Regularity in Logarithmic Algebraic Geometry: a different viewpoint
Abstract
Logarithmic algebraic geometry is the right context to deal with some problems in algebraic geometry. For instance, certain morphisms of schemes which are not smooth can be considered as morphisms of logarithmic schemes for some logarithmic structure making them log smooth. Thus, as it happens in algebraic geometry, log smoothness and log regularity are important in this field. The aim of this dissertation is to study regularity in logarithmic algebraic geometry from a different viewpoint. This allows us to obtain new results and also to remove unnecessary hypotheses from some known results. For that purpose, we will develop the necessary results on the logarithmic cotangent complex in order to characterize log regularity (and also log smoothness) in terms of this complex.
Full text
CENTRO INTERNACIONAL DE ESTUDOS DE DOUTORAMENTO E AVANZADOS DA USC (CIEDUS) TESE DE DOUTORAMENTO REGULARITY IN LOGARITHMIC ALGEBRAIC GEOMETRY: A DIFFERENT VIEWPOINT Jesús Conde Lago ESCOLA DE DOUTORAMENTO INTERNACIONAL PROGRAMA DE DOUTORAMENTO EN MATEMÁTICAS SANTIAGO DE COMPOSTELA ANO 2019
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DECLARACIÓN DO AUTOR DA TESE Regularity in Logarithmic Algebraic Geometry: a different viewpoint D. Jesús Conde Lago Presento a miña tese, seguindo o procedemento adecuado ao Regulamento, e declaro que: 1) A tese abarca os resultados da elaboración do meu traballo. 2) No seu caso, na tese faise referencia ás colaboracións que tivo este traballo. 3) A tese é a versión definitiva presentada para a súa defensa e coincide coa versión enviada en formato electrónico. 4) Confirmo que a tese non incorre en ningún tipo de plaxio doutros autores nin de traballos presentados por min para a obtención doutros títulos. En Santiago de Compostela, 2 de maio de 2019. Asdo. Jesús Conde Lago
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AUTORIZACIÓN DO DIRECTOR DA TESE Regularity in Logarithmic Algebraic Geometry: a different viewpoint D. Javier Majadas Soto INFORMA: Que a presente tese se corresponde co traballo realizado por D. Jesús Conde Lago, baixo a miña dirección, e autorizo a súa presentación, considerando que reúne os requisitos esixidos no Regulamento de Estudos de Doutoramento da USC, e que como director desta non incorro nas causas de abstención establecidas na lei 40/2015. En Santiago de Compostela, 2 de maio de 2019. Asdo. Javier Majadas Soto
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Contents Introduction iii 1 Monoids 1 2 Logarithmic rings 9 3 Derivations and differentials 13 3.1 The conormal module . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Module of differentials . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4 Logarithmic André-Quillen (co)homology 29 4.1 The logarithmic cotangent complex . . . . . . . . . . . . . . . . . 29 4.2 A spectral sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3 A complex to compute H2. . . . . . . . . . . . . . . . . . . . . . 50 5 Smoothness 61 5.1 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.2 Formal smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6 Regularity 79 6.1 Technical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Resumo en galego 95 Bibliography 103 Index 105 i
Introduction We study regularity in logarithmic algebraic geometry from a different viewpoint, which allows us to obtain new results and also to remove unnecessary hypotheses from some known results. Logarithmic algebraic geometry was introduced independently by P. Deligne [5], G. Faltings [7], and J. M. Fontaine and L. Illusie (developed by K. Kato in [13]). The definition of Fontaine-Illusie is the one usually used. A prelogarithmic scheme is a scheme Xwith an extra structure: a morphism of sheaves of commutative monoids α:M → (OX,·)for some sheaf of monoids M. If in addition αinduces an isomorphism α−1(O∗ X) = O∗ X, where O∗ Xis the sheaf of units, we say that it is a logarithmic scheme. Any prelogarithmic scheme induces a logarithmic scheme (replacing Mby the pushout of α−1(O∗ X),→ M and α−1(O∗ X)→ O∗ X). For instance, if i:U ,→Xis an open inclusion, M=ϕ−1(i∗O∗ U)where ϕ:OX→ i∗(OU)is the canonical map, then the inclusion M → (OX,·)gives a prelogarithmic structure on Xand thus there is an associated logarithmic structure. Probably the most important concept in this field is log smoothness. It is defined for a morphism of logarithmic schemes in a completely similar way as smoothness is defined for a morphism of schemes, and has good functorial properties as usual smoothness. Log smoothness morphisms work well in logarithmic algebraic geometry as smooth morphisms work well in algebraic geometry. The important fact is that certain morphisms of schemes which are not smooth can be considered as morphisms of logarithmic schemes for some logarithmic structure making them log smooth. For example, if Sis the spectrum of a discrete valuation ring, Xis a regular scheme and f:X→Sis a flat morphism with semi-stable reduction (i.e. the fiber over the closed point of Sis a divisor Dwith normal crossings in X), then the logarithmic structures associated as above to the open immersions i:U:= X−D ,→Xand j:{ξ} → S, with ξthe generic point of S, makes flog smooth. So, in some sense, logarithmic geometry is the right context to work with semi-stability, since log smoothness has nice functorial properties (for instance, semi-stability is not preserved under base change, unlike log smoothness). iii
2 1. Monoids The trivial monoid M={1}is clearly a zero object in Mon. Moreover, this category has kernels, since ker ϕwill be the kernel of ϕ. Definition 1.3. Acongruence Ron a monoid Mis an equivalence relation on M compatible with the multiplication of M, that is, (x, x0),(y, y0)∈R⇒(xy, x0y0)∈R. The quotient set M/R is a monoid with the multiplication induced by the one on M through the canonical map p:M→M/R, with KER p=R. If ϕ:M0→Mis a homomorphism of monoids, KER ϕis a congruence in M0. Therefore we have an injective homomorphism of monoids ϕ0:M0/KER ϕ→M that composed with the above canonical homomorphism M0→M/KER ϕgives us ϕ. Example 1.4. Let Mbe a monoid and Ha subgroup of M(i.e. a submonoid that is a group). The set R={(x, y)∈M×M| ∃h∈Hsuch that x=yh}is a congruence in M. If Mis also a (commutative) group, then the quotient monoid of Mby this congruence is just the usual (group) quotient of groups M/H. If Mis again a monoid but His only a monoid now, the set of invertible elements H∗of His a subgroup and we can define M/H0. When H=M, the unique invertible element of M/M∗is the class of the identity element of M. Example 1.5. An ideal Iof a monoid Mis a non empty set of Mthat verifies xa ∈I for all x∈M,a∈I. If Mhas zero element, it belongs to any ideal of M. While if 1∈I, then I=M. The intersection of ideals is an ideal as long as it would not be empty, and the union of ideals is always an ideal. If Yis a non empty subset of a monoid M, we call ideal generated by Yto the smallest ideal of Mthat contains Yand that is MY := {xy |x∈M, y ∈Y}. An ideal Iof Mis proper if I6=M(equivalently if 1/∈I). An ideal Iis prime if it is proper and furthermore xy ∈Iimplies that x∈Ior y∈Ifor every x, y ∈M. It means that an ideal Iis prime if and only if M−Iis a submonoid of M. We define the maximal ideal of a monoid Mas mM:= M−M∗. We will say that a homomorphism of monoids f:M→M0is local if f(mM)⊂mM0. Example 1.6. Let Abe a commutative ring, Ian ideal of A,Ma monoid and α:M→(A, ·)a homomorphism of monoids. The set α−1(I)is an ideal of M. Now, if Iis an proper ideal of A,α−1(I)is a proper ideal of M. And similarly, in the case Iis a prime ideal of A, then α−1(I)is also prime.
3 Example 1.7. The Rees congruence of the ideal Iis the congruence in M R= (I×I)∪∆(M) where ∆(M) = {(x, x)∈M×M}is the diagonal. That is, x∼yif x, y ∈Ior x=y. We will call Rees quotient of Mby Iand denote M/I to the quotient monoid. Note that M/I = (M−I)∪ {?} with product xy =(xy if x, y, xy ∈M−I(i.e. xy ∈M−I), ?if in any other case, where ?is the zero element of M/I. Definition 1.8 (Products).Let {Mi}i∈Ibe a family of monoids. The cartesian product of monoids M=Y i∈I Mi has monoid structure (component to component) and it is easy to check that it is a product in Mon. Note that if every Miis a group, then Mis a group; and therefore it is also the product in the category of groups. Definition 1.9 (Coproducts).Let {Mi}i∈Ibe a family of monoids. We define its coproduct as the monoid M i∈I Mi:= {(xi)∈Y i∈I Mi|xi= 1 for almost all i∈I} with the operation component to component. It is the coproduct in Mon and coincides with the product when Iis finite. Definition 1.10 (Equalizers).Let f, g:M1→M2be homomorphisms of monoids. Then M0={x∈M1|f(x) = g(x)}is a submonoid of M1and the inclusion M0,−→ M1is the equalizer of fand g. Definition 1.11 (Coequalizers).Let f, g:M1→M2be homomorphisms of monoids and we define M3=M2/R, where Ris the congruence generated by the set {(f(x), g(x)) |x∈M1}(i.e. the intersection of all congruences that contain {(f(x), g(x)) |x∈M1}). The canonical homomorphism M2→M3is the coequalizer of fand g.
4 1. Monoids Corollary 1.12. The category of monoids has limits and colimits. Moreover, since it has zero object (see Definition 1.2), it has also kernels and cokernels. Definition 1.13. If Xis a non empty set, we define free monoid over Xas the monoid NX:= M x∈X Nx where Nx:= (N,+) for every x∈X. The free monoid functor from Set to Mon is left adjoint to the forgetful functor. From this, for example, we deduce that the forgetful functor preserves limits. Therefore, in particular, if M0M1 M2M3 p q is a cartesian square (pullback) of monoids, we know that it is also a cartesian square of the underlying sets, and so M0={(x1, x2)∈M1×M2|p(x1) = q(x2)}. Now we will see with more detail the cocartesian squares (pushouts). Proposition 1.14. Let h1:P→Q1,h2:P→Q2be two homomorphisms of monoids, we denote their pushout by Q1⊕PQ2. Let Ebe the congruence on Q1⊕Q2 which define Q1⊕PQ2as a quotient of Q1⊕Q2. 1. If any of P,Q1or Q2is a group, then Eis the set of pairs ((q1, q2),(q0 1, q0 2)) of Q1⊕Q2such that there exist p, p0∈Pwith q1+h1(p0) = q0 1+h1(p)and q2+h2(p) = q0 2+h2(p0). 2. If Pis a group, then ((q1, q2),(q0 1, q0 2)) ∈Eif and only if there exists p∈P such that q0 1=q1+h1(p)and q0 2=q2−h2(p). 3. If P,Q1or Q2are groups, then Q1⊕PQ2is also a group, which is the pushout in the category of Abelian groups. Proof. See [18, Proposition I.1.1.5] and also [13, Remark in 1.3]. Definition 1.15. Let Mbe a monoid and Ta set. An action of Mon Tis a homomorphism of monoids θ:M−→ End(T)
5 where End(T)is the monoid not (necessarily) commutative of endomorphisms (of sets) of T(Definition 1.2). That is, θverifies that θ(xy) = θ(x)·θ(y)for any x, y ∈ Mand that θ(1) = idT. We will say that Tis an M-set and denote x·t:= θ(x)(t) for x∈M,t∈T. Ahomomorphism of M-sets is an application ϕthat makes commutative the diagram M×T1T1 M×T2T2 (idM, ϕ)ϕ Example 1.16. If f:M→Nis a homomorphism of monoids, then Nis an M-set with x·y:= f(x)·y for all x∈M,y∈N. Remark 1.17. The forgetful functor from the category of M-sets to Set has a left adjoint functor: if Bis a set, M×Bis an M-set via x(y, b) = (xy, b)(it is said that M×Bis the free M-set with basis B). Therefore the category of M-sets has limits, and they commute with the forgetful functor. Definition 1.18. Let Mbe a monoid and S⊂Ma submonoid. Consider the direct product M×Sand the congruence (x, s)∼(y, t) :⇔ ∃u∈S|syu =xtu. The quotient monoid is denoted by S−1Mand is called monoid of fractions. We will denote the class of (x, s)in S−1Mby x s, so that x s·y t=xy st and 1 = s s. We have a canonical homomorphism γ:M→S−1Mdefined as the composition M M ⊕S=M×S(M×S)/∼=S−1M, x(x, 1) that is, x7→ x/1. Proposition 1.19. Let Mbe a monoid, Sa submonoid and ϕ:M→Na homomorphism of monoids. Then there exists a homomorphism of monoids h:S−1M→N
6 1. Monoids making commutative the diagram M S−1M N ϕ γ h if and only if ϕ(s)is invertible in Nfor all s∈S. In this case, his unique and it is define as follows: h(x/s) := h(x)h(s)−1. Proposition 1.20. Let Mbe a monoid. Then M−1Mis an Abelian group that we denote by Mgp. Moreover, it verifies the following universal property: Given a group Gand a homomorphism of monoids ϕ:M→G, there exists a unique homomorphism of groups h:Mgp →Gmaking commutative the diagram M Mgp G ϕ γ h It is called group completion of M. Proof. It is clear that M−1Mis a group, since (x y)−1=y x, and the universal property follows from Proposition 1.19. Remark 1.21. From Proposition 1.20, we can deduce that (−)gp is a functor from Mon to Ab that is left adjoint of the forgetful adjoint. In particular, it preserves coproducts. Definition 1.22. Let Mbe a monoid. An element t∈Mis integral if for any x, y ∈ Msuch that tx =ty, then x=y. We say that a monoid Mis integral if every element of Mis integral. Definition 1.23. Let Mbe a monoid. If M∗={1},Mis said to be sharp. Proposition 1.24. Let Mbe a monoid and Sa submonoid. The canonical homomorphism γ:M→S−1Mis injective if and only if the elements of Sare integral in M.
7 Therefore, in the case Sis integral, we will have x/s =y/t ⇔xt =ys in S−1M. And moreover, if Mis integral, we can think Mas a subset of Mgp. Remark 1.25. Let Mbe a monoid with zero element 0and Sa submonoid with 0∈S. Then S−1M={1}If ϕ:M→Nis a homomorphism of monoids such that ϕ(s)is invertible for all s∈S, then ϕ(0) is invertible and so ϕ(x) = 1 for all x∈M(since ϕ(0) = ϕ(0 ·x) = ϕ(0)ϕ(x)). This means that ϕis the trivial morphism. For example, under this hypothesis Mgp is the (Abelian) trivial group. For example, (N,·)gp is the trivial group. However, if S=N− {0}, we obtain S−1(N, ·)=(Q≥0,·). Definition 1.26. If Mis a monoid and Ris a ring, R[M]will denote the usual monoid R-algebra. In particular, R[M]is a free R-module with basis M. Remark 1.27. The ring monoid functor M7→ Z[M]from the category of monoids to the category of commutative rings is left adjoint to the functor A7→ (A, ·)and so ir preserves colimits. In particular, if M0M1 M2M is a cocartesian square of monoids, then Z[M] = Z[M1]⊗Z[M0]Z[M2]. In other words, we can say that Z[M1⊕M0M2] = Z[M1]⊗Z[M0]Z[M2]. Proposition 1.28. Let Mbe a monoid. The ring Z[M]is Noetherian if and only if Mis finitely generated. Proof. See [8, Theorem 7.7]. Definition 1.29. Let Mbe a monoid. Mis fine if it is finitely generated and integral. Mis said saturated if it is integral and verifies that x∈Mgp and xn∈Mfor some n≥1implies x∈M.
8 1. Monoids Definition 1.30. A homomorphism of monoids h:M→Nis exact if the square M Mgp N Ngp h hgp is cartesian (i.e. a pullback). Definition 1.31. Let f:M→Pbe a homomorphism of integral monoids. fis said to be Kummer if it is injective and for every p∈P, there exists n∈Nand m∈M such that f(m) = pn. Definition 1.32. A homomorphism of monoids M→Nis integral if for any homomorphism of monoids M→Pwith Pintegral we have N⊕MPintegral [9, Definition 6.2.1].
Chapter 2 Logarithmic rings Definition 2.1. Aprelog ring (A, M, α)(simply (A, M)or Aif there is no confusion) consists of a commutative ring A, a commutative monoid Mand a multiplicative homomorphism of monoids α:M→A. In general, we will denote this structural map by αfor any prelog ring (if it is necessary, to avoid confusion, we will use αA or αM). A homomorphism of log rings f= (f], f[): (A, M, αA)−→ (B, N, αB) is a homomorphism of rings f]:A→Btogether with a homomorphism of monoids f[:M→Nsuch that αBf[=f]αA. A log ring is a prelog ring (A, M, α)such that the homomorphism α−1(A∗)→A∗induced by α(where A∗is the group of units of A) is an isomorphism. Definition 2.2. If (A, M, α)is a prelog ring, define Mlog as the pushout α−1(M)A∗ M Mlog α and let αlog :Mlog →Abe the monoid homomorphism induced by the homomorphisms α:M→Aand A∗,→A. Then (A, M, α)log := (A, Mlog, αlog)is a log ring. The functor (−)log from the category of prelog rings to the category of log rings is left adjoint to the forgetful functor. Definition 2.3. If (A, M, α)is a prelog ring and f]:A→Bis a ring homomorphism, we have a homomorphism of prelog rings f= (f], fb): (A, M, α)−→ (B, f∗(M), α∗) where (B, f∗(M), α∗) := (B, M, f ◦α)log. A homomorphism of log rings f: (A, M)−→ (B, N) is called strict if the canonical map f∗(M)→Nis a isomorphism.
10 2. Logarithmic rings Definition 2.4. A Noetherian local prelog ring ((A, m, k), M)is a Noetherian local ring (A, m, k), a monoid Msuch that the monoid ring Z[M]is Noetherian (i.e. M is finitely generated by Proposition 1.28) and a local homomorphism of monoids α:M→(A, ·). We say that (A, M)→(B, N)is a homomorphism (essentially) of finite type of Noetherian prelog rings if M,Nare finitely generated monoids, Ais a Noetherian ring, and A→Bis a homomorphism (essentially) of finite type. We have then that Z[M]→Z[N]is a homomorphism of finite type of Noetherian rings, and Mgp,Ngp are Z-modules of finite type. Lemma 2.5. Let (A, M)be a log ring, P→Ma homomorphism of monoids and π:A→Ba homomorphism of rings. If (A, P)log = (A, M), then (B, P)log = (B, M)log. Proof. Consider the pushouts of monoids α−1(A∗)A∗ PP⊕α−1(A∗)A∗=M A α β (πβ)−1(B∗)B∗ P⊕α−1(A∗)A∗P⊕α−1(A∗)A∗⊕(πβ)−1(B∗)B∗ B A β π
11 (πα)−1(B∗)B∗ PP⊕(πα)−1(B∗)B∗ B A α π and the maps P⊕α−1(A∗)A∗⊕(πβ)−1(B∗)B∗P⊕(πα)−1(B∗)B∗ ϕ ψ defined as follows for any p∈P,a∈A∗and b∈B∗: ψ(p, b) := ((p, 1), b), ϕ((p, a), b) := (p, π(a)b). Let us see that both homomorphisms are well defined. We will use Proposition 1.14. Assume that ((p, a), b) = ((p0, a0), b0)in P⊕α−1(A∗)A∗⊕(πβ)−1(B∗)B∗. Then by Proposition 1.14 there exists a pair (p1, a1),(p2, a2)∈(πβ)−1(B∗)⊂ P⊕α−1(A∗)A∗such that (i) (p, a)·(p2, a2) = (p0, a0)·(p1, a1), (ii) bπβ(p1, a1) = b0πβ(p2, a2). These conditions are equivalent to (i’) There exists s1, s2∈α−1(A∗)⊂Psuch that (i0 1)pp2s2=p0p1s1, (i0 2)aa2α(s1) = a0a1α(s2); (ii’) bπα(p1)π(a1) = b0πα(p2)π(a2). Let q1=p1s1and q2=p2s2in P. Clearly, q1, q2∈(πα)−1(B∗). In order to see that (p, π(a)b) = (p0, π(a0)b0)we will see that
18 3. Derivations and differentials Proposition 3.10 (Base Change).We consider a diagram of log rings (A1, M1) (B1, N1) (A2, M2) and let (B2, N2)be its pushout, i.e. B2=A2⊗A1B1,N2=M2⊕M1N1. Then the canonical homomorphism B2⊗B1Ω(B1,N1)|(A1,M1)−→ Ω(B2,N2)|(A2,M2) is an isomorphism. Proof. Since B2⊗B1−is right-exact, by the cocartesian square in Definition 3.9 B1⊗Z[N1]ΩZ[N1]|Z[M1]B1⊗ZNgp 1/Im(Mgp 1) ΩB1|A1Ω(B1,N1)|(A1,M1) we obtain another cocartesian square B2⊗Z[N1]ΩZ[N1]|Z[M1]B2⊗ZNgp 1/Im(Mgp 1) B2⊗B1ΩB1|A1B2⊗B1Ω(B1,N1)|(A1,M1) and then a commutative diagram where both front and rear squares are cocartesian: B2⊗Z[N1]ΩZ[N1]|Z[M1]B2⊗ZNgp 1/Im(Mgp 1) B2⊗Z[N2]ΩZ[N2]|Z[M2]B2⊗ZNgp 2/Im(Mgp 2) B2⊗B1ΩB1|A1B2⊗B1Ω(B1,N1)|(A1,M1) ΩB2|A2Ω(B2,N2)|(A2,M2) αγ β
3.2 Module of differentials 19 It is enough to prove that α, β, γ are isomorphisms. Since Z[N2] = Z[M2⊕M1N1] = Z[M2]⊗Z[M1]Z[N1] (Remark 1.27) we have ΩZ[N2]|Z[M2]=Z[N2]⊗Z[N1]ΩZ[N1]|Z[M1] by [2, Lemme 1.13], and therefore αis an isomorphism. Clearly βis an isomorphism too. Finally, from the exact sequence 1Mgp 2Mgp 2⊕Mgp 1Ngp 1Ngp 1/Im(Mgp 1) 1 a(a, 1) (a, b)¯ b we obtain an isomorphism Ngp 2 Im(Mgp 2)=Mgp 2⊕Mgp 1Ngp 1 Im(Mgp 2)=Ngp 1 Im(Mgp 1) and so γis an isomorphism. Corollary 3.11 (Localization).Let (A, M)→(B, N)be a homomorphism of prelog rings and S⊂N,T⊂(B, ·)submonoids such that α(S)⊂Twhere α:N→Bis the structural homomorphism. Then T−1Ω(B,N)|(A,M)= Ω(T−1B,S−1N)|(A,M). Proof. As in the previous proposition, applying the exact functor T−1(−)to the cocartesian square in Definition 3.9 we obtain a cocartesian square T−1B⊗Z[N]ΩZ[N]|Z[M]T−1B⊗ZNgp/Im(Mgp) T−1ΩB|AT−1Ω(B,N)|(A,M)
20 3. Derivations and differentials and consequently a commutative diagram where both front and rear squares are cocartesian: T−1B⊗Z[N]ΩZ[N]|Z[M]T−1B⊗ZNgp/Im(Mgp) T−1B⊗Z[S−1N]ΩZ[S−1N]|Z[M]T−1B⊗Z(S−1N)gp/Im(Mgp) T−1ΩB|AT−1Ω(B,N)|(A,M) ΩT−1B|AΩ(T−1B,S−1N)|(A,M) αγ β The homomorphism γis an isomorphism because (S−1N)gp =Ngp, and αand β are isomorphisms too. Hence the desired isomorphism follows. Lemma 3.12. The pushout of right-exact sequences of modules is also a right-exact sequence. Proof. Let us consider the commutative diagram C C0C00 0 A A0A00 0 D D0D00 0 B B0B00 0 where the rows A→A0→A00 →0,B→B0→B00 →0and C→C0→C00 →0 are right-exact sequence of modules, and the commutative diagrams A C B D A0C0 B0D0 A00 C00 B00 D00
3.2 Module of differentials 21 are cocartesian. We will show that the sequence D→D0→D00 →0of the pushouts is also a right-exact sequence. The surjectivity of D0→D00 and Im(D→D0)⊂ker(D0→D00)are obvious, while ker(D0→D00)⊂Im(D→D0)can be seen by diagram chasing: Let b0∈B0, c0∈C0such that d0=b0+c0, and let b00, c00, d00 be their images in B00, C00, D00 respectively. Since d00 = 0, there exists a00 ∈A00 such b00 and −c00 are both the image of a00. Let a0∈A0be such that its image in A00 is a00, and let ˜ b0,−˜ c0their images in B0and C0respectively. Then ˜ b0and ˜ c0go to b00 and c00 in C00, D00 respectively, so there exist b∈B,c∈Csuch that band cgo to b0−˜ b0,c0−˜ c0respectively. Then the image of b+c∈Din D0is b0−˜ b0+c0−˜ c0=d0−(˜ b0+˜ c0) = d0since ˜ b0and −˜ c0 are the images of a0∈A0. Thus it is done. Proposition 3.13. Let (A, M)→(B, N)→(C, P)be homomorphisms of prelog rings. There exists a natural exact sequence of C-modules C⊗BΩ(B,N)|(A,M)−→ Ω(C,P)|(A,M)−→ Ω(C,P )|(B,N)−→ 0. Proof. Using Lemma 3.12, we deduce the result from the definition of the module of differentials of prelog rings and from the exact sequences C⊗Z[N]ΩZ[N]|Z[M]−→ C⊗Z[P]ΩZ[P]|Z[M]−→ C⊗Z[P]ΩZ[P]|Z[N]−→ 0, C⊗BΩB|A−→ ΩC|A−→ ΩC|B−→ 0 and Ngp/ImNgp (Mgp)−→ Pgp/ImPgp (Mgp)−→ Pgp/ImPgp (Ngp)−→ 1, which is exact since Ngp/ImNgp (Mgp)−→ ImPgp (Ngp)/ImPgp (Mgp) is surjective and (clearly) ImPgp (Ngp)/ImPgp (Mgp)−→ Pgp/ImPgp (Mgp)−→ Pgp/ImPgp (Ngp)−→ 1 is exact. Proposition 3.14. Let (A, M)→(C, L)→(B, N)be homomorphisms of prelog rings with C→Band L→Nsurjective. There exists a natural exact sequence of B-modules N(B,N)|(C,L)−→ B⊗CΩ(C,L)|(A,M)−→ Ω(B,N)|(A,M)→0.
22 3. Derivations and differentials Proof. In the same way that the proof of the Proposition 3.13. It is deduced from the following exact sequences: B⊗Z[N]J/J2−→ B⊗Z[L]ΩZ[L]|Z[M]−→ B⊗Z[N]ΩZ[N]|Z[M]−→ 0, I/I2−→ B⊗CΩC|A−→ ΩB|A−→ 0 and B⊗ZW−→ B⊗ZLgp/ImLgp (Mgp)−→ B⊗ZNgp/ImNgp (Mgp)→0, where I= ker(C→B),J= ker(Z[L]→Z[N]) and W= ker(Lgp →Ngp). Note that the third sequence is exact because W= ker Lgp/ImLgp (Mgp)→Ngp/ImNgp (Mgp), as we deduce, for example, applying the Ker-Coker Lemma to the next diagram of exact sequences: 1Mgp Lgp Lgp/ImLgp (Mgp) 1 1Mgp Ngp Ngp/ImNgp (Mgp) 1 Proposition 3.15 (Künneth).Let (A, M)→(B, N),(A, M)→(C, P)be homomorphisms of prelog rings. There exists a natural isomorphism of B⊗AC-modules Ω(B⊗AC,N⊕MP)|(A,M)= (Ω(B,N)|(A,M)⊗AC)⊕(B⊗AΩ(C,P)|(A,M)). Proof. Applying the right-exact functors B⊗A−and C⊗A−to the cocartesian diagram that define Ω(B,N)|(A,M)(as in the proof of Proposition 3.10) and taking the direct sum of them, we obtain a pushout D⊗Z[N]ΩZ[N]|Z[M]⊕D⊗Z[P]ΩZ[P]|Z[M] D⊗ZNgp/Im(Mgp)⊕D⊗ZPgp/Im(Mgp) C⊗AΩB|A⊕B⊗AΩC|A C⊗AΩ(B,N)|(A,M)⊕B⊗AΩ(C,P)|(A,M) where D=B⊗AC.
3.2 Module of differentials 23 Now, we note that (C⊗AB)⊗Z[N]ΩZ[N]|Z[M]⊕Z[M](B⊗AC)⊗Z[P]ΩZ[P]|Z[M] = (C⊗AB)⊗Z[N⊕MP]ΩZ[N]|Z[M]⊕Z[M]ΩZ[P]|Z[M] = (C⊗AB)⊗Z[N⊕MP]ΩZ[N]⊗Z[M]Z[P]|Z[M] = (B⊗AC)⊗Z[N⊕MP]ΩZ[N⊕MP]|Z[M] (using Remark 1.27 and [2, Lemme 1.14]). Moreover, using the fact that the exact sequence 1Mgp Ngp ⊕Mgp Pgp Ngp/Im(Mgp)⊕Pgp/Im(Mgp) 1 a(a, 1) (a, b) (¯a,¯ b) is exact and Proposition 1.21, we have C⊗AB⊗ZNgp/Im(Mgp)⊕B⊗AC⊗ZPgp/Im(Mgp)= =C⊗AB⊗ZNgp/Im(Mgp)⊕Pgp/Im(Mgp) =C⊗AB⊗ZNgp ⊕Mgp Pgp/Mgp) =C⊗AB⊗Z(N⊕MP)gp/Mgp Finally, using [2, Lemme 1.14], C⊗AΩB|A⊕B⊗AΩC|A=ΩC|A⊕ΩB|A= ΩB⊗AC|A and we obtain a diagram of pushouts D⊗Z[N]ΩZ[N]|Z[M]⊕D⊗Z[P]ΩZ[P]|Z[M] D⊗ZNgp/Mgp⊕D⊗ZPgp/Mgp D⊗Z[M⊕MP]ΩZ[M⊕MP]|Z[M]D⊗Z(N⊕MP)gp/Mgp C⊗AΩB|A⊕B⊗AΩC|A C⊗AΩ(B,N)|(A,M)⊕B⊗AΩ(C,P )|(A,M) ΩB⊗AC|AΩ(B⊗AC,N⊕MP)|(A,M) == =
24 3. Derivations and differentials Examples 3.16. (i) Free prelog algebras. Let (A, M)be a prelog ring, Xand Y sets, N=M⊕NYwhere NYis a direct sum of copies of the (additive) monoid of natural numbers indexed by Y,B=A[X, Y ]the polynomial A-algebra on X∪Y,N→Bthe homomorphism which extends M→Ain the obvious way. We will say that (B, N)is a free (A, M)-prelog algebra. Then Ω(B,N)|(A,M)is a free B-module with basis X∪Y. It follows from the pushout that defines Ω(B,N)|(A,M), BX=B⊗Z[N]ΩZ[N]|Z[M]B⊗ZNgp/Mgp =BX BX∪Y= ΩB|AΩ(B,N)|(A,M) v uw (note that vis not an isomorphism). We know uis injective (it can be seen as the canonical inclusion) and so coker(u) = BY. Therefore wis also injective and coker(w) = BYagain. Thus, we obtain an exact sequence of B-modules 0−→ BXw −→ Ω(B,N)|(A,M)−→ BY−→ 0 that gives us the desired isomorphism, since it is split because BYis a free B-module. (ii) Let (A, M)→(B, N)a homomorphism essentially of finite type of Noetherian prelog rings (see Definition 2.4). Then Ω(B,N)|(A,M)is a B-module of finite type. It follows from the above Example 3.16.(i) and Proposition 3.14. (iii) Let M→Nbe a homomorphism of monoids, then Ω(Z[N],N)|(Z[M],M)=Z[N]⊗ZNgp/Im(Mgp) since, by definition, Ω(Z[N],N)|(Z[M],M)is the pushout of Z[N]⊗Z[N]ΩZ[N]|Z[M]Z[N]⊕ZNgp/Im(Mgp) ΩZ[N]|Z[M] =
3.2 Module of differentials 25 (iv) Let (B, N)be a prelog ring where Nis a group. Then the homomorphism ν:B⊗Z[N]ΩZ[N]|Z→B⊗ZNgp is surjective since ν(bn−1⊗dn) = b⊗n for n∈Ngp =Nand b∈B. Therefore, from the cocartesian square B⊗Z[N]ΩZ[N]|ZB⊗ZNgp 0 = ΩB|BΩ(B,N)|(B,1) we deduce that Ω(B,N)|(B,1) = 0. (v) Let (B, N)be a prelog ring and Ma subgroup of N. From the exact sequence in Proposition 3.13 applied to (B, 1) →(B, M)→(B, N)and the Example 3.16.(iv), we obtain an isomorphism of B-modules Ω(B,N)|(B,M)= Ω(B,N)|(B,1). (vi) Let (A, M)→(B, N)be a homomorphism of prelog rings with M→N surjective, then Ω(B,N)|(A,M)= ΩB|A. Since N⊕MN→Nis an isomorphism, the result follows from Example 3.7. (vii) Let (A, A∗)→(B, B∗)be the obvious homomorphism of prelog rings where A∗and B∗are the subgroups of units of Aand Brespectively. Then Ω(B,B∗)|(A,A∗)= ΩB|A. By Example 3.16.(iv) we have Ω(A,A∗)|(A,1) = 0 and so, from the exact sequence in Proposition 3.13 applied to (A, 1) →(A, A∗)→(B, B∗)we obtain Ω(B,B∗)|(A,A∗)= Ω(B,B∗)|(A,1). Using now Ω(B,B∗)|(B,1) = 0 and the exact sequence in Proposition 3.13 applied to (A, 1) →(B, 1) →(B, B∗)we obtain Ω(B,B∗)|(A,1) = Ω(B,1)|(A,1), which is isomorphic to ΩB|Aby Example 3.16.(vi).
26 3. Derivations and differentials 3.3 Derivations Definition 3.17. Let (A, M)→(B, N)a homomorphism of prelog rings, TaBmodule. An (A, M)-derivation of (B, N)into Tis a pair (D, e D), where Dis an A-derivation D:B→Tand e Dis a homomorphism of groups e D:Ngp/Im(Mgp)−→ (T, +) such that D(n) = ne D(¯n)for all n∈N(where ¯nis its image in Ngp/Im(Mgp)and we denote the image of n∈Nin Bagain by n). We denote by Der(A,M)((B, N), T) the set of (A, M)-derivations of (B, N)into T, which is a B-module with the induced structure by the structure of B-module of T. Proposition 3.18. With the above notation, there exists an isomorphism of B-modules HomB(Ω(B,N)|(A,M), T ) = Der(A,M)((B, N), T). Proof. Let (D, e D)∈Der(A,M)((B, N), T). From the diagram B⊗Z[N]ΩZ[N]|Z[M]B⊗ZNgp/Im(Mgp) ΩB|AΩ(B,N)|(A,M) T i ju v g f h where fis the homomorphism of B-modules induced by D:B→Tand gis the homomorphism of B-modules induced by e D:Ngp/Im(Mgp)→T, we obtain a homomorphism h: Ω(B,N)|(A,M)→Tmaking commutative the corresponding triangles (since fj =gi and the square is a pushout). Now, consider h∈HomB(Ω(B,N)|(A,M), T ). We define D∈DerA(B, T)by the composition BdB|A −→ ΩB|A v −→ Ω(B,N)|(A,M) h −→ T
3.3 Derivations 27 and e Dby the composition Ngp/Im(Mgp)−→ B⊗ZNgp/Im(Mgp)u −→ Ω(B,N)|(A,M) h −→ T. Since vand hare homomorphism of B-modules, Dis a derivation. Similarly, e Dis a homomorphism of groups. We also have hui(1 ⊗dn) = hu(n⊗¯n) = n·hu(1 ⊗¯n) = ne D(¯n) and hvj(1 ⊗dn) = D(n). Since ui =vj, we obtain D(n) = ne D(n). Now, we are able to state the version for derivations of Proposition 3.10, Corollary 3.11, Proposition 3.13 and Proposition 3.14. Corollary 3.19. (i) We consider a diagram of log rings (A1, M1) (B1, N1) (A2, M2) and let (B2, N2)be its pushout, i.e. B2=A2⊗A1B1and N2=M2⊕M1N1. Then the canonical homomorphism Der(A2,M2)((B2, N2), T)−→ Der(A1,M1)((B1, N1), T) is an isomorphism for any B2-module T. (ii) Let (A, M)→(B, N)be a homomorphism of prelog rings and S⊂N, T⊂(B, ·)submonoids such that α(S)⊂Twhere α:N→Bis the structural homomorphism. Then for any T−1B-module W Der(A,M)((B, N), W) = Der(A,M)((T−1B, S−1N), W). (iii) Let (A, M)→(B, N)→(C, P)be homomorphisms of prelog rings. There exists a natural exact sequence of C-modules for any C-module W 0→Der(B,N)((C, P), W)→Der(A,M)((C, P), W)→Der(A,M)((B, N), W). (iv) Let (A, M)→(C, L)→(B, N)be homomorphisms of prelog rings with C→Band L→Nsurjective. There exists a natural exact sequence of B-modules for any B-module W 0→Der(A,M)((B, N), W)→Der(A,M)((C, L), W)→HomB(N(B,N)|(C,L), W). Proof. All of them are deduced from the respective properties for the module of differentials using Proposition 3.18.
34 4. Logarithmic André-Quillen (co)homology Proposition 4.7. Let (A, M)be a prelog ring and (B, N)a free (A, M)-prelog algebra (Example 3.16.(i)). Then Hn((A, M),(B, N), W) = 0 = Hn((A, M),(B, N), W) for all n > 0and all B-modules W. Proof. (A, M)→(B, N)→(B, N)is a cofibration - trivial fibration. Proposition 4.8 (Base Change).Let (A, M)→(B, N)and (A, M)→(C, P)be homomorphisms of prelog rings, let (D, Q) = (B⊗AC, N ⊕MP), and let Wbe a B⊗AC-module. Assume that the following conditions hold: (i) TorA i(B, C)=0for all i > 0. (ii) TorZ[M] i(Z[N],Z[P]) = 0 for all i > 0. (iii) The canonical homomorphism Mgp −→ Ngp ⊕Pgp is injective (e.g. when Mgp →Ngp or Mgp →Pgp is injective). Then the canonical homomorphisms Hn((A, M),(B, N), W)−→ Hn((C, P),(D, Q), W), Hn((C, P),(D, Q), W)−→ Hn((A, M),(B, N), W) are isomorphisms for all n≥0. Proof. The canonical homomorphisms Hn(A, B, W)−→ Hn(C, D, W ), Hn(C, D, W )−→ Hn(A, B, W), Hn(Z[M],Z[N], W)−→ Hn(Z[P],Z[Q], W), Hn(Z[P],Z[Q], W)−→ Hn(Z[M],Z[N], W) are isomorphisms for all n≥0by [2, Proposition 4.54]. Since (−)gp is left adjoint, we have a pushout Mgp Ngp Pgp Qgp j u v w
4.1 The logarithmic cotangent complex 35 Since u⊕j:Mgp →Ngp ⊕Pgp is injective, we have isomorphisms of Abelian groups ker(u) = ker(w), coker(u) = coker(w). The result then follows from the fundamental exact sequences (Theorem 4.3). Proposition 4.9 (Jacobi-Zariski exact sequence).Let (A, M)→(B, N)→(C, L) be homomorphisms of prelog rings and WaC-module. There exist natural exact sequences · · · Hn((A, M),(B, N), W)Hn((A, M),(C, L), W)Hn((B, N),(C, L), W) · · · Hn−1((A, M),(B, N), W)· · · H0((B, N),(C, L), W) 0 and 0H0((B, N),(C, L), W)· · · Hn−1((A, M),(C, L), W) Hn((B, N),(C, L), W)Hn((A, M),(C, L), W)Hn((A, M),(B, N), W)· · · . Proof. See [19, Theorem 8.18]. Proposition 4.10 (Localization).Let (A, M)→(B, N)be a homomorphism of prelog rings WaB-module. Let S⊂N,T⊂(B, ·)be submonoids such that α(S)⊂Twhere α:N→Bis the structural homomorphism of (B, N). Then, for all n≥0: (i) Hn((B, N),(T−1B, S−1N), T−1W) = 0, Hn((B, N),(T−1B, S−1N), T−1W) = 0. (ii) Hn((A, M),(B, N), T−1W) = Hn((A, M),(T−1B, S−1N), T−1W), Hn((A, M),(B, N), T−1W) = Hn((A, M),(T−1B, S−1N), T−1W). (iii) Hn((A, M),(B, N), T−1W) = T−1Hn((A, M),(B, N), W). Proof. (i) It follows from Base Change (Proposition 4.8) applied to the pushout (B, N) (T−1B, S−1N) (T−1B, S−1N) (T−1B, S−1N)
36 4. Logarithmic André-Quillen (co)homology (ii) It follows from (i) and the Jacobi-Zariski exact sequence (Proposition 4.9). (iii) Property (iii) follows from the universal coefficient spectral sequence E2 p,q = TorB p(Hq(L(B,N)|(A,M)⊗BW), T −1B) = TorB p(Hq((A, M),(B, N), W), T −1B)⇒Hn((A, M),(B, N), T−1W). The following two results will not be used in the sequel. We include them for possible future references. Proposition 4.11. Let (R, Q)→(A, M),(A, M)→(B, N),(A, M)→(C, P)be homomorphisms of prelog rings, D=B⊗ACand L=N⊕MP. Assume that the following conditions (the same that we considered in Proposition 4.8) hold: (i) TorA i(B, C)=0for all i > 0, (ii) TorZ[M] i(Z[N],Z[P]) = 0 for all i > 0, (iii) The canonical homomorphism Mgp −→ Ngp ⊕Pgp is injective (e.g. when Mgp →Ngp or Mgp →Pgp is injective). Then, for any D-module Wwe have a natural exact sequence · · · → Hn((R, Q),(A, M), W )→Hn((R, Q),(B, N), W)⊕Hn((R, Q),(C, P), W )→ →Hn((R, Q),(D, L), W)→Hn−1((R, Q),(A, M), W)→ · · · · · · → H0((R, Q),(D, L), W )→0. Proof. Applying the Jacobi-Zariski exact sequence to the rows of the commutative diagram (R, Q) (A, M) (B, N) (R, Q) (C, P) (D, L) and noting that the right square satisfies the hypotheses of Proposition 4.8, we obtain the result as in [2, Proposition 5.22]. Similarly, we also have an analogue to [2, Proposition 5.21]:
4.1 The logarithmic cotangent complex 37 Proposition 4.12. Let (A, M)→(B, N),(A, M)→(C, P)be homomorphisms of prelog rings, D=B⊗ACand L=N⊕MP. Consider also a homomorphism of prelog rings (D, L)→(R, Q). Assume that the conditions of Proposition 4.8 hold again, as in the previous proposition. Then, for any R-module Wwe have a natural exact sequence · · · → Hn((A, M),(R, Q), W )→Hn((B, N),(R, Q), W )⊕Hn((C, P),(R, Q), W)→ →Hn((D, L),(R, Q), W)→Hn−1((A, M),(R, Q), W)→ · · · · · · → H0((D, L),(R, Q), W )→0. Remark 4.13. Let Hi →Xp →Nbe homomorphisms of simplicial monoids with H, N constants, Ha group and pi injective. We suppose that pis a trivial fibration in the category of the simplicial monoids. Since Nis a Kan complex (because it is constant) and pis a fibration, Xis a Kan complex too (the composition of fibrations is a fibration). We consider the induced actions of Hin Xand in Nand also the quotients X/H,N/H. We will assume that Hacts freely on X. Then Xπ →X/H is a fibration [17, Lemma 18.2] with fibre H, and then for any basepoint we have an exact sequence · · · → πn(H)→πn(X)→πn(X/H)→πn−1(H)→ · · · → π0(X/H)→1. We have πn(H) = πn(X) = 1 for all n > 0, and so we obtain πn(X/H) = 1 for all n > 1and an exact sequence 1−→ π1(X/H)−→ π0(H)−→ π0(X)−→ π0(X/H)−→ 1 which takes the form 1−→ π1(X/H)−→ Hpi −→ N−→ π0(X/H)−→ 1. Therefore we deduce πn(X/H) = (N/H if n= 0, 1if n > 0. That is, X/H →N/H is a weak equivalence, and therefore by [10, I.2.2.3] Z[X/H]→Z[N/H]is a weak equivalence. Lemma 4.14. Let Hbe an Abelian group and WaZ[H]-module. Then Hn(Z,Z[H], W) = TorZ n(H, W), Hn(Z,Z[H], W) = ExtZ n(H, W) for all n≥0. In particular, these modules vanish for all n≥2.
38 4. Logarithmic André-Quillen (co)homology Proof. Let X→Hbe a simplicial resolution of the Z-module H, that is, X→H is surjective, Xis a free simplicial Z-module with πn(X) = πn(H)for all n. Then Z[X]is a free simplicial Z-algebra, Z[X]→Z[H]is surjective and a weak equivalence by [10, I.2.2.3]. So Z→Z[X]→Z[H]is a free cofibration - trivial fibration factorization of simplicial rings [20, Proposition 2.5]. Therefore Hn(Z,Z[H], W) = Hn(ΩZ[X]|Z⊗Z[X]W), and similarly for cohomology. So it suffices to show that ΩZ[X]|Z=X⊗ZZ[X]. Each Xnis a free Z-module. Since Ω,⊗and Z[−]commute with colimits, we can assume that Xnis a free Zmodule of finite rank and by Künneth formula for the module of differentials [2, Lemme 1.14] we can assume Xn=Z: ΩZ[H1⊕H2]|Z= ΩZ[H1]⊗ZZ[H2]|Z = ΩZ[H1]|Z⊗ZZ[H2]⊕Z[H1]⊗ZΩZ[H2]|Z =H1⊗ZZ[H1]⊗ZZ[H2]⊕Z[H1]⊗ZH2⊗ZZ[H2] = (H1⊕H2)⊗Z(Z[H1]⊗ZZ[H2]) = (H1⊕H2)⊗ZZ[H1⊕H2]. So we only have to prove ΩZ[H]|Z=H⊗ZZ[H]when H=Z. We have Z[H] = Z[t, 1 t] = S−1Z[t]with S={1, t, t2, . . .}. Then ΩZ[H]|Z=S−1ΩZ[t]|Z=S−1Z[t] = Z⊗ZS−1Z[t] =H⊗ZZ[H]. Remark 4.15. Now we will give here a more explicit proof of Lemma 4.14 that can be useful for similar computations to those made in the proofs of some results in Chapter 6. For the case n= 0 we have to show that ΩZ[H]|Z=H⊗ZZ[H]. If H=Cris a finite cyclic group (of order r), then Z[H] = Z[x]/(xr−1) and the result follows from an immediate computation. If H=Zis an infinite cyclic group, Z[H] = Z[x, 1/x] = S−1Z[x], where S={1, x, x2, . . .}, and then ΩZ[H]|Z=S−1ΩZ[x]|Z= S−1Z[x] = Z⊗ZS−1Z[x] = H⊗ZZ[H]. If H1and H2are two groups such that ΩZ[Hi]|Z=Hi⊗ZZ[Hi]for i= 1,2, then by the Künneth formula for the module of differentials [2, Lemme 1.14] and Remark 1.27 we have ΩZ[H1⊕H2]|Z= ΩZ[H1]⊗ZZ[H2]|Z = ΩZ[H1]|Z⊗ZZ[H2]⊕Z[H1]⊗ZΩZ[H2]|Z =H1⊗ZZ[H1]⊗ZZ[H2]⊕Z[H1]⊗ZH2⊗ZZ[H2] = (H1⊕H2)⊗Z(Z[H1]⊗ZZ[H2]) = (H1⊕H2)⊗ZZ[H1⊕H2].
4.1 The logarithmic cotangent complex 39 So the result holds for Abelian groups of finite type. Finally write H= lim −→ i Hiwith HiAbelian groups of finite type. Then ΩZ[H]|Z= ΩZ[lim −→ i Hi]|Z= Ωlim −→ i Z[Hi]|Z= lim −→ i ΩZ[Hi]|Z = lim −→ i (Hi⊗ZZ[Hi]) = lim −→ i Hi⊗Zlim −→ i Z[Hi] =H⊗ZZ[H], given that Z[−], being a left adjoint, commutes with colimits. Now consider the case n≥2. If His cyclic, Z[H] = Z[x]/(xr−1) or Z[H] = Z[x, 1/x], and in both cases it is clear that Hn(Z,Z[H], W ) = 0 for all n≥0 (it follows for instance from [2, Théorème 6.19] and [2, Corollaire 5.27]). Since Z[H1⊕H2] = Z[H1]⊗ZZ[H2], the result also holds for finitely generated Abelian groups by [2, Corollaire 5.23], and since Z[−]preserves direct limits, the result in general follows from [2, Proposition 3.35]. For the case n= 1. Let H=Crbe a finite cyclic group, so that Z[H] = Z[x]/(xr−1). The the Jacobi-Zariski exact sequence associated to Z−→ Z[x]−→ Z[x]/(xr−1) can be written as 0−→ H1(Z,Z[H], W)−→ (xr−1) ⊗Z[x]Wδ −→ Z[x]⊗Z[x]W=W−→ · · · , where the map δsends (xr−1) ⊗ωto rω. On the other hand, the exact sequence of Abelian groups 0→rZ→Z→H→0induces an exact sequence 0−→ TorZ 1(H, W)−→ rZ⊗ZWδ0 −→ W−→ · · · where δ0(r⊗w) = rw. The isomorphism rZ⊗ZW=rZ⊗ZZ[x]⊗Z[x]W=rZ[x]⊗Z[x]Wϕ −→ (xr−1) ⊗Z[x]W, sending r⊗ωinto (xr−1) ⊗ωsatisfies δϕ =δ0and so gives H1(Z,Z[H], W) = TorZ 1(H, W). For H=Zthe infinite cyclic group, H1(Z,Z[H], W) = 0 = TorZ 1(H, W). Using again that Z[−]and Tor preserve coproducts and filtered colimits, the result follows from [2, Lemme 3.35].
40 4. Logarithmic André-Quillen (co)homology Lemma 4.16. Let (A, H)→(B, G)be a homomorphism of prelog rings, where H and Gare groups. Then Hn((A, H),(B, G), W) = Hn(A, B, W)for all n≥0and all B-modules W, and similarly for cohomology. Proof. We have a Jacobi-Zariski exact sequence Hn+1((B, H),(B, G), W)−→ Hn((A, H),(B, H), W ) −→ Hn((A, H),(B, G), W)−→ Hn((B, H),(B, G), W)−→ · · · Since Hn((A, H),(B, H), W) = Hn(A, B, W)for all n≥0by Example 4.2, it suffices to show that Hn((B, H),(B, G), W)=0for all n≥0. From the JacobiZariski sequence associated to (B, {1})→(B, H)→(B, G), we see that it suffices to show that Hn((B, {1}),(B, H), W ) = 0 for any group Hand all n≥0. Since Hn(B, B, W )=0for all n≥0, the fundamental exact sequence gives isomorphisms Hn((B, {1}),(B, H), W) = Hn−1(Z,Z[H], W) = 0 for n≥3by Lemma 4.14.(ii) and an exact sequence 0H2((B, {1}),(B, H), W) H1(Z,Z[H], W) TorZ 1(W, H)H1((B, {1}),(B, H), W) H0(Z,Z[H], W)W⊗ZH H0((B, {1}),(B, H), W) 0. κ1 κ0 Furthermore, since κ1and κ0are isomorphisms by Lemma 4.14 we deduce also Hi((B, {1}),(B, H), W) = 0 for i= 0,1,2. Proposition 4.17. [19, Theorem 8.20] Let (A, M)→(B, N)be a homomorphism of prelog rings with Mand Nintegral monoids and WaB-module. Then, for all n≥2, we have (i) Hn((A, M),(B, N), W) = Hn((A, M),(B, N)log, W) =Hn((A, M)log,(B, N)log, W ). (ii) Hn((A, M),(B, N), W) = Hn((A, M),(B, N)log, W) =Hn((A, M)log,(B, N)log, W ). Proof. We will prove (i). By the Jacobi-Zariski exact sequences, it suffices to show that for a prelog ring (A, M)with Mintegral, Hn((A, M),(A, M)log, W)=0for
4.1 The logarithmic cotangent complex 41 any A-module Wfor all n≥0. Let α:M→Abe the structural map. Consider the diagram of pushouts (A, α−1(A∗)) (A, α−1(A∗)gp) (A, α−1(A∗)gp/ker v) (A, A∗) (A, M) (A, M1) (A, M2) (A, Mlog) where v:α−1(A∗)gp →A∗is the canonical map. Applying base change (Proposition 4.8) to the pushout (A, α−1(A∗)) (A, α−1(A∗)gp) (A, M) (A, M1) (since TorZ[α−1(A∗)] n(Z[M],Z[α−1(A∗)gp]) = 0 for n > 0because, as we know, Z[α−1(A∗)] −→ Z[α−1(A∗)gp]being a localization is flat), we obtain Hn((A, M),(A, M1), W) = Hn((A, α−1(A∗)),(A, α−1(A∗)gp), W) = 0 where this last module vanishes by Proposition 4.10.(ii). Now consider to the pushout (A, α−1(A∗)gp) (A, α−1(A∗)gp/ker v) (A, M1) (A, M2) since Mis integral, the map α−1(A∗)gp →M1is injective, being a localization of the injective map α−1(A∗)→M. In particular, ker(v)→M1is injective. If we take a cofibration - trivial fibration factorization α−1(A∗)gp −→ X−→ M1
42 4. Logarithmic André-Quillen (co)homology in the category of simplicial monoids with Xn=α−1(A∗ 1)gp⊕Ne Xna free α−1(A∗)gpmonoid, then the action of ker(v)on Xis free, since α−1(A∗)gp is a group. Therefore from [10, I.2.2.3] and Remark 4.13: TorZ[α−1(A∗)gp] n(Z[M1],Z[α−1(A∗)gp/ker(v)]) = =Hn(Z[X]⊗Z[α−1(A∗)gp]Z[α−1(A∗)gp/ker(v)]) =Hn(Z[X/ ker(v)]) = 0 for n > 0. So base change (Proposition 4.8) gives an isomorphism Hn((A, M1),(A, M2), W) = Hn((A, α−1(A∗)gp),(A, α−1(A∗)gp/ker(v)), W) = 0 where the last module vanishes by Lemma 4.16 above. Finally, applying again base change to the pushout (A, α−1(A∗)gp/ker(v)) (A, A∗) (A, M2) (A, Mlog) (since α−1(A∗)gp/ker(v)→A∗is an injective homomorphism of groups, we know that the homomorphism Z[α−1(A∗)gp/ker(v)] →Z[A∗]is flat) we obtain similarly Hn((A, M2),(A, Mlog), W) = Hn((A, α−1(A∗)gp/ker(v)),(A, A∗), W) = 0 which vanishes again by Lemma 4.16. By the Jacobi-Zariski exact sequence associated to the sequence (A, M)→ (A, M1)→(A, M2), we obtain Hn((A, M),(A, M2), W) = 0, and then by the Jacobi-Zariski exact sequence associated to (A, M)→(A, M2)→(A, Mlog)we obtain Hn((A, M),(A, Mlog), W) = 0 as desired.
4.2 A spectral sequence 43 4.2 A spectral sequence Let C→Bbe a surjective homomorphism of rings. We have a convergent spectral sequence [20, Theorem 6.8] E2 p,q =Hp+q(SqLB|C)⇒TorC p+q(B, B), which when A→Bis a flat homomorphism and C=B⊗ABtakes the form E2 p,q =Hp(∧qLB|A)⇒HHp+q(B|A) where HH∗(B|A) = TorB⊗AB p+q(B, B)is the Hochschild homology of the A-algebra Band Sq,∧qdenote symmetric and exterior powers respectively. We will give here an analogous spectral sequence for the logarithmic cotangent complex. Let (C, Q)→(B, N)be a homomorphism of prelog rings such that C→Band Q→Nare surjective. Let (C, Q)→(R, P)→(B, N)be a free cofibration - trivial fibration factorization with R0=Cand P0=Q. Let I= ker(π:R⊗CB→B), J= ker(Z[P]⊗Z[Q]Z[N]→Z[N]) and T= ker((P⊕QN)gp →Ngp). Let D1 (B,N)|(C,Q)be the simplicial R⊗CB-module defined by the pushout U B ⊗ZT I D1 (B,N)|(C,Q) β α where U= (R⊗CB)⊗Z[P]⊗Z[Q]Z[N]J,βis the homomorphism induced by the map J→B⊗ZTof the proof of Proposition 3.1 and the map π:R⊗CB→B, and the homomorphism αis the obvious one. Proposition 4.18. In this situation, we have an exact sequence · · · −→ Hn(U)−→ TorC n(B, B)⊕Wn−→ HnD1 (B,N)|(C,Q)−→ −→ Hn−1(U)−→ · · · −→ H1D1 (B,N)|(C,Q)−→ 0 where Wn= ker(Qgp →Ngp)⊗ZBif n= 1, TorZ 1(ker(Qgp →Ngp), B)if n= 2, 0if 16=n6= 2.
50 4. Logarithmic André-Quillen (co)homology and additionally, since B⊗Z[P]⊗Z[Q]Z[N](J0/J2 0⊗Z[P]Z[N]) = B⊗Z[N]Z[N]⊗Z[P]⊗Z[Q]Z[P]J0/J2 0 =B⊗Z[N]J0/J2 0 (because the Z[P]⊗Z[Q]Z[P]-module J0/J2 0is already a Z[N]-module), we obtain a pushout B⊗Z[N](J0/J2 0)B⊗ZT B⊗RI0/I2 0D1 (B,N)|(C,Q)/D2 Finally, we deduce T= ker Pgp/ker(Qgp →Ngp)→Ngp=Pgp/Qgp given that Pgp 0=Qgp. Thus, the previous pushout gives us L(B,N)|(C,Q)by definition. 4.3 A complex to compute H2 Now we will adapt the definition of [15] to compute H2((A, M),(B, N),−). Let (A, M)→(B, N)be a homomorphism of prelog rings. Let (A, M)→ (R, P0)→(B, N)be a factorization with R→Band h:P0→Nsurjective homomorphisms, where P0=M⊕NX(with hextending M→N) and R= Z[P0]⊗Z[M]A[Y] = A[X∪Y](with R→Bextending A→B). We consider W0= ker(Pgp 0→Ngp),J= ker(Z[P0]→Z[N]) and I= ker(R→B). Let 0−→ V−→ Gπ −→ J−→ 0 be an exact sequence of Z[P0]-modules with Ga free Z-module, 0−→ U−→ Fτ −→ I−→ 0 and exact sequence of R-modules with Fa free R-module containing R⊗Z[P0]Gas a direct summand and τextending Gπ →J→I, and 0−→ W1−→ Q1−→ W0−→ 0 an exact sequence of Z-modules with Q1a free Z-module. Let U0={τ(x)y−τ(y)x∈U|x, y ∈F}, which is an R-module, and V0={π(x)y−π(y)x∈V|x, y ∈G}which is a Z[P0]-submodule of V.
4.3 A complex to compute H251 Consider the following commutative diagram of R-modules B⊗Z[N]V/V0B⊗ZW1 B⊗Z[P0]G=B⊗Z[N]G/JG B ⊗ZQ1 B⊗Z[P0]ΩZ[P0]|Z[M]B⊗ZPgp 0/Mgp U/U0 F/IF B⊗RΩR|A β2 α2 d2D2 β1 α1 d1D1 β0 α0 δ2 δ1 (4.3.1) where the homomorphisms are defined as follows: •δ2,d2and D2are induced by the inclusions U→F,V→Gand W1→Q1 respectively. •δ1is the composition F/IF →I/I2→B⊗RΩR|A, where the first map is the one induced by τand the second one by the canonical derivation R→ΩR|A. The map d1is analogously defined while D1is induced by the composition Q1→W0→Pgp 0→Pgp 0/Mgp 0. • The homomorphism β0is the one of Definition 3.9: β0(1 ⊗dp) := h(p)⊗¯p. • The exact sequences 0−→ V−→ G−→ J−→ 0, 0−→ W1−→ Q1−→ W0−→ 0 give a diagram of homomorphisms of Z[N]-modules V/JV G/JG J/J20 0Z[N]⊗ZW1Z[N]⊗ZQ1Z[N]⊗ZW00 vh
52 4. Logarithmic André-Quillen (co)homology where vh, which is defined by λ(p−p0)7−→ λh(p0)⊗p(p0)−1 (where h(p) = h(p0)), is the map of Definition 3.1 (see Remark 3.2). Since G/JG =G⊗Z[P0]Z[N]is a projective Z[N]-module, we have a commutative diagram V/JV G/JG J/J20 0Z[N]⊗ZW1Z[N]⊗ZQ1Z[N]⊗ZW00 f β0 2β0 1vh g We define β1=B⊗Z[N]β0 1. Since V0⊂JG,f(V0/JV )=0and then β0 2(V0/JV ) = 0 by the injectivity of g. So β0 2gives a map ˜ β0 2:V/V0→ Z[N]⊗ZW1and we define β2=B⊗Z[N]˜ β0. • Finally, α2, α1, α0are the obvious maps. The commutativity of the square (i.e. D1β1=β0d1) follows from the commutativity of the squares G/JG J/J2 Z[N]⊗ZQ1Z[N]⊗ZW0 β0 1vh and J/J2Z[N]⊗Z[P0]ΩZ[P0]|Z[N] Z[N]⊗ZW0Z[N]⊗ZPgp 0/Mgp vh The commutativity of the remaining squares is clear. Definition 4.22. We define the complex (LS(B,N)|(A,M))i, for i= 0,1,2, as the complex given by the pushout of (α2, β2),(α1, β1)and (α0, β0)of diagram (4.3.1).
4.3 A complex to compute H253 Proposition 4.23. In this context, we consider the complex K:= T⊗ZW1D2 −→ T⊗ZQ1D1 −→ T⊗ZPgp 0/Mgp for any Z-module T. Then: (i) H2(K) = TorZ 1T, ker(Mgp →Ngp) (ii) We have an exact sequence 0→T⊗Zker(Mgp →Ngp)→H1(K)→TorZ 1(T, Ngp/Im(Mgp →Ngp)) →0. (iii) H0(K) = T⊗ZNgp/Im(Mgp →Ngp) Proof. We have H2(K) = ker(B⊗ZW1→B⊗ZQ1). Applying B⊗Z−to the exact sequence of Z-modules 0−→ W1−→ Q1−→ W0−→ 0 we obtain an exact sequence 0 = TorZ 1(B, Q1)→TorZ 1(B, W0)→B⊗ZW1→B⊗ZQ1→B⊗ZW0→0 and then H2(K) = TorZ 1(B, W0). Consider the diagram of exact rows and columns defining the lower row 0 0 0 0 ker(Mgp →Ngp)Mgp Im(Mgp →Ngp) 0 0W0Pgp 0=Mgp ⊕ZXNgp 0 0W0/ker(Mgp →Ngp)ZXNgp/Im(Mgp →Ngp) 0 0 0 0 (4.3.2)
54 4. Logarithmic André-Quillen (co)homology Since the lower row is exact, TorZ iB, W0/ker(Mgp →Ngp)= TorZ i+1 B, Ngp/Im(Mgp →Ngp) and this last module vanishes for i+ 1 ≥2given that Zis a principal ideal domain. So, from the left column we deduce then TorZ 1B, ker(Mgp →Ngp)= TorZ 1(B, W0), that is, H2(K) = TorZ 1(B, W0) = TorZ 1B, ker(Mgp →Ngp). Consider now the diagram 0 ker(Mgp →Ngp) 0W1Q1W00 0W0/ker(Mgp →Ngp)ZXNgp/Im(Mgp →Ngp) 0 0 ϕ σ We have exact sequences 0→ϕ−1(ker(Mgp →Ngp)) →Q1→ZX→Ngp/Im(Mgp →Ngp)→0,(4.3.3) since ker(σ) = ϕ−1(ker(Mgp →Ngp)), and 0→W1→ϕ−1(ker(Mgp →Ngp)) →ker(Mgp →Ngp)→0.
4.3 A complex to compute H255 Therefore we have a vertical exact sequence of (horizontal) complexes 0 0 0 0 W1Q1ZXNgp/Im(Mgp →Ngp) ϕ−1(ker(Mgp →Ngp)) Q1ZXNgp/Im(Mgp →Ngp) ker(Mgp →Ngp) 0 0 0 0 0 0 0 Applying B⊗Z−we obtain a diagram of (horizontal) complexes E0:B⊗ZW1B⊗ZQ1B⊗ZZX E:B⊗Zϕ−1(ker(Mgp →Ngp)) B⊗ZQ1B⊗ZZX E00 :B⊗Zker(Mgp →Ngp) 0 (4.3.4) where the columns correspond to degrees 2, 1 and 0 respectively (from left to right). We have Hn(E) = TorZ nB, Ngp/Im(Mgp →Ngp) for n= 0,1by the exactness of (4.3.3), and the fact that Q1and ZXare free Zmodules.
56 4. Logarithmic André-Quillen (co)homology On the other hand, Ext1 Z(W0/ker(Mgp →Ngp), ϕ−1(ker(Mgp →Ngp))) = = Ext1 Z(Ngp/Im(Mgp →Ngp), ϕ−1(ker(Mgp →Ngp))) = 0 by the lower row of (4.3.2), and then the exact sequence 0−→ ϕ−1(ker(Mgp →Ngp)) −→ Q1σ −→ W0/Im(Mgp →Ngp)−→ 0 is split. Therefore 0−→ B⊗Zϕ−1(ker(Mgp →Ngp)) −→ B⊗ZQ1 is injective and then H2(E) = 0. Consider the left column of (4.3.4) and let Λ := ker B⊗Zϕ−1(ker(Mgp →Ngp)) →B⊗Zker(Mgp →Ngp) = Im B⊗ZW1→B⊗Zϕ−1(ker(Mgp →Ngp)) = Im(B⊗ZW1→B⊗ZQ1). We have a vertical exact sequence of (horizontal) complexes 0 0 0 ˜ E0: Λ B⊗ZQ1B⊗ZZX E:B⊗Zϕ−1(ker(Mgp →Ngp)) B⊗ZQ1B⊗ZZX E00 :B⊗Zker(Mgp →Ngp) 0 0 0
4.3 A complex to compute H257 and taking the homology exact sequence 0 = H2(E)→H2(E00) = B⊗Zker(Mgp →Ngp)→H1(˜ E0) = H1(E0)→ →H1(E) = TorZ 1(B, Ngp/Im(Mgp →Ngp)) →H1(E00)=0 we obtain the desired exact sequence 0→B⊗Zker(Mgp →Ngp)→H1(K)→TorZ 1(B, Ngp/Im(Mgp →Ngp)) →0. Finally, it is clear that H0(K) = H0(E) = B⊗ZNgp/Im(Mgp →Ngp). Let LSZ[N]|Z[M]be the complex of Z[N]-modules V/V0−→ G/JG −→ Z[N]⊗Z[P0]ΩZ[P0]|Z[M], LSB|Athe complex U/U0−→ F/IF −→ B⊗RΩR|A and Dthe complex W1−→ Q1−→ Pgp 0/Mgp. By [15], for any B-module T,H∗(LSZ[N]|Z[M]⊗Z[N]T)and H∗(LSB|A⊗BT)do not depend on the choices, and similarly H∗(D⊗ZT)by Proposition 4.23. By definition, we have a commutative diagram of complexes B⊗Z[N]LSZ[N]|Z[M]B⊗ZD LSB|ALS(B,N)|(A,M) β αε γ which is a pushout in each degree, and so coker(αi) = coker(εi). So we have a commutative diagram of complexes for any B-module T 0T⊗Z[N]LSZ[N]|Z[M]T⊗BLSB|Acoker(T⊗α) 0 0T⊗ZD T ⊗BLS(B,N)|(A,M)coker(T⊗ε) 0 T⊗α T⊗βT⊗γ T⊗ε
58 4. Logarithmic André-Quillen (co)homology Since α1, α0are split injective, so are ε1, ε0, and then both lines are exact except that T⊗Z[N](LSZ[N]|Z[M])2=T⊗Z[N]V/V0−→ T⊗BU/U0=T⊗B(LSB|A)2 and T⊗ZD2=T⊗ZW1−→ T⊗B(LS(B,N)|(A,M))2 are not necessarily injective. We deduce a commutative diagram of exact rows H2(T⊗Z[N]LSZ[N]|Z[M])H2(T⊗BLSB|A)H2(coker(T⊗α)) H2(T⊗ZD)H2(T⊗BLS(B,N)|(A,M))H2(coker(T⊗ε)) H1(T⊗Z[N]LSZ[N]|Z[M])· · · H0(coker(T⊗α)) 0 H1(T⊗ZD)· · · H0(coker(T⊗ε)) 0 By diagram chasing, we deduce an exact sequence H2(T⊗Z[N]LSZ[N]|Z[M])H2(T⊗BLSB|A)⊕H2(T⊗ZD)H2(T⊗BLS(B,N)|(A,M)) H1(T⊗Z[N]LSZ[N]|Z[M])·· · H0(T⊗BLS(B,N)|(A,M)) 0 (if x∈H2(T⊗BLSB|A)and y∈H2(T⊗ZD)are such that their images in H2(T⊗BLS(B,N)|(A,M))are equal, then x∈T⊗BU/U0is such that δ2(x) = 0, y∈B⊗ZWis such that D2(y) = 0, and γ(x) = ε(y). Therefore there exists z∈T⊗Z[N]V/V0such that α2(z) = xand β2(z) = y. Moreover, d2(z) = 0 given that α1d2(z) = δ2α2(z) = δ2(x)=0and α1is injective, and hence z∈ H2(T⊗Z[N]LSZ[N]|Z[M]). This proves exactness at H2(T⊗BLSB|A)⊕H2(T⊗ZD). Exactness at the remaining points are clear). By [2, Proposition 15.12] and Proposition 4.23, this exact sequence takes the form H2(Z[M],Z[N], T)H2(A, B, T)⊕H2(T⊗ZD)H2(T⊗BLS(B,N)|(A,M)) H1(Z[M],Z[N], T)H1(A, B, T)⊕H1(T⊗ZD)H1(T⊗BLS(B,N)|(A,M)) H0(Z[M],Z[N], T)H0(A, B, T)⊕H0(T⊗ZD)H0(T⊗BLS(B,N)|(A,M)) 0
4.3 A complex to compute H259 where •H2D⊗ZW) = TorZ 1(T, ker(Mgp →Ngp), •H1(D⊗ZW)appears in an exact sequence 0T⊗Zker(Mgp →Ngp)H1(D⊗ZW) TorZ 1T, coker(Mgp →Ngp)0, •H0(D⊗ZW) = T⊗Zcoker(Mgp →Ngp). Let (A, M)ϕ −→ (F, R)ψ −→ (B, N)be a factorization in the category of simplicial prelog rings with ϕa free cofibration and ψa trivial fibration. We have a commutative diagram of complexes T⊗Z[R]ΩZ[R]|Z[M]T⊗ZRgp/Mgp T⊗Z[N]LSZ[N]|Z[M]T⊗ZD T⊗FΩF|AT⊗BL(B,N)|(A,M) T⊗BLSB|AT⊗BLS(B,N)|(A,M) inducing a morphism from the exact sequence of Theorem 4.3 into the above exact sequence, and giving isomorphisms in two of each three terms by [2, Proposition 15.12]. Therefore, the remaining homomorphisms Hi(T⊗BL(B,N)|(A,M))−→ Hi(T⊗BLS(B,N)|(A,M)),for i= 0,1,2, are also isomorphisms.
66 5. Smoothness We will see first that this definition is independent of the choice of rsuch that π[(r) = n. Let r, r0∈Rbe such that π[(r) = π[(r0) = n. Then αF(r0), h2r0⊗r0r2 r1−nw−αF(r), h2r⊗rr2 r1−nw= =αF(r0)−αF(r), nh21⊗r0r2 r1 ·r1 rr2 =αF(r0)−αF(r), nh21⊗r0 r =0, nh21⊗r0 r−h0 1αF(r0)−αF(r) =0, h2ν(r0−r)−h1η(r0−r)= 0 since h2ν=h1η. Now assume that n, (r1 r2, w)=n, (r0 1 r0 2, w0)∈N×Ngp (Rgp ⊕TW)and we will see that αF(r), h2r⊗rr2 r1−nw=αF(r), h2r⊗rr0 2 r0 1−nw0. Let s1 s2∈Tbe such that r0 1 r0 2=r1 r2 s1 s2,w0=h2(1 ⊗s2 s1) + w. Then αF(r), h2r⊗rr0 2 r0 1−nw0=αF(r), h2r⊗rr2s2 r1s1−nh21⊗s2 s1−nw =αF(r), h2r⊗rr2s2 r1s1 +r⊗s1 s2−nw =αF(r), h2r⊗rr2 r1−nw. Finally, it is immediate to check that αCis a monoid homomorphism. Now we define τ: (C, P)−→ (B, N) by τ](f, w) := π](f)and τ[(n, (r1 r2, w)) := n. We will see now that τlog : (C, Plog)→(B, N)is strict. Consider the pushout (ταC)−1(B∗)B∗ P P ⊕(ταC)−1(B∗)B∗
5.1 Extensions 67 We will see that the canonical map λ:P⊕(ταC)−1(B∗)B∗→Nhas an inverse µ:N→P⊕(ταC)−1(B∗)B∗. Note first that ((1,(1, w)),1) = ((1,(1,0)),1) in (N×Ngp (Rgp ⊕TW)) ⊕(ταC)−1(B∗)B∗, since ταC(1,(1, w)) = 1 ∈B∗. Let n∈N. Let r∈Rsuch that π[(r) = n. We define µ(n) = (n, (r, 0)),1∈(N×Ngp (Rgp ⊕TW)) ⊕(ταC)−1(B∗)B∗. If π[(r) = π[(r0), then r r0∈Tand so (r0,0) = (r0r r0, h0 2(r0 r)) = (r, h0 2(r0 r)) in Rgp ⊕TWand so (n, (r0,0),1=n, r, h0 2r0 r,1=(n, (r, 0)),1. Therefore µdoes not depend on the choice of r. It is surjective since n, r1 r2 , w, b=bn, ˜ br1 r2 , w,1=bn, ˜ br1 r2 ,0,1 =(bn, (r0,0)),1 where r0=br1 r2∈Ris such that π[(r0) = bn, we have identified b∈B∗with its image via B∗=α−1 B(B∗)⊂Nand ˜ b∈Ris such that π[(˜ b) = b∈N. It is injective since λµ = idN. Now let us see that 1+Wacts freely on Plog. Since τ]:C→Bis surjective with square zero kernel, we have C∗= (τ])−1(B∗) = {(f, x)∈F⊕IW|π](f)∈B∗}. Let [(n, (r1 r2, w)),(f, x)] ∈N×Ngp (Rgp ⊕TW)⊕α−1 C(C∗)C∗=Plog. We have π[(r1) π[(r2)=n 1,π](f)∈B∗. Since (B, N)is a log ring, there exists m∈Nsuch that αB(m) = π](f). Let em∈Rbe such that π[(em) = m. We have αF(em)−f∈I and so (f, x) = (αF(em), x0)where x0=x−h0 1(αF(em)−f). Let g∈Cbe such that π](g) = τ](f)−1∈B∗. We have αCm, em 1,−gx0=αFem, h2em⊗em em+mgx0 = (αF(em), x0)=(f, x0). Therefore (n, r1 r2 , w,(f, x)=n, r1 r2 , w·m, em 1,−gx0,(1,0) =nm, emr1 r2 , w −gx0,(1,0).
68 5. Smoothness So we can write any element of Plog as [(n, (r1 r2, w)),(1,0)]. An element 1 + z∈ 1 + Wacts on Plog as the multiplication by [(1,(1 1,−z)),(1,0)] and so the action of 1 + Won Plog is free, since (r1 r2, w) = (r1 r2, w −z)∈Rgp ⊕TWimplies z= 0, the map T→Rgp being injective. If h∈θ(D, e D), then h0 2:T→Wfactorizes as T Rgp Rgp/Im(Mgp →Rgp)W h00 2e D and then the exact sequence 0−→ W−→ Rgp ⊕TWε −→ Ngp −→ 1 splits by s:Rgp⊕TW→W,s(t, w) = h00 2(t)+w. Therefore Rgp⊕TW=Ngp ×W and so P=N×W. Similarly, we can show that C=B⊕Wand then the extension obtained is the trivial one. Finally, we can easily see that the two maps defined between classes of extensions and H1((A, M),(B, N), W)are inverse, and natural on (B, N). 5.2 Formal smoothness Definition 5.9. Let (A, M)→(B, N)be a homomorphism of prelog rings and Jan ideal of B. We say that (B, N)is log formally smooth over (A, M)for the J-adic topology if for any commutative diagram of prelog rings (A, M) (B, N) (C, P) (C0, P0) f where (C, P)→(C0, P0)is an (A, M)-extension and f(Jt)=0for some t > 0, there exist a homomorphism of prelog rings h: (B, N)→(C, P)making commutative the triangles. Proposition 5.10. Let (A, M)→(B, N)be a homomorphism of prelog rings and J an ideal of B. The following are equivalent: (i) (B, N)is a log formally smooth (A, M)-algebra for the J-adic topology,
5.2 Formal smoothness 69 (ii) (B, N)log is a log formally smooth (A, M)-algebra for the J-adic topology, (iii) (B, N)log is a log formally smooth (A, M)log-algebra for the J-adic topology. Proof. It follows easily from the universal property of the functor (−)log and the diagram (A, M) (B, N) (A, M)log (B, N)log (C, P) (C0, P0) where (C, P)→(C0, P0)is an (A, M)-extension and f(Jt) = 0 for some t. Lemma 5.11. Let Bbe a ring, Jan ideal of B,t > 0an integer and {Fn:B/Jt−modules −→ B/Jt−modules} a direct system of half-exact functors (i.e. if 0→U→W→V→0is exact, then Fn(U)→Fn(W)→Fn(V)is exact). The following are equivalent: (i) lim −→ n Fn(W) = 0 for all B/J-modules W(considering Was a B/Jt-module via B/Jt→B/J), (ii) lim −→ n Fn(W) = 0 for all B/Jt-modules W. Proof. We have to show (i) ⇒(ii). Induction on t. The case t= 1 is trivial. Let t > 1, and assume the result valid for t−1. Let Wbe a B/Jt-module. In the exact sequence 0−→ JW −→ W−→ W/JW −→ 0 we have lim −→ n Fn(JW)=0given that JW is a B/Jt−1-module, and similarly we deduce lim −→ n Fn(W/JW) = 0. Since lim −→ n and Fnare half-exact, we conclude that lim −→ n Fn(W) = 0.
70 5. Smoothness Theorem 5.12. Let (A, M)→(B, N)be a homomorphism of prelog rings and J an ideal of B. The following are equivalent: (i) For each integer n≥1, let pn:B→B/Jnbe the canonical map. Then lim −→ n H1((A, M),(B/Jn, p∗ nN), W) = 0 for all B/J-modules W. (ii) (B, N)is log formally smooth over (A, M)for the J-adic topology. Proof. (i) ⇒(ii) Consider a diagram as in Definition 5.9. (A, M) (B, N) (C, P) (C0, P0) f τ and let I= ker τ](C→C0). For all n≥twe have a commutative square (Dn, Qn) (B/Jn, p∗ nN) (C, P) (C0, P0) θn gnfn τ where pn:B→B/Jnis the canonical projection, fnis the map induced by f, Dn=C×C0B/Jnand Qn=P×P0p∗ nN. Both lines are (A, M)-extensions by I(see Definition 5.5). Lemma 5.11.(i) tell us that lim −→ n H1((A, M),(B/Jn, p∗ nN), I)=0, and so by Theorem 5.8 there exists some s≥tsuch that θshas a section σ. The map (B, N)−→ (B/Js, p∗ sN)σ −→ (Ds, Qs)gs −→ (C, P) has the required properties of (ii). (ii) ⇒(i) Let ϕn: (D, Q)−→ (B/Jn, p∗ nN)be an (A, M)-extension of (B/Jn, p∗ nN)by W. By assertion (ii), there exists a homomorphism of prelog rings
5.2 Formal smoothness 71 φn: (B, N)→(D, Q)such that ϕnφn=pn. Since W2= 0 as ideal of D, we have φn(J2n)=0and so φnfactorizes as (B, N)p2n −→ (B/J2n, p∗ 2nN)ξ2n −→ (D, Q). We have a commutative diagram where the rows are (A, M)-extensions by W (D×B/JnB/J2n, Q ×p∗ nNp∗ 2nN) (B/J2n, p∗ 2nN) (D, Q) (B/Jn, p∗ nN) θ2n ξ2n The homomorphism ξ2ninduces a section of θ2n, showing that the element of H1((A, M),(B/Jn, p∗ nN), W)corresponding to the extension ϕn(see Theorem 5.8) goes to 0 in H1((A, M),(B/J2n, p∗ 2nN), W). Note that if (E, R)θ −→ (F, S)is an (A, M)-extension of (F, S)by Wand σa section of θ, then the map S×W→R, (s, w)7→ (1+w)σ(s)is an isomorphism with inverse r7→ (θ(r), w0)where w0∈W is such that (1 + w0)σθ(r) = r(there exists by Proposition 5.3). Lemma 5.13. Let (B, N)be a prelog ring, Jan ideal of B,pn:B→B/Jnthe canonical homomorphism and WaB/J-module. (i) Assume that J= 0 or Nis integral, then lim −→ n Hi((B, N),(B/Jn, p∗ nN), W)=0 for i= 0 and i= 1. (ii) If J= 0, or Bis a Noetherian ring and Nintegral, then lim −→ n Hi((B, N),(B/Jn, p∗ nN), W)=0 for all i≥0. Proof. If J6= 0, since (B/Jn, p∗ nN)=(B/Jn, N)log, we have by Proposition 4.17 and Example 4.2 Hi((B, N),(B/Jn, p∗ nN), W) = Hi((B, N),(B/Jn, N)log, W) =Hi((B, N),(B/Jn, N), W) =Hi(B, B/Jn, W) and so the result follows from [2, proof of Lemme 10.7 and of Théorème 10.14].
72 5. Smoothness Theorem 5.14. Let (A, M)→(B, N)be a homomorphism of prelog rings, Jan ideal of B. Assume J= 0, or BNoetherian and Nintegral. The following are equivalent: (i) (B, N)is an (A, M)-algebra log formally smooth for the J-adic topology. (ii) H1((A, M),(B, N), W)=0for all B/J-modules W. (iii) H1((A, M),(B, N), B/J) = 0 and Ω(B,N)|(A,M)⊗BB/J is a projective B/J-module. (iv) For any homomorphism (F, R)→(B, N)of prelog rings where (F, R)is a free (A, M)-prelog algebra and the homomorphisms F→Band R→Nare surjective, the homomorphism N(B,N)|(F,R)⊗BB/J −→ Ω(F,R)|(A,M)⊗FB/J is split injective. Proof. (i) ⇔(ii) We have an exact sequence induced by Jacobi-Zariski exact sequences (Proposition 4.9) lim −→ n H1((B, N),(B/Jn, p∗ nN), W)−→ lim −→ n H1((A, M),(B/Jn, p∗ nN), W)−→ −→ lim −→ n H1((A, M),(B, N), W)−→ lim −→ n H2((B, N),(B/Jn, p∗ nN), W). Then, Lemma 5.13 gives an isomorphism lim −→ n H1((A, M),(B/Jn, p∗ nN), W) = H1((A, M),(B, N), W ). So the result follows from Theorem 5.12. (ii) ⇔(iii) The universal coefficient spectral sequence Ep,q 2⇒Hp+q(HomB/J (L(B,N)|(A,M)⊗BB/J, W)) =Hp+q((A, M),(B, N), W), with Ep,q 2= Extp B/J (Hq(L(B,N)|(A,M)⊗BB/J), W), gives an exact sequence
5.2 Formal smoothness 73 0 Ext1 B/J (Ω(B,N)|(A,M)⊗BB/J, W)H1((A, M),(B, N), W) HomB/J (H1((A, M),(B, N), B/J), W ) Ext2 B/J (Ω(B,N)|(A,M)⊗BB/J, W), from which we obtain the desired equivalence. (iii) ⇔(iv) We have a Jacobi-Zariski exact sequence, taking by Proposition 4.6 the form H1((A, M),(F, R), B/J)H1((A, M),(B, N), B/J)N(B,N)|(F,R)⊗BB/J Ω(F,R)|(A,M)⊗FB/J Ω(B,N)|(A,M)⊗BB/J Ω(B,N)|(F,R)⊗BB/J = 0. The first module vanishes by Proposition 4.7 and furthermore Ω(F,R)|(A,M)⊗FB/J is a free B/J module by Example 3.16.(i). Then the result follows. Theorem 5.15. Let (A, M)→(B, N)be a homomorphism essentially of finite type of Noetherian prelog rings (see Definition 2.4) and Jan ideal of B. Assume J= 0 or Nis integral. The following are equivalent: (i) (B, N)is an (A, M)-algebra log formally smooth for the J-adic topology. (ii) H1((A, M),(B, N), W) = 0 for all B/J-modules W. If moreover Bis local and Ja proper ideal of B, these conditions are also equivalent to (iii) (B, N)is an (A, M)-algebra log formally smooth for the 0-adic topology. Proof. Conditions (i) and (ii) are equivalent by Theorem 5.14 and Proposition 4.5, and (iii) implies (ii) is clear by Theorem 5.14.(ii) or by definition. Therefore, let us assume the hypothesis (i). Then by Theorem 5.14.(ii) we have H1((A, M),(B, N), B/n) = 0 where nis the maximal ideal of B, and so by Proposition 4.5 we deduce H1((A, M),(B, N), W) = 0 for any B-module W. So we get (iii) by Theorem 5.14. We will see some examples where homological methods give short proofs of known results. If Cis a finitely generated Abelian group and Ris a ring, then Cis finite of order inversible in Rif and only if HomZ(C, W) = 0 for any R-module W, and the torsion part of Cis of order inversible in Rif and only if Ext1 Z(C, W)=0 for any R-module W. So the following two results are included in [18, Theorem IV.3.1.8, Proposition IV.3.1.13] in the case of finitely generated monoids (and taking J= 0). The following result is dual of [16, Lemma 2.6.2].
74 5. Smoothness Lemma 5.16. Let Abe a ring, Band Ctwo A-algebras and WaB⊗AC-module. Then the canonical homomorphism H1(C, B ⊗AC, W)−→ H1(A, B, W) is injective. Proof. Let Rbe a polynomial A-algebra such that B=R/I. We have an exact sequence 0−→ I⊗AC−→ R⊗ACp −→ B⊗AC−→ 0. If we denote J= ker p, then I⊗AC→Jis surjective. Consider now the exact sequences A→R→Band C→R⊗AC→B⊗AC, whose Jacobi-Zariski exact sequences of cohomology give us a commutative diagram with exact rows DerC(B⊗AC, W) DerC(R⊗AC, W) DerA(B, W) DerA(R, W) HomB⊗AC(J/J2, W)H1(C, B ⊗AC, W) 0 HomB(I/I2, W)H1(A, B, W) 0 γ It is enough to show that γ: HomB⊗AC(J/J2, W)→HomB(I/I2, W ) is injective. But this is clear, since I⊗AC→Jis surjective and we have the isomorphisms HomB⊗AC(J/J2, W) = HomR⊗AC(J, W)and HomB(I/I2, W) = HomR(I, W) = HomR⊗AC(I⊗AC, W). Proposition 5.17. Let M→Nbe a homomorphism of monoids with Nintegral, R a ring, A=R[M],B=R[N]and Jan ideal of B. If HomZ(ker(Mgp →Ngp), W) = 0 = Ext1 Z(coker(Mgp →Ngp), W) for any B/J-module W, then (A, M)→(B, N)is log formally smooth for the J-adic topology.
5.2 Formal smoothness 75 Proof. Let Wbe a B/J-module and consider the fundamental exact sequence (see Theorem 4.3) H0(R[M],R[N], W)⊕HomZ(coker(Mgp →Ngp), W )→H0(Z[M],Z[N], W)→ →H1((A, M),(B, N), W)→ →H1(R[M], R[N], W)⊕Ext1 Z(coker(Mgp →Ngp), W)⊕HomZ(ker(Mgp →Ngp), W)→ →H1(Z[M],Z[N], W). The canonical homomorphism H0(R[M], R[N], W)→H0(Z[M],Z[N], W )is an isomorphism by propositions 3.10 and 3.18, and the homomorphism H1(R[M], R[N], W)−→ H1(Z[M],Z[N], W) is injective by Lemma 5.16. The result follows from Theorem 5.14. Proposition 5.18. Let ϕ: (A, M)→(B, N)be a homomorphism of prelog rings and Jan ideal of B. Assume that Nis integral and HomZ(ker(Mgp →Ngp), W) = Ext1 Z(coker(Mgp →Ngp), W) = HomZ(coker(Mgp →Ngp), W) = 0 for any B/J-module W. The following are equivalent: (i) ϕis log formally smooth for the J-adic topology. (ii) Bis formally smooth over A⊗Z[M]Z[N]for the J-adic topology. Proof. From the fundamental sequence and Theorem 5.14 H0(A, B, W)H0(Z[M],Z[N], W) H1((A, M),(B, N), W)H1(A, B, W)H1(Z[M],Z[N], W) β0 β1 we see that (i) holds if and only if β1is injective and β0is surjective for all B/Jmodules W. Now, from the Jacobi-Zariski exact sequence H0(A, B, W)H0(A, A ⊗Z[M]Z[N], W) H1(A⊗Z[M]Z[N], B, W)H1(A, B, W)H1(A, A ⊗Z[M]Z[N], W) β0 0 β0 1 we see that (ii) holds if and only if β0 1is injective and β0is surjective.
82 6. Regularity Now we will see that TorZ[M] 1(A, Z[N]) = 0. We have H2(A/I, B, k) = 0 and then H3(A/I, B, k) = 0 by [2, Théorème 6.25]. Therefore in the diagram H2(Z[M],Z[N], k)H2(A, A/I, k) H2(Z[M],Z[N], k)H2(A, B, k) pr1β2 α2 pr1γ2 α2is an isomorphism. Since the lower map is surjective by hypothesis, the upper one is also surjective. From the exact sequence [2, Proposition 15.18] H2(Z[M],Z[N], k)H2(A, A/I, k) TorZ[M] 1(A, Z[N]) ⊗Ak H1(Z[M],Z[N], k)H1(A, A/I, k) pr1β2 pr1β1 we deduce TorZ[M] 1(A, Z[N]) ⊗Ak= 0 and then, since Ais Noetherian local, TorZ[M] 1(A, Z[N]) = 0. (ii) ⇒(i) By [2, Proposition 15.18] we have an exact sequence for each B-module W H2(Z[M],Z[N], W)H2(A, A/I, W) TorZ[M] 1(A, Z[N]) ⊗AW H1(Z[M],Z[N], W)H1(A, A/I, W) 0. θ2 θ1 By hypothesis, TorZ[M] 1(A, Z[N]) = 0, so θ2is surjective and θ1injective. Since Hn(A/I, B, W) = 0 for all n≥2, we deduce that in the diagram H2(Z[M],Z[N], W)H2(A, A/I, W) H2(Z[M],Z[N], W)H2(A, B, W) θ2 α2 ψ2 the map α2is bijective and so ψ2is surjective.
6.1 Technical results 83 In the diagram H2(A/I, B, W) = 0 H1(Z[M],Z[N], W)H1(A, A/I, W) H1(Z[M],Z[N], W)H1(A, B, W) θ1 α1 ψ1 θ1is injective and then ψ1is also injective. From the fundamental exact sequence H2(Z[M],Z[N], W)ψ2⊕ω2 −→ H2(A, B, W)⊕TorZ[M] 1(ker(Mgp →Ngp), W)−→ −→ H2((A, M),(B, N), W)−→ −→ H1(Z[M],Z[N], W)ψ1⊕ω1 −→ H1(A, B, W)⊕ker(Mgp →Ngp)⊗ZW we see that we are done if we prove that the map H2(Z[M],Z[N], W)−→ TorZ[M] 1(ker(Mgp →Ngp), W) vanishes. We have a surjective natural homomorphism [20, Theorem 6.16] TorZ[M] 2(Z[N], W)−→ H2(Z[M],Z[N], W), so it is sufficient to show that the composition TorZ[M] 2(Z[N], W)−→ H2(Z[M],Z[N], W)−→ TorZ 1(ker(Mgp →Ngp), W) is zero. We have isomorphisms TorZ[M] 1(a, W) = TorZ[M] 2(Z[N], W), TorZ 1(ker(Mgp →Ngp), W) = TorZ[M] 1(Z[M]⊗Zker(Mgp →Ngp), W) where a= ker(Z[M]→Z[N]). Since the map a/a2−→ Z[M]⊗Zker(Mgp →Ngp)
84 6. Regularity of Proposition 3.1 vanishes after tensoring by − ⊗Z[M]Band Wis a B-module, the map TorZ[M] 1(a, W)−→ TorZ[M] 1(Z[M]⊗Zker(Mgp →Ngp), W) is zero as desired. Definition 6.2. (see [14, Theorem 6.1]) Let ((A, m, k), M)be a Noetherian local prelog ring, Ja proper ideal of Aand B=A/J. Let N=M/α−1 M(J)be the quotient of the monoid Mby the ideal α−1 M(J)and let Ibe the ideal of Agenerated by αM(M)∩J. We say that Jis a log regular ideal if the equivalent conditions of Theorem 6.1 hold. We say that ((A, m, k), M)is a log regular local ring if mis a log regular ideal. Theorem 6.3. Let ((A, m, k), M)be a Noetherian local prelog ring and consider ((A, m, k), M0)another prelog structure inducing the same log structure. Assume that Mand M0are integral. For any proper ideal Jof A,Jis log regular in ((A, m, k), M)if and only if it is log regular in ((A, m, k), M0). Proof. It is sufficient to show the result for M0=Mlog. With the notation of Definition 6.2, we have I:=< αM(M)∩J >A=< αMlog (Mlog)∩J)>since for any element x= (m, u)∈M⊕α−1 M(A∗)A∗=Mlog, we have αMlog (x) = α(m)·uwhere uis a unit in A. In particular, ker(A/I →B)do not change replacing Mby Mlog. We also have ZMlog/α−1 Mlog (J)=ZM/α−1 M(J)⊗Z[M]Z[Mlog] =Z[N]⊗Z[M]Z[Mlog], so it is enough to show that TorZ[M] 1(A, Z[N]) = TorZ[Mlog] 1A, Z[N]⊗Z[M]Z[Mlog]. For simplicity, let us denote α=αM:M→A, and factorize α−1(A∗)→A∗first as α−1(A∗)u −→ α−1(A∗)gp v −→ A∗ and factorize again the second map as follows: α−1(A∗)u −→ α−1(A∗)gp v1 −→ α−1(A∗)gp/ker vv2 −→ A∗.
6.1 Technical results 85 Therefore, the pushout that defines Mlog α−1(A∗)A∗ M Mlog can be descompose in a commutative diagram of pushouts α−1(A∗)α−1(A∗)gp α−1(A∗)gp/ker(v)A∗ M M1M2Mlog u r v1 s v2 Note that sis injective since it is a localization of the injective homomorphism rand Mis integral. Now, we denote as N1N2 N3N4 any of previous pushouts and consider WaZ[Mlog]-module and P=N3/T with Tan ideal of N3. Let N3→X→Pa factorization cofibration-trivial fibration in the category of simplicial monoids. Then Z[X]is a projective Z[N3]-resolution of Z[P][10, I.2.2.3] and so TorZ[N3] ∗(W, Z[P]) = H∗(W⊗Z[N3]Z[X]) = H∗(W⊗Z[N4]Z[N4]⊗Z[N3]Z[X]) (1) = TorZ[N4] ∗(W, Z[N4]⊗Z[N3]Z[P]) where the equality (1) follows from: H∗(Z[N4]⊗Z[N3]Z[X]) = H∗(Z[N3⊕N1N2]⊗Z[N3]Z[X]) =H∗(Z[N3]⊗Z[N1]Z[N2]⊗Z[N3]Z[X]) =H∗(Z[N2]⊗Z[N1]Z[X]) (2) =Z[N2]⊗Z[N1]Z[P] =Z[N4]⊗Z[N3]Z[P]
86 6. Regularity where the equality (2) follows (depending of the pushout) from: • In the first pushout, since Z[N2]is a flat Z[N1]module (because uis a localization and then Z[u]is a localization too). • In the second one, it follows by Remark 4.13, since Z[N2]⊗Z[N1]Z[X] = Z[X/ ker(v)] is a resolution of Z[P/ ker(v)] = Z[N2]⊗Z[N1]Z[P]. Note that Pis M/T ⊕MM1=M1/e Tin this case (where e Tdenotes the ideal image of Tin M1) and ker(v)→α−1(A∗)gp →M1is injective. Moreover, since ˜ Tis an ideal of M1,ker(v)→M1/˜ Tis injective. On the other hand, the action of ker(v)on Xis free, since X=M1⊕Ne X, and M1is integral (the homomorphism uis integral by [9, Remark 6.2.5.(i)] and so M1is integral because Mit is also integral). • In the last pushout, it follows because Z[N2]is a free Z[N1]module (since v2 is injective homomorphism of groups). Finally, applying successively this process to the three pushouts, we obtain as desired TorZ[M] ∗(A, Z[N]) = TorZ[Mlog] ∗(A, Z[N]⊗Z[M]Z[Mlog]). Corollary 6.4. With the notation of Theorem 6.1, assume that Mgp is a free Abelian group. Then Jis log regular if and only if H2((A, M),(B, N), k) = 0. By Lemma 6.6, we can always work in this case (Mgp free). Lemma 6.5. Let p:G1→G2be a surjective homomorphism of Abelian groups, i:N→G2a homomorphism of monoids. Then (G1×G2N)gp =G1×G2Ngp. Proof. The homomorphism (G1×G2N)gp −→ G1×G2Ngp (g1, n1) (g2, n2)7−→ g1 g2 ,n1 n2 is well defined and has inverse g, n1 n27−→ (gh, n1) (h, n2) where h∈p−1(i(n2)).
6.1 Technical results 87 Lemma 6.6. Let (B, N)be a log ring. There exists a monoid Pand a homomorphism of monoids P→Nsuch that (B, N) = (B, P)log and Pgp is a free Abelian group. Moreover, if Nis integral, then Pcan be chosen integral too. Proof. [9, Corollary 12.1.35] Let Gbe a free Abelian group with a surjective homomorphism of groups ρ:G→Ngp. Let P:= G×Ngp N. We have a commutative diagram of pullbacks H:= (βτ)−1(B∗)B∗ P N G Ngp π τ ji ρ where β:N→Bis the structural homomorphism. So ker π(:= π−1({1})) = ker τ= ker ρand ker ρis a group. If P/ ker(τ)denotes the quotient by the action of ker(τ)on P, it is easy to check that P/ ker(τ) = Nand H/ ker(π) = B∗. Moreover, P⊕HB∗=P/ ker(τ)⊕H/ ker(π)B∗=N⊕B∗B∗=N and so (B, P)log = (B, N). Finally, Pgp =Gby Lemma 6.5, and so it is a free Abelian group. Finally, if Nis integral, that is, iis injective, then jis injective, and therefore P is integral. Lemma 6.7. Let ((A, m, k), M)be a Noetherian local prelog ring and the maximal ideal mM=M−M∗of M. Assume that Mis integral. If TorZ[M] 1(A, Z[M/mM]) = 0, then TorZ[M] 1(A, Z[M/I]) = 0 for any ideal Iof M. Proof. Since Mis Noetherian, there exists a maximal element of the set of ideals of Msuch that TorZ[M] 1(A, Z[M/I]) 6= 0, assuming this set is not empty. Clearly, I⊂mM(if not I=M), and then by hypothesis, I(mM. Let a∈mM−I. Let T={x∈M/xa ∈I}, so that we have an exact sequence 0−→ Z[M/T]·a −→ Z[M/I]−→ Z[M/I ∪Ma]−→ 0.
88 6. Regularity We have I⊂T. If I(T,TorZ[M] 1(A, Z[M/T]) = 0 = TorZ[M] 1(A, Z[M/I ∪Ma]) by the maximality of I. So TorZ[M] 1(A, Z[M/I]) = 0. If I=T, we have a surjective homomorphism of A-modules of finite type ·˜a: TorZ[M] 1(A, Z[M/I]) −→ TorZ[M] 1(A, Z[M/I]) where ˜a:= α(a)∈m⊂A. By Nakayama Lemma, TorZ[M] 1(A, Z[M/I]) = 0. Corollary 6.8. Let ((A, m, k), M)be a Noetherian local prelog ring and the maximal ideal mM=M−M∗of M. Assume that Mis integral. If TorZ[M] 1(A, Z[M/mM]) = 0, then TorZ[M] n(A, Z[M/mM]) = 0 for all n≥1. Proof. By Lemma 6.7, it is sufficient to show that if TorZ[M] n(A, Z[M/I]) for any ideal Iof Mfor some n≥1, then TorZ[M] n+1 (A, Z[M/I]) = 0 for any ideal Iof M. So assume TorZ[M] n(A, Z[M/I]) = 0 for any ideal Iof M. By [9, 6.1.27.(i)-(iii)], we have then TorZ[M] n(A, Z[T]) = 0 for any M-set T. Applying Tor to the exact sequence 0−→ Z[I]−→ Z[M]−→ Z[M/I]−→ 0, we obtain 0 = TorZ[M] n+1 (A, Z[M]) −→ TorZ[M] n+1 (A, Z[M/I]) −→ TorZ[M] n(A, Z[I]) = 0 and so TorZ[M] n+1 (A, Z[M/I]) = 0. Theorem 6.9. Let ((A, m, k), M)be a Noetherian local prelog ring and mM= M−M∗. The following are equivalent: (i) (A, M)is a log regular local ring. (ii) The map H2(A, k, k)⊕TorZ 1(ker(Mgp →(M/mM)gp), k)−→ H2((A, M),(k, M/mM), k) is zero on the first summand and an isomorphism on the second one. If moreover Mis integral, then these conditions imply: (iii) Hn((A, M),(k, M/mM), k) = 0 for any n≥3.
6.2 Applications 89 Proof. Equivalence of (i) and (ii) follows from Theorem 6.1. We will see (iii). By Corollary 6.8, TorZ[M] n(A, Z[M/mM]) = 0 for all n > 0and so by base change [2, Proposition 4.54] we have isomorphisms Hi(Z[M],Z[M/mM], k)−→ Hi(A, A/α(mM)A, k) for all i≥0. Since A/α(mM)Ais a regular ring, we have by [2, Proposition 6.26, Théorème 5.1] the isomorphisms Hi(A, A/α(mM)A, k)−→ Hi(A, k, k) for all i≥2. Therefore Hi(Z[M],Z[M/mM], k)−→ Hi(A, k, k) are isomorphisms for all i≥2, and then from the fundamental exact sequence we deduce Hn((A, M),(k, M/mM), k)=0 for all n≥3. Definition 6.10. We say that a local prelog ring ((A, m, k), M)is log complete intersection if H3((A, M),(k, M/mM), k) = 0. In particular, a log regular ring with Mintegral is log complete intersection. 6.2 Applications From the fundamental exact sequence (Theorem 4.3) we obtain the following theorem: Theorem 6.11. Let ((A, m, k), M)be a Noetherian local prelog ring with Man integral monoid. (i) If (A, M)is a log regular ring, then Ais a regular local ring if and only if ker(Z[M]→Z[M/mM]) is generated by a regular sequence. (ii) If Ais a regular local ring, then (A, M)is log complete intersection if and only if ker(Z[M]→Z[M/mM]) is generated by a regular sequence. Proof. It follows from Theorem 4.3, Theorem 6.9 and [2, Théorème 6.25], having in mind that we have seen at the end of the proof of Theorem 6.1 that the map H2(Z[M],Z[M/mM], k)−→ TorZ[M] 1(ker(Mgp →(M/mM)gp), k) vanishes.
90 6. Regularity Theorem 6.12. Let (A, M)be a Noetherian local prelog ring, Ja proper ideal of A,B=A/J and N=M/α−1 M(J), where αM:M→Ais the structural map. (i) If (A, M)is log regular and (B, N)is log complete intersection, then Jis a log regular ideal. (ii) If (B, N)is log regular and Jis a log regular ideal, then (A, M)is a log regular local ring. (iii) Assume that Mis integral and Mgp free (for instance if Mis saturated and sharp [18, I.1.3.5]). If (A, M)is log regular (or log complete intersection) and Jis log regular, then (B, N)is log complete intersection. Proof. (i) Assume first that α−1 M(J)6=∅, so Ngp ={1}. If (B, N)is log complete intersection, then the map α:H2((A, M),(B, N), k)−→ H2((A, M),(k, M/mM), k) is injective. In the commutative diagram H2(Z[M],Z[N], k)H2(Z[M],Z[M/mM], k) H2(A, B, k)⊕TorZ 1(Mgp, k)H2(A, k, k)⊕TorZ 1(Mgp, k) H2((A, M),(B, N), k)H2((A, M),(k, M/mM), k) ε⊕id β1⊕β2γ1⊕γ2 α we know that γ2is an isomorphism and γ1= 0. Therefore β2is surjective and β1= 0. But β2is injective (we have seen in the last lines of the proof of Theorem 6.1 that the map H2(Z[M],Z[N], k)−→ TorZ 1(Mgp, k)vanishes), and then Jis log regular by Theorem 6.1. Now if α−1 M(J) = ∅, then M=N, so ker(Mgp →Ngp) = {1}. Therefore J is a log regular ideal if and only if H2((A, M),(B, N), k) = 0. As in the previous diagram, we obtain that the map β1:H2(A, B, k)−→ H2((A, M),(B, N), k) vanishes. But since M=N, this map is an isomorphism (Example 4.2).
6.2 Applications 91 (ii) Assume first that α−1 M(J)6=∅. We have the above commutative diagram again, where now αis surjective (since TorZ 1(Ngp, k) = H2((B, N),(k, M/mM), k)) is zero, β1= 0 and β2is an isomorphism. This implies that γ2is an isomorphism since the map H2(Z[M],Z[M/mM], k)→TorZ 1(Mgp, k)is zero as above. Also, γ1= 0 by diagram chasing. If α−1 M(J) = ∅, the proof is similar to the proof of (i) when α−1 M(J)6=∅. (iii) By Corollary 6.4, Jis log regular if and only if H2((A, M),(B, N), k) = 0. Then the result follows from the Jacobi-Zariski exact sequence and Theorem 6.9.(iii). Remark 6.13. Theorem 6.12.(i) together with Definition 6.10 extends to the logarithmic setting the fact that a closed immersion between regular schemes is a regular immersion. Another extension of this fact was already proved by Kato [14, Theorem 4.2]: Theorem. Let (A, M)be a Noetherian log ring with Mintegral and Ja proper ideal of A. Let (B, N) = (A/J, p∗M)where p:A→Bis the canonical projection. If (A, M)and (B, N)are log regular, then Jis generated by a regular sequence. We can deduce this theorem easily from our results: Let P→Mbe a homomorphism of monoids with Pintegral and Pgp free such that (A, P)log = (A, M) (Lemma 6.6). By Lemma 2.5, (B, N) = (B, P)log. Also, by Theorem 6.3, (A, P ) and (B, P)are log regular. We have then an exact sequence, where kis the residue field of Aand B, H3((B, P),(k, P/mP), k)→H2((A, P),(B, P), k)→H2((A, P),(k, P/mP), k) and where the modules to the left and right are zero by Theorem 6.9.(iii) and Corollary 6.4. Therefore, by Example 4.2, H2(A, B, k) = H2((A, P),(B, P), k)=0, and so Jis generated by a regular sequence by [2, Théorème 6.25]. Example 6.14. Let (A, m, k), M)be a Noetherian local prelog ring with Mintegral saturated and sharp. Let ˆ Abe the completion of Afor the m-adic topology and consider the prelog ring (ˆ A, M). We have Hn((A, M),(ˆ A, M), k)=0for all n≥0 by Example 4.2 and [2, Remarque 10.22]. Then from the Jacobi-Zariski exact sequence we obtain Hn((A, M),(k, M/mM), k) = Hn(( ˆ A, M),(k, M/mM), k) for all n≥0. In particular, (A, M)is log regular (resp. log complete intersection) if and only if so is (ˆ A, M). Let R:= C[[M]][[x1, . . . , xn]] →ˆ Abe a surjective
98 Resumo en galego A continuación describiremos os contidos da tese. Nos capítulos 1 e 2 dáse unha rápida revisión das definicións e notacións de monoides e aneis logarítmicos que serán empregadas durante os seguintes capítulos. No Capítulo 3 estudamos as diferenciais logarítmicas e as derivacións logarítmicas. Non hai nada esencialmente novo nesta parte, pero dado que non atopamos referencias na literatura para algúns destes resultados, polo que preferimos incluír unha exposición dos mesmos. A Sección 4.1 céntrase no desenvolvemento do complexo logarítmico cotanxente de Gabber. O estudo realizado en [19] non é suficiente para os nosos propósitos, dado que carece de resultados que necesitamos en certos momentos, outros son incluídos pero con hipóteses máis fortes que as que podemos asumir (cambio de base ou da localización). Esta sección contén tamén un dos dous principais resultados técnicos, ao que chamaremos triángulo fundamental (Teorema 4.3): se (A, M)→(B, N)é un homomorfismo de aneis prelogarítmicos e Wé un B-módulo, temos un triángulo distinguido na categoría derivada de Z-módulos LZ[N]|Z[M]⊗Z[N]W−→ (LB|A⊗BW)⊕(XN|M⊗ZW)−→ L(B,N)|(A,M)⊗BW−→ onde os dous complexos Lda esquerda son os complexos cotanxentes “usuais” e o L da dereita é o complexo cotanxente logarítmico, mentres que XN|Mé un complexo que aparece no triángulo distinguido Mgp −→ Ngp −→ XN|M−→ . Este resultado é fundamental e será empregado de comezo a fin, xa que nos permitirá relacionar propiedades do homomorfismo logarítmico (A, M)→(B, N) con propiedades dos homomorfismos de aneis A→BeZ[M]→Z[N]a través da sucesión exacta fundamental ··· Hn(Z[M],Z[N],W)Hn(A, B, W )⊕Hn(XN|M⊗ZW)Hn((A, M),(B, N), W) Hn−1(Z[M],Z[N], W)··· H0((A, M),(B, N), W) 0, onde •Hn(XN|M⊗ZW)=0para n≥3, •H2(XN|M⊗ZW) = TorZ 1(ker(Mgp →Ngp), W), •H1(XN|M⊗ZW)é un B-módulo que aparece na sucesión exacta (que escinde de xeito non canónico) 0→ker(Mgp →Ngp)⊗ZW→H1(XN|M⊗ZW)→TorZ 1(Ngp/Im(Mgp →Ngp), W)→0, •H0(XN|M⊗ZW) = Ngp/Im(Mgp →Ngp)⊗ZW.
99 Incluímos tamén unha demostración do feito de que a loguificación non cambia a (co)homoloxía do complexo cotanxente logarítmico cando as estruturas son íntegras. Este resultado aparece en [19]. Non sabemos se as hipóteses de integridade son necesarias. Nalgúns puntos do Capítulo 6 non poderemos ir máis lonxe por mor destas hipóteses de integridade (mesmo para estruturas íntegras, xa que aparecen cocientes non íntegros nas probas). Esta é a razón pola cal incluímos unha demostración alternativa deste teorema de Olsson: non fomos quen de evitar as hipóteses de integridade, pero pensamos que a nosa proba está un pouco máis preto de poder evitalas. A Sección 4.2 non é necesaria para o resto do contido da memoria, pero ten un interese independente. Nela damos unha versión logarítmica e análoga da sucesión espectral de Quillen. Se C→Bé un homomorfismo sobrexectivo de aneis logarítmicos, Quillen dá unha sucesión espectral converxente [20,22] E2 p,q =Hp+q(SqLB|A)⇒TorC p+q(B, B) onde SqLB|Aé a q-ésima potencia simétrica do complexo cotanxente. Se ademais C=B⊗AB, onde Bé unha A-álxebra plana, esta sucesión espectral toma a forma E2 p,q =Hp(ΛqLB|A)⇒HHp+q(B|A) onde Λqdenota a q-ésima potencia exterior e HH é a homoloxía de Hochschild. Nós damos unha sucesión espectral similar no caso logarítmico (Teorema 4.21). Aínda que a sucesión espectral non logarítmica ten un papel fundamental nos traballos de Quillen [20, 22], o noso único interese para obter a versión logarítmica é que o seu abutment (que definimos de forma explícita na memoria) dá outro candidato para ser a (co)homoloxía logarítmica de Hochschild. O Capítulo 4 finaliza coa Sección 4.3, onde adaptamos os métodos de [15] e [2, Proposition 15.12] co obxectivo de obter o complexo máis pequeno que sexa útil para calcular a segunda (co)homoloxía de grupos do complexo cotanxente logarítmico. O Capítulo 5 dedícase ao estudo (co)homolóxico da lisitude logarítmica. A idea é evitar as hipótesis de finitude nos morfismos e queremos tamén evitar as hipóteses de integridade tanto como sexa posible, polo cal os primeiros pasos feitos en [19] e [12] non nos son de axuda. Necesitamos polo tanto comezar dende o inicio. Procederemos da forma habitual: primeiro caracterizamos os elementos de H1como extensións de álxebras logarítmicas. No caso íntegro está feito en [19], pero en xeral a propia definición de extensión está máis involucrada, e así a nosa demostración é diferente e substancialmente máis longa ca no caso íntegro. Posteriormente deducimos algúns resultados que caracterizan a lisitude logarítmica a través da anulación do complexo cotanxente logarítmico. Este proceso é o estándar para a lisitude logarítmica, pero necesitamos ter en conta as topoloxías, é dicir, caracterizar a lisitude logarítmica
100 Resumo en galego formal (dado que é necesaria cando os homomorfismos non son de tipo finito), dando lugar a demostracións máis longas. Por exemplo, obtemos os teoremas 5.14 e 5.15. Este capítulo remata con algunhas aplicacións de menor importancia: a Proposición 5.18 e a Proposición 5.19. En particular, a Proposición 5.19 é unha ampla xeneralización da parte de lisitude de [11, Theorem 0.1]. Para unha mellor comparación destes resultados, engadimos o seguinte corolario desta proposición: Corolario 5.20. Sexan (A, M)→(B, N)e(A, M)→(C, P)homomorfismos de aneis prelogarítmicos con Spec(C)→Spec(A)sobrexectivo e supoñamos que Z[M]→Z[P]eA→Cson planos, e tamén que Mgp →Pgp é inxectivo. Consideramos un ideal Jde B. Entón (B, N)é unha (A, M)-álxebra formalmente lisa para a topoloxía J-ádica se e só se (B⊗AC, N ⊕MP)é unha (C, P )-álxebra formalmente lisa para a topoloxía J⊗AC-ádica. Finalmente, no Capítulo 6, estamos preparados para estudar a regularidade logarítmica. Comezamos coa caracterización desta en termos do complexo cotanxente logarítmico, que é o noso segundo principal resultado técnico (Teorema 6.3 a Teorema 6.9): Teorema. Sexa ((A, m, k), M)un anel (pre)logarítmico local noetheriano e mMo conxunto de non unidades de M. (i) (A, M)é un anel logarítmico local regular se e só se o conúcleo do homomorfismo canónico inxectivo TorZ 1(ker(Mgp →(M/mM)gp), k)−→ H2((A, M),(k, M/mM), k) se anula. (ii) Existe unha estrutura prelogarítmica (A, P)en Aque induce a mesma estrutura logarítmica que (A, M)e tal que (A, P)é log regular se e só se H2((A, P),(k, P/mP), k) = 0. (iii) Se Mé un monoide íntegro, o monoide Pen (ii) tamén pode ser escollido íntegro, e polo tanto (A, P)é log regular se e só se (A, M)é log regular. (iv) Se Mé un monoide íntegro, entón que (A, M)sexa log regular implica que Hn((A, M),(k, M/mM), k) = 0 para todo n≥3. En particular, o apartado (i) do teorema anterior dedúcese dun resultado máis xeral, que non só é válido para o ideal maximal m, senón que tamén é certo para calquera ideal de A. Isto permite introducir o concepto de ideal log regular. Este xoga un papel similar ao da regularidade de ideais no caso non logarítmico. Por exemplo,
101 probamos que se (A, M)é un anel local log regular e Jun ideal propio de A, entón Jé un ideal log regular se A/J, M/α−1(J)é log intersección completa (onde α:M→Aé a aplicación canónica), e a implicación contraria é certa baixo certas hipóteses máis relaxadas. A definición é a seguinte: Sexa ((A, m, K), M)un anel prelogarítmico local noetheriano, Jun ideal propio de A,α:M→Ao homomorfismo estrutural e N=M/α−1(J)o monoide cociente. Sexa Io ideal de Axerado por α(M)∩J. Dise que Jé un ideal log regular se se verifican as dúas condicións seguintes: (i) O ideal ker(A/I →A/J)de A/I está xerado por unha sucesión regular. (ii) TorZ[M] 1(A, Z[N]) = 0. Esta definición ten unha cómoda caracterización homolóxica (Teorema 6.1): Teorema. Baixo as condicións anteriores, son equivalentes: (i) Jé log regular. (ii) A aplicación H2(A, A/J, k)⊕TorZ 1(ker(Mgp →Ngp), k)→H2((A, M),(A/J, N), k) na sucesión fundamental exacta é a aplicación nula no primeiro sumando e un isomorfismo no segundo. (iii) Para todo A/J-módulo W, a aplicación H2(A, A/J, W)⊕TorZ 1(ker(Mgp →Ngp), W)→H2((A, M),(A/J, N), W) na sucesión fundamental exacta é a aplicación nula no primeiro sumando e un isomorfismo no segundo. Á vista destes resultados, estamos xa en disposición de estudar a regularidade logarítmica. Por exemplo, a partir do triángulo fundamental chegamos de xeito inmediato á relación entre regularidade logarítmica e regularidade dun anel logarítmico: Teorema 6.11. Sexa ((A, m, k), M)un anel prelogarítmico local noetheriano con Mun monoide íntegro. (i) Se (A, M)é un anel log regular, entón temos que Aé un anel regular se e só se ker(Z[M]→Z[M/mM]) está xerado por unha sucesión regular. (ii) Se Aé un anel local regular, entón cúmprese que (A, M)é log intersección completa se e só se ker(Z[M]→Z[M/mM]) está xerado por unha sucesión regular. Este derradeiro capítulo remata cos teoremas 6.15 e 6.16, que xa foron enunciados ao comezo deste resumo.
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Index Conormal module (logarithmic), 16 Cotangent complex (logarithmic), 30 Essentially of finite type (homomorphism of prelog rings), 10 Exact homomorphism of monoids, 8 Extension of prelog rings, 61 Fine monoid, 7 Group completion, 6 Homomorphism of monoids, 1 of prelog rings, 9 Integral homomorphism of monoids, 8 monoid, 6 Kummer homomorphism of monoids, 8 Log complete intersection ring, 89 Log formally smooth (homomorphism of prelog rings), 68 Log regular ideal, 84 Log regular local ring, 84 Log ring, 9 Logification of a prelog ring, 9 Module of differentials (logarithmic), 17 Monoid, 1 Monoid algebra, 7 Noetherian local prelog ring, 10 Prelog ring, 9 Saturated monoid, 7 Sharp monoid, 6 Strict homomorphism of log rings, 9 (A, M)log, 9 Hn((A, M),(B, N), W), 30 Hn((A, M),(B, N), W), 30 LB|A, 30 L(B,N)|(A,M), 30 Mgp, 6 NX, 4 R[M], 7 Ω(B,N)|(A,M), 17