logarithmic sites. Then we will need to check the sheaf condition. This is done in the following way; we find an appropriate class of morphisms that generate the logarithmic topologies and then the sheaf condition becomes easy to verify. In what follows, we write the details of the arguments appearing in the liter- ature. The root of the ideas can be found in [Kat89a]. The theory of Kummer log etale cohomology was formalized by Nakayama in [Nak97] and the descent statements can be found in [Niz08], and [Hag03]. For the theory of logarithmic cohomology for the full log etale topology we refer the reader to [Nak17].
Conventions 4.1. We will use X to denote an fs log scheme, X to denote the underlying scheme and MX to denote the log structure. In order to avoid confu- sion we emphasize that we will denote with X´et the (small) etale site associated to X. This is natural to do and we explain why in Example 4.19. We also assume that our log schemes have underlying schemes which are of finite type over a field of characteristic zero.
Before defining the Kummer log etale topology we introduce some terminology.
Definition 4.2. (Strict log etale neighborhoods). Let X be an fs log scheme and pick x ∈ X.A strict log etale neighborhood of x is a strict log etale morphism U → X such that the underlying morphism of schemes U → X is an etale neighborhood of x, see [Sta18, Sect. 02LE].
Remark 4.3. (Strict log etale covers). Let f : Y → X be a morphism of fs log schemes where the underlying morphism of schemes f is surjective. We say that (f : Y → X) is a strict log etale cover of X, if f is strict, see Subsection 1.7, and log etale. A standard way to construct strict log etale covers is the following: Consider an etale covering of schemes (g : V → X). Then, in Example 1.60 we constructed ∗ the associated strict morphism of log schemesg ˜ :(V, glog MX ) → X. Note that sinceg ˜ is strict andg ˜ := g is etale we use Proposition 2.12 to conclude thatg ˜ is log etale.
Remark 4.4. Since we defined the notion of a strict log etale cover we can rephrase Remark 2.19. A morphism of log schemes f : Y → X satisfies a property P classically etale locally on X and Y means that there exists a strict log etale
51 cover (Ui → X)i∈I and for every i ∈ I a strict log etale cover (Vi,j → Ui ×X Y )j∈J such that the induced morphisms Vi,j → Ui satisfies the property P for all i ∈ I and j ∈ J.
4.1 The Kummer log etale topology Definition 4.5. ([Nak97, Definition 2.2]). Let X be an fs log scheme, we define fs the Kummer log etale site Xket to be the full subcategory of Log /X with objects (f : Y → X) where Y is an fs log scheme over X and f is Kummer log etale (Definition 1.62) and whose coverings are of the form (ui : Yi → Y )i∈I where ui are Kummer log etale morphisms over X and Y = ∪i∈I ui(Yi) (set theoretically).
Caution 4.6. The fiber product on Xket is the fs fiber product described in Remark 1.52.
Remark 4.7. We also have that for any two objects (Y → X), (Z → X) in Xket, every morphism in MorXket (Y,Z) is Kummer log etale, [Ill02, Section 2.1]. In order to prove that the above definition gives actually a Grothendieck site we need show the stability of surjectivity under fs base change for surjective Kummer log etale morphisms. This is proved by the Fourth point lemma which is due to Nakayama and can be found in [Nak97], [Ill02], [Niz08].
Proposition 4.8. (Fourth point lemma). Let f : Y → X be a morphism of log schemes that is Kummer and surjective, then for any log scheme Z → X the fs morphism Y ×X Z → Z is surjective. In fact, for any y ∈ Y , z ∈ Z having the fs same image in X there exists an element t ∈ Y ×X Z mapping to y, z. Remark 4.9. Since Kummer log etale morphisms are stable under fs fiber prod- ucts (Remark 1.69, Lemma 2.11) and the ”Fourth point lemma” guarantee the stability of surjectivity under fs fiber products, we conclude that surjective Kum- mer log etale morphisms are stable under fs-pullbacks and indeed the Kummer log etale topology is well defined.
We want to find nice refinements for the Kummer log etale coverings. For this, we state the following.
Proposition 4.10. ([Niz08, Lemma 2.8]). Let f : Y → X be a morphisms of fs log schemes and let (P, θ) be a chart for X. Assume that f is log etale. Then, (cl) etale locally on X and Y , there exists a chart (P, Q, φ) for f including (P, θ) and satisfies the log etale condition, see Remark 2.22. Furthermore we can require that θgp : P gp → Qgp is injective.
Remark 4.11. We note that the monoid Q in the above chart is always integral. This is true, because Q is defined as a fiber product of integral monoids, see the proof of [Kat89a, Lemma 3.1.6].
52 Remark 4.12. Note that the above proposition has in mind general log etale morphisms and not only Kummer log etale morphisms. This is the basic reason that we do not see a conclusion stating ”θ is exact...” or ”θ is Kummer...”. If that was the case then choosing f to be a log blow-up and applying the above proposition would imply that f is an isomorphism since log blow-ups which are exact are isomorphisms [Nak17, Proposition 2.6 (4)]. Having said that, we will prove in the first part of the proof of the following proposition that, in the case that f is Kummer then θ is exact and a fortiory Kummer.
Remark 4.13. We say some things about the importance of Proposition 4.10. Note that, not every chart for a log etale morphism satisfies the log etale condition from Remark 2.22. For a counterexample we set X = Y = (Spec(C[x±1]), triv) and consider f : X → Y the identity morphism on the level of log schemes. We equip both log schemes with the following charts: On Y we consider the trivial chart and on X we take the chart P = (N, θ) where θ : N → C[x±1] is defined by n 7→ xn. Note that the logarithmification P log gives the trivial log structure on X, since the image of θ is included in (C[x±1])∗. So we can consider the induced chart for f given by (triv, N, inc) where inc : {0} ,→ N is the natural inclusion. If this chart was satisfying the log etale condition then we would get that the morphism of schemes X → Y × 1 is etale which is false for dimension reasons. AC The following proposition is the main ingredient that we will use for the exis- tence of good refinements for Kummer log etale covers.
Proposition 4.14. ([Niz08, Proposition 2.15]). Let f : Y → X be a Kummer log etale morphism and let y ∈ Y and f(y) := t ∈ X. Let P be a chart for MX neat at t, see Definition 1.55. Then after replacing X and Y by some strict log etale neighborhoods of y, t and there exists a commutative diagram:
h T Xn
g θn f Y X
where y is in the image of g, h is strict log etale, θn is a standard Kummer Galois cover of X and n is invertible on X. Moreover, h is surjective if f is surjective.
Proof. Pick y, t, on Y and X respectively, as above. By Proposition 4.10 we replace X,Y by strict log etale neighborhoods of y, t respectively and extend the chart P to a chart (P, Q, θ) for f which satisfies the log etale condition. Consider the map θ : P → Q. We argue that this is a Kummer homomorphism. In order to prove that we will use the characterization of Kummer homomorphisms of Lemma 1.17. Note that since the chart P is neat at some point, it is integral and
53 the same is true for Q, see Remark 4.11. By Proposition 4.10, we can choose θgp to be injective, which implies that θ is injective. Using the equivalent characterization of log etale morphisms of Theorem 2.20, the homomorphism θgp has finite cokernel, which is equivalent for θgp to be Q-surjective. According to Lemma 1.17, in order to prove that θ is of Kummer type we just need to prove that θ is exact. Let us pick geometric points t,¯ y¯ above t, y respectively. Note that by the assumption of neatness, φ : P → MX,t¯ is an isomorphism. Consider the following commutative diagram:
ψ Q MY,y¯
¯[ θ fy¯ φ P MX,t¯
¯[ By Lemma 1.17, Kummer morphisms are exact. Since fy¯ and ψ ◦ θ are Kummer, they are exact as well. Now we use Lemma 1.16 (2) to conclude that θ is also exact. Hence θ is of Kummer type and by Remark 1.13 we have a factorization P → Q → Pn for some n invertible on X. Consider the following commutative diagram:
h c T Xn Spec(Z[Pn])
k e
g b XQ Spec(Z[Q]) ψ i d f a Y X Spec(Z[P ])
where XQ := X ×Spec(Z[P ]) Spec(Z[Q]), Xn := X ×Spec(Z[P ]) Spec(Z[Pn]) and T :=
Y ×XQ Xn. Note that the map ψ is the strict log etale morphism given by the log etale condition, see Remark 2.22. The upper left rectangle is Cartesian hence h is strict log etale since ψ is strict log etale by the log etale condition. Since the right square are cartesian, the composition i ◦ k = θn : Xn → X is a Kummer Galois cover of X. Also note that the morphisms e, d are surjective, since they are standard Kummer Galois covers, and hence k, i are also surjective and the same is true for g. This completes the first part of the statement. For the second part assume that f is surjective. We will prove that h is also ∼ surjective. By using the universal properties of fiber products T := Y ×XQ Xn = Y ×X Xn. Hence, h is the fs pullback of a surjective morphism and by the fourth
54 point lemma h is surjective. Remark 4.15. We can use the first part of the above proposition in order to give an equivalent definition for Kummer morphisms of fs log schemes. A morphism of fs log schemes f : Y → X is of Kummer type if and only if (cl) etale locally on X and Y there exists a chart (P, Q, θ) for f such that θ is of Kummer type. Lemma 4.16. Let f : Y → X be a Kummer log etale morphism of fs log schemes which is surjective. Then for every x ∈ X there exists a point y ∈ Y such that f(y) = x and strict log etale neighborhoods U, V of x, y respectively, such that f|V : V → U is surjective Kummer log etale, U and V are affine and f|V satisfies the log etale condition, see Remark 2.22. Proof. Pick x ∈ X, then by Proposition 1.57 we can find a strict log etale neigh- borhood Ux of x and a neat chart P for x. Consider the Kummer log etale surjective morphism f : Y ×X Ux → Ux, and a point y ∈ Y ×X Ux above x. Then by Proposition 4.10 we can replace Ux and Y ×X Ux with strict log etale neighborhoods of x, y respectively and extend P the chart to a chart for f which satisfies the log etale condition. Now we can further restrict to a strict log etale 0 0 neighborhood Ux of x such that Ux → Ux is an affine Zariski neighborhood of x 0 0 and to a strict log etale neighborhood Vx,y of y such that Vx,y → Vx,y is a Zariski neighborhood of y with f : Vx,y → Ux being surjective. Note that f : Vx,y → Ux will still satisfy the log etale condition. Proposition 4.17. (Good refinements). Let (f : Y → X) a be Kummer log etale cover. Then we can refine this cover as follows:
θ hi,j i,nj φi (Ti,j → Ui,nj → Ui → X)i∈I,j∈J where for every i ∈ I, j ∈ J the morphisms φi, hi,j are strict log etale morphisms and θi,nj are standard Kummer Galois covers. Moreover, we can pick the index sets I,J to be finite sets. Proof. To see this, by Lemma 4.16, for every x ∈ X and y ∈ Y we can find strict log etale neighborhoods Ux,Vx,y of x, y such that for every x ∈ X, y ∈ Y we have the following:
1. the scheme Ux is affine,
2. the restriction f|Vx,y : Vx,y → Ux is surjective. Then by Proposition 4.14 we can further refine to:
hx,y Tx,y Ux,ny
g x,y θx,ny
Vx,y Ux
55 for some ny invertible on Ux. Now consider the covering
hx,y θx,ny φx Uf := (Tx,y → Ux,ny → Ux → X)x∈X,y∈Y
fin and note that since X is quasi-compact, we can find a finite subcover Uf of Uf . Definition 4.18. For a Kummer log etale morphism f : Y → X we call the fin refinement Uf of the above proposition a good refinement for f. Good refinments will be essential for our arguments in log etale descent. In the rest of this subsection we give some examples of logarithmic sites and functors of logarithmic sites.
Example 4.19. (The strict log etale site). Let X be a log scheme and consider the category of strict log etale morphisms of fs log schemes over X which we denote with Xslet. We equip Xslet with the strict log etale topology which has covers strict log etale covers, see Remark 4.3. Then by Proposition 2.12 for every element (f : Y → X) of Xslet, the under- lying morphism f is etale and every etale morphism g : Z → X gives an strict 0 ∗ log etale morphism (g, g ):(Z, glog MX ) → X. Hence, the sites (X)´et and Xslet are isomorphic by the forgetting the log structure functor, and we identify them through this isomorphism. Having said that, we will denote the strict log etale site of X by X´et and for any morphism of log schemes f : Y → X we denote the induced functor on the etale sites by f´et : X´et → Y´et. We continue with some basic examples of functors between logarithmic sites.
Example 4.20. Let X be an fs log scheme and consider the functor X : X´et → Xket which is defined by the assignment:
˜ ∗ X :(f : Y → X) 7→ (f :(Y, flog MX ) → X) where f˜ is the strict morphism of log schemes induced by f, see Example 1.60. Since f˜ is strict and the underlying morphism of schemes f˜ = f is etale we have by Proposition 2.12 that the morphism of log schemes f˜ is log etale, and it is of Kummer type since it is strict, see Lemma 1.63. Hence X is well defined. This functor is continuous because it maps etale covers over X to strict log etale covers, and preserves the fiber products since only strict morphisms are involved, see Proposition 1.68. We call X as a functor of sites the strict inclusion and as a morphism of sites it can be found in the literature as the natural projection X : Xket → X´et. When confusion does not arise, we abbreviate the index and write this functor as .
56 Remark 4.21. Let X be an fs log scheme equipped with the trivial log structure. ∼ We prove that Xket = X´et. Consider an object f : Y → X in Xket. Then by the definition of Kummer morphisms of log schemes, for every geometric pointx ¯ on X: [ f x¯ :(MX )f(x) → (MY )x¯ is a Kummer homomorphism of monoids. Since MX is trivial we can write the above homomorphism as: [ f x¯ : {1} → (MY )x¯ n The Kummer condition implies that, there exists an n ∈ N≥1 such that (MY )x¯ is the trivial monoid. By Remark 1.6, a finitely generated integral monoid of finite order is an abelian group, hence:
∗ (MY )x¯ = ((MY )x¯) = {1} since the monoid (MY )x¯ is sharp. This means that MY is the trivial log structure on Y and f is etale. This proves our claim. Example 4.22. (Usual morphisms of log sites). Let f : X → Y be a morphism of fs log schemes. Then, by fs base changing along f we get a morphism of sites fket : Yket → Xket defined by:
0 fs 0 fs fket :(u : Y → Y ) 7→ (u ×X f : Y ×Y X → X)
The functor fket preserves surjective Kummer log etale morphisms by Remark 4.9 and maps Kummer log etale coverings over Y to Kummer log etale coverings over X. By definition, it commutes with fs pullbacks, hence it is continuous. Example 4.23. (Ket restriction). Let f : Y → X be a Kummer morphism of fs log schemes. Then the continuous functor of sites fket has a left adjoint ˜ fket : Yket → Xket defined by: ˜ 0 0 fket :(u : Y → Y ) 7→ (f ◦ u : Y → X) This is well defined since composition of Kummer morphisms is Kummer. We call this morphism the ket restriction functor from Yket to Xket which is con- tinuous and cocontinuous [Sta18, Sect. 00XZ], and it is left adjoint to fket [Sta18,
Sect. 03CE]. We sometimes use the notation F |Yket for the restriction of the sheaf F ∈ Sh(Xket) on Yket.
4.2 The log etale topology The following definition is from [Nak17]. Let X be an fs log scheme.
Definition 4.24. We define the log etale site of X to be the site Xlet which has as objects the log etale morphisms over X, and morphisms the X-morphisms of fs log schemes. The covering families are morphisms f over X such that f is universally surjective with respect to the fs-pullback.
57 Definition 4.25. A surjective morphism of fs log schemes f : Y → X is univer- sally surjective with respect to the fs fiber product if for any morphism of fs log fs schemes g : Z → X the projection Z ×X Y → Z is again surjective. Remark 4.26. We defined the full log etale topology to have as coverings uni- versally surjective morphisms. The reason is that surjectivity is not preserved by fs base change, see [Nak17, Section 2.8]. Some examples of universally surjective morphisms are Kummer log etale covers, see Proposition 4.8, and log blow-ups, see Lemma 3.22. Remark 4.27. Nakayama in [Nak17, Proposition 3.9] proved that the full log etale topology is generated by Kummer log etale morphisms and log blow-ups. The main idea is that for any log etale morphism f : Y → X, where Y is quasi- compact, we can find, (cl) etale locally on X,Y , a chart (θ, P, Q) for f and a log ideal I of P such that the following commutative diagram is fs-cartesian:
fs h Y ×X XI XI
π
Y X and h is of Kummer type. Note that this enables us to construct a refinement of f similar as in Remark 4.17 which we will use for full log etale descent in Theorem 5.21. In general, we can factor similarly any morphism of fs log schemes with the difference that h will not be of Kummer type but exact [Nak17, Lemma 3.10]. In our case, since the objects of Xlet are log etale morphisms every morphism over X will be also log etale and hence we can factor any universally surjective cover as a composition of Kummer log etale and a log blow up cover. Example 4.28. Let X be an fs log scheme, then we have the following functor of sites ρX : X´et → Xlet which is defined by: ˜ ∗ ρX :(f : Y → X) 7→ (f :(Y, flog MX ) → X) We argue similarly as in Example 4.20 to show that this is a continuous functor of sites. Example 4.29. Let X be an fs log scheme, then we have the following functor of sites κX : Xket → Xlet which is defined by:
κX :(f : Y → X) 7→ (f : Y → X)
Since every ket morphism is a let morphism, see Remark 4.26, κX respects the coverings of Xket and since the underlining functor of categories κX is fully faithful it respects the fs-fiber products, hence is a continuous functor.
58 Remark 4.30. Assembling the functors of Examples 4.20, 4.28, 4.29 we have the following commutative diagram of sites:
X X´et Xket
ρX κX
Xlet
Note that this diagram is natural in X.
Example 4.31. (Usual functor of sites). Let f : X → Y be a morphism of fs log schemes and consider the functor flet : Ylet → Xlet defined by:
0 fs 0 fs flet :(u : Y → Y ) 7→ (u ×Y f : Y ×Y X → X)
Arguing similarly with Example 4.22 we get that flet is a continuous functor.
Example 4.32. (Let restriction). Let π : Y → X be an element of Xlet. Then we have another functorπ ˜let : Ylet → Xlet defined by:
0 0 π˜let :(u : Y → Y ) 7→ (π ◦ u : Y → X)
Note that since π, u are log etale their composition is also log etale by Proposition 2.10, hence the functor is well defined and maps the coverings of Ylet to coverings of Xlet. Arguing exactly as in Example 4.23,π ˜let is continuous and cocontinuous and left adjoint to πlet. We sometimes use the notation F |Ylet for the restriction of the sheaf F ∈ Sh(Xlet) on Ylet. Remark 4.33. To sum up, using Proposition 4.17 and Remark 4.27:
1. The Kummer log etale topology is generated by strict log etale covers and standard Galois covers.
2. The full log etale topology is generated by ket coverings and log blow ups.
59 5 Sheaves on Log sites
5.1 Intoduction Notation 5.1. What we say in this subsection is true for any logarithmic topol- ogy, so pick τ ∈ {ket, let} and we denote with λ the corresponding functor λ : X´et → Xτ i.e., λ = for τ = ket, defined in Example 4.20, and λ = ρ for τ = let, defined in Example 4.28. 5.2. (Cohomology of modules for log sites). Let X be an fs log scheme and let τ be a logarithmic topology. Then, we can consider the following presheaf of rings on Xτ : f OXτ :(Y → X) 7→ Γ(Y , OY´et ) which is a sheaf as we will prove in Corollary 5.12. Assume for now that OXτ is a sheaf of rings on Xτ . Then, we can consider the ringed topos (Sh(Xτ ), OXτ ) and the ringed topos (Sh(pt),R) where R := Γ(Xτ , OXτ ). Now we apply the usual theory of ringed toposes from [Sta18, Sect. 03A4]. We have a morphism of ringed toposes:
Γ(Xτ , −)
Mod(OXτ ) Mod(R)
Sτ
The functor Sτ is the left adjoint of Γ(Xτ , −) and it is defined by mapping a −1 −1 module M ∈ Mod(R) to S M := S M ⊗ −1 O where S : Ab → Ab(X ) τ τ Sτ R Xτ τ τ is the inverse image functor on the level of sheaves of abelian groups. Note that this functor is naturally equivalent to the functor which associates to each module M ∈ Mod(R), the sheafification of the following presheaf:
(f : U → X) 7→ M ⊗R OXτ (U) where (f : U → X) ∈ Obj(Xτ ), see Paragraph 8.4 for this proof.
As usual, for an element F ∈ Mod(OXτ ) we define the n-th cohomology groups of Xτ with coefficients in F to be the the n-th higher direct images of Γ(Xτ , −) i.e.: n n H (Xτ ,F ) := R Γ(Xτ , −)(F ) By definition this is the n-th cohomology of the complex: • Γ(Xτ ,I ) • where I is an injective resolution of F in Mod(Xτ ). Such resolutions exists since Mod(Oτ ) has enough injectives [Sta18, Sect. 01DU]. We note also that by [Sta18,
Sect. 03FD], calculating the cohomology groups for a sheaf of OXτ -modules F gives us the same as when we calculate the cohomology groups for the underlying sheaf of abelian groups.
60 5.3. Let X be an fs log scheme. The functor λ : X´et → Xτ induces a morphism between the toposes of abelian groups and a morphism of toposes of modules:
λ λ∗ ∗
Sh(Xτ ) Sh(X´et) Mod(OXτ ) Mod(OX´et )
λ−1 λ∗
Where λ−1 is exact and since λ∗ operates by tensoring with the structure sheaf OXτ it is only right exact. Let R := Γ(OXτ , OXτ ) as above and let Q :=
Γ(X´et, OX´et ). Then by functionality we have the following commutative diagram:
λ∗ Mod(Xτ ) Mod(X´et)
Γ(Xτ , −) Γ(X´et, −)
Mod(R) ∼ Mod(Q)
where the bottom line is an isomorphism since λ∗ OXτ = OX´et implies that
Γ(Xτ , OXτ ) = Γ(X´et, OX´et ). Since the left adjoint Sτ of Γ(Xτ , −) is unique up a natural equivalence we ∗ have that Sτ is naturally equivalent to λ ◦ S´et where S´et : Mod(Q) → Mod(OX´et ) is the left adjoint of Γ(X´et, −).
For a quasi-coherent sheaf F on X´et, we can extend it to a presheaf Fτ on Xτ by the assignment: ∗ Fτ :(f : Y → X) 7→ Γ(Y , f´et F ) where τ is the ket or the let topology.
Definition 5.4. For τ ∈ {ket, let}, we call Fτ the extension presheaf of F on Xτ . Caution 5.5. Note that in [Niz08, Proposition 2.19] and [Hag03, Proposition 3.1], they define the extension presheaf in the Zariski topology with Zariski inverse images etc. We note again that by [Sta18, Sect. 03DX (1)] using the etale topology makes no difference since we are interested in quasi-coherent sheaves and vector bundles.
Remark 5.6. As we will see later, the extension presheaves associated to OX´et and Ωlog satisfy descent in the various logarithmic topologies. The statement for X´et the Kummer log etale topology is proved in Corollary 5.12 and for the log etale topology in Corollary 5.22.
61 Remark 5.7. We note that the presheaf Ωlog is equal to: Xket
(f : Y → X) 7→ Γ(Y , Ωlog ) Y´et In order to see this, we have by definition that:
f Ωlog (Y → X) = Γ(Y , f ∗ Ωlog ) Xket ´et X´et The above description is obtained by Proposition 2.14, since for any log etale morphism we have the isomorphism f ∗ Ωlog ∼= Ωlog . This is the log etale analogue ´et X´et Y´et of [Mil80, Proposition 1.3]. We note that the same is true for Ωlog since every Xlet object (f : Y → X) in Xlet is log etale. The following Subsection is devoted in proving ket descent for the extension presheaf. Specifically for τ = ket and F ∈ Qcoh(X´et) we will show in Theorem 5.11 that the extension presheaf Fket is actually sheaf on Xket. This will let us transfer quasi-coherent sheaves from X´et to Xket. Then we close Subsection 5.2 by giving the details of the proofs of Corollaries 3.5, 3.6 and 3.7 from [Hag03]. This will be done by extending the standard theory of modules [Sta18, Sect. 01I6] on ket site. For τ = let the picture is more restrictive. We will prove in Theorem 5.21 that when X is log regular then, for any E ∈ Vect(X´et), the extension presheaf Elet is actually a sheaf. This will let us transfer vector bundles (i.e. locally free sheaves of finite rank) from X´et to Xlet. Then, we will prove the analogues of Corollaries 3.5, 3.6, 3.7 from [Hag03] but for the let topology and for vector bundles.
5.2 ket descent for the extension presheaf
We will show that for any F ∈ Qcoh(X´et) the extension presheaf Fket satisfies ket descent influenced by ideas from [Mil80] and [Vis04]. As we show in Proposition 4.17, we can find good refinements for Kummer log etale covers and in Theo- rem 5.11 we show how to reduce the descent argument by just checking descent for strict log etale covers and standard Kummer Galois covers. In the following propositions we fill in the details for strict log etale descent and standard Kummer Galois descent. We prove also that the extension presheaf is separated. Proposition 5.8. Let X be an fs log scheme. Then, for any quasi-coherent sheaf F in X´et the extension presheaf Fket on Xket satisfies strict log etale descent.
Proof. Consider an object (f : Z → X) in Xket, a strict log etale cover g : Y → Z over X, and a quasi-coherent sheaf F in X´et. Note that the sequence:
fs 0 → Fket(Z) → Fket(Y ) → Fket(Y ×Z Y ) is isomorphic, by definition, to:
∗ ∗ ∗ fs ∗ ∗ 0 → Γ(Z, f´et F ) → Γ(Y , g´et (f´et F )) → Γ(Y ×Z Y , (g ◦ pr1,g)´et (f´et F ))
62 Since g is strict and log etale, the underlying morphism of schemes g is etale, so fs ∼ by Proposition 1.68 we have that Y ×Z Y = Y ×Z Y . Having said that, note that the last short sequence is just the sheaf condition sequence for the etale ∗ cover g : Y → Z and sheaf f´et F ∈ Qcoh(Z´et), which is exact by faithfully flat descent. Now we treat the standard Kummer Galois covers.
Proposition 5.9. ([Hag03, Lemma 3.3]). Let X be an fs log scheme endowed with a global chart P , such that X is affine. Pick a homomorphism of Kummer type θ : P → Q and consider the standard Kummer Galois cover:
˜ fs θ : XQ := X ×Spec(Z[P ]) Spec(Z[Q]) → X of X associated to θ. Then, the following sequence:
˜ ∗ fs ˜ ∗ 0 → Γ(X,F ) → Γ(XQ, θ´et F ) ⇒ Γ(XQ ×X XQ, (θ ◦ pr1,θ˜)´et F ) is exact for any quasi-coherent sheaf F on X´et. Details of the proof. By Lemma 1.70:
XQ = X ×Spec(Z[P ]) Spec Z[Q] fs gp gp XQ ×X XQ = X ×Spec(Z[P ]) Spec(Z[Q ⊕ (Q /P )]) Since X is affine:
˜ ∗ Γ(XQ, θ´et F ) = Γ(X,F ) ⊗Z[P ] Z[Q] fs ˜ ∗ gp gp Γ(XQ ×X XQ, (θ ◦ pr1,θ˜)´et F ) = Γ(X,F ) ⊗Z[P ] Z[Q ⊕ (Q /P )]
We set M := Γ(X,F ), and we have the following sequence:
gp gp 0 → M → M ⊗Z[P ] Z[Q] ⇒ M ⊗Z[P ] Z[Q ⊕ (Q /P )] where the first morphism just a 7→ a ⊗ 1 and the last morphisms are defined by the morphisms induced by the projections 1⊗a 7→ 1⊗(a, a) and 1⊗a 7→ 1⊗(a, 1) respectively. This sequence is exact as it is stated in [Kat89a, Section 4.1]. For the descent argument we refer to [Niz08, Lemma 3.28].
In Propositions 5.8 and 5.9 we proved that the extension presheaf Fket, for a quasi-coherent sheaf F ∈ Qcoh(X´et), satisfies descent for strict log etale covers and standard Kummer Galois covers Y → Z where Z has affine underlying scheme. In order to prove descent for a general Kummer log etale cover f : Y → Z over Xket, we are going to reduce it in the cases dealt in Propositions 5.8 and 5.9 by Proposition 4.17. First we show that the extension presheaf is separated.
63 Proposition 5.10. Let Fket be the extension presheaf of a quasi-coherent sheaf F on X´et. Then Fket is separated.
Proof. Let (f : Y → Z) be a Kummer log etale cover over Xket and consider a good refinement: fin Uf := (Ti,j → Ui,nj → Ui → Z)i∈I,j∈J as defined in Proposition 4.17. For the moment let us write this cover as (Ti,j → Ui → Z)i∈I,j∈J . Then consider the following commutative diagram:
β Fket(Z) F (Y )
α
Q F (U ) Q F (T ) i∈I ket i γ i∈I,j∈J ket i,j
Since (Ui → Z)i∈I is a strict log etale covering and Fket satisfies strict log etale descent by Proposition 5.8, α is injective. In order to prove that β is injective we just need to prove injectivity for γ. Note that if for every i ∈ I the morphism γi : Q Fket(Ui) → j∈J Fket(Ti,j) is injective then the same is true for γ. Changing back Ti,j → Ui as the composition Ti,j → Ui,j → Ui we have the following commutative diagram: γi Q Fket(Ui) j∈J Fket(Ti,j)
∼= Q Q j∈J Fket(Ui,nj ) j∈J Fket(Ti,j)
Note that we just need to prove that the morphisms Fket(Ui) → Fket(Ui,nj ),
Fket(Ui,nj ) → Fket(Ti,j) are injective for all i ∈ I, j ∈ J. Since for every j ∈ J the morphism Ui,nj → Ui is a standard Kummer Galois cover with Ui affine, we use standard Kummer Galois descent from Proposition 5.9, and Fket(Ui) → Fket(Ui,nj ) is injective. By strict log etale descent the morphism Fket(Ui,nj ) → Fket(Ti,j) is also injective. Hence, for every i ∈ I the morphism γi is injective and the extension presheaf is separated.
Theorem 5.11. Let X be an fs log scheme. For any quasi-coherent sheaf F the extension presheaf Fket satisfies Kummer log etale descent.
Proof. Let (f : Y → Z) be a cover in Xket. Consider a good refinement of f as in
64 Proposition 4.17 and consider the following commutative diagram:
` hi,j ` i,j Ti,j i,j Ui,nj
g i,j θi,nj ` ` i,j Vi,j i Ui
Y Z f
where (Ui → Z)i∈I and (Vi,j → Y )i∈I,j∈J are strict log etale covers, Ui is affine and Ui is equipped with a chart which satisfies the log etale condition. Consider the following commutative diagram:
0 0
fs Fket(Z) Fket(Y ) Fket(Y ×Z Y )
Q F (U ) Q F (V ) Q F (V ×fs V ) i ket i i,j ket i,j i,j,k ket i,j Ui i,k
Q fs Q fs i,k Fket(Ui ×Z Uk) i,j,k,l Fket(Vi,j ×Y Vk,l)
We make the following observations:
1. The first two columns are exact by strict log etale descent, see Proposition 5.8.
2. The first and the bottom vertical morphisms are injective since Fket is sep- fs fs arated, see Proposition 5.10, and Vi,j ×Y Vk,l → Ui ×Z Uk is surjective and Kummer log etale.
By a diagram chase we can reduce the exactness of the first row to the exactness of the second row. Furthermore, arguing similarly as [Vis04], we can reduce the exactness of the middle row to the exactness of each piece: Y Y F (U ) → F (V ) → F (V ×fs V ) ket i ket i,j ket i,j Ui i,k j j,k
65 ` ` We refine j∈J Vi,j → Ui by j∈J Ti,j → Ui and consider the following commuta- tive diagram:
F (U ) Q F (T ) Q F (T ×fs T ) ket i j ket i,j j,k ket i,j Ui i,k
Q j Fket(Vi,j)
Q F (V ×fs V ) j,k ket i,j Ui i,k
Assume that the top row is exact, then by a quick diagram chasing we can prove that the column is exact. The only thing that is now left to prove is that the first row is exact. Arguing similarly as above, it suffices to show that the each morphism Ti,j → Ui satisfies fs the sheaf condition. We set Wi,j := Ti,j × Ti,j (note that since Ti,j → Ui,nj Ui,nj is strict we can skip the superscript fs) and consider the following commutative diagram:
0 0
Fket(Ui) Fket(Ti,j) Fket(Ti,j ×Ui Ti,j)
w fs 0 Fket(Ui,nj ) Fket(Wi,j) Fket(Wi,j × Wi,j) Ui,nj
F (U ×fs U ) F (W × W ) ket i,nj Ui i,nj ket i,j Ti,j i,j
We make the following observations:
1. The first column is exact from standard Kummer Galois descent.
2. The second column is exact by strict log etale descent since the morphism W := T × T → T is strict log etale and surjective. i,j i,j Ui,nj i,j i,j
3. The second row is exact since Wi,j → Ti,j → Ui,nj is surjective and strict log etale as a composition of surjective strict log etale morphisms.
4. The bottom horizontal morphism is injective since Fket is separated and W × W → U ×fs U is surjective Kummer etale. i,j Ti,j i,j i,nj Ui i,nj
66 Now by diagram chasing we get that the first row is exact as well. Going back up we have proved that Fket satisfies Kummer etale descent. Corollary 5.12. Taking F = O or F = Ωlog in Theorem 5.11 we have that X´et X´et the extension presheaves O , Ωlog are sheaves on X . Xket Xket ket Proof. Note that since X is an fs log scheme, hence coherent, the sheaf Ωlog is X´et quasi-coherent by Proposition 2.16 and so we can apply Theorem 5.11. Remark 5.13. Since the extension presheaf satisfies ket descent we can define the functor: ()ket : Qcoh(X´et) → Mod(Xket) which maps an object F ∈ Qcoh(X´et) to Fket and maps a morphism φ : F → G in Qcoh(X´et) to φket : Fket → Gket which is defined for an object (f : Y → X) in Xket by: ∗ φket(Y ) = Γ(Y , f´etφ)
Our goal is to prove that the functor ( )ket : Qcoh(X´et) → Mod(Xket) is ∗ naturally isomorphic to the functor : Qcoh(X´et) → Mod(Xket).
∗ Theorem 5.14. The functors ()ket, : Qcoh(X´et) → Mod(Xket) are naturally equivalent. The idea of the proof goes as follows: First we prove that these two functors agree for globally presented modules on X´et. Then, for the quasi-coherent case, we reduce the argument to our first case since every quasi-coherent sheaf on X´et is etale locally, globally presented.
∗ Lemma 5.15. The functors ()ket, : Qcoh(X´et) → Mod(Xket) are naturally equivalent for globally presented modules on X´et
Proof. Let us set R = Γ(X´et, OX´et ) = Γ(Xket, OXket ). Note that we can do this since ∗(OXket ) = OX´et . By definition, every globally presented module on Mod(X´et) is in the essential image of the functor S´et : Mod(R) → Mod(X´et). Hence, in order to prove this statement we need to show that the functors:
∗ ()ket ◦ S´et, X ◦ S´et : Mod(R) → Mod(Xket)
∗ are naturally equivalent. We note that since both Sket and X ◦ S´et are both left adjoint to the global sections functor Γ(Xket, −) we just need to identify ( )ket ◦S´et and Sket. For this, notice that for every M ∈ Mod(R), both of the functors can be described as the sheaf associated to the presheaf:
U 7→ M ⊗R OX´et (U) where (U → X) ∈ Obj(Xket), see Paragraph 8.4 for this description. Hence the two functors agree.
67 Now we treat the quasi-coherent case.
Proof of Theorem 5.14. Consider F ∈ Qcoh(X´et) and an etale cover (f : U → X) ∗ where F |U´et is of global presentation. Note that the covering (V := (U, flog MX ) → X) is a ket covering of X since V → X is strict and the underlying morphism is ∗ etale. Pick (f : Y → X) in Xket. Then by taking the sheaf condition for X F and Fket we have the following commutative diagram:
∗ ∗ ∗ 0 F (Y ) F (VY ) F (VY ×Y VY )
a ∼ ∼
0 Fket(Y ) Fket(VY ) Fket(VY ×Y VY )
where the right square comes from Lemma 5.15 since F |U´et is globally presented. Then a is an isomorphism by the universal property of the equalizers.
We note the following corollaries.
Corollary 5.16. ([Hag03, Corollary 3.6]). For any quasi-coherent sheaf F over X´et, the natural morphism: ∗ F → ∗ F is an isomorphism.
∗ Proof. By Theorem 5.14, F is naturally isomorphic to Fket. On the other hand, checking on local sections ∗Fket = F . Corollary 5.17. ([Hag03, Corollary 3.7]). The functor:
∗ : Qcoh(X´et) → Mod(Xket) is fully faithful.
Proof. Let F,G ∈ Qcoh(X´et). By Corollary 5.16 and by adjunction:
∼ ∗ ∼ ∗ ∗ HomMod(X´et)(F,G) = HomMod(X´et)(F, ∗ G) = HomMod(Xket)( F, G) which proves that ∗ is fully faithful.
68 5.3 let descent for the extension presheaf
Let X be an fs log scheme and consider the log etale site Xlet. Our main interest is to prove that the extension presheaves:
OXlet :(Y → X) 7→ Γ(Y , OY ) Ωlog :(Y → X) 7→ Γ(Y , Ωlog ) Xlet Y´et are sheaves on Xlet. Note that the sheaf of logarithmic differential forms has the above form by Remark 5.7. In the previous section, Theorem 5.11, we showed that these presheaves satisfy Kummer log etale descent. Since by Remark 4.33 the full log etale topology is generated by Kummer log etale morphisms and log blow-ups we only need to treat the case of log blow-ups. fs Consider a log blow-up π : Y → X. By Lemma 3.23, the fiber product Y ×X Y is isomorphic to Y hence, a presheaf F ∈ Pr(Xlet) satisfies log-blow up descent if the following sequence is exact:
0 → F (X) → F (Y ) ⇒ F (Y ) where the last arrows are the same and hence, their difference is zero. So we need to find presheaves F that satisfy F (X) ∼= F (Y ), a condition which is not trivial. We make the following remark.
Remark 5.18. Consider a toric variety XΣ associated to a fan Σ and consider a ˜ proper toric morphism π : XΣ˜ → XΣ i.e., Σ is just a subdivision of Σ. Then it is always true that O → Rπ O is a quasi-isomorphism [CLS11, Theorem 9.25], XΣ ∗ XΣ˜ which implies that Γ(X , O ) ∼ Γ(X , O ). In particular, if π : X → X was Σ XΣ = Σ˜ XΣ˜ Σ˜ Σ a covering for some topology τ on (XΣ)τ the associated extension presheaf OXΣ,τ would satisfy the following:
O (X ) := Γ(X , O ) ∼ Γ(X , O ) =: O (X ) XΣ,τ Σ Σ XΣ = Σ˜ XΣ˜ XΣ,τ Σ˜ which is exactly what we need but for log blow-ups. As we saw in Theorem 3.26, we can have similar quasi-isomorphisms for log blow-ups when we restrict to the case of log regular log schemes.
In view of the above remark and in the case where the log structure is Zariski, we can use the fact that log blow ups are rational and Theorem 3.26 to prove log blow-up descent. Since we are interested in etale log structures, we can replace Theorem 3.26 in the above process by Theorem 3.27 and the same arguments work for etale log structures. Also note that we can go a bit further. We can use Remark 5.19 to prove log blow-up descent for the extension presheaf Elet of any locally free sheaf of finite rank E coming from the etale site.
69 Recall from Section3 that the theory of log regular log schemes splits as follows: Toroidal embedings without {Zariski log schemes} self-intersections
Toroidal embedings with {Etale log schemes} self-intersections
Consider a locally free sheaf of finite rank E on X´et, and recall Definition 5.4 of the extension presheaf:
∗ Elet :(f : Y → X) 7→ Γ(Y , f´et E)
This is actually a sheaf as we will prove in Theorem 5.21. First, we make the following remark.
Remark 5.19. Let π : Y → X be a log blow up where X is log regular and consider a locally free sheaf of finite rank E over X´et. By Theorem 3.27 we have a quasi-isomorphism OX → Rπ´et∗ OY and since E is flat as an OX -module, tensoring with E will preserve the quasi-isomorphism. Then, we use the projection formula [Sta18, Sect. 01E8] to get a quasi-isomorphism:
∗ E → Rπ´et∗(π´et E)
By taking homology we have the following assertions:
∼ ∗ 1. E = π´et∗π´et E as OX -modules,
p ∗ 2. R π´et∗(π´et E) = 0 for all p ≥ 1. The second assertion guarantee the degeneration of the Leray spectral sequence ∗ for π´et∗ and π´et E. This gives us:
i ∗ ∼ i ∗ ∼ i H (Y´et, π´et E) = H (X´et, π´et∗π´et E) = H (X´et,E) for all i ≥ 0.
Proposition 5.20. Let X be a log regular fs log scheme and E a locally free sheaf of finite rank on X. The extension presheaf Elet on Xlet satisfies log blow-up descent. In particular, the presheaves O , Ωlet satisfy log blow-up descent. Xlet Xlet
Proof. Let (f : Z → X) be an object in Xlet and pick a log blow up π : Y → Z over X. By Theorem 3.27 we have a quasi-isomorphism OZ → Rπ´et ∗ OY´et . We ∗ set M := f´etE and we use Remark 5.19 to get the quasi-isomorphism:
qis ∗ M → Rπ´et∗(π´et M)
70 ∼ ∗ ∼ ∗ In particular, M = π´et∗(π´etM) which implies that Γ(Z,M) = Γ(Y , π´etM). Writing everything down, we get that: ∗ ∼ ∗ Elet(f : Z → X) := Γ(Z, f´et E) = Γ(Y , (f ◦ π)´et E) =: Elet(f ◦ π : Y → X) which proves log blow-up descent. Since X is log regular, we have from Proposition 2.16 that Ωlog is a locally free X´et sheaf of finite rank. Having said that, we conclude that the extension presheaves O , Ωlog satisfy log blow up descent. Xlet Xlet Theorem 5.21. Assume the setup of the above proposition. Then, the extension presheaf Elet satisfies log etale descent. Proof. Let Y → Z be a log etale cover, then by Remark 4.27 we have a factoriza- tion: ` hi,j ` i,j Ti,j i,j Ui,Ij
gi,j πi,Ij
` ` i,j Vi,j i Ui
Y Z f such that:
1.( Ui → Z)i and (Vi,j → Y )i,j are strict log etale covers,
2. each Ui,Ij → Ui is the log blow up of the log ideal Ij in Ui,
3. and Ti,j := Vi,j ×Ui Ui,Ij → Ui,Ij is Kummer log etale. Then by using same ideas as we did in the proof of Theorem 5.11 we can reduce it to the case Vi,j → Ui. So we replace Z,Y with Ui,Vi,j and consider the following commutative diagram:
0 0
Elet(Z) Elet(Y ) Elet(Y ×Z Y )
fs fs fs 0 Elet(ZI ) Elet(ZI ×Z Y ) Elet(ZI ×Z Y ×Z Y )
fs 0 Elet(ZI ) Elet(ZI ×Z Y )
71 where the two first columns are exact by log blow-up descent, see Proposition 5.20, and the two last rows are exact Kummer log etale descent, see Theorem 5.11. By a diagram chasing we prove exactness of the first row and conclude that Elet satisfies full log etale descent. Corollary 5.22. Using the above theorem we have that, in the case where X is log regular fs log scheme the extension presheaves O , Ωlog are sheaves on X Xlet Xlet let
Since the extension presheaf is actually a sheaf on Xlet, we can define the functor:
()let : Vect(X´et) → Mod(Xlet)
Similarly as in the ket case, Theorem 5.14, we will identify the functor ( )let with ∗ the restricted functor ρ : Vect(X´et) → Mod(Xlet). We argue similarly as in the ket case. First we show that the functors agree for free OX´et -modules of finite rank. Then, since vector bundles are locally free sheaves of finite rank we can reduce the argument for vector bundles to the case of free sheaves of finite rank. The next lemma can be seen as an analogue of the technical Lemma 5.15. Note that until the end of this subsection the fs log scheme X is log regular.
∗ Lemma 5.23. The functors ()let, ρ : Vect(X´et) → Mod(Xlet) are naturally iso- morphic for free sheaves of finite rank. Proof. The argument of the proof is identical to that of Lemma 5.15. We notice ∗ again that the functors ρX ◦S´et and Slet are naturally equivalent as the left adjoint of Γ(Xlet, −). Then we note that both of the functors ( )let◦S´et and Slet are defined by sending M ∈ Mod(R) to the sheaf associated to:
U 7→ M ⊗R OX´et ∗ where (U → X) ∈ Obj(Xlet), see Paragraph 8.4. Hence the functors ( )let, ρ agree on free sheaves of finite rank.
Now by using he sheaf condition, as we did in the proof of Theorem 5.14, we have the following.
∗ Theorem 5.24. The functors ()let, ρ : Vect(X´et) → Mod(Xlet) are naturally isomorphic. We also have the following analogues of the corollaries in the end of the previous subsection. We note again the the main arguments of the proofs are identical. Corollary 5.25. (Analogue of Corollary 5.16 for the let topology). For every vector bundle E on X´et, the natural morphism:
∗ E → ρ∗ρ E is an isomorphism.
72 Corollary 5.26. (Analogue of Corollary 5.17 for the let topology). The functor:
∗ ρ : Vect(X´et) → Mod(Xlet) is fully faithful.
5.4 Classical and locally classical sheaves In this section we introduce some terminology that we will use in the rest sub- sections. We remind that, all log schemes of this section are Etale log schemes. Following W. Niziol in [Niz08, Definition 3.6] and Hagihara in [Hag03, Definitions in Sections 3.3 and 3.6], we give the following definition.
Definition 5.27. Let τ ∈ {ket, let} be a logarithmic topology and let λ : X´et → Xτ be the associated continuous functor of sites. A module F ∈ Mod(Xτ ) is called classical (resp. classical quasi-coherent module, classical vector bundle) ∗ if it is in the essential image of λ : Mod(X´et) → Mod(Xτ ) (resp. in the essential ∗ ∗ image of λ : Qcoh(X´et) → Mod(Xτ ), λ : Vect(X´et) → Mod(Xτ )) and it is called τ-locally classical (resp. τ-locally classical quasi-coherent module, τ-locally classical vector bundle) if there exist some τ-covering (Y → X) such that the restriction F |Yτ is a classical module (resp. classical quasi-coherent module, classical vector bundle) on Yτ . Remark 5.28. The definitions that we give here are a bit different from the ones given by W. Niziol and K. Hagihara, but are of the same spirit. Specifically we just call classical (resp. locally classical) what comes (resp. locally comes) from the etale site and we add ”quasi-coherent” or ”vector bundle” to specify what kind of modules we pull back (resp. locally pull back). Remark 5.29. Note that the notion of ket-locally classical sheaves (resp. let- locally classical vector bundles) on Xket (resp. on Xlet) is equivalent to the notion of quasi-coherent sheaves on Xket (resp. locally free sheaves of finite rank on Xlet). We prove this in Paragraph 8.3. We continue with some technical remarks that we will need below. Remark 5.30. Let X be an fs log scheme. For every Kummer log etale morphism f : Y → X and for every quasi-coherent module F ∈ Qcoh(X´et): −1 ∗ ∼ ∗ ∗ fket X F = fketX F −1 ∗ meaning that the functors fket , fket agree for classical quasi-coherent modules on Xket. In order to see this pick an object (g : V → Y ) in Yket. Then: −1 ∗ ∗ [fket X F ](g : V → Y ) := [X F ](f ◦ g : V → X) ∗ := Γ(V , (f ◦ g)´et F ) ∗ ∗ = Γ(V , g´et (f´et F )) ∗ ∗ =: [Y f´et F ](g : V → Y )
73 −1 ∗ ∼ ∗ ∗ which means that fket X F = Y f´et F . Also note that by functoriality of the pullbacks of modules, we have:
∗ ∗ ∼ ∗ ∗ fketX F = Y f´et F which completes the proof.
Remark 5.31. We also have an analogue of the above remark for the full log etale topology. Let X be log regular and pick an element (f : Y → X) in Xlet. Then, for every vector bundle E ∈ Vect(X´et):
−1 ∗ ∼ ∗ ∗ flet ρX E = fletρX E
−1 ∗ meaning that the functors flet , flet agree for classical vector bundles on Xlet. The proof is identical as the above. The reasons that we restrict to vector bundles is because we only proved log etale descent for vector bundles coming from X´et and ∗ so we can express ρX E as Elet. This argument is what we basically used in the ket case.
The following lemma is independent of the logarithmic topology τ that we use. So let use pick τ ∈ {ket, let} and let λ : X´et → Xτ be the associated continuous functor of sites, see Notation 5.1.
Lemma 5.32. Let f : Y → X be a morphism of fs log schemes and consider a ∗ classical module F on Xτ . Then fτ F is classical on Yτ .
Proof. This is just the functoriality of pullbacks of modules. Let F ∈ Mod(Xτ ) ∼ ∗ with F = λX G for G ∈ Mod(X´et) then:
∗ ∗ ∗ ∼ ∗ ∗ fτ F = fτ λX G = λY f´etG
∗ which means that fτ F is classical on Yτ . The next lemmata deals with restrictions of locally classical modules along log etale morphisms.
Lemma 5.33. Let f : Y → X be an element in Xket. If F is a ket-locally classical −1 quasi-coherent sheaf on Xket then F |Yket := fket F is a ket-locally classical quasi- coherent sheaf on Yket. In particular, the lemma holds for classical quasi-coherent sheaves.
Proof. Note that we only need to prove the ket locally classical case since the classical case is ket locally classical case applied to the strict log etale cover {idX : X → X}. Let f : Y → X be a ket morphism and consider a ket locally classical quasi- coherent module F ∈ Mod(Xket) and a ket covering (g : V → X) such that
74 ∼ ∗ F |Vket = V G for some G ∈ Qcoh(V´et). We want to show that F |Yket is a ket lo- cally classical quasi-coherent module in Yket. Consider the following commutative diagram of sites: h fs ket (Y ×X V )ket Vket
eket gket
Yket Xket fket
−1 fs Now we take the restriction F |Yket := fket F and restrict further to (Y ×X V )ket := −1 −1 eket(fket F ), then by the functoriality of the restriction functors:
(F |Y )| fs = (F |V )| fs ket (Y ×X V )ket ket (Y ×X V )ket Since F | is a classical quasi-coherent module on V we have that F | = Vket ket Vket ∗ V (G) for some G ∈ Qcoh(V´et) and:
−1 −1 ∗ ! ∗ ∗ ∼ ∗ ∗ h (F |V ) = h (V (G)) = hket(V G) = fs (h´etG) ket ket Y ×X V
where in ! we used Remark 5.30 since G ∈ Qcoh(V´et). Hence, F |Yket is a ket locally fs classical quasi-coherent module over Yket (achieved) by the covering (Y ×X V → Y ).
Lemma 5.34. Let X be log regular and let f : Y → X be an element of Xlet.
Then, if E is a let-locally classical vector bundle on Xlet then E|Ylet is a let-locally classical vector bundle on Ylet. In particular, the lemma holds for classical vector bundles.
Proof. The proof is similar as the proof of Lemma 5.33 where we use Remark 5.31 instead of Remark 5.30. We close this section with a technical remark that we will use in the proof of Theorem 6.13.
Remark 5.35. Let π : Y → X be a log blow-up where X is log regular. We will prove that for every classical vector bundle E on Xket:
−1 ∗ ∗ κY ∗ ◦ πlet ◦ κX (E) = πketE We can prove this by identifying the local sections. Note that, in the view of −1 Example 4.32, we can write the restriction functor πlet as the pushforwardπ ˜let ∗ whereπ ˜let is the left adjoint of πlet. Let (f : Z → Y ) be an element of Yket, and
75 ∼ ∗ 0 0 E = X (E ) for a vector bundle E in Vect(X´et). Then:
−1 ∗ −1 ∗ [κY ∗ ◦ πlet ◦ κX (E)](f : Z → Y ) := [πlet ◦ κX (E)](f : Z → Y ) ∗ := [κX (E)](π ◦ f : Z → X) ∗ 0 := Γ(Z, (π ◦ f)´etE ) ∗ ∗ 0 = Γ(Z, f´et(π´etE )) ∗ ∗ 0 := [Y ◦ π´et(E )](f : Z → Y ) ∗ = [πketE](f : Z → Y ) which proves our statement.
76 6 Cohomological descent for Log sites
6.1 Comparison of ket cohomology and etale cohomology In algebraic geometry, affine schemes and affine morphisms have a cohomological characterization. Namely, as soon as we have a notion of quasi-coherence, we can characterize affine schemes by the property: X is an affine scheme if and i only if for all F ∈ Qcoh(XZar) we have H (XZar,F ) = 0 for all i > 0, [Sta18, Sect. 01XB]. Similarly we can define the notion of affine morphism of schemes from the vanishing of the higher direct images i.e. a morphism of schemes f : X → Y is affine if and only if for every quasi-coherent sheaf F on X the higher direct i images R f∗F vanish for all i > 0, [Sta18, Sect. 01XC]. This is a very helpful characterization because together with the Leray spectral sequence it gives us comparison between different cohomologies i.e., if f : X → Y is affine then: i ∼ i H (XZar,F ) = H (YZar, f∗F ), for all i ≥ 0 In this section we try similar ideas in the logarithmic category.
Conventions 6.1. We note that for many of the arguments coming next, we will need the intersection of affine opens to be affine. So all log schemes in this section are assumed to have separated underlying schemes [Sta18, Sect. 01KW].
Definition 6.2. An fs log scheme X is said to be classically affine if, for every ket-locally classical quasi-coherent module F ∈ Mod(Xket) we have that i H (Xket,F ) = 0 for all i > 0. A theorem of K.Kato gives a nice class of classically affine log schemes.
Proposition 6.3. ([Niz08, Proposition 3.27]). Let A be an fs log scheme such that the underlying scheme A is affine and let F ∈ Mod(Aket) be a ket-locally classical quasi-coherent module. Then A is classically affine as a log scheme i.e., i H (Aket,F ) = 0 for all i > 0. Since classical affine log schemes have trivial higher cohomology we can com- pare the classical etale cohomology and the ket cohomology.
Proposition 6.4. Let X be an fs log scheme and : X´et → Xket be the strict inclusion of Example 4.20 and let F be a ket-locally classical quasi-coherent module on Xket. Then: i ∼ i H (Xket,F ) = H (X´et, ∗F ) Proof. We will use the Leray spectral sequence [Sta18, Sect. 0732] for F, . Then q the result comes by the fact that R ∗F = 0 for all q > 0. The higher direct image sheaf is just the sheafification of the presheaf:
i ∗ Obj(X´et) 3 (f : Y → X) 7→ H ((Y, flog MX )ket,F )
77 Using that X is separated we can find an affine open covering with affine intersec- tions [Sta18, Sect. 01KW]. So we can reduce it to the case when Y is affine open subscheme of X, and we need to prove that the cohomology groups: i ∗ i ∗ H ((Y, f M ) ,F ) ∼= H ((Y, f M ) ,F | ∗ ) log X ket log X ket (Y,flog MX )ket are zero for all i > 0. But this is true from Proposition 6.3 combined with the ∗ fact that the restriction of F on (Y, flog MX )ket is ket locally classical by Lemma 5.33.
Remark 6.5. Since every quasi-coherent sheaf on on Xket is ket locally classi- cal quasi-coherent by Paragraph 8.3, the above comparison is true for all quasi- coherent sheaves on Xket. Remark 6.6. If X is classically affine log scheme then the underlying scheme i X is an affine scheme. This is true because H (X´et,F ) = 0 for all i > 0 and F ∈ Qcoh(X´et). In order to see this take a quasi-coherent sheaf F ∈ Qcoh(X´et), ∗ then F is a classical on Xket, hence: i ∗ H (Xket, F ) = 0 for all i > 0 i ∗ Using Proposition 6.4, the cohomology group H (X´et, ∗ F ) = 0 and by Corollary ∗ 5.16 we have that F → ∗ F is an isomorphism which implies: i H (X´et,F ) = 0 for all i > 0 Hence the underlying scheme of X is affine. This identifies classically affine log schemes with log schemes that the underlying scheme is affine. From now on we will call such log schemes affine log schemes. Theorem 6.7. (Relative version of Proposition 6.4). Let f : X → Y be a mor- phism of fs log schemes where the underlying morphism of schemes f is affine. Then, for every ket-locally classical quasi-coherent module F ∈ Mod(Xket) we have that: i ∼ i H (Xket,F ) = H (Yket, f∗F )
Proof. We will argue similarly as in Proposition 6.4. Let fket : Yket → Xket be the induced continuous functor of Example 4.22, and pick F ∈ Mod(Xket) ket locally classical. The i-th higher direct images of (fket)∗ : Mod(Xket) → Mod(Yket) are given by the sheafification of the presheaf: i fs Obj(Yket) 3 U 7→ H ((U ×Y X)ket,F ) fs ∼ −1 Note that if U → Y is a strict open immersion then U ×Y X = U ×Y X = f (U) −1 by Proposition 1.68, and F |f (U)ket is a ket locally classical quasi-coherent module −1 on f (U)ket by Lemma 5.33. If U is an affine log scheme then by assumption f −1(U) is also affine and by Proposition 6.3: i −1 H (f (U)ket,F |f −1(U)) = 0 i for all i > 0. Hence, R (fket)∗F = 0 for all i > 0 and by Leray we are done.
78 Remark 6.8. Assume the same setup and the same assumptions as in Theo- rem 6.7, then for every object (U → Y ) in Yket the pullback morphism fU : fs U ×Y X → U has trivial higher direct images for ket locally classical quasi- fs p coherent modules on (U ×Y X)ket i.e., R (fU )ket ∗F = 0 for all ket-locally classical fs quasi-coherent modules F ∈ Mod((U ×Y X)ket) and so:
i fs ∼ i H ((U ×Y X)ket,F ) = H (Uket, (fU )ket ∗F )
To prove this note the following:
1. Since affine morphisms of schemes are preserved under pullbacks we have 0 that f : U ×Y X → U is affine.
fs fs 2. Since (U ×Y X) := U ×X Y and U ×Y X = U ×X Y , by Remark 1.71 the 00 fs morphism of schemes f : U ×Y X → U ×Y X is affine.
fs 3. Hence, the underlying morphism of schemes of fU : U ×Y X → U is affine 0 00 morphism and it is given by fU = f ◦ f .
fs Now we can apply Theorem 6.7 to fU : U ×Y X → U and get the comparison in cohomologies. To sum up, we proved the following: Let f : X → Y of log schemes, where f is affine. Then, for all elements (U → Y ) in Yket the higher p fs direct images R (fU )ket ∗F vanish for ket locally classical sheaves on (U ×Y X)ket . Remark 6.9. It seems tempting to prove an inverse of Theorem 6.7 but there are some difficulties. Assume that f : X → Y is a morphisms of fs log schemes p such that R fket ∗F = 0 for all p > 0 and F ket locally classical quasi-coherent on Xket. Then the idea is to use our assumption together with Proposition 6.3 ∗ applied to Yket and fket ∗F , where F = X G for some G ∈ Qcoh(X´et) , to conclude p the vanishing of R f´et ∗(G) for p > 0. This will prove that f is affine. The main obstacle here is that we do not know if fket ∗F is ket locally classical quasi-coherent on Yket. One way to solve this is by considering a base change iso- ∗ ∼ ∗ morphism fket ∗X = Y f´et ∗ as in [Hag03, Proposition 3.15]. But that proposition is too restrictive since it requires f to be a strict morphism of log schemes and Y to satisfy a similar condition to log regularity.
79 6.2 Comparison of let cohomology and etale cohomology In this section we give a cohomological comparison between the etale, Kummer log etale sites and the log etale site. K.Kato in [Kat89a] and Nakayama in [Nak17] give a description of Sh(Xlet) as the projective limit of toposes of Sh((XI )ket) where I varies along the set of all log ideals of the log scheme X. Using that we can have a really good picture of how the functor κ∗ : Ab(Xket) → Ab(Xlet) behaves with cohomology. For reasons of completeness we state the following proposition.
Proposition 6.10. ([Nak17, Proposition 5.6]). Let X be an fs log scheme charted by an fs monoid P . Let us denote with I(P ) the set of all log blow-ups πI : XI → X of X where I is a non empty ideal of P . The following limit and colimit are running over all elements π : Y → X in I(P ).
1. We have an equivalence of toposes:
←−−−lim Sh(Yket) = Sh(Xlet) π:Y →X
where ←−−−lim is the projective limit of toposes [SGA72, Definition 8.1.1].
2. Let F be a sheaf of abelian groups on Xlet. Then, for any q ≥ 0 we have that: q q −1 R κX∗F =−−−−−→ colim R πket ∗(κY ∗πlet F ) π:Y →X Consider a log blow-up π : Y → X. Then we have the following diagram:
πlet ∗
Ab(Ylet) Ab(Xlet)
κY ∗ −1 κX∗ πlet πket ∗ Ab(Yket) Ab(Xket)
−1 where πlet =π ˜let ∗ is the restriction functor induced by the functor of Example 4.32. Note that the above diagram is not commutative. We are going to use the second part of the above theorem to prove coho- mological descent between the Kummer log etale cohomology and the log etale cohomology. The following lemma will be needed in Proposition 6.12
Lemma 6.11. Let X be a log regular log scheme, π : Y → X a log blow up and E a classical vector bundle on Xket. Then:
i ∗ ∼ i H (Yket, πketE) = H (Xket,E)
80 for every i ≥ 0. In particular, we have that:
Hi(Y , O ) ∼= Hi(X , O ),Hi(Y , Ωlog ) ∼= Hi(X , Ωlog ) ket Yket ket Xket ket Yket ket Xket for every i ≥ 0.
∗ 0 0 Proof. First note that since E is classical E = X E for a vector bundle E in ∗ ∈ Vect(X´et). By Lemma 5.32 we know that πketE is classical on Yket and in ∗ ∼ ∗ ∗ 0 particular πketE = Y π´etE . Then:
1 2 1 i ∗ i ∗ ∗ 0 ∼ i ∗ 0 ∼ i 0 ∼ i H (Yket, πketE) = H (Yket, Y π´et E ) = H (Y´et, π´et E ) = H (X´et,E ) = H (Xket,E) for every i ≥ 0, where use in 1 we use Proposition 6.4 and in 2 we use Remark 5.19. This proves the general case. Taking for E = O or E = Ωlog gives the Xket Xket rest.
Proposition 6.12. Let π : Y → X be a log blow up then, for any classical vector bundle E on X´et: q ∗ R πket ∗(πketE) = 0 for all q > 0. In particular, it holds for the extension sheaves O and Ωlog on Xket Xket Xket.
Proof. Consider the presheaf on Xket:
i fs ∗ (U → X) 7→ H ((U ×X Y )ket, πketE)
Since π : Y → X is a log blow-up and log blow-ups are preserved under fs-base change we use Lemma 6.11 and we get:
i fs ∗ ∼ i H ((U ×X Y )ket, πketE) = H (Uket,E)
Hence the above presheaf is locally zero and the higher direct images vanish.
Theorem 6.13. (let cohomology vs ket cohomology). Let X be a log regular fs log scheme and consider a classical vector bundle E on Xket. Then:
i ∗ ∼ i H (Xlet, κX E) = H (Xket,E) for every i ≥ 0. In particular:
Hi(X , O ) ∼= Hi(X , O ),Hi(X , Ωlog ) ∼= Hi(X , Ωlog ) let Xlet ket Xket let Xlet ket Xket for every i ≥ 0.
81 Proof. In order to prove this theorem we may assume that X is affine and globally charted. We will use the Leray spectral sequence associated to κX : Xket → Xlet ∗ and to the sheaf κX E. In order for the Leray spectral sequence to degenerate we need the higher direct images to vanish. So we are using Proposition 6.10 (2):
q ∗ q −1 ∗ R κX∗(κX E) = colim−−−−−→ R πket ∗(κY ∗πlet κX E) π:Y →X where π : Y → X varies over the log blow-ups of log ideals of X. By Remark 5.35 we have the description −1 ∗ ∼ ∗ κY ∗πlet κX E = πketE Hence we just need to prove that:
q ∗ R πket ∗(πketE) = 0 for all q > 0 which we already proved in Proposition 6.12. We have the analogues statement for the etale topology. Theorem 6.14. (let cohomology vs etale cohomology). Let X be a log regular fs log scheme and consider a vector bundle E on X´et. Then:
i ∗ ∼ i H (Xlet, ρ E) = H (X´et,E) for every i ≥ 0. In particular:
Hi(X , O ) ∼= Hi(X , O ),Hi(X , Ωlog ) ∼= Hi(X , Ωlog ) let Xlet ´et X´et let Xlet ´et X´et for every i ≥ 0.
i ∗ Proof. We will prove that the higher direct images R ρ∗(ρ E) vanish for i > 0. For this we use the Grothendieck spectral sequence for the composition of the functors ∗ ◦ κ∗ = ρ∗. Note that we can do that this κ∗ sends injective objects to limp sheaves which means that it sends κ∗-acyclic objects to ∗-acyclic objects [Sta18, Sect. 0731]. Note that the Grothendieck spectral sequence has second page [Sta18, Sect. 0734]: p,q p q ∗ E2 = R ∗(R κ∗(ρ E)) q ∗ In Theorem 6.13 we proved that the higher direct images R κ∗(ρ E) vanish for ∗ ∗ q > 0 and since κ∗ρ E = E the second page of the spectral sequence is: Rp (∗E) if q = 0 Ep,q = ∗ 2 0 if q > 0
i ∗ From Proposition 6.4 the higher direct images R ∗( E) also vanish for q > 0 since ∗E is classical. Having said that, the Grothendieck spectral sequence i degenerates at the E2-page and R ρ∗E = 0 for i ≥ 0. This gives us the comparison statement.
82 We close this subsection by emphasizing [Nak17, Proposition 5.16] in the fol- lowing remark. Namely, for any log blow-up π : Y → X the morphism of toposes πlet ∗ : Sh(Ylet) → Sh(Xlet) is an equivalence.
−1 Remark 6.15. Consider the restriction functor πlet : Sh(Xlet) → Sh(Ylet) of −1 ∼ −1 ∼ Example 4.32. We will show that πlet ◦ πlet ∗ = idSh(Ylet) and πlet ∗ ◦ πlet = idSh(Xlet). For this we use the fact that we can write the inverse image functor as a push- −1 forward functor i.e., πlet =π ˜let ∗. First note that for any element (g : U → Y ) in Ylet:
πlet ◦ π˜let(g : U → Y ) = πlet(π ◦ g : U → X) fs fs = ([π ◦ g] ×X Y : U ×X Y → Y ) =! (g : U → Y )
−1 ∼ which implies that πlet ◦ πlet ∗ = idSh(Ylet), and ! comes from Remark 3.24. For the other implication note that:
fs fs π˜let ◦ πlet(f : V → X) =π ˜let(f ×X Y : V ×X Y → Y ) fs fs = (π ◦ (f ×X Y ): V ×X Y → X)
So for a sheaf F ∈ Sh(Xlet) the unit morphism of the adjunction:
−1 fs F (V → X) → [πlet ∗ ◦ πlet F ](V → X) = F (V ×X Y → X)
fs which is an isomorphism since V ×X Y → V is a log blow-up and F satisfies log blow-up descent. Hence π induces an equivalence of toposes.
83 7 Applications and concluding remarks
7.1 Applications: Algebraic log de Rham cohomology for fs log schemes As an application of Theorem 6.13 we will define an algebraic log de Rham co- homology for fs log schemes on the log etale site. First we prove some descent arguments for the complex of logarithmic differential forms. Note that in what follows we will use heavily the language of filtered complexes and derived filtered categories as developed by Deligne in [Del71]. The reference that we follow is [Sta18, Sect. 015O]. For the convenience of the reader we include a paragraph in the Appendix with the definitions that we will use.
Conventions 7.1. In the rest of this section our schemes are of finite type over a field k of characteristic zero. The log schemes are fine and saturated, defined over (Spec(k), triv).
7.2. (The log de Rham complex). Let X be a log regular fs log scheme and consider the log de Rham complex over X´et:
Ωlog,• : O →d Ωlog →d Ωlog,2 →d ... X´et X´et X´et X´et where Ωlog,n := ∧nΩlog is the exterior algebra of the sheaf of log differential forms X´et X´et on X and Ωlog,1 := Ωlog . Note that since the differentials d :Ωlog,p → Ωlog,p+1 ´et X´et X´et X´et X´et are not OX´et -linear, the log de Rham complex is a complex of sheaves of abelian groups. We equip the log de Rham complex with the stupid filtration, which is a decreasing filtration defined by:
τ p(Ωlog,•) : 0 → ... → 0 → Ωlog,p → Ωlog,p+1 → ... X´et X´et X´et
Note that the graded pieces of the filtered complex (Ωlog,•, τ) are: X´et
Grp Ωlog,• = Ωlog,p[−p] τ X´et X´et where Ωlog,p[−p] is the complex which equal to Ωlog,p and zero everywhere else. X´et X´et The following lemma is a consequence of the fact that the sheaf of logarithmic forms is a vector bundle.
Lemma 7.3. Let X be a log regular log scheme. Then there exits a natural number n ∈ such that Ωlog,n = 0. N≥0 X´et Proof. Since X is log regular the etale sheaf Ωlog is a locally free sheaf of rank r by X´et Proposition 2.16. Thus, we can find an etale cover f : V → X such that f −1Ωlog = ´et X´et
84 Or . Note that since Ωlog,p is an etale sheaf we have that f ∗ Ωlog,p = f −1Ωlog,p. V´et X´et ´et X´et ´et X´et Taking exterior powers commutes with the restriction [Sta18, Sect. 01CI]:
f −1Ωlog,n = f −1(∧nΩlog ) ∼= ∧nf −1Ωlog ∼= ∧n Or ´et X´et ´et X´et ´et X´et V´et Hence, by taking n > r and the above sheaf is trivial. This completes the proof.
By Lemma 7.3, the log de Rham complex is bounded from above. This gives us the following corollary.
Corollary 7.4. The stupid filtration τ on the log de Rham complex is biregular i.e. for every i ∈ there exists n, m such that τ nΩlog,i = 0 and τ mΩlog,i = Ωlog,i. N X´et X´et X´et Let π : Y → X be a log blow-up. Note that Y is log smooth hence, what we said above for the log de Rham complex of X also applies for the log de Rham complex of Y . We want to compare the hyercohomology groups:
i(X , Ωlog,•), i(Y , Ωlog,•) H ´et X´et H ´et Y´et For this we will use the language of filtered derived categories and Hodge theoretic techniques. Consider the log de Rham complex of Y equipped with the stupid filtration (Ωlog,•, σ) which we denote with σ. We want to construct a Y´et filtered quasi-isomorphism:
φ˜ : (Ωlog,•, τ) → (Rπ Ωlog,•, Rπ σ) X´et ∗ Y´et ∗ We give the details in the following paragraph.
7.5. (Construction of the filtered quasi-isomorphisms). Let the setup and notation be as above. Consider the following composition of filtered homomorphism in + FComp (X´et):
φ (Ωlog,•, τ) → (π (Ωlog,•), π σ) π→∗s (π (I•), π F ) X´et ∗ Y´et ∗ ∗ ∗ where:
1. The morphism φ :Ωlog,• → π Ωlog,• is constructed by using the unit iso- X´et ∗ Y´et ∗ morphism id → π∗π of Remark 5.19 and it is an isomorphism of filtered complexes.
2. The morphism s : (Ωlog,•, τ) → (I•,F ) is a filtered injective resolution in Y´et the sense of Paragraph 8.2, and in particular a filtered quasi-isomorphism. In particular, since π∗ is left exact the morphism π∗s is compatible with the filtrations.
85 Note that in DF +(X ), the filtered complex (Rπ Ωlog,•, Rπ σ) is represented by ´et ∗ Y´et ∗ • the filtered complex (π∗I , π∗F ). Since both of the morphisms π∗s and φ are compatible with the filtrations, the same is true for the composition (π∗s) ◦ φ. Hence, it induces a well defined morphism in the filtered derived category which we will denote by φ˜. We note that for a different choice of filtered injective resolution s0 the resulting morphisms in the filtered derived category will be the same up to unique isomorphism [Sta18, Sect. 05TU]. Hence, φ˜ is a filtered homomorphism of filtered complexes. Lemma 7.6. The filtered homomorphism constructed above:
φ˜ : (Ωlog,•, τ) → (Rπ Ωlog,•, Rπ σ) X´et ∗ Y´et ∗ is a filtered quasi-isomorphism. Proof. Assume the setup of the above paragraph. In order to prove our statement we examine the induced morphism on the graded pieces Grp(φ˜). Consider the + following commutative diagram in Comp (X´et):
0 τ p+1Ωlog,• τ pΩlog,• Grp Ωlog,• 0 X´et X´et τ X´et
Grp φ
0 π σp+1Ωlog,• π σpΩlog,• Grp π Ωlog,• 0 ∗ Y´et ∗ Y´et π∗σ ∗ Y´et
p Gr π∗s
0 π F p+1I• π F pI• Grp π I• 0 ∗ ∗ π∗F ∗
We note the following: 1. The morphism Grp φ : Grp Ωlog,• → Grp π Ωlog,• is equal to: τ X´et π∗σ ∗ Y´et φ [−p]:Ωlog,p[−p] → π Ωlog,p[−p] p X´et ∗ Y´et where [−p] is the p-shift and φ :Ωlog,p → π Ωlog,p is the natural isomorphism p X´et ∗ Y´et ∗ induced by the unit id → π∗π . 2. The graded pieces of (Rπ Ω•,log, Rπ σ) are given by: ∗ Y´et ∗ Grp Rπ Ωlog,• := Grp π I• ∼= π (Grp I•) Rπ∗σ ∗ Y´et π∗F ∗ ∗ F where the last isomorphism is from [Sta18, Sect. 015U]. Since s is a filtered quasi-isomorphism it induces a quasi-isomorphism on the graded pieces:
Grp(s):Ωlog,p[−p] → Grp I• Y´et F
86 Since by the definition of filtered injective resolutions the morphism Grp(s) is an injective resolution of Ωlog,p[−p], π (Grp I•) represents Rπ Ωlog,p[−p] in the Y´et ∗ F ∗ Y´et p p derived category. Thus, the composition Gr φ ◦ Gr π∗s represents the natural morphism φ [−p]:Ωlog,p[−p] → Rπ (Ωlog,p[−p]) which is a quasi-isomorphism by p X´et ∗ Y Remark 5.19. Hence φ˜ is a filtered quasi-isomorphism. Since φ˜ is a filtered quasi-isomorphism we use [Del71, I.4.5] to conclude that φ˜ induces an isomorphism on the induced spectral sequences associated to the filtered complexes (Ωlog,•, τ) and (Rπ Ωlog,•, σ): X´et ∗ Y´et
Ep,q := Hq(X , Rπ Ωp ) ∼= Hq(Y , Ωp ) ⇒ p+q(Y , Ωlog,•) 1 ´et ∗ Y´et ´et Y´et H ´et Y´et
0Ep,q := Hq(X , Ωp ) ⇒ p+q(X , Ωlog,•) 1 ´et X´et H ´et X´et Gathering all the above we proved the following.
Theorem 7.7. Let X be a log regular log scheme and consider a log blow-up π : Y → X. Then, the natural morphism φ˜ :Ωlog,• → Rπ Ωlog,• constructed in X´et ∗ Y´et Paragraph 7.5, induces an isomorphism:
i(X , Ωlog,•) ∼= i(Y , Ωlog,•) H ´et X´et H ´et Y´et on the hypercohomologies.
In the next theorem we give a generalization of the theorem of Deligne [Del71, Proposition 3.1.8] for etale log regular fs log schemes with proper underlying scheme.
Theorem 7.8. Let X be a log regular fs log scheme over (Spec(C), triv) where X is proper as scheme over Spec(C), and let Xtriv be the trivial locus of the log structure, see Notation 1.36. Then we have a canonical isomorphism:
i(X , Ωlog,•) ∼= Hi (X , ) H ´et X´et sing triv C Proof. Let π : Y → X be a resolution of singularities via log blow-ups of X such that Y is Zariski log regular. In particular π is log etale as a morphism of log schemes since it is a log blow-up. Note that we have such resolution of singularities from [GR04, Proposition 12.6.52, Corollary 12.6.53]. We also note that by [GR04, Remark 12.5.31] the divisorial log structure of Y comes from a simple normal crossings divisor. By Theorem 7.7:
1 i(X , Ωlog,•) ∼= i(Y , Ωlog,•) ∼= Hi (Y , ) H ´et X´et H ´et Y´et sing triv C where the isomorphism 1 comes from [Del71, Proposition 3.1.8]. Then, by [GR04, ∼ Corollary 12.6.53] we have that Ytriv = Xtriv which gives us the isomorphism i ∼ i Hsing(Ytriv, C) = Hsing(Xtriv, C). This completes the proof.
87 Remark 7.9. The general case (without the assumption of properness) was orig- inally conjectured by T. Oda and proved by A. Ogus, [Ogu18, Theorem 4.2.5 (1)], in a very straightforward way. Specifically, he shows that the natural morphism:
Ωlog,• → Rj Ω X/C ∗ Xtriv/C is a quasi-isomorphism, hence:
i ∼ i HdR(X) = HdR(Xtriv) where the latter is isomorphic to Hsing(Xtriv, C) by the classical comparison iso- morphism. We note that our proof is completely different since we use log blow-ups and logarithmic sites. Now we compare the hypercohomology of the log de Rham complex on the etale site and the log etale site. We define our setup in the following paragraph. 7.10. (Log de Rham complex for the let topology). Let X be a log regular fs log scheme over (Spec(k), triv). Similarly as in the introduction of this section we define the log de Rham complex on X to be the complex Ωlog,• where: let Xlet Ωlog,p := ρ∗Ωlog,p Xlet X´et and the differentials are defined locally in the following way: For every object (f : Y → X) in Xlet we define: d (f : Y → X):Ωlog,p(f : Y → X) → Ωlog,p+1(f : Y → X) let Xlet Xlet to be equal to:
d(f : Y → X) : Γ(Y , Ωlog,p) → Γ(Y , Ωlog,p+1) Y´et Y´et Note that this assignment is natural since the construction of Ωlog and the exterior powers is functorial in the category of log schemes. We note that the reason that we define the differentials in this way is because the derivations d :Ωlog,p → Ωlog,p+1 are not O -linear so we can not just pull X´et X´et X´et them back via ρ∗. Theorem 7.11. Let X be a log regular fs log scheme. Then, we have the following isomorphism: i(X , Ωlog,•) ∼= i(X , Ωlog,•) H let Xlet H ´et X´et Proof. The proof is similar to the proof of Theorem 7.7. Consider the log de Rham complex on X equipped with the stupid filtration (Ωlog,•, σ). Choose a let Xlet filtered injective resolution s : (Ωlog,•, σ) → (I•,F ) in the sense of Paragraph 8.2 Xlet and consider the induced filtered morphism of filtered complexes:
φ˜ : (Ωlog,•, τ) → (Rρ Ωlog,•, Rρ σ) X´et ∗ Xlet ∗
88 where τ is the stupid filtration on Ωlog,•. Arguing similarly as in Theorem 7.7 X´et we get that φ˜ is compatible with the filtrations. Note that the graded piece of (Rρ Ωlog,•, Rρ σ) is equal to ∗ Xlet ∗ Grp Rρ Ωlog,• := Grp ρ I• ∼= π Grp I• Rρ∗σ ∗ Xlet ρ∗F ∗ ∗ F On the other hand, the graded pieces of (Ωlog,•, τ) are: Xlet Grp Ωlog,• = Ωlog,p[−p] σ Xlet Xlet and since s is a filtered injective resolution the induced map on the graded pieces: Grp(s):Ωlog,p[−p] → Grp I• Xlet F is a quasi-isomorphism. In particular, Grp I• is an injective resolution of Ωlog,p[−p] F Xlet which means that π Grp I• represents the complex Rρ Ωlog,p[−p] in the derived ∗ F ∗ Xlet + ˜ category D (X´et). Hence φ induces a morphism on the graded pieces: φ˜ :Ωlog,p[−p] → Rρ Ωlog,p[−p] p X´et ∗ Xlet ˜ We prove that φp is a quasi-isomorphism. For this, note that we only need to prove the same statement but without shifts. So we take cohomology of the above complexes, without shifts, and the induced morphism on the higher direct images is: 0 → Riρ Ωlog,p ∗ Xlet for i > 0 and for i = 0 is the isomorphism Ωp → ρ (Ωlog,p) induced by X´et ∗ Xlet ∗ the unit id → ρ∗ρ from Corollary 5.25. By Theorem 6.14, the higher direct images Riρ Ωlog,p vanish hence, the morphism on the graded pieces is a quasi- ∗ Xlet isomorphism. This means that φ˜ is a filtered quasi-isomorphism and it induces an isomorphism on the spectral sequences. This completes the proof. We note that by the Theorem 7.11 we can give a slightly different proof of Theorem 7.8. Alternative proof for Theorem 7.8. Consider a resolution of singularities π : Y → X via log blow-ups as in the proof of Theorem 7.8. Then, by Theorem 7.11 we have that: i(X , Ωlog,•) ∼= i(X , Ωlog,•), i(Y , Ωlog,•) ∼= i(Y , Ωlog,•) H let Xlet H ´et X´et H let Ylet H ´et Y´et Since by Remark 6.15 the full log etale topos of X is equivalent to the log etale topos of Y , we get an isomorphism: i(X , Ωlog,•) ∼= i(Y , Ωlog,•) H let Xlet H let Ylet Then, by using [Del71, Proposition 3.1.8] and [GR04, Corollary 12.6.53] we have that: i(Y , Ωlog,•) ∼= Hi (Y , ) ∼= Hi (X , ) H let Ylet sing triv C sing triv C Connecting together all the isomorphisms we get a proof for Theorem 7.8.
89 Remark 7.12. We note this second proof to emphasize the following fact: The log smooth scheme X behaves as a smooth scheme and this is manifested by passing to the log etale toposes. This is an instance of the classical yoga of logarithmic geometry stating that log smooth log schemes behave as smooth schemes. As we can see, the two proofs are similarly hard since in both of them we used the same mathematical machinery hodge theory, derived filtered categories etc. Finally, we need to note that the main geometrical input that we used was the fact that log regular log schemes have rational singularities and that log etale morphism preserve the sheaves of logarithmic differential forms, Proposition 2.14. By what we already established we can define the algebraic log de Rham cohomology for fs log schemes using the log etale site. Definition 7.13. Let X be an fs log scheme. We define the i-th algebraic log de Rham cohomology group of X to be:
Hi (X) := i(X , Ωlog •) log dR H let Xlet where the hypercohomology is defined on Xlet. Note that we can always define the hypercohomology groups i(X , Ωlog •) H let Xlet irregardless if X is log regular or not. The main point of Theorem 7.8 is that we can actually ”compute” the algebraic log de Rham cohomology for log regular fs log schemes. Note that the second part of the following theorem is a reformulation of Theorem 7.8.
Theorem 7.14. Let X be log regular fs log scheme over (Spec(C), triv) such that X is proper as a scheme over Spec(C). Then: i ∼ i Hlog dR(X) = Hsing(Xtriv, C) In particular, when X is a scheme which is smooth proper and equipped with a normal crossings divisor D, the algebraic log de Rham cohomology agrees with the usual log de Rham cohomology used in [Del71] i.e.:
i log,• ∼ i • H ((X, MD), Ω ) = (X, Ω hlog Di) log dR (X,MD)let H where Ω•hlog Di is the logarithmic de Rham complex defined in [Del71, 3.1.2].
7.2 Concluding remarks The main starting point of this thesis was the finding of an analogue of the h- topology for the category of log schemes for questions of cohomological nature. Two starting ideas for this was: 1. To explore the category of h-log schemes. These should be log schemes (X, M) where the sheaf of monoids M is defined in the h-topology.
90 2. To use a log version of the valuation criterion of properness in order to define log h morphisms of log schemes. This idea is due to Shane Kelly.
The reason that we used the log etale topology is the following: The log etale topology contains log blow-ups and for log regular log schemes we can resolve singularities via log blow-ups. Hence, the log etale topology makes log regular log schemes locally smooth, an effect similar to what h-topology does on schemes, assuming the resolutions of singularities of Hironaka. Thus, the log etale topology is similar enough to the h-topology, as long as we work over a log regular base. Having this in mind, we expect the log etale site of a log regular log scheme X to be similar to the small h-site of X. Hence, the log etale topology should be treated first. What is missing from the big picture is a direct comparison of the h-site and the log etale site. The main obstacle is that the fiber products of Xlet and Xh are not compatible and hence, a comparison between the two sites is not straight forward. More needs to be done in this direction.
91 8 Appendix
In the first part of this Appendix, we give some notes and references about the definitions and the theory that we use. In the last part, we complete some proofs about some statements appearing in this document.
8.1 (Sites and toposes). Whatever regards the theory of sites, toposes, modules and cohomology used in this document, we use [Sta18, Sect. 00UZ, Sect. 03A4 and Sect. 01FQ]. For the convenience of the reader we include some definitions and conventions that we use.
1. Since our categories of interest are defined by over categories, for example fs log schemes over an fs log scheme X, we represent the objects of our sites via the structure morphisms and we say: ”Let (f : Y → X) be an object of Obj(Xket).”. Whenever confusion does not arise we may skip the structure morphism and represent our objects only by the domain of the structure morphism.
2. For a pair of adjoint functors F : C → D,G : D → C we write G a F to indicate that G is left adjoint to F i.e: for any objects A in C and B in D we have natural isomorphisms in both variables:
∼ ϕB,A : HomC(GB, A) → HomD(B,FA)
A mnemonic trick to use is that the left adjoint G appears on the left.
3.A site [Sta18, Sect. 00VH] is a category C equipped with a set Cov(C) of families of morphisms (Ui → U)i∈I for every U ∈ Obj(C), which we will call coverings or covers, and satisfy some axioms. In general, we refer to the collection of coverings as a topology on C and we usually denote the topology by τ. Sometimes to relax the notation of a covering (Vi → U)i∈I we omit the index in the covering and we say ”Let (V → U) be a covering...”. We note that this does not bring any problems since we ` can replace a covering (Vi → U)i∈I by the covering ( Vi → U) and set ` i V := i∈I Vi. We can do this since the presheaves that we use already satisfy Zariski descent. We also note that the above definition of sites is similar, if not the same, to the notion of a category equipped with a Grothendieck pretopology. We refer to [Sta18, Sect. 00ZN] for the connections of notion of sites that we give here and the notion of Grothendieck sites and Grothendieck topologies. We note that in this document we use small sites [Sta18, Sect. 020T, Sect. 021B].
4.A topos, is a category which can be written as Sh(Cτ ) for a site (C, τ). Classical examples of toposes appearing in algebraic geometry are the Zariski topos Sh(XZar) and the etale topos Sh(X´et).
92 5.A continuous functor of sites [Sta18, Sect. 00WV] f : C → D is given by the honest functor in the level of categories and a continuous morphism of sites is given traditionally in the other direction f : D → C. We justify the word traditionally. Let f : X → Y be a continuous map of topological spaces. Note that the set Open(X) of open subsets of X can be view as a partially ordered set where the ordering is the inclusion of sets. Since f is continuous we get an induced map f˜ : Open(Y ) → Open(X) defined by mapping an open subset U of Y to f˜(U) := f −1(V ). A continuous functor of sites looks like f˜ and not like f. Hence, when we work on sites coming from geometric data, traditionally we write the morphism in the opposite direction of the honest functor in the level of categories.
6. For a site C we denote with Ab(C) (resp. PAb(C)) the category of sheaves (resp. presheaves) of abelian groups on C [Sta18, Sect. 03CM] and by Comp(C) we denote the category of complexes of sheaves abelian groups on C. We also use Comp+(C) to denote complexes that are bounded below.
7. Let (C, O) be a ringed site. By a module on C [Sta18, Sect. 03CW] a sheaf of O-modules on and we denote the category of modules over (C, O) by Mod(O). In the same spirit, we denote with PMod(O) the category of presheaves of O-modules. When the ringed site (C, O) is the Zariski site of a ring R we write Mod(R) for the category of sheaves of modules over (Spec(R)Zar, OSpec(R)) which is equivalent to the category of modules over R, hence the notation is justified.
8. We say that a module F ∈ Mod(O) is of global presentation [Sta18, Sect. 03DE (6)] if it has the following presentation:
O⊕I → O⊕J → F → 0
for some index sets I,J.
9. We say that a module F ∈ Mod(O) is quasi-coherent [Sta18, Sect. 03DL (6)] if for every U in C there exists a cover (V → U) in C such that F |V is of global presentation
10. We say that a module F ∈ Mod(O) is a vector bundle [Sta18, Sect. 03DL (2)] if it is locally free of finite rank i.e., for every U in C there exist a cover (V → U) such that F |V is a free O |V module of finite rank. In the next paragraph we fix the notation and give some definitions that we use for the theory of filtered derived categories.
8.2 (Filtered Derived categories). Let A be an abelian category with enough injectives.
93 1. We define the category of filtered objects [Sta18, Sect. 0121] of A, which we denote with Fil(A) to be the category with objects (A, F ) where A ∈ Obj(A) and F is a decreasing fltration on A. The morphisms in Fil(A) are morphisms in A which are compatible with the filtration i.e., f :(A, F ) → (B,G) such that f(F n(A)) ⊂ Gn(B).
2. In the same spirit, we define the category of finite filtered objects f [Sta18, Sect. 05RY] of A to be the full subcategory Fil (A) of Fil(Xτ ) gen- erated by filtered objects (A, F ) where the filtration is finite i.e., there exists n, m ∈ N such that F n(A) = A and F m(A) = 0. An object (I,F ) in Filf (A) is called filtered injective [Sta18, Sect. 015P] if all the graded p pieces GrF (A) are injective objects in A. We say that a morphism α :(A•,F ) → (B•,G) in K(Filf (A)) is a filtered quasi-isomorphism if for every p the induced morphism on the graded p p • p • pieces Gr (α) : GrF (A ) → GrG(B ) is a quasi-isomorphism. 3. Having said that, we define the derived filtered category DF(A) to be the category K(Filf (A)) localized at filtered quasi-isomorphisms [Sta18, Sect. 05S2]. We also define the derived filtered category of bounded below complexes to be the full subcategory DF+(A) of DF(A) generated by complexes of elements in Filf (A) that are bounded below. We note that the derived category DF+(A) has enough injectives in the following sense [Sta18, Sect. 05TW]: For every complex of filtered objects (K•,F ) in K+(Filf (A)) we can find a filtered quasi-isomorphism s :(K•,F ) → • n (I ,G) where each piece of the complex with the induced filtration (I ,Fn) is graded injective. We will call such resolution a filtered injective reso- lution.
8.3 (Quasi-coherent vs ket locally quasi-coherent, Vector bundles vs let locally vector bundles). We prove in this paragraph that the notion of a ket locally classical quasi-coherent module on Xket is equivalent to the notion of quasi-coherent modules on Xket. This is used in Remark 6.5.
Let X be an fs log scheme and pick a module F in Mod(OXket ). Suppose that F is ket locally classical quasi-coherent i.e., there exists a ket-covering (f : V → X) ∼ ∗ such that F |Vket = V G for some quasi-coherent sheaf G over V´et. Since G is quasi-coherent we can further refine by a strict log etale cover (V 0 → V ) such that ∼ ∗ 0 0 0 F |V 0 = V 0 G where G ∈ Qcoh(V´et) is of global presentation. The key concept here is that we can use a strict log etale cover to do that. To see this, since G is 0 quasi-coherent on V we can find an etale cover (f : V → V ) such that G| 0 is of ´et V´et global presentation. Then, we equip the scheme V 0 with the inverse log structure 0 ∗ of V via f, see Example 1.60. Then, by taking G := f´etG:
∗ 0 ∗ ∗ 1 ∗ ∗ 2 −1 ∗ 0 G := 0 f G = f G = f G = F | 0 V V ´et ket V ket V Vket
94 where 1 comes from the functoriality of the pullback of modules and 2 comes from ∗ Remark 5.30. Since V 0 is right exact we can pull pack the global presentation of 0 G to a global presentation for F | 0 . Hence F is of global presentation, achieved Vket by the covering (V 0 → V → X).
Now, for every (U → X) in Xket the module F |Uket is ket locally of global 0 fs presentation, achieved by the covering (V ×V U → V ×X U → U). This makes F a quasi-coherent module on Xket. For the other direction, assume that F is quasi-coherent on Xket. Then, there exists a ket cover (V → X) such that F |V is of global presentation i.e, there exist φ : Or → Om such that F | ∼ coker φ. Since ∗ : Qcoh(V ) → Mod(X ) is Vket Vket Vket = ´et ket fully faithful by Corollary 5.17, we can find an ψ : Or → Om such that ∗ ψ = φ V´et V´et V ∗ and since V is right exact: ∗ ∼ ∼ V coker ψ = coker ψ = F |Vket This means that F is a ket locally classical quasi-coherent. Note that the arguments for the equivalence of vector bundles and let locally classical vector bundles on Xlet are exactly the same. The only difference is that we use Remark 5.31 instead of Remark 5.30 and Corollary 5.26 instead of Corollary 5.17. In the following paragraph we give a proof about a statement mentioned in Paragraph 5.2. 8.4 (Equivalence of description of sheaves). We prove a statement we use in Paragraph 5.2. It is also used in Lemma 5.15 and Lemma 5.23. Let τ ∈ {´et, ket, let} be a topology and consider an fs log scheme X and the associated site Xτ . Note that when τ = let we require X to be log regular since we want the extension presheaf OXlet to be a sheaf. For the moment assume that the extension presheaf OXτ is a sheaf on Xτ and we set R := Γ(Xτ , OXτ ).
Then consider the functor Sτ : Mod(R) → Mod(OXτ ) from Paragraph 5.2. We −1 denote with Sτ : Ab → Ab(Xτ ) the inverse image functor on the level of sheaves −1,p of abelian groups and by Sτ : Ab → PAb(Xτ ) the inverse image functor on −1,p ] −1 −1,p the level of presheaves of abelian groups with (Sτ ) =: Sτ . Note that Sτ associates to each abelian group M the constant presheaf of M on Xτ .
We are going to prove that the functor Sτ : Mod(R) → Mod(OXτ ) is the functor which associates to each module M ∈ Mod(R) the sheaf associated to the presheaf:
(f : U → X) 7→ M ⊗R OXτ (U) where (f : U → X) is an object of Xτ . We argue as follows. Note that since OXτ is a sheaf, for every object (f : U → X) in Xτ we have a morphism
φf:U→X : Γ(Xτ , OXτ ) → OXτ (U) := Γ(U, OU´et ) −1,p So, we can define the morphism of presheaves of rings φ : Sτ R → OXτ such that φ(f : U → X) := φf:U→X . By the universal property of sheafification this extends
95 ] −1 ] to a morphism of sheaves of rings φ : Sτ R → OXτ . By using φ we can define −1 a morphism of ringed sites (Xτ , OXτ ) → (Xτ ,Sτ R), see [Sta18, Sect. 03AD]. Consider now the following commutative diagram:
φ] ∗ −1 Mod(OXτ ) Mod(Sτ R)
Inc Inc
−1,p PMod(OXτ ) PMod(Sτ R) φ∗ where both of the functors Inc are the right adjoints of the sheafification functors ] [Sta18, above Sect. 03EI], and φ∗, φ∗ are the pushforward morphisms and right adjoint to the base change morphisms defined in [Sta18, above Sect. 03CU and above Sect.03CZ]. Taking the associated diagram of the left adjoints:
−1,p ] ] −1,p p ] −1 ∼ ((Sτ M) ⊗S R OXτ ) = (Sτ M ⊗ −1,p OXτ ) τ Sτ R
−1,p ] −1 Since (Sτ M) = Sτ M the left hand-side is equal to Sτ (M). On the other hand −1,p since Sτ assigns the constant presheaf, the right hand is the sheaf associated to:
(f : U → X) 7→ M ⊗R OXτ (U) This means that the two descriptions agree.
96 References
[ACM+16] Dan Abramovich, Qile Chen, Steffen Marcus, Martin Ulirsch, and Jonathan Wise. Skeletons and fans of logarithmic structures. In Nonarchimedean and tropical geometry, Simons Symp., pages 287– 336. Springer, [Cham], 2016.
[CLS11] David A. Cox, John B. Little, and Henry K. Schenck. Toric vari- eties, volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2011.
[Del71] Pierre Deligne. Th´eoriede Hodge. II. Inst. Hautes Etudes´ Sci. Publ. Math., (40):5–57, 1971.
[EV92] H´el`ene Esnault and Eckart Viehweg. Lectures on vanishing theorems, volume 20 of DMV Seminar. Birkh¨auserVerlag, Basel, 1992.
[EZuT14] Fouad El Zein and LˆeD ung Tr´ang. Mixed Hodge structures. In Hodge theory, volume 49 of Math. Notes, pages 123–216. Princeton Univ. Press, Princeton, NJ, 2014.
[GK15] Ofer Gabber and Shane Kelly. Points in algebraic geometry. J. Pure Appl. Algebra, 219(10):4667–4680, 2015.
[GM71] Alexander Grothendieck and Jacob P. Murre. The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme. Lecture Notes in Mathematics, Vol. 208. Springer- Verlag, Berlin-New York, 1971.
[GR04] Ofer Gabber and Lorenzo Ramero. Foundations for almost ring theory – release 7. Available on the internet at https://arxiv. org/abs/math/0409584, 2004.
[Gro65] A. Grothendieck. ´el´ements de g´eom´etriealg´ebrique.IV. ´etudelocale des sch´emaset des morphismes de sch´emas.II. Inst. Hautes Etudes´ Sci. Publ. Math., (24):231, 1965.
[Gro67] A. Grothendieck. ´el´ements de g´eom´etriealg´ebrique.IV. ´etudelocale des sch´emaset des morphismes de sch´emas IV. Inst. Hautes Etudes´ Sci. Publ. Math., (32):361, 1967.
[Hag03] Kei Hagihara. Structure theorem of Kummer ´etale K-group. K- Theory, 29(2):75–99, 2003.
[Har77] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York- Heidelberg, 1977. Graduate Texts in Mathematics, No. 52.
97 [HJ14] Annette Huber and Clemens J¨order. Differential forms in the h- topology. Algebr. Geom., 1(4):449–478, 2014.
[Ill02] Luc Illusie. An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic ´etalecohomology. Ast´erisque, (279):271– 322, 2002. Cohomologies p-adiques et applications arithm´etiques, II.
[Ill15] Luc Illusie. From pierre deligne’s secret garden: looking back at some of his letters. Japanese Journal of Mathematics, 10(2):237–248, Sep 2015.
[INT13] Luc Illusie, Chikara Nakayama, and Takeshi Tsuji. On log flat de- scent. Proc. Japan Acad. Ser. A Math. Sci., 89(1):1–5, 2013.
[IT14] Luc Illusie and Michael Temkin. Expos´eVIII. Gabber’s modifica- tion theorem (absolute case). Ast´erisque, (363-364):103–160, 2014. Travaux de Gabber sur l’uniformisation locale et la cohomologie ´etale des sch´emas quasi-excellents.
[Kat89a] Kazuya Kato. Logarithmic degeneration and Dieudonne theory. Preprint, 1989.
[Kat89b] Kazuya Kato. Logarithmic structures of Fontaine-Illusie. In Alge- braic analysis, geometry, and number theory (Baltimore, MD, 1988), pages 191–224. Johns Hopkins Univ. Press, Baltimore, MD, 1989.
[Kat91] Kazuya Kato. Logarithmic structures of Fontaine - Illusie II — Logarithmic flat topology. Preprint, 1991.
[Kat92] Kazuya Kato. Logarithmic Dieudonne theory. Preprint, 1992.
[Kat94] Kazuya Kato. Toric singularities. Amer. J. Math., 116(5):1073–1099, 1994.
[Kat96] Fumiharu Kato. Log smooth deformation theory. Tohoku Math. J. (2), 48(3):317–354, 1996.
[Kat99] Fumiharu Kato. Exactness, integrality, and log modifications. Avail- able on the internet at https://arxiv.org/abs/math/9907124, 1999.
[KKMSD73] G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat. Toroidal embeddings. I. Lecture Notes in Mathematics, Vol. 339. Springer-Verlag, Berlin-New York, 1973.
98 [Kol97] J´anos Koll´ar. Singularities of pairs. In Algebraic geometry—Santa Cruz 1995, volume 62 of Proc. Sympos. Pure Math., pages 221–287. Amer. Math. Soc., Providence, RI, 1997.
[Mat89] Hideyuki Matsumura. Commutative ring theory, volume 8 of Cam- bridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 1989. Translated from the Japanese by M. Reid.
[Mat02] Kenji Matsuki. Introduction to the Mori program. Universitext. Springer-Verlag, New York, 2002.
[Mil80] James S. Milne. Etale´ cohomology, volume 33 of Princeton Mathe- matical Series. Princeton University Press, Princeton, N.J., 1980.
[Nak97] Chikara Nakayama. Logarithmic ´etalecohomology. Math. Ann., 308(3):365–404, 1997.
[Nak17] Chikara Nakayama. Logarithmic ´etalecohomology II. Advances in Mathematics, 314:663 – 725, 2017.
[Nik60] Kazantzakis Nikos. The Saviors of God: Spiritual Exercises. En- glish translation by Kimon Friar. New York: Touchstone/ Simon & Schuster, 1960.
[Nik15] Kazantzakis Nikos. Askitiki. Kazantzakis Editions, revised version: 2015.
[Niz06] Wies lawa Nizio l. Toric singularities: log-blow-ups and global reso- lutions. J. Algebraic Geom., 15(1):1–29, 2006.
[Niz08] Wies lawa Nizio l. K-theory of log-schemes. I. Doc. Math., 13:505– 551, 2008.
[Oda88] Tadao Oda. Convex bodies and algebraic geometry, volume 15 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties, Translated from the Japanese.
[Oda93] Tadao Oda. The algebraic de Rham theorem for toric varieties. Tohoku Math. J. (2), 45(2):231–247, 1993.
[Ogu18] Arthur Ogus. Lectures on Logarithmic Algebraic Geometry. to be published, U.C. Berkeley, 2018.
[Ols03] Martin C. Olsson. Logarithmic geometry and algebraic stacks. Ann. Sci. Ecole´ Norm. Sup. (4), 36(5):747–791, 2003.
99 [Sam10] Keerthi Madapusi Sampath. Log p-divisible groups. Available on the internet at https://pdfs.semanticscholar.org/6dff/ 95faa93ecb3ec2e0c161436d0eb9077183c8.pdf, 2010.
[SGA72] Th´eoriedes topos et cohomologie ´etaledes sch´emas.Tome 2. Lecture Notes in Mathematics, Vol. 270. Springer-Verlag, Berlin-New York, 1972. S´eminairede G´eom´etrieAlg´ebriquedu Bois-Marie 1963–1964 (SGA 4), Dirig´epar M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat.
[Sta18] The Stacks Project Authors. Stacks project. Available on the inter- net at https://stacks.math.columbia.edu, 2018.
[Tam94] G¨unter Tamme. Introduction to ´etalecohomology. Universitext. Springer-Verlag, Berlin, 1994. Translated from the German by Man- fred Kolster.
[Uli15] Martin Ulirsch. Tropical geometry of logarithmic schemes. Brown university, Phd thesis, 2015.
[Vis04] Angelo Vistoli. Notes on grothendieck topologies, fibered categories and descent theory. Available on the internet at https://arxiv. org/abs/math/0412512, 2004.
100