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Cohomological Descent for Logarithmic Differential Forms in the Log Etale

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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. Wies lawa 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.
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  • Modules Over Algebraic Cobordism

    Modules Over Algebraic Cobordism

    MODULES OVER ALGEBRAIC COBORDISM ELDEN ELMANTO, MARC HOYOIS, ADEEL A. KHAN, VLADIMIR SOSNILO, AND MARIA YAKERSON Abstract. We prove that the 1-category of MGL-modules over any scheme is equivalent to the 1-category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite P1-loop spaces, we deduce that very effective MGL-modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers. Along the way, we describe any motivic Thom spectrum built from virtual vector bundles of nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes with the corresponding tangential structure. In particular, over a regular equicharacteristic base, we show that Ω1 MGL is the A1-homotopy type of the moduli stack of virtual finite flat P1 local complete intersections, and that for n > 0, Ω1 Σn MGL is the A1-homotopy type of P1 P1 the moduli stack of finite quasi-smooth derived schemes of virtual dimension −n. Contents 1. Introduction 2 1.1. Modules over algebraic cobordism 2 1.2. Framed correspondences and the motivic recognition principle 2 1.3. Geometric models of motivic Thom spectra 3 1.4. Analogies with topology 4 1.5. Related work 5 1.6. Conventions and notation 5 1.7. Acknowledgments 6 2. Twisted framed correspondences 6 2.1. Stable vector bundles and K-theory 6 2.2. Presheaves of twisted framed correspondences 7 2.3. Descent and additivity 9 2.4. Comparison theorems 10 2.5. Base change 15 3. Geometric models of motivic Thom spectra 16 3.1.