The Logarithmic Cotangent Complex

The logarithmic cotangent complex

Martin C. Olsson Received: date / Revised version: date – c Springer-Verlag 2005

Abstract. We deﬁne the cotangent complex of a morphism of ﬁne log schemes, prove that it is functorial, and construct under certain restrictions a transitivity triangle. We also discuss its relationship with deformation theory.

1. Introduction

1.1. The goal of this paper is to develop a logarithmic version of the theory of cotangent complexes for morphisms of schemes ([6]). Our interest in the development of such a theory comes from two sources. First, we want to generalize the log smooth deformation theory due to K. Kato ([9], 3.14, see also [8]) in order to understand more complicated deformation theory problems in the logarithmic category (see section 5). Such deformation theory problems arise for example in the study of log Gromov–Witten theory ([12], [19]). Secondly, we are hopeful that the development of a theory of log cotangent complexes can be useful in the study of intersection theory on degenerations of varieties. In fact, the beginnings of a theory of log cotangent complexes can be found in the recent work of K. Kato and T. Saito ([10]) on logarithmic localized intersection product, where it is used to study a proper scheme over a complete discrete valuation ring whose reduced closed ﬁber is a divisor with normal crossings. In this paper, we associate to every morphism of ﬁne log schemes f : X → Y ([9]) a projective system

≥−n−1 ≥−n ≥0 LX/Y = (· · · −→ LX/Y −→ LX/Y −→ · · · −→ LX/Y ),

≥−n [−n,0] where each LX/Y is an essentially constant ind-object in D (OX ) (the derived category of sheaves of OX -modules with support in [−n, 0]). We call the system LX/Y the cotangent complex of f, and we show that it has the following properties (1.1 (i))–(1.1 (v)):

M. Olsson School of Mathematics, Institute for Advanced Study , Einstein Drive, Princeton, NJ 08540 USA 2 Martin C. Olsson

≥−n−1 ≥−n (1.1 (i)). For any n ≥ 0, the natural map τ≥−nLX/Y → LX/Y is an isomorphism. ∗ (1.1 (ii)). If f is strict (i.e. the map f MY → MX is an isomorphism), ◦ ◦ then (τ≥−nL ◦ ◦) represents LX/Y , where X (resp. Y ) denotes the un- X/Y derlying scheme of X (resp. Y ). 1 (1.1 (iii)). If f is log smooth, then the sheaf of log diﬀerentials ΩX/Y represents LX/Y . (1.1 (iv)). If X0 −−−−→a X   0  (1.1.1) f y yf Y 0 −−−−→b Y is a commutative diagram of ﬁne log schemes, then there is a natural map

∗ La LX/Y −→ LX0/Y 0 . (1.1.2)

If (1.1.1) is cartesian and if f is log ﬂat, then (1.1.2) is an isomorphism, and if in addition the composite X0 → Y 0 → Y satisﬁes condition (T ) below (1.4), then the map

0∗ ∗ Lf LY 0/Y ⊕ La LX/Y −→ LX0/Y (1.1.3) is an isomorphism. (1.1 (v)). Given a composite

f g X −−−−→ Y −−−−→ Z (1.1.4) satisfying condition (T ) below (1.4), there is a natural map

∗ LX/Y −→ Lf LY/Z [1]. (1.1.5) such that the resulting triangle

∗ ∗ Lf LY/Z −→ LX/Z −→ LX/Y −→ Lf LY/Z [1] (1.1.6) is distinguished (in the sense of (1.8 (iii)).

1.2. As an application of this log cotangent complex, we explain how to compute crystalline cohomology of “log complete intersections” in terms of the log cotangent complex. This application is modelled on the approach of Illusie in ([7], VIII). The logarithmic cotangent complex 3

1.3. As shown by an argument of W. Bauer (section 7), there does not exist a reasonable theory of cotangent complexes for log schemes for which one has a distinguished triangle (1.1.6) for every composite as in (1.1.4). In this paper, we construct the distinguished triangle (1.1.6) when (1.1.4) satisﬁes the following condition (which holds for example if g is log ﬂat (3.14) or if f is an integral morphism (3.15)): 1.4 (Condition (T)). There exists a family of commutative diagrams

π Xi XX ×Z Zi X ×Y Yi Xi f ? ? ? fi + πYi YY ×Z Zi Yi (1.4.1) g ? ? g π i Zi + ZZi such that ◦ ◦ ◦ (1.4 (i)). The schemes Xi, Y i, and Zi are all aﬃne.

(1.4 (ii)). The maps πXi , πYi , and πZi are all strict, and their underlying morphisms of schemes are ﬂat and locally of ﬁnite presentation. ` ◦ ◦ (1.4 (iii)). The induced map i Xi → X is surjective. (1.4 (iv)). There exists charts

βXi : QXi −→ MXi , βYi : QYi −→ MYi , βZi : QZi −→ MZi and injective maps

θ1 : QZi −→ QYi , θ2 : QYi −→ QXi such that the diagrams

βXi βYi QXi −−−−→ MXi QYi −−−−→ MY x x x x θ  f b θ  gb 2  i 1  i β β Yi ∗ Zi ∗ QYi −−−−→ fi MYi QZi −−−−→ gi MZi commute and j Tor (OZ ⊗ [Q ] [QX ], OY [G]) = 0 for all j > 0. OZ ⊗ [Q ] [QY ] i Z Zi Z i i i Z Zi Z i (1.4.2) 4 Martin C. Olsson

gp gp Here G := Coker(Q → Q ) and OY [G] is viewed as a OZ ⊗ [QY ] Zi Yi i i Z[QZi ]Z i algebra via the map

∗ OZ ⊗ [QY ] −→ OY [G], t ⊗ e(q) 7−→ g (t)βQ (q) · gq¯, (1.4.3) i Z[QZi ] Z i i Yi where gq¯ denotes the image of q in G. Remark 1.5. As explained to us by Gabber, the vanishing condition (1.4.2) is equivalent to

j Tor (OZ ⊗ [Q ] [QX ], OY ) = 0 for all j > 0. (1.5.1) OZ ⊗ [Q ] [QY ] i Z Zi Z i i i Z Zi Z i A proof of this equivalence is given at the end of section 4. 1.6. On the other hand, Gabber has shown that if one does not require 1 LX/S ' ΩX/S for X → S log smooth, then there does exist a theory of log cotangent complexes for which one has a distinguished triangle for all composites X → Y → Z. We explain Gabber’s approach in section 8. 1.7. The paper is organized as follows. In section 2 we construct certain algebraic stacks LogΓ classifying commutative diagrams of log structures of a fixed form (see (2.1) for Γ the precise definition of Log ). These stacks generalize the stacks LogS studied in ([14]), and are the main technical tool used in the construction of the functoriality morphism (1.1.2) and the transitivity triangle (1.1.6). We define LX/Y in section 3 and verify properties (1.1 (i))–(1.1 (v)), assuming a key proposition (3.10) concerning condition (T ). This proposition is proven in section 4. In section 5 we discuss the relationship between the logarithmic cotangent complex and deformation theory of log schemes generalizing ([6], III.1.2.3, III.2.1.7, III.2.2.4) to log schemes. This section relies heavily on the results of ([13]). In section 6 we apply the cotangent complex to study log versions of results in ([7], VIII) as mentioned above. In section 7 we discuss an argument of W. Bauer which shows that there does not exist a reasonable theory of cotangent complexes for log schemes for which one has a distinguished triangle (1.1.6) for every composite as in (1.1.4). Thus some condition, such as condition (T ), is needed in order to have (1.1 (v)). Finally we discuss in section 8 an alternate approach to defining a G log cotangent complex LX/S for a morphism of log schemes X → S due to Gabber. This approach has the advantage that one always has a dis- G tinguished triangle. Furthermore, there is a natural map LX/S → LX/S whose cone is cohomologically concentrated in degrees ≤ −3. This implies The logarithmic cotangent complex 5

G that for the standard deformation theory problems one can use LX/S instead of LX/S. We have also included an appendix which contains some basic results about simplicial monoids which are needed in the study of Gabber’s cotangent complex.

1.8. Notation and prerequisites

(i). Logarithmic geometry: We assume that the reader is familiar with logarithmic geometry at the level of the first two sections of ([9]) and with the paper ([14]). Throughout the paper, a log structure on a scheme X means a log structure on the étalesite Xet. If α : M → OX is a pre-log structure on a scheme X, then we denote by (Ma, α) (or just Ma) the associated log structure, and if (M, α) is a ∗ log structure on X then we denote by M the sheaf M/OX . If X is a log scheme, we usually denote its log structure by MX and its ◦ underlying scheme by X. If f : X → Y is a morphism of log schemes, we ◦ ◦ ◦ denote the underlying morphism of schemes by f : X → Y . For a fine log ◦ scheme X, we denote by LogX the algebraic stack over X whose objects are fine X-log schemes and whose morphisms (T1 → X) → (T2 → X) are b ∗ morphisms of X-log schemes g : T1 → T2 for which g : g MT2 → MT1 is an isomorphism ([14]). If f : T → X is a morphism of fine log schemes, ◦ then we denote by Lf : T → LogX the associated 1-morphism. Recall that a chart for a fine log structure M on a scheme X is a map P → M from the constant sheaf associated to a fine (i.e. finitely generated and integral) monoid P such that P a → M is an isomorphism ([9], 2.9 (1)). If P is a fine monoid, and R a ring with a map P → R, then we write Spec(P → R) for the log scheme with underlying scheme Spec(R) and log structure associated to the map P → R. In particular, for a fine monoid P , we obtain a log scheme Spec(P → Z[P ]). For p ∈ P , we denote by e(p) ∈ Z[P ] the image. If no confusion seems likely to arise, we may also abuse notation and write Spec(Z[P ]) for the log scheme Spec(P → Z[P ]). We denote by SP the stack-theoretic quotient of Spec(Z[P ]) by the natural gp action of the group scheme Spec(Z[P ]), and by MSP the natural log structure on SP ([14], section 5). If S is a scheme, we write S[P ] for the scheme S[P ] := S × Spec( [P ]) (1.8.1) Spec(Z) Z and MS[P ] for the log structure on S[P ] induced by the log structure on Spec(Z[P ]). 6 Martin C. Olsson

A chart for a morphism of fine log schemes f : X → S is a 5-tuple (Q, P, , βQ, βP , θ), where βQ : Q → MS and βP : P → MX are charts and θ : Q → P is a morphism of fine monoids, such that the induced diagram of fine log schemes

βP X −−−−→ Spec(P → Z[P ])   f  (1.8.2) y yθ

βQ S −−−−→ Spec(Q → Z[Q]) commutes ([9], 2.9 (2)). If no confusion seems likely to arise, we sometimes write (Q, P, θ) for a chart (Q, P, βQ, βP , θ). In general, given a morphism of ﬁne monoids θ : Q → P and a chart β : Q → M for a ﬁne log structure M on a scheme X, we let XQ[P ] denote the scheme

X [P ] := X × Spec( [P ]). (1.8.3) Q Spec(Z[Q]) Z

The natural map P → OXQ[P ] induces a log structure MXQ[P ] on XQ[P ], and there is a natural morphism of log schemes

(XQ[P ], MXQ[P ]) −→ (X, M). (ii). Algebraic stacks: We follow the conventions of ([11]), except we do not assume that our stacks are quasi-separated. More precisely, by an algebraic stack we mean a stack X in the sense of ([11], 3.1) satisfying the following: (1.8.4). The diagonal ∆ : X −→ X × X is representable and of finite type; (1.8.5). There exists a surjective smooth morphism X → X from a scheme. Similarly, we call an algebraic stack X in the above sense a Deligne- Mumford stack if there exists a cover X → X as in (1.8.5) which is étale and surjective. The reader should be aware that there is a mistake in ([11]) regarding the functoriality of the lisse-étaletopos. This mistake is corrected in ([15]). The reader is assumed to be familiar with algebraic stacks.

(iii). The category D0(A): If A is an abelian category, we denote by D0(A) the category of projective systems

K = (· · · −→ K≥−n−1 −→ K≥−n −→ · · · −→ K≥0), The logarithmic cotangent complex 7 where each K≥−n ∈ D+(A) (the derived category of bounded below complexes of objects in A), and the maps

≥−n ≥−n ≥−n−1 ≥−n K −→ τ≥−nK , τ≥−nK −→ τ≥−nK are all isomorphisms. We denote by D0b(A) the full subcategory of D0(A) consisting of objects K for which K≥−n ∈ Db(A) for all n, and D0≤0(A) the subcategory of objects with Hi(K≥−n) = 0 for all i > 0 and all n. The shift functor (–)[1] : D+(A) → D+(A) is extended to D0(A) by deﬁning

≥−n−1 ≥−n 0 0 K[1] := (· · · → K [1] → K [1] → · · · → K [1] → (τ≥1K )[1]) for K ∈ D0(A) ([11], 17.4 (3)). We say that a triangle of D0(A)

u v w K1 −−−−→ K2 −−−−→ K3 −−−−→ K1[1] (1.8.6) is distinguished if for every n ≥ 0, there exists a commutative diagram

u v0 w0 K1,≥−n −−−−→ K2,≥−n −−−−→ L −−−−→ K1,≥−n[1]         (1.8.7) idy idy yβ y u v w K1,≥−n −−−−→ K2,≥−n −−−−→ K3,≥−n −−−−→ K1,≥−n+1[1] where the top row is a distinguished triangle in D(A) and the map

τ≥−nL −→ K3,≥−n (1.8.8) induced by β is an isomorphism (this differs from the definition in ([11], 17.4 (3)); see the discussion of this in ([13], 2.15)). If A has enough in- jectives, then for any I ∈ A and K ∈ D0b(A), one can define groups Extj(K,I), and a distinguished triangle (1.8.6) in D0b(A) induces in a natural way a long exact sequence

j j j j+1 · · · → Ext (K3,I) → Ext (K2,I) → Ext (K1,I) → Ext (K3,I) → · · · of groups ([13], 2.18). For each integer j and K ∈ D0b(A), there is a natural isomorphism

j j ≥−n Ext (K,I) ' ExtD(A)(K ,I) for n > j. For more details about this see ([13], 2.16–2.18). 8 Martin C. Olsson

1.9. Acknowledgements

It is a pleasure to thank L. Illusie for his encouragement and numerous enlightening conversations. The main results of this paper grew out of our attempt to understand his comments and questions; in particular, we found the construction of the functoriality morphism (1.1 (iv)) and the transitivity triangle (1.1 (v)) by trying to understand an argument of Illusie using toric stacks. This paper grew out of the author’s thesis written under A. Ogus. We would like to thank Professor Ogus for his advice and encouragement. Finally we are very grateful to O. Gabber who provided a large number of helpful comments and corrections to the ﬁrst version of this paper. Moreover, section 8 as well as (4.5)–(4.9) is entirely due to Gabber (though of course any mistakes are due to the present author). Several key ideas in this paper were found while the author visited Universit´ede Paris-Sud. We would like to thank this institution and its members for providing a very pleasant and productive working environ- ment. During this research the author was partially supported by an NSF post–doctoral research fellowship.

2. Universal commutative diagrams of log structures

2.1. Let Γ be a finite category (i.e. Γ has finitely many objects and any two objects of Γ have only finitely many morphisms between them), and define LogΓ to be the fibered category over the category of schemes whose fiber over some scheme T is the groupoid of functors

Γ −→ (category of ﬁne log structures on T ).

We shall be especially interested in the following two examples:

i Example 2.2 (The stacks L ). For i ≥ 0, let Ci be the category with objects {0, . . . , i} and

∅ if j > j0 Hom(j, j0) = unital set if j ≤ j0.

Diagramatically, Ci can be described as follows:

•0 −→ •1 −→ · · · −→ •i−1 −→ •i.

Ci i i In this case, Log is the classifying stack of data ({Mj}j=0, {sj}j=1), where Mj is a log structure and sj : Mj−1 → Mj is a morphism of log structures. In what follows, we will denote the stack LogCi simply The logarithmic cotangent complex 9

i 0 1 ∗ 0 by L . Note that L = Log(Spec( ),O ) and that L = Log(L ,M 0 ), Z Spec(Z) L 0 where ML0 denotes the universal log structure on L (see ([14], 5.1) for the definition of a log structure on an algebraic stack and (loc. cit., 5.9) 0 1 for the definition of Log 0 ). In particular, L and L are algebraic (L ,ML0 ) stacks locally of finite presentation over Z by (loc. cit., 5.9). Note also that there is a natural map

1 0 0 τ : L −→ L × L , (M1 → M2) 7→ (M1, M2).

Example 2.3 (The stack L). Let denote the category with four objects {0, 1, 2, 3} and morphisms given by

{∗} if j = j0 or (j, j0) ∈ {(0, 1), (0, 2), (0, 3), (1, 3), (2, 3)} Hom(j, j0) = ∅ otherwise.

Diagramatically, can be described by the commutative diagram

•0 −−−−→ •1     y y

•2 −−−−→•3, and Log is the classifying stack of commutative diagrams of ﬁne log structures M0 −−−−→ M1     y y (2.3.1)

M2 −−−−→M3. In what follows we denote Log simply by L.

Theorem 2.4. For any finite category Γ , the fibered category LogΓ is an algebraic stack locally of finite presentation over Z.

Proof. Label the objects of Γ by {0, . . . , r} for some r, and let {al} be the set of morphisms in Γ . For a morphism al, we denote by il (resp. jl) 0 the source (resp. target) of al. Let L be the stack which to any scheme r T associates data ({Mj}j=1, {ρl}), where Mj is a ﬁne log structure on T and ρl : Mil → Mjl is a morphism of log structures. There is a natural inclusion j : LogΓ −→ L0. (2.4.1) Lemma 2.5. The map (2.4.1) is representable and locally of ﬁnite presentation. 10 Martin C. Olsson

r Proof. A collection of data ({Mj}j=1, {ρl}) over some scheme T defines Γ an object of Log (T ) if and only if ρl = id when al is the identity map on some object in Γ and ρl0 ◦ ρl = ρl00 whenever al0 ◦ al = al00 in Γ . For each 0 0 r such equality E, let LE ⊂ L be the substack of objects ({Mj}j=1, {ρl}) for which E holds. Order the equalities {E1,...,Eq} in some way. Then for any T → L0, we have Γ 0 0 0 0 0 0 0 T ×L Log = (T ×L LE1 ) ×T (T ×L LE2 ) ×T · · · ×T (T ×L LEq ), 0 0 so to prove the lemma it suffices to show that LE → L is representable and locally of finite presentation for each E.

Suppose E is an equality of two morphisms ρl0 , ρl1 : Mi → Mj (ρl1 = id is allowed), and let 0 1 1 F : L −→ L ×τ,L0×L0,τ L r be the functor which sends data ({Mj}j=1, {ρl}) to the pair (ρl0 : Mi →

Mj, ρl1 : Mi → Mj). Let 1 1 1 ∆ : L −→ L ×τ,L0×L0,τ L be the diagonal map. The functor ∆ is representable and locally of finite presentation since it is faithful ([11], 8.1.2) and L1 and L0 are algebraic 0 stacks locally of finite presentation over Z ([14], 5.9). Since LE is isomorphic to the cartesian product of L0   yF 1 ∆ 1 1 L −−−−→L ×τ,L0×L0,τ L , 0 0 it follows that LE → L is representable and locally of finite presentation. ut Thus to prove (2.4) it suffices to show that L0 is an algebraic stack lo- 0 cally of finite presentation over Z. This follows from the fact that L is isomorphic to the cartesian product of stacks Q 1 l L Q  y τ Qr 0 G Q 0 0 j=0 L −−−−→ l(L × L ), where G is the functor induced the map

(M0,..., Mr) 7→ (..., (Mil , Mjl ),... ). ut The logarithmic cotangent complex 11

2.6. If ρ : Γ 0 → Γ is a functor between ﬁnite categories, then composition with ρ induces a functor

0 ρ∗ : LogΓ −→ LogΓ .

If : Γ 00 → Γ 0 is a second functor, then there is an equality of functors (not just a natural isomorphism) (ρ ◦ )∗ = ∗ ◦ ρ∗. We shall be especially interested in the following four examples of functors between the Ci and .

Example 2.7. To give a functor Ci → Cj is equivalent to giving an order- preserving map {0, . . . , i} → {0, . . . , j}. Therefore, the standard simplicial category ∆ (see for example ([11], 12.4)) is naturally equivalent to the category whose objects are the Ci and whose morphisms are the functors between them. Thus the algebraic stacks Li form in a natural way a simplicial algebraic stack. For any integer j satisfying 0 ≤ j ≤ i + 1, let δj : {0, . . . , i} → {0, . . . , i + 1} be the unique injective order-preserving map whose image is does not contain j and let

∗ i+1 i δj : L −→ L (2.7.1)

∗ be the induced functor. The functor δj is the one induced by the map

sj−1 sj sj+1 sj+2 si+1 (··· → Mj−1→Mj → Mj+1 → · · · → Mi+1)   y sj−1 sj+1◦sj sj+2 si+1 (··· → Mj−1 → Mj+1 → · · · → Mi+1).

Example 2.8. Let ρ1 : C1 → (resp. ρ2 : C1 → ) be the unique functor sending 0 to 0 and 1 to 1 (resp. 2), and let

∗ 1 ρi : L −→ L , i = 1, 2 (2.8.1)

∗ ∗ denote the resulting morphisms of stacks. The functor ρ1 (resp. ρ2) is that induced by the map which sends a commutative diagram

a M0 −−−−→M1    c (2.8.2) by y d M2 −−−−→M3 to a : M0 → M1 (resp. b : M0 → M2). 12 Martin C. Olsson

Example 2.9. Let q1 : C1 → (resp. q2 : C1 → ) be the unique functor sending 0 to 1 (resp. 2) and 1 to 3, and let

∗ 1 qi : L −→ L , i = 1, 2 (2.9.1)

∗ ∗ be the resulting morphisms of stacks. The functor q1 (resp. q2) sends a commutative diagram (2.8.2) to c : M1 → M3 (resp. d : M2 → M3).

Example 2.10. Let τi : C2 → (i = 1, 2) be the unique functor sending 0 to 0, 1 to i, and 2 to 3, and let

∗ 2 τi : L → L , i = 1, 2 (2.10.1)

∗ denote the resulting morphism of stacks. The functor τi sends a commutative diagram (2.8.2) to M0 → Mi → M3.

The following proposition will be used in the subsequent section (see (1.8 (ii)) for our conventions about Deligne-Mumford stacks).

Proposition 2.11. (i). If j ≤ i, then the morphism (2.7.1) is relatively Deligne-Mumford and ´etale. (ii). The morphisms (2.9.1) and (2.10.1) are relatively Deligne-Mumford and ´etale.

Remark 2.12. Recall ([11], 7.3.3) that a morphism F : X → Y of algebraic stacks is relatively Deligne-Mumford if for every 1-morphism Y → Y from a scheme, the fiber product X×Y Y is a Deligne-Mumford stack. Sup- pose P is a property of morphisms of Deligne-Mumford stacks f : U → V which is preserved under arbitrary base change V → V and is local on the base in the étaletopology. Then it makes sense to say that a relatively Deligne-Mumford morphism F : X → Y has property P: F has P if and only if for any 1-morphism Y → Y the projection X ×Y Y → Y has property P. In particular, we can speak of an étalerelatively Deligne-Mumford morphism of algebraic stacks.

Proof (Proof of (2.11)). Consider ﬁrst a lemma:

Lemma 2.13. Let h : T0 ,→ T be a closed immersion of schemes defined by a nilpotent ideal. (i). If : M1 → M2 is a morphism of fine log structures on T , then there are no non-trivial automorphisms ι of M1 for which = ◦ ι and h∗(ι) = id. ∗ (ii). If M is a fine log structure on T and if 0 : N0 → h M is a morphism of fine log structures on T0, then there is a unique morphism of fine log The logarithmic cotangent complex 13 structures : N → M inducing 0. Moreover, the diagram N −−−−→ M     y y 0 N0 −−−−→M0 is cartesian.

Proof. To see (i), note that if ι is an automorphism of M1 with = ◦ ι ∗ and h (ι) = id, then the map ¯ι : M1 → M1 must be the identity since the étalesites of T0 and T are equivalent ([3], I.8.3). Thus, for any local section m ∈ M1, ι(m) = λ(um) + m for some unique unit um. But then (ι(m)) = λ(um) + (m), and hence the condition that ◦ ι = implies that um = 1. As for (ii), the projection map pr1 : M×h∗M N0 → M is one lifting of (N0, 0) (we leave it to the reader to verify that M ×h∗M N0 is a fine log structure), and for any other lifting (N , ) there exists by the definition ∗ of cartesian product a unique map ρ : N → M ×h M0 N0, reducing to ∗ the specified isomorphism h N'N0 over T0, such that pr1 ◦ ρ = . Since a morphism of fine log structures M1 → M2 is an isomorphism if and ∗ only if the map M1 → M2 is an isomorphism, the fact that h (ρ) is an isomorphism implies that ρ is also an isomorphism. ut ∗ ∗ ∗ Note first that since qi = δ0 ◦ τi , it suffices to consider the δj and τi, and by symmetry it suffices to prove (ii) for τ1. ∗ ∗ To prove that δj (resp τ1 ) is relatively Deligne-Mumford, note that the diagonal i+1 i+1 i+1 L −→ L ×Li L (resp. L −→ L ×L2 L) is of finite presentation since the diagonals of Li+1 and Li (resp. L and L2) are of finite presentation. Hence by ([11], 8.1) it suffices to show that ∗ ∗ the diagonal of δj (resp. τ1 ) is formally unramified. In the case at hand, this amount to the following: given a closed immersion h : T0 ,→ T defined by a nilpotent ideal I, and a collection of log structures and maps between them on T (resp. a commutative diagram of log structures on T )  a  M0 → M1 sj−1 sj sj+1 sj+2 si+1     c  (··· → Mj−1→Mj → Mj+1 → · · · → Mi+1) resp. yb y  , d M2 → M3 there does not exist a non-trivial automorphism ι of Mj for which sj = ∗ ι ◦ sj, sj+1 = sj+1 ◦ ι, and h (ι) = id (resp. there does not exist a non- trivial automorphism ι : M2 → M2 for which ι ◦ b = b, d = d ◦ ι, and for which j∗(ι) is trivial). This follows from (2.13 (i)). 14 Martin C. Olsson

i Since the stacks L and L are locally of finite presentation over Z, to ∗ ∗ prove that δj (resp. τ1 ) is étaleit suffices by ([14], 4.9) (though the result there is stated only for representable morphisms the proof generalizes immediately to give the result for relatively Deligne-Mumford morphisms) to show that it is formally étalein the sense of (loc. cit., 4.5). In the case ∗ ∗ of δj (resp. τ1 ), the infinitesimal lifting criterion of (loc. cit., 4.5) amounts to the following: given a closed immersion h : T0 ,→ T defined by a square zero ideal I, and an object

sj−1 sj sj+1 sj+2 si+1 α0 = (··· → Mj−1→Mj → Mj+1 → · · · → Mi+1)  a  M0 → M1     c  resp. α0 = yb y  , d M2 → M3

i+1 of L (T0) (resp. L(T0)) together with a lifting

s˜j−1 ρ s˜j+2 s˜i+1 a˜ c˜ β = (··· → Mfj−1→Mfj+1 → · · · → Mfi+1) (resp. β = Mf0→Mf1→Mf3)

∗ ∗ of δj (α0) (resp. τ1 (α0)) to T , there exists a unique lifting α of α0 to Li+1(T ) (resp. L(T )) inducing β. This follows from (2.13 (ii)) which shows that

s˜j−1 ρ×sj pr1 s˜j+2 s˜i+1 α := (··· → Mfj−1 → Mfj+1×Mj+1,sj+1 Mj →Mfj+1 → · · · → Mfi+1)

 a˜  Mf0 → Mf1       resp. α := yc˜◦a˜×b yc˜ pr1 Mf3 ×M3 M2 → Mf3 is the unique such lift. ut

3. Deﬁnition of the cotangent complex and (1.1 (i))–(1.1 (v))

◦ 3.1. Let f : X → Y be a morphism of ﬁne log schemes, and let Lf : X → LogY be the associated 1-morphism (1.8). The stack LogY is canonically isomorphic to the ﬁber product of

L1  δ∗ y 1 ◦ M Y −−−−→LY 0, The logarithmic cotangent complex 15

◦ ◦ ◦ ◦ and the map Lf : X → LogY is the map induced by the map f : X → Y ◦ ∗ 1 and the object (f MY → MX ) ∈ L (X). In ([11], 17.3) (see also ([15], §10)), the cotangent complex LX /Y is defined for any morphism of algebraic stacks F : X → Y. This cotangent 0 complex LX /Y is an object in Dqcoh(Xlis-et), where Dqcoh(Xlis-et) denotes the full subcategory of the derived category of sheaves of OXlis-et –modules in the lisse-étale topos of X consisting of complexes with quasi–coherent cohomology sheaves. ◦ 0 Definition 3.2. The cotangent complex LX/Y ∈ D (Xet) of f is the re- ◦ striction of L ◦ to the étale site of X. X/LogY Remark 3.3. By ([15], 6.14), the restriction functor ◦ ◦ 0 0 Dqcoh(Xlis-et) → Dqcoh(Xet) is an equivalence of categories. Furthermore, by the construction of the derived pullback in ([15], §9), this equivalence is compatible with pull- ◦ 0 backs. We can therefore also view LX/Y as an object in D (Xlis-et), and this will sometimes be necessary for technical reasons. In particular, in (3.10.1), (3.11.2), (3.11.3), (3.12.3), (3.12.4), and (3.12.5) below we view LX/Y as defined in the lisse-étale topos. Remark 3.4. If the morphism f is log flat, then ([11], 17.8) shows that LX/Y is represented by the usual cotangent complex L ◦ ◦ ◦[−1] of X/X× X LogY ◦ ◦ ◦ the diagonal map ∆ : X → X ×LogY X. 3.5 (Verification of (1.1 (i))). This follows from ([11] 17.3 (0)).

3.6 (Verification of (1.1 (ii))). If f is strict, then the map Lf factors ◦ ◦ as s ◦ f , where s : Y → LogY is the map induced by the identity map on Y . By ([14], 3.19 (ii)) the map s is an open immersion, and hence the result follows from ([11], 17.3 (1)). 3.7 (Verification of (1.1 (iii))). If f is log smooth, then by ([14], 4.6 1 (ii)) the map Lf is smooth, and hence by ([11], 17.5.8) the sheaf Ω ◦ X/LogY represents LX/Y . The result therefore follows from the following lemma. Lemma 3.8. Let f : X → Y be a morphism of fine log schemes and ◦ let Lf : X → LogY be the induced morphism of stacks. Then there is 1 a natural isomorphism between the logarithmic differentials ΩX/Y and 1 Ω ◦ . X/LogY 16 Martin C. Olsson

◦ ◦ Proof. For each OX -module I, let X[I] be the scheme over X obtained from the sheaf of algebras OX [I] whose underlying sheaf of OX -modules is OX ⊕ I and whose algebra structure is given by

(a, i) · (a0, i0) = (aa0, ai0 + a0i).

◦ ◦ Denote by j : X,→ X[I] the closed immersion deﬁned by the map ◦ ◦ OX [I] → OX sending (a, i) to a, and let r : X[I] → X be the map obtained from the map OX → OX [I] sending a to (a, 0). We deﬁne X[I] ◦ ∗ to be the log scheme (X[I], r MX ). The maps j and r then extend naturally to morphisms of log schemes (which we denote by the same letters) j : X → X[I] and r : X[I] → X. Let

F1 (resp. F2):(OX -modules) −→ (Sets) be the functor which to any OX -module I associates the set of sections ◦ ◦ ◦

ϕ : X[I] → X[I] ×Y X (resp. ϕ : X[I] → X[I] ×Lf ◦r,LogY X) of the ◦ ◦ ◦ projection X[I]×Y X → X[I] (resp. X[I]×Lf ◦r,LogY X → X[I]) for which ϕ◦j = j ×id. Giving an element ϕ ∈ F1(I) (resp. ϕ ∈ F2(I)) is equivalent to giving a morphism of Y -log schemes ϕ : X[I] → X (resp. a morphism b ∗ of Y -log schemes ϕ : X[I] → X for which ϕ : ϕ MX → MX[I] is an isomorphism) for which ϕ◦j = id. Since a morphism of ﬁne log structures M1 → M2 is an isomorphism if and only if the map M1 → M2 is an b ∗ isomorphism, the map ϕ : ϕ MX → MX[I] is an isomorphism for any morphism ϕ : X[I] → X for which ϕ ◦ j = id. Hence there is a natural isomorphism of functors F1 ' F2. On the other hand, by ([9], 3.9) (resp. ([3], III.5.1)) there is a natural isomorphism

1 1 F1 ' Hom(ΩX/Y , –), (resp. F2 ' Hom(Ω ◦ , –)), X/LogY

1 1 and hence Yoneda’s lemma gives an isomorphism ΩX/Y ' Ω ◦ . ut X/LogY

3.9. Before discussing (1.1 (iv)) and (1.1 (v)), it will be useful to explain the role of condition (T ) in this paper. Thus suppose given a composite as in (1.1.4), and consider the resulting commutative diagram The logarithmic cotangent complex 17

◦ pr ◦ h - 2 - 2 X S L × ∗ ∗ 0 Z δ1 ◦δ2 ,L ◦ pr ∗ f 1 δ2 × id ? + ◦ + ◦ Lg - 1 Y L ×δ∗,L0 Z 1 (3.9.1) ◦ g ? pr ◦ ◦ + Z Z.

∗ Here S is the cartesian product of Lg and δ2 × id, and h is the map in- ◦ 2 ∗ ∗ duced by the morphism from X to L coming from the triple f g MZ → ∗ f MY → MX on X. Note that S is canonically isomorphic to LogY 2 via the map induced by the map LogY → L sending ` : T → Y to ◦ ∗ ∗ ∗ 2 (` g MZ → ` MY → MT ) ∈ L (T ). We use a separate notation, however, to indicate that we are thinking of S as sitting inside the above commutative diagram. Part (ii) of the following proposition, whose proof we defer until the next section, is the key consequence of condition (T ) which we use:

Proposition 3.10. (i). The natural map

∗ ∗ ∗ ∗ Lf L = Lh Lpr L ◦ ◦ −→ Lh L ◦ (3.10.1) Y/Z 1 1 2 Y/L × ∗ 0 Z S/L × ∗ ∗ 0 Z δ1 ,L δ1 ◦δ2 ,L induces an isomorphism on H0, and a surjection on H1. (ii). If (1.1.4) satisﬁes condition (T ), then (3.10.1) is an isomorphism.

3.11 (Veriﬁcation of (1.1 (iv))). Suppose given a commutative diagram as in (1.1.1), and consider the resulting commutative diagram

◦ ◦ X0 −−−−→a X    L  hy f y (3.11.1) ◦ ∗ ◦ 0 ρ1 1 Y × 1 ∗ L −−−−→ Y × 0 ∗ L , L ,ρ2 L ,δ1 where h is the map induced by the commutative square ∗ ∗ ∗ a f MY −−−−→ a MX     y y 0∗ f MY 0 −−−−→ MX0 18 Martin C. Olsson

◦ on X0. From (3.11.1) and ([11] 17.3 (2)) we obtain a morphism

∗ ∗ La L ' La L ◦ −→ L ◦ ◦ . (3.11.2) X/Y 1 0 0 X/Y ×L0,δ∗ L X /Y × 1 ∗ L 1 L ,ρ2 Since the map ◦ ◦ ∗ 0 0 2 τ : Y × 1 ∗ L −→ Y × 1 ∗ L ' Log 0 2 L ,ρ2 L ,δ2 Y is ´etaleby (2.11 (ii)), there are natural isomorphisms

L 0 0 ' L ◦ ' L ◦ ◦ (3.11.3) X /Y 0 0 0 X /LogY 0 X /Y × 1 ∗ L L ,ρ2 and hence (3.11.2) provides the desired map (1.1.2). In order to prove the remaining statements in (1.1 (iv)), suppose now that (1.1.1) is cartesian. Denote by T the ﬁber product of the diagram

◦ X  L y f ◦ ∗ ◦ 0 ρ1 1 Y × 1 ∗ L −−−−→ Y × 0 ∗ L . L ,ρ2 L ,δ1

Lemma 3.12. (i) The substack V ⊂ L classifying diagrams as in (2.3.1) which are co-cartesian (in the category of ﬁne log structures) is an open substack. ◦ ◦ (ii). The natural map X0 → T induces an isomorphism between X0 and

◦ ◦ 0 (Y × 1 ∗ V) × ◦ X. (3.12.1) L ,ρ2 ∗ 1 ρ1,Y × 0 ∗ L L ,δ1 Proof. To see (i), suppose given a diagram (2.3.1) over some scheme T . We have to show that there exists an open subset U ⊂ T representing the condition that the diagram is co-cartesian. Deﬁne M := M1 ⊕M0 M2 (pushout in the category of log structures). The log structure M is in general not ﬁne, but M is coherent ([9], 1.6 and 2.6). We claim that the underlying scheme T 0 of the associated integral log scheme (T, M)int ([9], 2.7) is equal to T . Indeed, the square (2.3.1) induces maps

int (T, M3) −→ (T, M) −→ (T, M), and hence the natural map j : T 0 → T admits a section. On the other hand, by ([9], proof of 2.7) the map j is a closed immersion, and hence j 0 int is an isomorphism. It follows that the pushout M := (M1 ⊕M0 M2) The logarithmic cotangent complex 19 in the category of fine log structures on T exists, and we denote by : 0 M → M3 the resulting morphism of log structures. The condition that the diagram (2.3.1) is cocartesian is then equivalent to the condition that is an isomorphism. This in turn is equivalent to the condition that 0 ¯ : M → M3 is an isomorphism, which is an open condition by ([14], 3.5). As for (ii), it follows from the definitions of the stacks V, L0, and L1 ◦ ◦ 0 that the stack (3.12.1) is the stack over the category of Y × ◦ X-schemes Y ◦ ◦ 0 which to any h × g : T → Y × ◦ X associates the groupoid of triples Y (M, a, b), where M is a fine log structure and

∗ ∗ a : g MX −→ M, b : h MY 0 −→ M are morphisms of ﬁne log structures such that the resulting diagram

∗ ∗ g f MY −−−−→MX    a y y ∗ b h MY 0 −−−−→ M commutes and is co-cartesian. By the universal property of the fiber product in the category of fine log schemes, giving data (M, a, b) as above, is equivalent to giving a pair (M, ρ), where M is a fine log structure on T ◦ ◦ 0 0 and ρ :(T, M) → X is a strict morphism of log schemes over X × ◦ Y . Y Thus (ii) follows from ([14], 3.19 (i)). ut

If f : X → Y is log ﬂat, then it follows from ([11], 17.3 (4)) and (3.12 (ii)) that the map (3.11.2) is an isomorphism, and hence the map (1.1.2) is also an isomorphism. In addition, ([11], 17.3 (5)) applied to the diagram

◦ ◦ ◦ 0 j pr2 X −−−−→ T −−−−→ X × ◦ (Y × 0 ∗ ∗ L) 1 ∗ L ,δ1 ◦ρ1 Y ×L0 L ,ρ1   pr   1y y ◦ ∗ ◦ 0 ρ1 Y × 1 ∗ L −−−−→ Y × 0 ∗ ∗ L, L ,ρ2 L ,δ1 ◦ρ1 20 Martin C. Olsson where the square is cartesian and j is an open immersion, shows that the natural map ∗ ∗ Lj Lpr2L ◦ ◦ ◦ (X× ◦ (Y ×L0,δ∗◦ρ∗ L))/(Y ×L0,δ∗◦ρ∗ L) Y × L1,ρ∗ 1 1 1 1 L0 1 ⊕

∗ ∗ Lj Lpr L ◦ ◦ 1 0 (Y × 1 ∗ L)/(Y × 0 ∗ ∗ L) (3.12.2) L ,ρ2 L ,δ1 ◦ρ1   y

L ◦ ◦ 0 X /(Y × 0 ∗ ∗ L) L ,δ1 ◦ρ1 is an isomorphism. Since the map

∗ ∗ 1 δ1 ◦ τ1 : L −→ L is ´etale(2.11), there is a natural isomorphism

L 0 = L ◦ ◦ ' L ◦ ◦ . (3.12.3) X /Y 0 1 0 X /Y × 0 L X /(Y × 0 ∗ ∗ L) L L ,δ1 ◦ρ1 Next consider the commutative diagram with cartesian squares ◦ ◦ 0 Y × 1 ∗ L −−−−→ Y × 0 ∗ ∗ L L ,ρ2 L ,δ1 ◦ρ1    τ ∗ y y 2 ◦ ◦ ◦ 0 2 2 Y × ◦ (Y × 0 ∗ ∗ L ) −−−−→ Y × 0 ∗ ∗ L 1 (L ,δ1 ◦δ2 ) L ,δ1 ◦δ2 (Y × 0 ∗ L ) L ,δ1    δ∗ y y 2 ◦ ◦ 0 Lg 1 Y −−−−→ Y × 0 ∗ L , L ,δ1 ∗ 0 0 where τ2 is étale by (2.11 (ii)). If X → Y → Y satisfies condition (T ), then (3.10), ([11], 17.3 (4)), and ([11], 17.5.8) imply that there is a natural isomorphism 0∗ ∗ ∗ Lf L 0 ' Lj Lpr L ◦ ◦ . (3.12.4) Y /Y 1 0 (Y × 1 ∗ L)/(Y × 0 ∗ ∗ L) L ,ρ2 L ,δ1 ◦ρ1 ◦ ◦ 1 In addition, since Lf : X → Y ×L0 L is flat (since f is log flat and ([14], 4.6)), there is a natural isomorphism ∗ ∗ ∗ La LX/Y ' Lj Lpr2L ◦ ◦ ◦ . (X× ◦ (Y ×L0,δ∗◦ρ∗ L))/(Y ×L0,δ∗◦ρ∗ L) ( Y × L1),ρ∗ 1 1 1 1 L0 1 (3.12.5) The logarithmic cotangent complex 21

Combining (3.12.3), (3.12.4), and (3.12.5), we see that the map (1.1.3) is naturally identiﬁed with (3.12.2), and hence (1.1.3) is an isomorphism.

3.13 (Veriﬁcation of (1.1 (v))). Suppose given a composite (1.1.4), and consider the resulting commutative diagram (3.9.1). By ([11], 17.3 (3)), there is a natural distinguished triangle

∗ ∗ Lh L ◦ −→ L ◦ ◦ −→ L ◦ −→ Lh L ◦ [1] 2 2 2 S/(L ×L0 Z ) X/L ×L0 Z X/S S/(L ×L0 Z ) (3.13.1) As mentioned in (3.9), there is a natural equivalence S ' LogY , and hence there is a natural isomorphism

L ◦ ' LX/Y . (3.13.2) X/S

◦ ◦ ∗ 2 1 Moreover, since the map δ1 : L ×L0 Z → L ×L0 Z is ´etale(2.11 (i)), there is a natural isomorphism

L ◦ ◦ ' L . (3.13.3) 2 X/Z X/L ×L1 Z Combining (3.10), (3.13.2), (3.13.3), and (3.13.1) we obtain the distinguished triangle (1.1.6).

We conclude this section by giving two important examples of com- f g posites X→Y →Z for which condition (T ) holds.

Example 3.14. Condition (T ) holds when the morphism g : Y → Z is log ﬂat. To see this, note that by ([14], 4.6 (iv)) we can assume that we have a chart (QZ ,QY , βZ , βY , θ1) for g such that θ1 : QZ → QY is injective and the map

◦ ◦ π : Y −→ Z × Spec( [Q ]) (3.14.1) Spec(Z[QZ ]) Z Y induced by the chart is ﬂat. By ([14], 2.2), we can, after replacing X by an ´etale cover, extend (QY , βY ) to a chart (QY ,QX , βY , βX , θ2) for f such that θ2 : QY → QX is injective. In this case, the map (1.4.3) is the composite of the map

ρ : O ⊗ [Q ] −→ O ⊗ [Q ][G], t ⊗ e(q) 7−→ t ⊗ e(q) · g Z Z[QZ ] Z Y Z Z[QZ ] Z Y q¯ (3.14.2) with the map π × id

O ⊗ [Q ][G] 'O ⊗ [Q ]⊗ [G] → O ⊗ [G] 'O [G]. Z Z[QZ ] Z Y Z Z[QZ ] Z Y Z Z Y Z Z Y (3.14.3) 22 Martin C. Olsson

Since (3.14.1) is flat, the map (3.14.3) is also flat, and hence to verify condition (T ) it suffices to show that (3.14.2) is flat. For this note that the isomorphism O ⊗ [Q ][G] → O ⊗ [Q ][G], Z Z[QZ ] Z Y Z Z[QZ ] Z Y −1 t ⊗ e(q) · g 7→ t ⊗ e(q) · gq¯ g identifies ρ with the flat map O ⊗ [Q ] −→ O ⊗ [Q ][G], t ⊗ e(q) 7−→ t ⊗ e(q) · g , Z Z[QZ ] Z Y Z Z[QZ ] Z Y 0 where g0 ∈ G denotes the identity element. Example 3.15. Condition (T ) holds if the morphism f : X → Y is integral ([9], 4.3). To see this, we can by ([14], 2.2) assume that we have a chart (QZ ,QY , βZ , βY , θ1) with θ1 injective. By the following lemma (3.16), we can, after replacing X and Y by étalecovers, find a morphism 0 0 0 0 of charts (QY , βY ) → (QY , βY ) for which the resulting map θ1 : QZ → QY 0 0 0 0 is still injective, and an extension (QY , P, , βY , βX , θ2) of (QY , βY ) to a 0 chart for f with θ2 : QY → P integral and injective. In this case, the map Z[QY ] → Z[P ] is flat ([9], 4.1), and hence (1.4.2) and condition (T ) hold. Lemma 3.16. Let f : X → Y be an integral morphism of fine log schemes, and let βQ : Q → MY be a chart. Then étalelocally on X and Y there 0 0 exists a chart (Q , P, βQ0 , βP , θ) for f with θ : Q → P integral and injec- 0 gp gp 0gp tive, and a map γ : Q → Q for which βQ = βQ0 ◦ γ and γ : Q → Q is an isomorphism. Remark 3.17. In fact the proof will show that we can take the map γ : Q → Q0 to be a localization. Since a localization morphism is integral, it follows that in the above we may take Q0 = Q.

Proof. Letx ¯ → X be a geometric point andy ¯ := f(¯x). Let Fy¯ = −1 ∗ βQ (OY,y¯) ⊂ Q, and let QFy¯ be the localization of Q by Fy¯ ([14], 3.5

(i)). The map βQ factors through QFy¯ , and hence after replacing Q by

QFy¯ and Y by an étale neighborhood ofy ¯, we may assume that the map Q → MY,y¯ is exact ([9], 4.6). gp Let ρ : G → MX,x¯ be a map from a finitely generated abelian group gp G for which the mapρ ¯ : G → MX,x¯ is surjective. Let P be the fiber product of the diagram G ⊕ Qgp   b yρ⊕f ◦βQ gp MX,x¯ −−−−→ MX,x¯, The logarithmic cotangent complex 23 and let θ : Q → P be the map induced by Q → G ⊕ Qgp, q 7→ (0, q). If βP : P → MX,x¯ denotes the projection, then the data (Q, P, βQ, βP , θ) defines a chart for f in some étaleneighborhood ofx ¯ by ([9], 2.10). Thus to prove the lemma it suffices to show that θ is integral. The integrality of θ amounts to the following: given (g1, q1), (g2, q2) ∈ P and q3, q4 ∈ Q for which

(g1, q1 + q3) = (g2, q2 + q4) in P , there exists (g, q) ∈ P and q5, q6 ∈ Q such that

q3 + q5 = q4 + q6, (g1, q1) = (g, q + q5), (g2, q2) = (g, q + q6). (3.17.1)

b Since the map f : MY,y¯ → MX,x¯ is integral, there exists q5, q6 ∈ Q and (g, q) ∈ P such that the images of q3 + q5 and q4 + q6 in MY,y¯ are equal, and for which the images of (g1, q1) and (g, q + q5) (resp. (g2, q2) and (g, q2 + q6)) in MX,x¯ are equal. Since βQ : Q → MY,y¯ is exact, there ∗ exists f ∈ Q such that q3 + q5 = q4 + q6 + f. After replacing q6 by q6 + f, we may there assume that q3 + q5 = q4 + q6 in Q. Similarly, there exists ∗ f2 ∈ P such that (g, q + q5) + f2 = (g1, q1). Then

(g2, q2) = (g1, q1) − (0, q4 − q3) = (g, q) + (0, q5 + q3 − q4) + f2 = (g, q) + (0, q6) + f2, and hence after replacing (g, q) by (g, q)+f2 we obtain elements (g, q) ∈ P and q5, q6 ∈ Q for which (3.17.1) hold. ut

4. Proof of (3.10)

In this section we prove (3.10). Part (i) of (3.10) follows from the following general result:

Lemma 4.1. Let f X ←−−−−X 0    g (4.1.1) y y Y ←−−−− Y0 be a cartesian diagram of algebraic stacks. Then for any morphism g : T → X 0 of stacks, the natural map

i ∗ ∗ i ∗ H (Lg Lf LX /Y ) → H (Lg LX 0/Y0 ) is an isomorphism for i ≥ 0 and a surjection for i = −1. 24 Martin C. Olsson

Proof. Note ﬁrst that if the result holds for g equal to the identity map g : X 0 → X 0, then it also holds for any morphism g. We therefore prove the result for the maps

i ∗ i H (Lf LX /Y ) → H (LX 0/Y0 ). (4.1.2) Furthermore, it suffices to prove the result after making a smooth base change on Y. We may therefore assume that Y is a scheme. It also suffices to prove the result after making a smooth base change on Y0, and hence we may also assume that Y0 is a scheme. Let x : X → X be a smooth cover with X a scheme, and let x0 : X0 → 0 0 ˜ 0 X denote the scheme X ×Y Y . Let f : X → X denote the projection. To prove that (4.1.2) is an isomorphism for i ≥ 0 and surjective for i = −1, it suffices to verify that this holds after pulling back to X0. Consider the morphism of distinguished triangles

˜∗ ∗ ˜∗ 1 +1 Lf Lx LX /Y −−−−→ Lf LX/Y −−−−→ ΩX/X −−−−→    a  c y by y (4.1.3) 0∗ 1 +1 Lx LX 0/Y0 −−−−→ LX0/Y0 −−−−→ ΩX0/X 0 −−−−→ . Since the formation of differentials commutes with base change, the morphism c is an isomorphism. Consideration of the associated morphism of long exact sequences then shows that it suffices to prove the Lemma with X instead of X . We are therefore reduced to the case when (4.1.1) is a diagram of schemes. For the case of schemes note first that we may work étalelocally on X and Y. We may therefore assume that X is equal to a filtering inductive limit of affine Y–schemes of finite type. By ([6], II.1.2.3.4) it therefore suffices to consider the case when f is of finite type and there exists an 0 0 embedding j : X ,→ Z over Y with Z → Y smooth. Set Z := Z ×Y Y , and let I (resp. I0) be the ideal of X (resp. X 0) in Z (resp. Z0). By ([6], III.3.1.3), we have

∗ ∗ ∗ 2 1 τ≥−1g LX /Y ' τ≥−1g τ≥−1LX /Y ' τ≥−1Lg (I/I → ΩZ/Y |X ), and 0 02 1 τ≥−1LX 0/Y0 ' (I /I → ΩZ0/Y0 |X 0 ). Since g∗I/I2 → I0/I02 is surjective, it follows that

i ∗ i H (g LX /Y ) → H (LX 0/Y0 ) is an isomorphism for i = 0 and surjective for i = −1. ut The logarithmic cotangent complex 25

To prove (3.10 (ii)), assume that we have a family of coverings as in (1.4.1) satisfying (1.4 (i))–(1.4 (iv)). fi gi Lemma 4.2. It suﬃces to prove the (3.10 (ii)) for the triples Xi→Yi→Zi.

Proof. Let i : Xi → X denote the projection to X. Since the family {i : Xi → X} is an fppf cover, it suﬃces to show that the map ∗ ∗ ∗ ∗ ∗ L Lh Lpr L ◦ ◦ −→ L Lh L ◦ (4.2.1) i 1 1 i 2 Y/L × ∗ 0 Z S/L × ∗ ∗ 0 Z δ1 ,L δ1 ◦δ2 ,L ◦ ◦ 2 is an isomorphism. Let S := Y × ◦ (L × 0 Z ), and let π : S → S i i 1 L i i (L ×L0 Z i) be the natural projection. We then have a commutative diagram

◦ L2 × Z 1 L0 Hi ¡ Z H ¡ Z H pr 2 pr Z H L 2,Si ¡ ∗ H δ2 × id Z H pr ◦ hi ¡ 1,Si ◦ . Z HHj ◦ X - S - Y L2 × Z i i ¡ i H Z L0 H Lg π i Z f × id π × id ¡ Yi H i pr × id H ZZ~ ? ?¡ 2 ? H ◦ . ◦ h × id◦ pr1, × id ◦ ◦ Lg × idHj ◦ X × Z - S × Z S - Y × Z - 1 ◦ i ◦ i ◦ i L × 0 Z i Z .Z Z L

∗ δ2 × id pr ◦ prS pr ◦ prL1 X Y ? ? ? ? ◦ ◦ ◦ h - pr1,S - .Lg - 1 X S Y L ×L0 Z ©

pr2,S (4.1.2) & % which gives rise to a morphism of distinguished triangles ∗ ∗ ∗ ∗ Lf Lπ L ◦ ◦ ◦ → Lf L ◦ ◦ → Lf L ◦ ◦ ◦ i Yi 1 i 1 i (Y × ◦ Z i)/(L ×L0 Z i) Y i/(L ×L0 Z i) Y i/(Y × ◦ Z i) Z   Z α  γ y yβ y ∗ ∗ ∗ Lι L ◦ ◦ → Lh L ◦ → Lh L ◦ , 2 i 2 i (S× ◦ Z i)/(L ×L0 Z i) Si/(L ×L0 Z i) Si/(S× ◦ Z i) Z Z (4.2.3) ◦ ◦ where ι : Xi → S × ◦ Zi denotes the map (π × id) ◦ hi. Since Z ◦ ◦ Si ' (S × ◦ Zi) × ◦ ◦ Y i Z (Y × ◦ Z i),πYi Z 26 Martin C. Olsson

◦ ◦ and πYi : Y i → Y is ﬂat, ([11], 17.3 (4)) implies that γ is an isomorphism. ◦ ◦

Moreover, since πZi : Zi → Z is ﬂat, ([11], 17.3 (4)) also shows that there are natural isomorphisms

∗ ∗ ∗ ∗ ∗ Lf Lπ L ◦ ◦ ◦ ' L Lh Lpr L ◦ ◦, i Yi 1 i 1,S 1 (Y × ◦ Z i)/(L ×L0 Z i) Y/L ×L0 Z Z ∗ ∗ ∗ L Lh L ◦ ' Lι L ◦ ◦ i 2 2 S/L ×δ∗◦δ∗,L0 Z (S× ◦ Z i)/(L ×L0 Z i) 1 2 Z identifying α with the map (4.2.1). Hence α is an isomorphism if β is an isomorphism. ut

Thus to prove (3.10 (ii)), we may assume that we have charts as in (1.4 (iv)). In this case, we have a commutative diagram

◦ 0 ◦ ◦ h qX 2 X −−−−→ SQ × Z −−−−→ L × 0 Z X SQZ L      δ∗×id fy `y y 2 ◦ L0 ◦ ◦ g qY 1 Y −−−−→ SQ × Z −−−−→L × 0 Z, Y SQZ L where qY is the map ([14], 5.24), qX is the map induced by the triple ∗ ∗ pr MZ −→ ` M ◦ −→ M , Z SQY SQ ×S Z Y QZ

` is the map induced by the map QY → QX ([14], discussion preceding 0 5.20), and h is the map obtained from the chart (QY ,QX , βY , βX , θ2) for f. The map qY is étaleby (loc. cit., 5.24), and since ◦ ◦ ∗ 2 1 δ1 × id : L ×L0 Z −→ L ×L0 Z is étaleby (2.11 (i)), it follows from ([14], 5.24) that the map qX is étale. Let P denote the fiber product of the diagram ◦ SQ × Z X SQZ   y (4.2.4) ◦ ◦ Y −−−−→ SQ × Z. Y SQZ

Since the maps qX and qY are ´etale, it follows from ([11], 17.5.8) and ([11], 17.3 (3)) that there are natural isomorphisms

L ◦ ◦ ' L ◦ ◦, 1 Y/(SQ ×S Z ) Y/L × 0 Z Y QZ L The logarithmic cotangent complex 27

∗ ∗ Lι L ◦ ' Lh L ◦, 2 P/(SQ ×S Z ) S/L × ∗ ∗ 0 Z X QZ δ1 ◦δ2 ,L where ι : X → P is the map induced by h0. Thus in order to prove (3.10.1) it suﬃces to show that on P the natural map

∗ Lpr1L ◦ ◦ −→ L ◦ (4.2.5) Y/(SQ ×S Z ) P/(SQ ×S Z ) Y QZ X QZ ◦ ◦ is an isomorphism, and since ZQ [QY ] → SQ × Z is faithfully ﬂat Z Y SQZ by the following lemma (4.3 (ii)), it suﬃces to verify that (4.2.5) is an ◦ isomorphism after base changing to ZQZ [QY ].

Lemma 4.3. Let Y be a fine log scheme, βQ : Q → MY a chart, and let θ : Q → P be a morphism of fine monoids. (i). Suppose (Q, P, βQ, θ) is extended to a chart (Q, P, βQ, βP , θ) for a morphism of log schemes f : X → Y , and define G := Coker(Qgp → P gp). Then the fibered product of the diagram ◦ Y Q[P ]   (4.3.1) y ◦ ◦

X −−−−→ Y ×SQ SP ◦ ◦ ◦ is naturally isomorphic to X[G], with the projection to X (resp. Y Q[P ]) induced by the map

OX −→ OX [G], x 7→ x · g0 ∗ (resp. OY [P ] −→ OX [G], x · e(p) 7→ f (x)αX (βP (p))gp¯), where g0 (resp. gp¯) denotes the zero element in G (resp. the image of p in G). ◦

(ii). If θ is injective, then any 1-morphism X → Y ×SQ SP factors fppf ◦ ◦ ◦ locally on X through Y Q[P ]. The map Y Q[P ] → Y ×SQ SP is faithfully flat and surjective. (iii). If θ0 : P → P 0 is an injective morphism of fine monoids and G0 := Coker(P gp → P 0gp), then the fiber product of the diagram

◦ Y Q[P ]   (4.3.2) y ◦ ◦ 0 Y ×SQ SP −−−−→ Y ×SQ SP 28 Martin C. Olsson

◦ 0 is isomorphic to the stack theoretic quotient of Y Q[P ] by the natural ◦ action of the group scheme Y [G0].

◦ ◦ 0 0 Remark 4.4. The action of Y [G ] on Y Q[P ] referred to in (4.3 (iii)) is ◦ ◦ 0 0 the following. The scheme Y Q[P ] (resp. Y [G ]) represents the functor ◦ which to any Y -scheme T associates the set of morphisms of sheaves of 0 0 monoids ρ : P → OT for which ρ(θ ◦ θ(q)) = αY (βQ(q)) for all q ∈ Q ◦ 0 ∗ 0 (resp. group homomorphisms τ : G → OT ). The action of τ ∈ Y [G ] on ◦ 0 0 0 0 Y Q[P ] is obtained by sending ρ to the map P → OT sending p ∈ P to τ(¯p0) · ρ(p0), wherep ¯0 ∈ G0 denotes the image of p0.

Proof (Proof of (4.3)). To prove (i), it suﬃces to consider the case when ◦ ◦ X = Y Q[P ]. Let V denote the ﬁber product of (4.3.1). By ([14], 5.22), ◦ ◦ V represents the functor which to any scheme h : T → Y Q[P ] × ◦ Y Q[P ] Y ∗ ∗ ∗ ∗ associates the set of isomorphisms : h pr1MYQ[P ] → h pr2MYQ[P ] for which the diagrams ∗ ∗ −1 −1 h pr1MYQ[P ] h pr1 MYQ[P ]

(4.4.1)

∗ s MY P ¯ @ @ @ @ @ ? @ ? @R @R ∗ ∗ −1 −1 h pr2MYQ[P ] h pr2 MYQ[P ]

◦ commute, where s : T → Y denotes the structure morphism. Let βi : P → ∗ ∗ h pri MYQ[P ] (i = 1, 2) denote the natural chart. The commutativity of the second diagram in (4.4.1) implies that for each p ∈ P there exists a unique unit u(p) such that

β2(p) = λ(u(p)) + (β1(p)), (4.4.2) and the commutativity of the ﬁrst triangle in (4.4.1) implies that u(θ(q)) = 1 for all q ∈ Q. Moreover, because is an isomorphism of log structures, we must have α2(β2(p)) = u(p) · α1(β1(p)) (4.4.3) ∗ ∗ for all p ∈ P , where αi : h pri MYQ[P ] → OT denotes the logarithm map. The logarithmic cotangent complex 29

In other words, given an isomorphism as above, we obtain a homo- ∗ morphism ρ : G → OT , and conversely, a homomorphism ρ for which (4.4.3) holds induces an isomorphism by formula (4.4.2). Thus there is a natural isomorphism

◦ ◦ Y Q[P ] × ◦ Y Q[P ] ' Spec(OY ⊗ [Q] Z[P ⊕ P ][G]/(0, p) = (p, 0) · e(gp)) Y × S Z SQ P ◦ ' Y Q[P ][G], where gp denotes the image of p in G. From this the statement about the two projections in (i) also follows. To see (ii), note that by (i) it suffices to prove the first statement. If ◦ ◦ ◦ s : X → Y is a Y -scheme, then to give a 1-morphism X → Y ×SQ SP is by ([14], 5.20) equivalent to giving a triple (M, γ, η), where M is a fine log structure on X, γ : P → M is a map which fppf locally lifts to a ∗ chart, and η : s MY → M is a morphism of fine log structures such that the diagram Q −−−−→θ P   ¯  γ (4.4.4) βQy y −1 η¯ s MY −−−−→ M commutes. After replacing X by an fppf cover, we may assume that we have a chartγ ˜ : P → M lifting γ. The resulting diagram

Q −−−−→θ P   β   (4.4.5) Qy yγ˜ ∗ η s MY −−−−→M may not commute, but the commutativity of (4.4.4) ensures that there ∗ exists a unique homomorphism ρ : Q → OX such that for all q ∈ Q, η(βQ(q)) = λ(ρ(q)) +γ ˜(θ(q)). Since θ : Q → P is injective, we have an exact sequence

pg ∗ gp ∗ 1 ∗ · · · −→ Hom(P , OX ) −→ Hom(Q , OX ) −→ Ext (G, OX ) −→ · · · , and by the same reasoning as in ([14], proof of 2.1), the image of ρ in 1 ∗ Ext (G, OX ) vanishes fppf locally on X. Thus after replacing X by an 0 gp ∗ 0 fppf cover, there exists a map ρ : P → OX such that ρ = ρ ◦ θ. Replacingγ ˜ by p 7→ λ(ρ0(−p)) +γ ˜(p), we can therefore assume that (4.4.5) commutes. In this case, (M, γ, η) is isomorphic to the pullback ◦ of (MYQ[P ], γYQ[P ], ηYQ[P ]) (the data on Y Q[P ] deﬁning the natural map 30 Martin C. Olsson

◦ ◦ ◦

Y Q[P ] → Y ×SQ SP ; see ([14], 5.20)) via the map X → Y Q[P ] induced byγ ˜. To see (iii), note that the fiber product of (4.3.2) is isomorphic to ◦ ◦ ◦ 0 0 0 Y Q[P ] ×SP SP . Let Y Q[P ] → Y Q[P ] ×SP SP be the natural map which is faithfully flat by (ii). To give the isomorphism in (iii), it suffices by ([11], 3.2) to give an isomorphism

◦ ◦ ◦ 0 0 0 0 Y Q[P ] × ◦ Y Q[P ] ' Y Q[P ][G ] (Y [P ]× ) Q SP SP 0 ◦ ◦ (4.4.6) 0 0 = Y [P ] × ◦ ◦ Y [P ] Q 0 0 Q [Y Q[P ]/Y [G ]] inducing an isomorphism of groupoids ([11], 2.4.3)

◦ ◦ ◦ 0 0 0 (Y Q[P ] × ◦ Y Q[P ] Y Q[P ]) (Y [P ]× ) ⇒ Q SP SP 0   'y ◦ ◦ ◦ 0 0 0 (Y [P ] × ◦ ◦ Y [P ] Y [P ]), Q 0 0 Q ⇒ Q [Y Q[P ]/Y [G ]] where the ﬁrst (resp. second) groupoid is that obtained from the map ◦ ◦ ◦ ◦ ◦ Y [P 0] → Y [P ] × × (resp. Y [P 0] → [Y [P 0]/Y [G0]]). Such an Q Q SP SP 0 Q P isomorphism (4.4.6) is provided by (i). ut

By (4.3), the base change of (4.2.4) via the map

◦ ◦ ZQ [QY ] −→ Z × SQ Z SQZ Y is isomorphic to the diagram

◦ ◦ 0 [ZQZ [QX ]/Z[G ]]   (4.4.7) yb ◦ a ◦ Y [G] −−−−→ ZQZ [QY ],

0 gp gp gp gp where G (resp. G ) denotes (QY /QZ ) (resp. (QX /QY )) and the map a (resp. b) is induced by the map

O ⊗ [Q ] −→ O [G], Z Z[QZ ] Z Y Y ◦ ◦ (4.4.8) ∗ t ⊗ e(q) 7−→ g (t)βQY (q) · gq¯ (resp. ZQZ [QX ] → ZQZ [QY ]). The logarithmic cotangent complex 31

◦ 0 Let P denote the fiber product of (4.4.7). Since the map ZQZ [QY ] → ◦ Z × SQ is faithfully flat, faithfully flat descent and ([11], 17.3 (4)) SQZ Y imply that in order to show that (4.2.5) is an isomorphism it suffices to show that the map ∗ Lpr L ◦ ◦ −→ L ◦ ◦ (4.4.9) 1 0 0 Y [G]/Z QZ [QY ] P /[Z QZ [QX ]/Z [G ]] ◦ ◦ ◦ 0 is an isomorphism. Now because the map ZQZ [QX ] → [ZQZ [QX ]/Z[G ]] is faithfully flat, to verify that (4.4.9) is an isomorphism, it suffices to show that the natural map ∗ Lpr1L ◦ ◦ −→ L ◦ (4.4.10) Y [G]/Z QZ [QY ] P00/Z QZ [QX ] ◦ 00 is an isomorphism, where P denotes the fiber product Y [G] × ◦ Z QZ [QY ] ◦

ZQZ [QX ]. By ([6], II.2.2.3), the map (4.4.10) is an isomorphism if Torj (O ⊗ [Q ], O [G]) = 0, for all j > 0. (4.4.11) O ⊗ [Q ] Z [QZ ] Z X Y Z Z[QZ ]Z Y Z and hence the proof of (3.10) is complete. ut We conclude this section by explaining the equivalence of the two vanishing conditions (1.4.2) and (1.5.1). The following argument is due to Gabber. Let G be an abelian group and A a G–graded commutative ring. Lemma 4.5. The forgetful functor (G-graded A–modules) → (ungraded A–modules) has an exact right adjoint.

Proof. Deﬁne M[G] := M ⊗Z Z[G] with G–grading induced by the G– grading on Z[G], and let a ∈ Ah act on M[G] by the given action on M times the action of h on Z[G]. Then for any G–graded A–module L = ⊕gLg, to give a morphism f : L → M[G] of graded G–modules, is equivalent to giving for each g ∈ G a map of abelian groups fg : Lg → M · 1g such that for a ∈ Ah and l ∈ Lg, if fg(l) = m · 1g then fg+h(al) = (am) · 1g+h. Such a collection of morphisms is in turn equivalent to a morphism of ungraded modules L → M. ut Lemma 4.6. Let N be an A–module and M a G–graded A–module. Then there is a natural isomorphism

ι : M ⊗A (N[G]) → (M ⊗A N)[G] of G–graded A–modules. 32 Martin C. Olsson

Proof. To define the map ι it suffices to give a map of ungraded A– modules M ⊗A (N[G]) → M ⊗A N. We take the map which sends m ⊗ (n · 1g) to m ⊗ n ∈ M ⊗A N. To prove that the resulting map ι is an isomorphism, note that the map is functorial in N and also compatible with direct sums. Let F 1 → F 0 → N be a presentation of N with F 1 and F 0 free. By the exactness of the functor M 7→ M[G] and the right exactness of tensor product, we then have a commutative diagram with exact rows 1 0 M ⊗A (F [G]) −−−−→ M ⊗A (F [G]) −−−−→ M ⊗A (N[G]) −−−−→ 0    ι ι ι y y y 1 0 (M ⊗A F )[G] −−−−→ (M ⊗A F )[G] −−−−→ (M ⊗A N)[G] −−−−→ 0. Consideration of this diagram then reduces the proof of the fact that ι is an isomorphism to the case when N is a free module. By the compatibility with direct sums it therefore suffices to consider the case when N = A. In this case, the map ι on the g0–th graded piece can be identified with the map

⊕g∈GMg · 1g0−g → M · 1g0 sending m · 1g0−g ∈ Mg · 1g0−g to m · 1g0 . ut Corollary 4.7. Let M be a G–graded A–module and N an A–module. Then for any integer j there is a canonical isomorphism of G–graded A–modules j j TorA(M,N[G]) ' TorA(M,N)[G]. Proof. Let F · → N be a free resolution of N in the category of ungraded A–modules. Then F ·[G] → N[G] is a free resolution in the category of graded A–modules. Thus

j · TorA(M,N[G]) ' Hj(M ⊗A (F [G])). On the other hand, by the preceding Lemma

· · Hj(M ⊗A (F [G])) ' Hj((M ⊗A F )[G]) which since M 7→ M[G] is exact is isomorphic to

· j Hj(M ⊗A F )[G] ' TorA(M,N)[G]. ut Corollary 4.8. Let G ⊂ H be an inclusion of abelian groups, and M an j j H–graded module. Then TorA(M,N[G]) = 0 if and only if TorA(M,N) = 0. The logarithmic cotangent complex 33

Proof. Write M = ⊕h∈H Mh. For each coset S ∈ H/G deﬁne MS ⊂ M to be ⊕h∈SMh. Then M = ⊕SMS. Moreover, the choice of an element hS ∈ S gives MS the structure of a G–graded A–module. To prove the corollary it suﬃces to consider each MS, which reduces the proof to the case G = H. ut

4.9. We apply this as follows. Let θ : Q → P be a morphism of integral monoids with θgp : Qgp → P gp injective, set G = P gp/Qgp, and let π : P → G be the projection. Then Z[P ] is a G–graded ring with Z[P ]g = ⊕p,π(p)=gZ · 1p. Since Z[Q] maps to Z[P ]0, for any ring homomorphism [Q] → R the tensor product R ⊗ [P ] is again a G–graded ring. Z Z[Q] Z In the situation of (1.4), deﬁne G = Qgp/Qgp and H = Qgp /Qgp. Set Yi Zi Xi Zi A = OZ ⊗ [QY ], M = OZ ⊗ [QX ], and N = OY . i Z[QZi ] Z i i Z[QZi ] Z i i

Then the A–module structure on OYi [G] described in (1.4) is precisely the above–deﬁned graded A–module structure on N[G]. The equivalence of (1.4.2) and (1.5.1) therefore follows from (4.8).

5. Deformation theory of log schemes

In this section we record some consequences of the results of ([13]) and the transitivity triangle (1.1 (v)) about the relationship between the logarithmic cotangent complex and deformation theory generalizing ([6], III.1.2.3, III.2.1.7, III.2.2.4) to log schemes.

5.1 (Log version of ([6], III.1.2.3)). Let f : X → Y be a morphism ◦ of ﬁne log schemes and let I be a quasi-coherent sheaf on X. Deﬁne a Y -extension of X by I to be a commutative diagram of log schemes

j - XX0 (5.1.1) f f 0

? © Y, where j is an exact closed immersion deﬁned by a square-zero ideal, together with an isomorphism j : I ' Ker(OX0 → OX ). The set of Y - extensions of X by I form in a natural way a category ExalY (X,I). A 0 0 morphism of Y -extensions (j1 : X,→ X1) → (j2 : X,→ X2) is a mor- 0 0 phism of log schemes ψ : X1 → X2 over Y such that j2 = ψ ◦ j1 and such that the induced isomorphism

j2 ψ I −−−−→ Ker(OX2 −→ OX ) −−−−→ Ker(OX1 −→ OX ) 34 Martin C. Olsson

is equal to j1 . There is a tautological equivalence of categories (see ([13], 2.2) for the meaning of the right hand side)

◦ Exal (X,I) ' Exal (X,I), Y LogY induced by the functor which sends a diagram (5.1.1) to

◦ ◦ j ◦0 X - X (5.1.2) L L0 f f ? © LogY .

Hence from ([13], 1.1) and the deﬁnition of LX/Y we obtain the following result (see (loc. cit., 2.16–2.18) for the meaning of the right hand side of (5.2.1)):

Theorem 5.2. If ExalY (X,I) denotes the set of isomorphism classes in ExalY (X,I), then there is a natural bijection

1 ExalY (X,I) ' Ext (LX/Y ,I). (5.2.1)

Remark 5.3. One can show ([13], 2.12) that the category ExalY (X,I) has a natural structure of a Picard category ([11], 14.4). Hence its set of isomorphism classes has a natural structure of a group, and the isomorphism (5.2.1) is a group isomorphism.

5.4 (Log version of ([6], III.2.1.7)). Let Y0 ,→ Y be an exact closed immersion of fine log schemes defined by a square-zero ideal I ⊂ OY , and let f0 : X0 → Y0 be a log flat morphism. Define a log flat deformation of X0 to Y to be a cartesian square

j X0 −−−−→ X     (5.4.1) f0y yf

Y0 −−−−→ Y with f log flat. The set of log flat deformations of X0 to Y form a category 0 Def(f0): a morphism of log flat deformations (j : X0 ,→ X) → (j : X0 ,→ X0) is a morphism of Y -log schemes ψ : X → X0 such that ψ ◦ j = j0. The logarithmic cotangent complex 35

5.5. To give a log ﬂat deformation as in (5.4.1) is equivalent, by the deﬁnition of LogY and ([14], 4.6 (iv)), to giving a 2-commutative diagram

◦ j ◦ X0 −−−−→ X   L  L (5.5.1) f0 y y f

LogY0 −−−−→ LogY , where Lf is ﬂat. In fact there is an equivalence of categories Def(f0) '

Def(Lf0 ), where Def(Lf0 ) is the category obtained from ([13], 3.1) applied to (5.5.1). Thus from (loc. cit., 1.4) we obtain the following:

Theorem 5.6. Let J denote the ideal of LogY0 in LogY . (i). There exists a canonical class o ∈ Ext2(L , L∗ J) whose vanishing X0/Y0 f0 is necessary and sufficient for the existence of a log flat deformation of X0 to Y . (ii). If o = 0, then the set of isomorphism classes of flat deformations of X to Y is a torsor under Ext1(L , L∗ J). 0 X0/Y0 f0 (iii). The automorphism group of any log flat deformation of X0 to Y is canonically isomorphic to Ext0(L , L∗ J). X0/Y0 f0 Remark 5.7. For a generalization of this result, see (8.31).

5.8 (Log version of ([6], III.2.2.4)). Suppose given a commutative diagram of solid arrows i X - X 0 .. .. Z J .. J Z .. f J Z f J .. 0 .. J Z J .. Z J .. J Z j. (5.8.1) J J ZZ~ j - J Y0 Y J J J J 0 J g0 J h gh J JJ^ ? J^ ? k - Z0 Z, where i (resp. j, k) is an exact closed immersion deﬁned by a square-zero ideal I ⊂ OX (resp. J ⊂ OY , K ⊂ OZ ).

1 ∗ Theorem 5.9. There is a canonical class o ∈ Ext (f0 LY0/Z0 ,I) whose vanishing is necessary and suﬃcient for the existence of a morphism f : X → Y ﬁlling in (5.8.1). If o = 0, then the set of such maps f is a torsor 0 ∗ under the group Ext (f0 LY0/Z0 ,I). 36 Martin C. Olsson

Proof. The existence of a map f ﬁlling in (5.8.1) is tautologically equivalent to the existence of a map Lf ﬁlling in the diagram of solid arrows

◦ i ◦ X - X 0 .. .. Z J .. J Z .. L J Z L J .. f f0 .. J Z J .. Z J .. J Z j. (5.9.1) Z~ J J j - J LogY0 LogY J J J J Lh0J Log(g0) J Lh Log(g) J JJ^ ? J^ ? k - LogZ0 LogZ , where Log(g) and Log(g0) denote the morphisms of stacks induced by g and g0 ([14], page 20). Note that Log(g) and Log(g0) are representable since they are faithful ([11], 8.1.2). Let U → LogZ be a smooth cover, • and let U → LogZ be the associated simplicial algebraic space ([11], 12.4). Base-changing the diagram (5.9.1) to U • we obtain a diagram of simplicial algebraic spaces

i• X• - X• 0 .. .. Z J .. J Z .. L• J Z L• J .. f f0 .. J Z J .. Z J .. J Z j. (5.9.2) • J J ZZ~ • j - • J W0 W J J J J L• g• L• g• h0J 0 J h J JJ^ J^ ? • ? • k - • U0 U , and by ([13], 2.7) the existence of the dotted arrow Lf is equivalent to • the existence of the dotted arrow Lf . From ([13], 2.21) and ([6], III.2.2.4) 0 1 •∗ ∗ we then obtain a class o ∈ Ext (LL L • • , π I) whose vanishing is f0 W0 /U0 X • • necessary and suﬃcient for the existence of the arrow Lf (here πX : X → X denotes the natural map). Now by the construction of the cotangent complex L and ([11], 13.5.4) (see also ([15], 6.19)), there is a LogY0 /LogZ0 natural isomorphism

1 •∗ ∗ 1 ∗ Ext (LL L • • , π I) ' Ext (LL L ,I). f0 W0 /U0 X f0 LogY0 /LogZ0 The logarithmic cotangent complex 37

On the other hand, by (3.10 (i)) the cone of the map

∗ ∗ Lf L → LL L ◦ , 0 Y0/Z0 f0 2 LogY /(L × ∗ ∗ 0 Z 0) 0 δ1 ◦δ2 ,L is cohomologically concentrated in degrees ≤ −2. Consequently the natural map

i ∗ i ∗ Ext (LL L ◦ ,I) → Ext (Lf L ,I) (5.9.3) f0 2 0 Y0/Z0 LogY /(L × ∗ ∗ 0 Z 0) 0 δ1 ◦δ2 ,L is an isomorphism for i = 0 and an injection for i = 1. On the other hand, since the map Log(g0) factors as

◦ ∗ ◦ 2 δ1 1 Log −−−−→L × ∗ ∗ 0 Z −−−−→L × ∗ 0 Z = Log , Y0 δ1 ◦δ2 ,L 0 δ1 ,L 0 Z0 where the second map is étaleby (2.11 (i)), there is a natural isomorphism Lf ∗L ' LL∗ L . Hence we get from o0 and the injective 0 Y0/Z0 f0 LogY0 /LogZ0 1 ∗ map (5.9.3) (with i = 1) a class o ∈ Ext (Lf0 LY0/Z0 ,I) whose vanishing is necessary and sufficient for the existence of the map f. We leave it to the reader to verify, using the technique of ([13], proof of 2.24), that this class is independent of the choice of U, thereby proving the first statement in (5.9). To prove the second statement, note that by ([11], 17.7) and (3.8) there is a canonical isomorphism Ext0(Lf ∗L ,I) ' Hom(f ∗Ω1 ,I), 0 Y0/Z0 0 Y0/Z0 and hence the second statement follows from ([9], 3.9). ut

6. Application: The derived de Rham complex

6.1. In this section we apply the general theory to the study of the de Rham and crystalline cohomology of morphisms of log schemes. For simplicity we restrict our attention to the following local situation. Let f : X = (Spec(B),MX ) → S = (Spec(A),MS) be a morphism of aﬃne ﬁne log schemes, and (βP : P → MX , βQ : Q → gp gp gp MS, θ : Q → P ) a chart for f such that θ : Q → P is injective. gp gp Let G denote P /Q , and let SP/Q denote the stack theoretic quotient of Spec(Z[P ]) by the natural action of the diagonalizable group scheme D(G) = Spec(Z[G]). The stack SP/Q is an algebraic stack over Z[Q]. If MQ denotes the canonical log structure on Spec(Z[Q]), then by ([14], 5.20) the stack SP/Q can be viewed as the stack which to any g : T → Spec(Z[Q]) 38 Martin C. Olsson

∗ associates the groupoid of morphisms of log structures g MQ → M on T together with a commutative diagram of sheaves of monoids

Q −−−−→θ P   can  y yβ −1 g M Q −−−−→ M, where can denotes the canonical projection and β fppf–locally on T lifts to a chart for M. Denote by S the ﬁber product S × S . P/Q,S Spec(Z[Q]) P/Q By ([14], 5.24), the natural map SP/Q,S → LogS is ´etale. In particular, there is a natural isomorphism

LX/S ' (τ≥nL ◦ )n. X/SP/Q,S

6.2. Let Spec(A0) → SP/Q,S be a smooth surjection, and let Spec(A•) be the 0–coskeleton (note that the diagonal of SP/Q,S is aﬃne). Deﬁne

Spec(B•) := Spec(B) ×SP/Q,S Spec(A•). The map B → B• is an equivalence of cosimplicial algebras since by descent theory the total complex of B• computes the coherent cohomology of the affine scheme Spec(B). More generally, for any B–module M the map M → M ⊗B B• is an equivalence. Let P•• → B• be the canonical free resolution of B• over A• (i.e. for every m ∈ N we have a simplicial Am–module P•m over Bm), and let Ω· be the functor P••/A• ∆o × ∆ → (differental graded A–algebras) sending ([n], [m]) to the de Rham complex of Pn,m over Am. We define · the derived de Rham complex of X/S, denoted LΩX/S, to be the complex obtained by taking the total complex of Ω· . As explained in ([7], P••/A• VIII.2.1.1.4) there is a canonical filtration F m on Ω· which makes P••/A• · LΩX/S a filtered complex. 6.3. Let CF (A) denote the category of complexes of A–modules L with a decreasing filtration F mL such that F aL = L for some a. A morphism u : L → M in CF (A) is defined to be a quasi–isomorphism if for every m the map F mu : F mL → F mM is a quasi–isomorphism. We denote by DF (A) the resulting localized category. gr gr Let Mod (A) denote the category of Z–graded A–modules, let C (A) gr denote the category of Z–graded complexes of A–modules, and let D (A) denote the category obtained by localizing along quasi–isomorphisms. The functor gr : CF (A) → Cgr(A) The logarithmic cotangent complex 39 sending a filtered complex to its associated graded preserves quasi–isomorphisms, and hence induces a functor

gr : DF (A) → Dgr(A).

− gr We can also consider the exterior power functor ΛB : C (B) → C (B) deﬁned in ([6], I.4.1.3). This functor also preserves quasi–isomorphisms, and hence as in (loc. cit., I.4.2.2) induces a functor

− gr LΛB : D (B) → D (B). This functor can be extended to the category D0≤0(B) (1.8 (iii)) as follows. Let D0gr(B) denote the category obtained by taking A in (1.8 (iii)) equal to the category of Z–graded B–modules. For an object (K≥−n)n ∈ D0≤0(B) and any r the pro–object lim τ LΛ K is essentially con- ←−n ≥r B ≥−n stant, and we deﬁne

0≤0 0gr LΛB : D (B) → D (B) by sending (K ) to the pro–object (lim τ LΛ K ) . ≥−n n ←−n ≥r B ≥−n r Proposition 6.4. There is a natural isomorphism in D0gr(A)

· grLΩX/S ' LΛBLX/S[−∗],

gr where for an object L = ⊕Lr ∈ C (B) we write L[−∗] for the object ⊕rLr[−r] (this operation passes to the derived category). Proof. First note the following:

Lemma 6.5. The diagram

gr LΛB• − D (A•) ←−−−− D (B•)  x  ⊗L B Toty  B • LΛ Dgr(A) ←−−−−B D−(B) is canonically 2–commutative.

Proof. Let o E : ∆ × ∆ → ModB be a functor such that for all ([r], [s]) ∈ ∆ × ∆o the B–module E([r], [s]) is ﬂat. Deﬁne

o gr F1 : ∆ × (∆ × ∆ ) → Mod (A), ([m], [r], [s]) 7→ (ΛBE([r], [s])) ⊗B Bm, 40 Martin C. Olsson and o gr F2 : ∆ × ∆ → Mod (A), ([r], [s]) 7→ ΛBE([r], [s]).

We also view F2 as a functor ∆ × ∆ × ∆o → Modgr(A) which is constant in the ﬁrst variable. Then by the construction of the left derived functor in ([6], I.4.2.2) there are canonical isomorphisms

L Tot(F1) ' Tot(LΛB• ◦ ⊗BB•)(Tot(E)), and Tot(F2) ' LΛB(Tot(E)), where Tot(−) denotes the total complex functor. It therefore suﬃces to show that the map F2 → F1 of functors ∆ × ∆ × ∆o → Modgr(A) obtained from B → B•, induces a quasi–isomorphism on total complexes. For this in turn it suﬃces to show that for any r, s ∈ N the natural map

ΛBE([r], [s]) → ([m] 7→ (ΛBE([r], [s])) ⊗B Bm) is an equivalence. As mentioned earlier this follows from the fact that B → B• is an equivalence of cosimplicial algebras. ut L By the construction of LX/S, the object LX/S ⊗B B• can be represented · by the complex LB•/A• , and by construction LΩX/S is the total complex of the filtered complex LΩ· over A . By (6.5) it therefore suffices to B•/A• • exhibit a natural isomorphism grLΩ· ' LΛ L [−∗] B•/A• B• B•/A• gr in D (A•). Such an isomorphism is provided by ([7], VIII.2.1.1.5) applied to the topos of presheaves on ∆o. ut Relation with crystalline cohomology. 6.6. As in ([7], VIII.2.1.3), let CF ∧(A) denote the category of complexes of pro–A–modules L with a decreasing filtration F mL such that F aL = L for some a, and let DF ∧(A) denote the localization of CF ∧(A) along quasi–isomorphisms (see loc. cit. for details). There is a natural completion functor DF (A) → DF ∧(A),L 7→ L∧ sending a filtered module L to L∧ := “ lim ”L/F mL ←− with the induced filtration. The logarithmic cotangent complex 41

6.7. Let (X/S)cris denote the topos associated to the site made up of PD–immersions U,→ T over X → S, where U → X is strict and étale ◦ ◦ and U,→ T is a quasi–PD–nilpotent immersion (in other words, the logarithmic analogue of the site defined in ([2], Appendice)). As in ([7], VIII.2.2.7) define J ⊂ OX/S to be the canonical PD–ideal and

O∧ = “ lim ”O /J [n] ∈ D+F ∧(O ). X/S ←− X/S X/S n

As in (loc. cit.) the global section functor deﬁnes a functor

+ + RΓ : D F (OX/S) → D F (A).

6.8. Recall ([10], 4.4.2 and 4.4.4) that the morphism f : X → S is a log complete intersection if there exists a factorization

j g X −−−−→ W −−−−→ S, (6.8.1)

◦ ◦ where j is a strict closed immersion, g is log smooth, and X,→ W is ◦ ◦ deﬁned by a regular ideal. By (loc. cit., 4.4.4) if X,→ W is a regular immersion for one choice of factorization of f then for any factorization ◦ ◦ (6.8.1) the map X,→ W is a regular immersion.

Lemma 6.9. Let f : X → S be a log complete intersection, and choose factorization as in (6.8.1). Then

2 d ∗ 1 LX/S ' (I/I →j ΩW/S),

◦ ◦ where I denotes the ideal of X in W , I/I2 is placed in degree −1, and d 1 denotes the map induced by the diﬀerential OW → ΩW/S.

Proof. Because j is strict, there is a distinguished triangle

∗ 1 ∗ 1 j Ω → LX/S → L ◦ ◦ → j Ω [1]. (6.9.1) W/S X/W W/S

2 Since j is a regular immersion L ◦ ◦ ' I/I [1] ([6], III.3.2.4), and hence X/W (6.9.1) implies that LX/S is isomorphic to the complex

2 ϕ ∗ 1 I/I →j ΩW/S (6.9.2) for some map ϕ. 42 Martin C. Olsson

We claim that ϕ is equal to the negative of the map deﬁned by the diﬀerential. To see this, note that by functoriality of the distinguished triangle there is a commutative diagram

ϕ ∗ 1 L ◦ ◦ −−−−→ j Ω [1] X/W W/S x x     (6.9.3) ψ ∗ L ◦ ◦ −−−−→ j L ◦ ◦[1]. X/W W/S

2 0 ∗ Hence it suffices to show that the map ψ : I/I → H (j L ◦ ◦) ' W/S ∗ 1 j Ω ◦ ◦ obtained from the distinguished triangle associated to the com- W/S ◦ ◦ ◦ posite X → W → S is equal to the map defined by the negative of the differential. This follows from ([6], III.1.2.9). ut Theorem 6.10. Let f : X → S be a morphism as in (6.1), and assume that f is a log complete intersection. Then there is a canonical isomorphism in DF ∧(A) ∧ ·∧ RΓ (OX/S) ' LΩX/S.

Proof. Set W = Spec(A ⊗Z[Q] Z[P ]), and let j : X,→ W be the natural strict closed immersion over S deﬁned by the charts. Since f is a log ◦ ◦ complete intersection the morphism X,→ W is a regular immersion ([10], 4.4.4.2). Let S denote SP/Q,S, and let Spec(A0) → S be a smooth cover with ◦ 0–coskeleton Spec(A•). Set Spec(B•) = X ×S Spec(A•) and Spec(C•) = ◦ W ×S Spec(A•). Let j· : Spec(B•) ,→ Spec(C•) denote the resulting closed immersion over Spec(A•). Deﬁne

CC•/A• : ∆×∆ → (A–algebras), ([r], [s]) 7→ Cr⊗Ar · · ·⊗Ar Cr (s + 1 factors), and let K ⊂ CC•/A• denote the kernel of the natural map CC•/A• → B• (where B• is viewed as constant in the second factor of ∆ × ∆). Let C∧K ∈ DF ∧(A ) denote the PD–completion in the sense of ([7], C•/A• • VIII.1.2.5). Also deﬁne CW/S : ∆ → (A–algebras) to be the functor sending [s] ∈ ∆ to the coordinate ring of W ×S · · ·×S W ∧K (s + 1 factors). Let CW/S denote the PD–completion along the kernel of the natural map CW/S → B. The logarithmic cotangent complex 43

Lemma 6.11. The natural map C∧K → C∧K (where C∧K is viewed W/S C•/A• W/S as constant in the ﬁrst factor of ∆ × ∆) induces a quasi–isomorphism on total complexes. Proof. For any ﬁxed s, the map

([r] 7→ Spec(CC•/A• ([r], [s]))) → Spec(CW/S([s])) is a smooth hypercover, and hence CW/S[s] → CC•/A• ([−], [s]) is an equivalence. We therefore have a cocartesian diagram

CW/S[s] −−−−→CC•/A• ([−], [s])     y y

B −−−−→ B•, where the horizontal arrows are equivalences. From this and the fact that the formation of divided power envelopes commutes with ﬂat base change ([2], I.2.7.1) the result follows. ut

∧ Lemma 6.12. There is a natural isomorphism between RΓ (OX/S) and ∧K the total complex of CW/S. Proof. This follows from the same argument as in the classical case ([2], ∼ V.1.2.5). The point is that if W• denotes the simplicial sheaf which to any [s] associates the sheaf associated to the pro–object of Cris(X/S) defined by the PD–completion of X,→ Spec(CW/S[s]) (with log structure ∼ ∼ defined by the map to S), then W• is the 0–coskeleton of the map W0 → e (where e denotes the inital object of the topos), and this map is a covering. ∧K ∼ Furthermore, if π :(X/S)cris|Ws → Ws,et denotes the morphism of topoi ∧K for which π∗ restricts a sheaf to Ws,et , then since π∗ is exact we have ∧ ∧K ∼ RΓ ((X/S)cris|Ws , OX/S) 'CW/S[s]. From this and cohomological descent ([1], V.2.3.4) the result follows. ut To prove (6.10), it therefore suffices to construct an isomorphism C∧K ' LΩ·∧ . C•/A• B•/A• Let o P•• : ∆ × ∆ → (A–algebras) be the functor for which Ps• is the canonical free resolution of Bs over As. Then there is a natural surjection P•• → B•. Let K denote the kernel and let C∧K and C∧K be the PD–completions as in ([7], VIII.2.2.3). P••/A• P•0/A• By ([7], VIII.2.2.5) there is a canonical isomorphism C∧K ' C∧K in P•0/A• C•/A• ∧ DF (A•). 44 Martin C. Olsson

There is a natural map C∧K → C∧K P•0/A• P••/A• which induces a quasi–isomorphism on the associated total complexes. This follows from ([6], proof of VIII.2.2.6) which shows that for any ﬁxed s the map C∧K → C∧K Ps0/As Ps•/As is a quasi–isomorphism.

Let J denote the kernel of the projection CP••/A• → P••, and let C∧J denote the PD–completion. There is a natural map P••/A• C∧J → C∧K P••/A• P••/A• which again by ([7], proof of VIII.2.2.6) is a quasi–isomorphism (since this can be checked for any fixed s as above). It follows that the prove the theorem, it suffices to construct an equivalence C∧J ' Ω· . P••/A• P••/A• Such an equivalence is provided by ([7], VIII.2.1.3.4). ut Example 6.13. Let k be a field (of any characteristic), let r be an integer, r and let X = Spec(k[t]/t ) with log structure induced by the map N → k[t]/tr sending 1 to t. Also let Spec(k) denote the spectrum of k with the trivial log structure. Let W = Spec(N → k[N]), and let j : X,→ W be the strict closed immersion defined by the chart. Then by (6.9) we have

∗ r 0 r LX/Spec(k) ' (j (t ) −−−−→ (k[t]/t · d log t)), where j∗(tr) is placed in degree −1. Let E denote j∗(tr) and let Ω denote k[t]/tr · d log t. Then by ([6], I.4.3.2.1) we have n n 0 n−1 LΛ LX/Spec(k) ' (Γ E→Γ (E) ⊗ Ω), where Γ nE denotes the n–th graded piece of the divided power algebra on E and Γ nE is placed in degree −n. By (6.4) we therefore have n · n 0 n−1 gr LΩX/Spec(k) ' (Γ E→Γ (E) ⊗ Ω) with Γ nE placed in degree 0. Combining this with (6.10) it follows that n ∧ n 0 n−1 gr RΓ (OX/Spec(k)) ' (Γ E→Γ (E) ⊗ Ω). Of course this can also be veriﬁed by direct calculation using the divided power envelope of j. The logarithmic cotangent complex 45

7. The non-existence of a distinguished triangle in general

7.1. In this section we present an argument of W. Bauer, communicated to us by L. Illusie, which shows that there does not exist a theory of cotangent complexes for log schemes, (X → Y ) 7→ LX/Y , satisfying the following conditions: (7.1.1). If X → Y is strict, then LX/Y = L ◦ ◦. X/Y 1 (7.1.2). If X → Y is log smooth, then LX/Y ' ΩX/Y . (7.1.3). For every commutative diagram as in (1.1.1), there is a natural ∗ 0 map La LX/Y → LX0/Y 0 , which in the case when f and f are strict is ∗ equal to the map La L ◦ ◦ → L ◦ ◦ of ([6], II.1.2.3). X/Y X0/Y 0 (7.1.4). For every composite as in (1.1.4) there is a distinguished triangle ∗ ∗ Lf LY/Z −→ LX/Z −→ LX/Y −→ Lf LY/Z [1]. 7.2. Suppose such a theory existed, and let f : X → Y be a log ´etale morphism. Consider any strict morphism b : Y 0 → Y , and let X0 −−−−→a X   0  f y yf Y 0 −−−−→b Y be the resulting cartesian square. Note that since b is strict the scheme ◦ ◦ ◦ 0 0 X is equal to X × ◦ Y . By (7.1.2), LX/Y and LX0/Y 0 are both 0. Hence Y the distinguished triangles ∗ ∗ La LX/Y −→ LX0/Y −→ LX0/X −→ La LX/Y [1] 0∗ 0∗ Lf LY 0/Y −→ LX0/Y −→ LX0/Y 0 −→ Lf LY 0/Y [1] obtained from (7.1.4) show that the natural maps 0∗ Lf LY 0/Y −→ LX0/Y ,LX0/Y −→ LX0/X are isomorphism. Hence from (7.1.1) we conclude that the map 0∗ Lf L ◦ ◦ −→ L ◦ ◦ (7.2.1) Y 0/Y X0/X

◦ ◦ is an isomorphism for every map of schemes Y 0 → Y . However, the underlying morphisms of schemes of log ´etalemorphisms do not have good enough properties for the map (7.2.1) to always be an isomorphism as ◦ ◦ the following example shows (in particular the map X → Y need not be ﬂat): 46 Martin C. Olsson

2 Example 7.3. Let β : P ⊂ N denote the submonoid generated by (2, 0), (0, 2), (1, 1), and let k be a field of characteristic 0. Define X := 2 2 Spec(N → k[N ]), Y := Spec(P → k[P ]), and let f : X → Y be the mor- ◦ phism defined by β. By ([9], 3.4), f is log étale. Let Y 0 := Spec(k), and ◦ ◦ let i : Y 0 ,→ Y be the map obtained from the map k[P ] → k obtained by ◦ sending all non-zero elements of P to 0. Then the fiber product X0 equals 2 2 2 Spec(k[u, v]/(u , uv, v )) with the map to Spec(k[N ]) given by sending (1, 0) to u and (0, 1) to v. We claim that in this example the map (7.2.1) is not an isomorphism. ◦ To compute L ◦ ◦, let j : Y,→ Z := Spec(k[a, b, c]) be the closed Y 0/Y immersion defined by a 7→ (2, 0), b 7→ (0, 2), and c 7→ (1, 1) so that j is the closed immersion defined by (ab − c2). Then j and the composite j0 : ◦ ◦ 0 Y ,→ Y,→ Z are regular immersions, and hence if IY (resp. IY 0 ) denotes ◦ ◦ the ideal of Y (resp. Y 0) in Z, we have by ([6], III.3.2.7) isomorphisms

2 ∗ 1 2 0∗ 1 L ◦ ' (IY /IY −→ j ΩZ/k),L ◦ ' (IY 0 /IY 0 −→ j ΩZ/k), Y /k Y 0/k

2 2 where IY /IY (resp. IY 0 /IY 0 ) is in degree −1. From the transitivity triangle ∗ ∗ Li L ◦ −→ L ◦ −→ L ◦ ◦ −→ Li L ◦ [1] Y /k Y 0/k Y 0/Y Y /k we conclude that

2 2 3 L ◦ ◦ ' (IY /IY ⊗ k −→ IY 0 /IY 0 ) ' (k −→ k ), Y 0/Y where k3 is in degree −1, and the map is the zero map. To compute L ◦ ◦, consider the distinguished triangle X0/X

k ⊗ L ◦ ◦ −→ L ◦ −→ L ◦ −→ k ⊗ L ◦ ◦[1] X0/X Spec(k)/X Spec(k)/X0 X0/X

◦ ◦ 0 0 obtained from the composite Spec(k) = X red ,→ X ,→ X. Since

◦ Spec(k) ,→ X is a regular immersion, L ◦ is isomorphic to a 2-dimensional k- Spec(k)/X vector space V placed in degree −1. Hence

−i −i−1 H (k ⊗ L ◦ ◦) ' H (L ◦ ) X0/X Spec(k)/X0 The logarithmic cotangent complex 47 for i ≥ 2. Now if (7.2.1) is an isomorphism in this case, then this gives −3 dimkH (L ◦ ) = 1. On the other hand, by ([17], 7.6) the dual Spec(k)/X0 −i of ⊕iH (L ◦ ) is isomorphic to the free graded skew Lie algebra Spec(k)/X0 generated by the dual of V in degree 1. We leave it to the reader to verify that the degree 3 part of this graded Lie algebra has dimension 2, and hence the map (7.2.1) must not be an isomorphism in this case.

Remark 7.4. Using a variant of the above argument one can show that the map (7.2.1) must be an isomorphism whenever f : X → Y is log smooth and Y 0 → Y is strict.

8. Gabber’s approach

Let R be a topos with enough points (the assumption of having enough points is probably not needed, but we include it for simplicity).

Deﬁnition 8.1. A prelog ringed structure in R is a triple (A, M, α), where A is a commutative ring with unit in R, M is a commutative integral monoid with unit in R, and α : M → A is a morphism of monoids. A prelog ringed structure (A, M, α) is a log ringed structure if the natural map α−1(A∗) → A∗ is an isomorphism. We often denote a prelog ringed structure (A, M, α) simply by M → A.

8.2. Fix a prelog ringed structure N → A, and let C denote the category of morphisms (N → A) → (M → B) of prelog ringed structures in R. Also deﬁne D to be the category R × R of pairs of objects of R. There is a forgetful functor

U : C → D, (M → B) 7→ (M,B), which has a left adjoint T : D → C deﬁned by

X a T (X,Y ) := (α : N ⊕ N → A{X Y }), where for a set S we write A{S} for the free algebra on S ([6], I.1.5.5.4) (in loc. cit. the notation A[S] is used but we adopt the above notation so as not to cause confusion with monoid algebras). The map α is deﬁned to be the given map N → A on N, and the map sending the generator ` 1x for x ∈ X to the generator of A{X Y } corresponding to x. 8.3. For any object (M → B) ∈ C we deﬁne its canonical free resolution as in ([6], I.1.5) using the functors (T,U). We denote the resulting simplicial object in C by P(N→A)(M → B). The underlying ring of 48 Martin C. Olsson

◦ P(N→A)(M → B) is denoted P (N→A)(M → B). There is also a canonical projection P(N→A)(M → B) → (M → B). For any commutative diagram of prelog ringed structures (M → B) −−−−→ (M 0 → B0) x x     (N → A) −−−−→ (N 0 → A0) there is a natural induced morphism 0 0 P(N→A)(M → B) → P(N 0→A0)(M → B ). 8.4. For a morphism of prelog ringed structures (N → A) → (M → B) 1 we deﬁne Ω(M→B)/(N→A) to be the quotient of the module 1 gp gp ΩB/A ⊕ B ⊗Z (Coker(N → M )) by the submodule generated by the image of the map 1 gp gp M → ΩB/A⊕B⊗Z(Coker(N → M )), m 7→ (dα(m), 0)−(0, α(m)⊗m). Observe that if (M → B) = T (X,Y ) for a pair (X,Y ) ∈ D, then there is a canonical isomorphism 1 (Y ) (X) Ω(M→B)/(N→A) ' B ⊕ B , (8.4.1) where for a set S we write B(S) for the free B–module on S. Deﬁnition 8.5. For a morphism of prelog ringed structures (N → A) → G (M → B), the Gabber cotangent complex, denoted L(M→B)/(N→A), is the simplicial B–module G 1 L := Ω ⊗ ◦ B. (M→B)/(N→A) P(N→A)(M→B)/(N→A) P (N→A)(M→B)

Lemma 8.6. The augmentation P(N→A)(M → B) → (M → B) induces an isomorphism G 1 H0(L(M→B)/(N→A)) ' Ω(M→B)/(N→A). Proof. The natural map

Coeq((P(N→A)(M → B))1 ⇒ (P(N→A)(M → B))0) → (M → B) is an isomorphism, since this can be veriﬁed after applying the forgetful functor U to the category D in which case it follows from ([6], I.1.5.3). Thus what is needed is that Coeq(Ω1 Ω1 ) (P(N→A)(M→B))1/(N→A) ⇒ (P(N→A)(M→B))0/(N→A) The logarithmic cotangent complex 49

' Ω1 Coeq((P(N→A)(M→B))1⇒(P(N→A)(M→B))0)/(N→A) 1 which is immediate from the deﬁnition of Ω−/−, and the corresponding property of Kahler diﬀerentials ([6], II.1.4.2). ut Lemma 8.7. For (X,Y ) ∈ D, the map

G 1 a (X) a (Y ) LT (X,Y )/(N→A) → ΩT (X,Y )/(N→A) ' A{X Y } ⊕ A{X Y } is a quasi–isomorphism.

Proof. By ([6], I.1.5.3 (i)) the simplicial object P(N→A)(T (X,Y )) → T (X,Y ) is homotopically trivial, and hence the map Ω1 → P(N→A)(T (X,Y ))/(N→A) 1 ΩT (X,Y )/(N→A) is also an equivalence. The result therefore follows from ([6], I.3.3.4.6). ut Deﬁnition 8.8. (i) ([6], I.2.2.1) A morphism f : X → Y of simplicial objects in R is an equivalence if for every point t of the topos R the induced morphism ft : Xt → Yt of simplicial sets is an equivalence (i.e. induces a weak equivalence on geometric realizations). (ii) A morphism (N· → A·) → (M· → B·) of simplicial prelog ringed structures in R is an equivalence if the induced morphisms of simplicial sheaves of sets N· → M· and A· → B· are equivalences in the sense of (i). Proposition 8.9. Fix a commutative diagram of simplicial prelog ringed structures 0 0 (M· → B·) −−−−→ (M· → B·) x x     (8.9.1) 0 0 (N· → A·) −−−−→ (N· → A·), with the horizontal arrows equivalences. Then the following canonical maps of bisimplicial objects are line–by–line quasi–isomorphisms (in the sense of ([6], I.1.2.2)): 0 0 0 0 (i). P(N·→A·)(M· → B·) → P(N· →A·)(M· → B·), 1 1 (ii). Ω → Ω 0 0 0 0 , P (M·→B·)/(N·→A·) P 0 0 (M· →B· )/(N· →A·) (N·→A·) (N· →A·) G G (iii). L → L 0 0 0 0 . (M·→B·)/(N·→A·) (M· →B· )/(N· →A·) Proof. For (i) we show by induction on n ≥ −1 that the map 0 0 in : P(N→A)(M → B)n → P(N 0→A0)(M → B )n is a quasi–isomorphism. The case n = −1 is by assumption, so assume that n ≥ 0 and that the result holds for n − 1. Write MPn → Pn (resp. 0 0 0 0 0 0 MPn → Pn) for P(N→A)(M → B)n (resp. P(N →A )(M → B )n). Then 0 0 P = A{P } ⊗ A{M },P = A{P } ⊗ A{M 0 }, n n−1 A Pn−1 n n−1 A Pn−1 50 Martin C. Olsson and (M 0 ) (MP ) P n−1 0 n−1 MPn = N ⊕ N ,MPn = N ⊕ N . As in the proof of ([6], II.1.2.6.2) the Whitehead theorem (loc. cit., I.2.2.3) 0 and Kunneth (loc. cit., I.3.3.5.2) implies that the Pn → Pn is an equiva- 0 lence. That the map MPn → MPn is an equivalence follows from (A.3). For (ii), note that the arrow in degree n ≥ 0 can by (8.4.1) be identiﬁed with the map

0 (MP ) 0(P ) (M 0 ) (Pn−1) n−1 n−1 0 Pn−1 Pn ⊕ Pn → Pn ⊕ (Pn) which is a quasi–isomorphism by the Whitehead Theorem ([6], I.2.2.3) and Kunneth (loc. cit., I.3.3.5.2). Statement (iii) follows from a similar argument. ut

8.10. Let (N· → A·) → (M· → B·) be a morphism of simplicial prelog ringed structures. Then P(N·→A·)(M· → B·) is a bisimplicial prelog ringed structure, and we write P ∆ (M → B ) for its diagonal. By the (N·→A·) · · Eilenberg-Zilber-Cartier Theorem ([6], I.1.2.2) the normalized complex 1 of Ω ∆ is quasi–isomorphic to the total complex of P (M·→B·)/(N·→A·) (N·→A·) Ω1 . We deﬁne P(N·→A·)(M·→B·)/(N·→A·)

G,∆ 1 L := Ω ∆ ⊗ ◦ B·. (M·→B·)/(N·→A·) P(N →A )(M·→B·)/(N·→A·) ∆ · · P (N·→A·)(M·→B·) By ([6], I.1.2.2) the natural map

LG,∆ → LG (M·→B·)/(N·→A·) (M·→B·)/(N·→A·) is an equivalence.

The transitivity triangle

8.11. Let (N· → A·) → (M· → B·) → (L· → C·) (8.11.1) be a diagram of prelog ringed structures. Set P = P ∆ (M → B ) and (N·→A·) · · ∆ Q = PP (L· → C·), so that there is a commutative diagram of simplicial prelog ringed structures

(N· → A·) −−−−→ P −−−−→ Q       idy y y

(N· → A·) −−−−→ (M· → B·) −−−−→ (L· → C·). The logarithmic cotangent complex 51

◦ Lemma 8.12. The induced sequence of Q–modules ◦ 1 1 1 0 → Ω ⊗ ◦ Q → Ω → Ω → 0 (8.12.1) P/(N·→A·) P Q/(N·→A·) Q/P is exact. Proof. By the construction of the canonical free resolution, for any integer n there exists an object (Xn,Yn) ∈ D such that ◦ Xn a Qn = (MPn ⊕ N → P n{Xn Yn}).

Therefore Qn is non–canonically isomorphic to the colimit of the diagram

(Nn → An) −−−−→ Pn   y

Xn ` (Nn ⊕ N → A{Xn Yn}). ◦ 1 1 From this it follows that Ω is isomorphic to Ω ⊗ ◦ Q ⊕ Q/(N·→A·) P/(N·→A·) P 1 ΩQ/P and that the sequence (8.12.1) is split. ut 8.13. We thus have a commutative diagram ◦ G,∆ G,∆ G,∆ L ⊗ ◦ Q −−−−→ L −−−−→ L P/(N·→A·) P Q/(N·→A·) Q/P       y y y ◦ 1 1 1 0 −−−−→ Ω ⊗ ◦ Q −−−−→ Ω −−−−→ Ω −−−−→ 0, P/(N·→A·) P Q/(N·→A·) Q/P where the vertical arrows are quasi–isomorphisms (the left arrow is a ◦ ◦ quasi–isomorphism since Q is term–by–term free over P ). Since the modules are free, this diagram induces after tensoring with C· a commutative diagram G,∆ G,∆ G,∆ L ⊗ ◦ C· → L ⊗ ◦ C· → L ⊗ ◦ C· P/(N·→A·) P Q/(N·→A·) Q Q/P Q ↓ ↓ ↓ 1 1 1 0 → Ω ⊗ ◦ C· → Ω ⊗ ◦ C· → Ω ⊗ ◦ C· → 0, P/(N·→A·) P Q/(N·→A·) Q Q/P Q where the bottom line is exact, and the vertical arrows are equivalences. On the other hand, by functoriality there is also a commutative diagram G,∆ G,∆ G,∆ L ⊗ ◦ C· → L ⊗ ◦ C· → L ⊗ ◦ C· P/(N·→A·) P Q/(N·→A·) Q Q/P Q ↓ ↓ ↓ LG,∆ ⊗ C → LG,∆ → LG,∆ , (M·→B·)/(N·→A·) B· · (L·→C·)/(N·→A·) (L·→C·)/(M·→B·) 52 Martin C. Olsson where the vertical arrows are quasi–isomorphisms by (8.9). Combining all this we obtain the following: Theorem 8.14. For any diagram of prelog ringed structures (N → A) → (M → B) → (L → C) (8.14.1) there is a canonical distinguished triangle G G G +1 L(M→B)/(N→A) ⊗B C → L(L→C)/(N→A) → L(L→C)/(M→B)→ in the derived category D(C) of C–modules. This triangle is functorial in the triple (8.14.1). Remark 8.15. In the Theorem we abusively have identiﬁed the category of simplicial C–modules with a full subcategory of the category of complexes of C–modules using the normalization functor ([6], I.1.4.6), and therefore view the cotangent complexes as objects of the category of complexes of C–modules. We continue to make this abuse in what follows. Some calculations Theorem 8.16. Let (N → A) → (M → B) be a morphism of prelog ringed structures, and let (N a → A) → (M a → B) be the induced morphism of log ringed structures. Then the natural map G G L(M→B)/(N→A) → L(M a→B)/(N a→A) is a quasi–isomorphism. The proof is in several steps (8.17)–(8.22). Lemma 8.17. Let (α : N → A) be a prelog ringed structure in R, and let f : A → B be a morphism of rings. Let (N → B) be the prelog ringed structure obtained by composing α with f. Then the natural map G G LB/A ' L(0→B)/(0→A) → L(N→B)/(N→A) is a quasi–isomorphism, where LB/A denotes the cotangent complex de- ﬁned in ([6]).

Proof. Let PA(B) → B be the canonical free resolution of the A–algebra B. The map A → PA(B) induces an equivalence of simplicial prelog ringed structures (N → PA(B)) → (N → B). By (8.9) and ([6], II.1.2.6.2), we therefore have equivalences LG,∆ → LG ,L∆ → L . (N→PA(B))/(N→A) (N→B)/(N→A) PA(B)/A B/A This reduces the proof to the case when B ' A{Y } for some Y ∈ R. In this case the result follows from (8.7) and ([6], II.1.2.4.4). ut The logarithmic cotangent complex 53

Lemma 8.18. (i) Let M be a monoid in R, H ⊂ M a subgroup, A a ring in R, and β : H → A∗ a homomorphism. Set A := A ⊗ [M]. Then e β,Z[H] Z there is a natural isomorphism in D(Ae) LG ' (M gp/H) ⊗L A. (M→Ae)/(H→A) Z e (ii) Let N → M be a morphism of monoids in R with N gp → M gp injective, and let B be a ring. Then

LG ' (M gp/N gp) ⊗L B[M]. (M→B[M])/(N→B[N]) Z

Proof. For (i), let MonR,H denote the category of commutative integral monoids in R under H. The forgetful functor U : MonR,H → R has a left X adjoint given by X 7→ H ⊕ N . In particular, we can deﬁne the canonical free resolution M → M of M in Mon . Set A := A ⊗ [M ]. By · R,H e· Z[H] Z · ([6], I.2.2.3), the map Z[M·] → Z[M] is an equivalence, and since each Z[Mn] as well as Z[M] is ﬂat over Z[H] this implies that the map Ae· → Ae is an equivalence. We therefore have an equivalence (M· → Ae·) → (M → Ae), and hence by (8.9) a quasi–isomorphism

G,∆ G L e → L e . (M·→A·)/(H→A) (M→A)/(H→A) On the other hand, for every n ≥ 0 we have

(Mn → Aen) ' T (Mn−1, ∅), and therefore by (8.7) we have isomorphisms in the derived category

G,∆ G gp L e ' L e ' (M· /H) ⊗ A[M·]. (M·→A·)/(H→A) (M·→A·)/(H→A) Z gp gp By (A.5), the map M· → M is an equivalence, and hence the map gp gp gp M· /H → M /H is also an equivalence. It follows that M· /H is a free gp gp resolution of M /H, and hence the normalized complex of (M· /H) ⊗Z gp L A[M·] is isomorphic to (M /H) ⊗Z Ae·. Finally since Ae· → Ae is an equivalence, we obtain (i) from ([6], I.3.3.4.6). For (ii), note that there is a distinguished triangle

G G G +1 L(N→B[N])/(0→B) ⊗ B[M] → L(M→B[M])/(0→B) → L(M→B[M])/(N→B[N])→

G which by (i) shows that L(M→B[M])/(N→B[N]) is isomorphic to

Cone(N gp ⊗L B[M] → M gp ⊗L B[M]) ' (M gp/N gp) ⊗L B[M]. ut 54 Martin C. Olsson

Lemma 8.19. Let α : M → B be a prelog ringed structure and set F := −1 ∗ gp gp α (B ). Let MF ⊂ M be the submonoid generated by M and F . Then the map LG → LG (M→B)/(0→B) (MF →B)/(0→B) is a quasi–isomorphism.

Proof. The commutative diagram (0 → B) −−−−→ (M → B[M]) −−−−→ (M → B)       idy y y

(0 → B) −−−−→ (MF → B[MF ]) −−−−→ (MF → B) induces a morphism of distinguished triangles (where for the right hand terms we use (8.17))

G G +1 L(M→B[M])/(0→B) −−−−→ L(M→B)/(0→B) −−−−→ LB/B[M] −−−−→    a  c y by y LG −−−−→ LG −−−−→ L −−−−→+1 . (MF →B[MF ])/(0→B) (MF →B)/(0→B) B/B[MF ]

Since B[M] → B[MF ] is a localization morphism, the map labelled c is a quasi–isomorphism ([6], II.2.2.1). By (8.18) the map a is also a quasi– isomorphism from which it follows that c is a quasi–isomorphism. ut

Lemma 8.20. Let (M → B) be a prelog ringed structure and H ⊂ M a subsheaf of groups. Then the natural map

G G L(M→B)/(0→B) → L(M→B)/(H→B) is a quasi–isomorphism.

Proof. Consideration of the distinguished triangle associated to the composite (0 → B) → (H → B) → (M → B) G shows that it suﬃces to show that L(H→B)/(0→B) = 0. For this consider the natural map (H → B[H]) → (H → B) over (0 → B) which induces a morphism of distinguished triangles

LB[H]/B → LB/B → LB/B[H] → LB[H]/B[1] ↓ ↓ ↓ ↓ G G G L(H→B[H])/(0→B) → L(H→B)/(0→B) → LB/B[H] → L(H→B[H])/(0→B)[1]. The logarithmic cotangent complex 55

G Since LB/B = 0, to prove the vanishing of L(H→B)/(0→B) it therefore suﬃces to show that the map G LB[H]/B → L(H→B[H])/(0→B) (8.20.1) is an isomorphism. Let H· → H be the canonical free resolution of the sheaf of groups H. By (8.9) and ([6], II.1.2.6) applied to the equivalences

(H· → B[H·]) → (H → B[H]),B[H·] → B[H], it suffices to consider the case when H is the free group on a set X. In this case the assertion amounts to the statement that the map 1 1 ΩB[H]/B → Ω(H→B[H])/(0→B) is an isomorphism, which follows from the definition of the log differentials (8.4). ut Lemma 8.21. Let (α : M → B) be a prelog ringed structure, and (M a → B) the associated log ringed structure. (i). The natural map G G L(M→B)/(0→B) → L(M a→B)/(B∗→B) is a quasi–isomorphism. G (ii). L(M a→B)/(M→B) is quasi–isomorphic to the zero complex. −1 ∗ Proof. For (i), let F = α (B ), and let MF denote the submonoid of M gp generated by M and F gp. By (8.19) and (8.20) the maps G G G L → L → L gp (M→B)/(0→B) (MF →B)/(0→B) (MF →B)/(F →B) are all quasi–isomorphisms. Hence it suffices to show that the natural G G map L gp → L a ∗ is a quasi–isomorphism. This (MF →B)/(F →B) (M →B)/(B →B) follows from noting that there is a morphism of distinguished triangles (using (8.18))

gp gp L G (M /F ) ⊗ B −−−−→ L gp −−−−→ LB/B⊗ gp [M ] (MF →B)/(F →B) Z[F ]Z F       'y y y' a ∗ gp L G (M /B ) ⊗ B −−−−→ L a ∗ −−−−→ LB/B⊗ ∗ [M a], (M →B)/(B →B) Z[B ]Z where the right hand vertical arrow is a quasi–isomorphism since B⊗ ∗ Z[B ] a [M ] ' B/B ⊗ gp [M ]. Z Z[F ] Z F Statement (ii) follows from (i) by considering the distinguished triangle associated to the composite (0 → B) → (M → B) → (M a → B). ut 56 Martin C. Olsson

8.22. We can now complete the proof of (8.16). Let N 0 → B be the log structure associated to the composite N → B, and consider the morphism of distinguished triangles

G G G +1 LB/A −−−−→ L(M→B)/(N→A) −−−−→ L(M→B)/(N→B) −−−−→    a  c y by y G G G +1 L(N 0→B)/(N a→A) −−−−→ L(M a→B)/(N a→A) −−−−→ L(M a→B)/(N 0→B) −−−−→ . From (8.21 (ii)) and the distinguished triangle associated to (N a → A) → (N a → B) → (N 0 → B),

G G it follows that L(N 0→B)/(N a→A) ' L(N a→B)/(N a→A) which by (8.17) is quasi–isomorphic to LB/A. It follows that the arrow a is an isomorphism in the derived category. To prove that b is an isomorphism, it therefore suﬃces to show that c is an isomorphism. This reduces the proof to the case when A = B. The case when A = B follows from (8.21) by considering the morphism of distinguished triangles

G G G +1 L(N→B)/(0→B) −−−−→ L(M→B)/(0→B) −−−−→ L(M→B)/(N→B) −−−−→       y y y G G G +1 L(N a→B)/(B∗→B) −−−−→ L(M a→B)/(B∗→B) −−−−→ L(M a→B)/(N a→B) −−−−→ . This completes the proof of (8.16). ut Finally we note the following calculation which will be used below. Lemma 8.23. Let (M → B) be a prelog ringed structure in R, and N → M a morphism of monoids in R with N gp → M gp injective. Then the natural map B ⊗ LG → LG Z[M] (M→Z[M])/(N→Z[M]) (M→B)/(N→B) is a quasi–isomorphism.

Proof. Let M· → M be the canonical free resolution of M in the category of monoids under N. Then

(M· → B) → (M → B) is an equivalence so we have LG ' LG,∆ , (M→B)/(N→B) (M·→B)/(N→B) The logarithmic cotangent complex 57

LG ' LG,∆ . (M→Z[M])/(N→Z[M]) (M·→Z[M])/(N→Z[M]) Deﬁne B := B ⊗ [M ] and [M ]∼ := [M] ⊗ [M ]. There is e· Z[N] Z · Z · Z Z[N] Z · then a natural morphism of distinguished triangles G,∆ G,∆ G,∆ B ⊗ L(M → [M ]∼)/(N→ [M]) → B ⊗ L(M → [M])/(N→ [M]) → B ⊗ L [M]/ [M ]∼ · Z · Z · Z Z Z  Z · a  c y yb y G,∆ G,∆ G,∆ B ⊗ L e → L → L e. (M·→B·)/(N→B) (M·→B)/(N→B) B/B There are natural isomorphisms

L e ' L e [1],L ∼ ' L ∼ [1], B/B· B·/B Z[M]/Z[M·] Z[M·] /Z[M] ∼ and since Z[M·] /Z[M] is flat it follows from ([6], II.2.2.1) that c is a quasi–isomorphism. The map a is also a quasi–isomorphism since by (8.7) and the construction of the canonical free resolution we have G gp gp L e ' (M /N ) ⊗ Ben, (Mn→Bn)/(N→B) n and G gp gp ∼ L ∼ ' (M /N ) ⊗ [Mn] . (Mn→Z[Mn] )/(N→Z[Mn]) n Z It follows that the map b is also a quasi–isomorphism, as desired. ut Application to log schemes 8.24. We apply the preceding discussion to log schemes as follows. G For a morphism of log schemes f : X → Y , we define LX/Y to be G the complex obtained by applying the functor L−/− to the morphism of prelog ringed structures −1 −1 (f MY → f OY ) → (MX → OX ) ◦ in the étaletopos of X. If g : Z → X is another morphism of log schemes, then by the con- G struction of the complex LX/Y there is a natural quasi–isomorphism −1 G G g L → L −1 −1 −1 −1 . X/Y (g MX →g OX )/((f◦g) MY →(f◦g) OY ) From the distinguished triangle (8.14) applied to −1 −1 −1 −1 ((f ◦ g) MY → (f ◦ g) OY ) → (g MX → g OX ) → (MZ → OZ ) we therefore obtain a distinguished triangle ∗ G G G ∗ G g LX/Y → LZ/Y → LZ/X → g LX/Y [1]. 58 Martin C. Olsson

Proposition 8.25. Let f : X → Y be a morphism of ﬁne log schemes. G Then the homology sheaves of LX/Y are quasi–coherent OX –modules. If Y is locally noetherian and f locally of ﬁnite type then these homology sheaves are coherent.

Proof. The assertion is étalelocal on X and Y , and hence we may assume that there exist finitely generated monoids Q and P , charts β : Q → MY and γ : P → MX , and an injective morphism θ : Q → P such that the diagram of sheaves γ P −−−−→ MX x x   θ  β −1 Q −−−−→ f MY commutes. We may further assume that Y = Spec(A) is affine so that there is a commutative diagram

j X −−−−→ Spec(P → A[P ])     fy y Y −−−−→ Spec(Q → A[Q]), where the horizontal arrows are strict. From the resulting triangle

∗ G G G +1 f LY/Spec(Q→A[Q]) → LX/Spec(Q→A[Q]) → LX/Y → and ([6], II.2.3.7), it follows that it suﬃces to consider the case when Y = Spec(Q → A[Q]). Similarly, consideration of the distinguished triangle

∗ G G G +1 j LSpec(P →A[P ])/Spec(Q→A[Q]) → LX/Spec(Q→A[Q]) → LX/Spec(P →A[P ])→ further reduces the proof to the case when X = Spec(P → A[P ]). In this case the result follows from (8.18 (ii)). ut

G Comparison of LX/Y and LX/Y .

G 8.26. Let f : X → Y be a morphism of ﬁne log schemes, and let τ≥·LX/Y G denote the projective system {τ≥−nLX/Y } in D(OX ). G There is a natural map τ≥·LX/Y → LX/Y deﬁned as follows. Let u : U → LogY be a smooth surjection, and let U• be the 0–coskeleton of u. The maps to LogY give U• the structure of a simplicial log scheme The logarithmic cotangent complex 59

(U•,MU• ) over Y . Let V• denote X ×LogY U•, and let MV• denote the pullback of MX to V•. There is then a commutative diagram

j (V•,MV• ) −−−−→ (U•,MU• )   π  y y X −−−−→ Y which induces a morphism π∗LG → LG ' L , (8.26.1) X/Y (V•,MV• )/(U•,MU• ) V•/U• where the right hand isomorphism is because j is strict. On the other hand, by the construction of the cotangent complex LX/Y , the complex b b LV•/U• induces via the equivalence Dqcoh(OX ) → Dqcoh(OV• ) ([15], 6.14) the cotangent complex LX/Y . By (8.25) and (loc. cit.) we therefore obtain G a morphism τ≥·LX/Y → LX/Y . We leave to the reader the veriﬁcation that this morphism is independent of the choice of u : U → LogY .

G Theorem 8.27. For every n ≤ −3, the cone of τ≥nLX/Y → τ≥nLX/Y is cohomologically concentrated in degrees ≤ −3.

Proof. The assertion is ´etale local on X and Y . Therefore,we may assume that there exist charts βQ : Q → MY , βP : P → MX , and a morphism θ : Q → P giving a chart for the morphism f. Furthermore, we may assume that Qgp → P gp is injective with torsion free cokernel. Consider the morphism of triangles

G G +1 L [P ]/ [Q] → LSpec(P → [P ])/(Q→ [P ]) → LSpec(P → [P ])/Spec(Q→ [P ]) → Z Z Z Z Z  Z α ↓ ↓ y G G G +1 L ◦ ◦ → L → L ◦ →. X/Y ∗ X/Y X/(X,f MY ) By (8.23) the map α induces a quasi–isomorphism

G G OX ⊗ [P ] L → L ◦ . Z Spec(P →Z[P ])/Spec(Q→Z[P ]) ∗ X/(X,f MY )

G Combining this with (8.18 (ii)), it follows that LX/Y is isomorphic to the homotopy pushout of the diagram O ⊗ L −−−−→O ⊗ (P gp/Qgp) X Z[P ] Z[P ]/Z[Q] X Z   y

L ◦ ◦. X/Y 60 Martin C. Olsson

Deﬁne A0 := O ⊗ [P ], so that there is a commutative diagram Y Z[Q] Z X −−−−→ Spec(P → A0)     y y Y ←−−−− Spec(Q → A0), where the horizontal arrows are strict. Then the composite X → Spec(P → A0) → Y satisﬁes condition (T ), and hence there is a natural morphism of distinguished triangles

∗ +1 g L ◦ −−−−→ L ◦ ◦ −−−−→ L ◦ −−−−→ Spec(A0)/Y X/Y X/Spec(A0)       y y y ∗ 1 +1 g Ω 0 −−−−→ LX/Y −−−−→ L ◦ −−−−→ . Spec(P →A )/Y X/Spec(A0)

It follows that LX/Y is isomorphic to the homotopy pushout of the diagram ∗ gp gp g L ◦ −−−−→ (P /Q ) ⊗ OX Spec(A0)/Y Z   y

L ◦ ◦. X/Y G It follows that Cone(LX/Y → LX/Y ) is isomorphic to

Cone(OX ⊗ [P ] L [P ]/ [Q] → OX ⊗A0 L ◦)[1]. Z Z Z Spec(A0)/Y Theorem (8.27) therefore follows from (4.1). ut Corollary 8.28. Theorems (5.2), (5.6), and (5.9) hold with the cotangent complex replaced by the Gabber cotangent complex. Corollary 8.29. If Y is log flat over Spec(Z) with the trivial log struc- G ture, or if the morphism f is integral, then the map τ≥·LX/Y → LX/Y is an isomorphism. Proof. If Y is log flat over Z we can choose the local charts Q and P in ◦ the above proof so that Y → Spec(Z[Q]) is flat, and if f is integral then we can choose the morphism θ : Q → P so that Z[Q] → Z[P ] is flat ([9], 4.1). In either case it follows by ([6], II.2.2.1) that the natural map

OX ⊗ [P ] L [P ]/ [Q] → OX ⊗A0 L ◦ Z Z Z Spec(A0)/Y is a quasi–isomorphism. ut The logarithmic cotangent complex 61

8.30. Corollary (8.28) can be used to generalize (5.6) as follows. Consider a commutative diagram of solid arrows of ﬁne log schemes

i - X0 ...... X . . . . f0 . f . ? ? (8.30.1) j - Y0 Y

? © S, where j is an exact closed immersion deﬁned by a square–zero ideal J ⊂ ∗ OY . Let I be a quasi–coherent sheaf on OX0 , and let u : f0 J → I be a morphism.

Theorem 8.31. (i). There is a canonical obstruction

o ∈ Ext2(LG ,I) X0/Y0 whose vanishing is necessary and suﬃcient for the existence of a I– extension i : X0 ,→ X over Y as in (8.30.1), such that the induced map ∗ f0 J → I is equal to u. (ii). If o = 0, then the set of isomorphism classes of such I–extensions form a torsor under the group Ext1(LG ,I). X0/Y0 (iii). The group of automorphisms of such a I–extension over Y is canonically a torsor under Ext0(LG ,I). X0/Y0

◦ Proof. Consider the morphisms of prelog ringed structure on X0,et

−1 −1 −1 −1 (f0 MY → f0 OY ) → (f0 MY0 → f0 OY0 ) → (MX0 → OX0 ) which gives rise to the distinguished triangle (note that Y0 ,→ Y is strict)

∗ G G ∗ f L ◦ ◦ → L → L → f L ◦ ◦[1]. (8.31.1) 0 X0/Y X0/Y0 0 Y 0/Y Y 0/Y

0 ∗ −1 ∗ By ([6], III.1.2.8.1) we have H (f0 L ◦ ◦) = 0 and H (f0 L ◦ ◦) ' J. Y 0/Y Y 0/Y We therefore obtain a map

1 G 1 ∗ ∗ ExalY (X,I) ' Ext (LX/Y ,I) → Ext (f LY0/Y ,I) ' Hom(f0 J, I). (8.31.2) 62 Martin C. Olsson

This map sends an I–extension j : X0 ,→ X over Y to the induced map ∗ f0 J → I. This follows from the corresponding statement for schemes without log structure by noting that there is a commutative diagram

' 1 G ∂ 1 ∗ ExalY (X,I) −−−−→ Ext (LX/Y ,I) −−−−→ Ext (f L ◦ ◦,I) Y 0/Y    a   y by yid ◦ 0 ' 1 ∂ 1 ∗ Exal ◦(X,I) −−−−→ Ext (L ◦ ◦,I) −−−−→ Ext (f L ◦ ◦,I), Y X/Y Y 0/Y

◦ 0 where ∂ is obtained from the distinguished triangle associated to X0 → ◦ ◦ Y 0 → Y , the map a is obtained by forgetting the log structures, and the G map b is induced by the functoriality morphism L ◦ ◦ → LX/Y . X0/Y 0 ∗ Since H (f0 LY0/Y ) = 0 we obtain from the long exact sequence of Ext–groups associated to (8.31.1), a short exact sequence

1 1 ∗ 2 0 → Ext (LX0/Y0 ,I) → Ext (LX0/Y ,I) → Hom(f0 J, I) → Ext (LX0/Y0 ,I), (8.31.3) and isomorphisms

Hom(Ω1 ,I) ' Ext0(L ,I) ' Ext0(L ,I) ' Hom(Ω1 ,I). X0/Y X0/Y X0/Y0 X0/Y0 (8.31.4) 2 ∗ We deﬁne o ∈ Ext (LX0/Y0 ,I) to be the image of u ∈ Hom(f0 J, I) under the boundary map. From the above description of (8.31.2) it follows that o has the desired properties. Statement (ii) follows from consideration of the exact sequence (8.31.3), and (iii) follows from the isomorphism (8.31.4). ut

Appendix A. Simplicial Monoids

A.1. Let ∆ denote the category of ﬁnite ordered sets with order preserving maps. The category ∆ is equivalent to the category with objects o [n] = {0, . . . , n} (n ∈ N) and order preserving maps. We write ∆ for the opposite category of ∆. Let Mon denote the category of commutative integral monoids with o unit, and let Mon∆ denote the category of simplicial monoids. o Let SSet denote the category of simplicial sets, and let U : Mon∆ → SSet be the forgetful functor. The functor U has a left adjoint T : SSet → ∆o Mon which sends a simplicial set X· to the simplicial monoid [n] 7→ X Y N n , where for a set Y we write N the free monoid on the set Y . The logarithmic cotangent complex 63

Deﬁnition A.2. A morphism f : N· → M· of simplicial monoids is an equivalence if the morphism U(f): U(N·) → U(M·) is an equivalence in SSet ([6], I.1.1.5).

Proposition A.3. Let g : X· → Y· be an equivalence in SSet. Then the ∆o induced morphism T (g): T (X·) → T (Y·) in Mon is an equivalence. Proof. The category Mon has finite limits. A morphism f : M → N in Mon is an effective epimorphism in the sense of ([18], §4) if and only if f surjective. This follows from the definition of an effective epimorphism. The category Mon also has a set of small projective generators given by [n] the monoids N for [n] ∈ ∆, and is closed under inductive limits. By ([18], Chapter II, §4, Theorem 4), there exists a cofibrantly gen- o erated simplicial closed model category structure on Mon∆ such that f : M· → N· is an equivalence (resp. fibration) if and only if for every projective monoid P the map of simplicial sets

Hom(P,M·) → Hom(P,N·) is a weak equivalence (resp. ﬁbration). The functor U can also be viewed as

M· 7→ Hom(N,M·). In particular U takes equivalences (resp. fibrations) to weak equivalences (resp. fibrations). It follows that U is a right Quillen functor, and hence by ([5], 1.3.4) the functor T is a left Quillen functor. Since any simplicial set is cofibrant ([5], 3.2.2) it follows from Ken Brown’s Lemma ([5], 1.1.12) that the functor T takes weak equivalences to weak equivalences. ut

o A.4. Let Ab be the category of abelian groups, and Ab∆ the category of simplicial objects in Ab. The inclusion Ab ⊂ Mon induces an inclusion ∆o ∆o Ab ⊂ Mon . We deﬁne a morphism f : H· → G· to be an equivalence o if the induced morphism in Mon∆ is an equivalence. o o The inclusion Ab∆ ⊂ Mon∆ has a left adjoint which sends a simpli- gp gp cial monoid M· to [n] 7→ Mn , where Mn → Mn is the universal map from Mn to a group.

Theorem A.5. Let M ∈ Mon be a monoid, and let M· → M be an o equivalence in Mon∆ (where M is viewed as a constant simplicial object). gp gp ∆o Then the induced morphism M· → M is an equivalence in Ab .

gp gp Proof. We have to show that the natural map π0(|M· |) → M is an gp gp isomorphism, and that for every v ∈ M0 the groups πn(|M· |, v) are zero for n ≥ 1, where | − | denotes geometric realization. 64 Martin C. Olsson

gp For the statement about π0, note that π0(|M· |) is equal to the co- gp gp equalizer of the two maps M1 ⇒ M0 which by the universal property of the group associated to a monoid is equal to gp gp Coeq(M1 ⇒ M0) ' M .

For the statement about higher πn’s, we proceed in several steps. gp gp 0 Lemma A.6. If πn(|M· |, v) = 0 for every v ∈ M0, then πn(|M· |, v ) = 0 gp 0 for every v ∈ M0 . 0 Proof. Choose λ ∈ M0 such that v + λ is equal to some element v ∈ M0, and for every m let λm denote the image of λ under the map M0 → Mm gp gp induced by the unique morphism [m] → [0]. Let g : M· → M· be the map which in degrees m is addition by λm. Then g is an isomorphism and induces an isomorphism gp 0 gp πn(|M· |, v ) ' πn(|M· |, v). ut gp Fix v0 ∈ M0, and consider the relative homotopy groups πn(|M· |, |M·|, v0) defined for n ≥ 1 ([4], p. 343). Consider the associated long exact sequence (loc. cit., 4.3) gp gp · · · → πn(|M·|, v0) → πn(|M· |, v0) → πn(|M· |, |M·|, v0) → · · · . (A.6.1) gp This sequence combined with the injectivity of π0(|M·|) → π0(|M· |) gp gp implies that the map π1(|M· |, v0) → π1(|M· |, |M·|, v0) is surjective. Since the groups πn(|M·|, v0) are zero for n ≥ 1, this in turn shows that for any m ∈ M0 the map gp gp λm : πn(|M· |, |M·|, v0) → πn(|M· |, |M·|, v0 + m) induced by translation by m is an isomorphism (since translation induces gp an isomorphism on πn(|M· |, v0)). To prove the Theorem it therefore suffices to show that for any n ≥ 1 gp and class α ∈ πn(|M· |, |M·|, v0) there exists an element m ∈ M0 such gp that λm(α) = 0. For then πn(|M· |, |M·|, v0) = 0 for all n ≥ 1 which by gp the long exact sequence (A.6.1) implies that πn(|M· |, v0) = 0 for n ≥ 1. n n n−1 Let D ⊂ R denote the unit disk with boundary S and fix a point n−1 s0 ∈ S . Then a class α as above is represented by a map of triples n n−1 gp ϕ :(D ,S , s0) → (|M· |, |M·|, v0). gp Observe that for n = 1 this follows from the fact that π1(|M· |, v0) → gp π1(|M· |, |M·|, v0) is surjective. The logarithmic cotangent complex 65

By the simplicial approximation Theorem ([4], 2C.1) (though it is stated there only for maps of spaces it can be extended to the present situation as well), there exists a simplicial decomposition (A, B, s) of n n−1 (D ,S , s0) and a map of simplicial complexes gp f :(A, B, s) → (M· ,M·, v0) such that ϕ is homotopic to |f|. By ([16], 2.2) there exists a map g : A → M· such that (f + g)(A) ⊂ M·. Let m0 ∈ M0 be the image of g(s), and let

λm0 : A → M· be the constant map induced by m0. Since |M·| ' M, the maps g and λm0 are homotopic. Thus there exists a map H : |A|×I → |M·| such that H||A|×{0} = g, H||A|×{1} = λm0 , and H(s × I) = m0. The map

n n−1 gp |f| + |g| :(D ,S , s0) → (|M· |, |M·|, v0 + m0) gp is homotopic to |f+g|, which deﬁnes the trivial class in πn(|M· |, |M·|, v0+ m0). On the other hand, deﬁne F to be the composite

n H×f gp + gp D × I −−−−→|M·| × |M· | −−−−→|M· |.

Then F |s0×I = v0 + m0, F |Dn×{0} = |f| + |g|, F |Dn×{1} = |f| + |λm0 |. We gp thus ﬁnd that |f|+|λm0 | deﬁnes the trivial class in πn(|M· |, |M·|, v0+m0) which completes the proof of the Theorem. ut

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