§6. Local and global Tor functors; the Künneth formula

Translator's note. This section is translated in two parts for length: §6.1–§6.6 here, §6.7–§6.9 in the companion file. They will be concatenated for the final volume.

6.1. Introduction

6.1.1.

Let $f:X\to Y$ be a morphism of preschemes and let $\mathcal{F}$ be a quasi-coherent ${\mathcal{O}}_X$-module. In the study of the "higher direct images" $R^n{f}_{\ast }(\mathcal{F})$, one is led to consider the following general problem: given a "base change" morphism $g:Y^{\prime }\to Y$, we set $X^{\prime }=X_{(Y^{\prime })}=X{\times }_Y{Y}^{\prime }$, ${\mathcal{F}}^{\prime }=\mathcal{F}{\otimes }_{{\mathcal{O}}_Y}{\mathcal{O}}_{Y^{\prime }}$, $f^{\prime }=f_{(Y^{\prime })}:X^{\prime }\to Y^{\prime }$, and we propose to obtain information on the higher direct images $R^n{f}_{\ast }^{\prime }({\mathcal{F}}^{\prime })$ (assuming the $R^n{f}_{\ast }(\mathcal{F})$ known). One sees easily (cf. for example (7.7.2)) that one is reduced to studying the variations of $R^n{f}_{\ast }(\mathcal{F}{\otimes }_{{\mathcal{O}}_Y}\mathcal{G})$ for a variable quasi-coherent ${\mathcal{O}}_Y$-module $\mathcal{G}$, in other words the functor $\mathcal{G}\mapsto R^n{f}_{\ast }(\mathcal{F}{\otimes }_{{\mathcal{O}}_Y}\mathcal{G})$. If $\mathcal{F}$ is flat over $Y$, the functor $\mathcal{G}\mapsto \mathcal{F}{\otimes }_{{\mathcal{O}}_Y}\mathcal{G}$ is exact $(0_I,\mathrm{6.7.4})$, and consequently the composite functor $\mathcal{G}\mapsto R^n{f}_{\ast }(\mathcal{F}{\otimes }_{{\mathcal{O}}_Y}\mathcal{G})$ is again a cohomological functor. But this is no longer the case in general; in order to be able to apply the cohomological methods, one is led to substitute for $\mathcal{G}\mapsto R^n{f}_{\ast }(\mathcal{F}{\otimes }_{{\mathcal{O}}_Y}\mathcal{G})$ other functors which this time are always cohomological functors. These functors, which generalize the "Tor" functors of module theory, are defined in n°s 6.3 to 6.7; there are moreover two such generalizations, one "local" and the other "global", related by spectral sequences that will be discussed in n° 6.7; as an application of these spectral sequences, one obtains in particular, under certain conditions, a "Künneth formula" expressing $R^n(f_1\times f_2{)}_{\ast }({\mathcal{F}}_1{\otimes }_Y{\mathcal{F}}_2)$ by means of the higher direct images $R^p{f}_{1\ast }({\mathcal{F}}_1)$ and $R^q{f}_{2\ast }({\mathcal{F}}_2)$. Other spectral sequences (6.8) generalize the associativity spectral sequences of the "Tor" functor of modules; finally, the base-change problem itself leads to spectral sequences (6.9).

6.1.2.

In addition, one observes (6.10) that the cohomological functors $\mathcal{G}\mapsto {\mathcal{T}}_i(\mathcal{G})$ so defined are locally (on $Y$) of the type $\mathcal{G}\mapsto {\mathcal{H}}_i({\mathcal{L}}_{\bullet }{\otimes }_Y\mathcal{G})$, where ${\mathcal{L}}_{\bullet }$ is a complex of locally free ${\mathcal{O}}_Y$-modules (defined up to homotopy) and ${\mathcal{H}}_{\bullet }$ denotes homology. There is then an interest in forgetting the particular situation that gave

rise to ${\mathcal{T}}_i$, and in studying in general the functors of the preceding form $\mathcal{G}\mapsto {\mathcal{H}}_i({\mathcal{L}}_{\bullet }{\otimes }_Y\mathcal{G})$ (where one moreover, if appropriate, makes finiteness hypotheses on the ${\mathcal{L}}_i$ or the ${\mathcal{H}}_i({\mathcal{L}}_{\bullet })$): this is the object of §7, whose reading, for the essentials, is independent of §6. The most important properties of these functors concern the exactness properties of a component ${\mathcal{T}}_i$ of ${\mathcal{T}}_{\bullet }$; we shall give various criteria allowing such properties to be established; as an application, one will obtain conditions allowing one to assert (with the notations of (6.1.1)) that the functor $\mathcal{G}\mapsto R^n{f}_{\ast }(\mathcal{F}{\otimes }_{{\mathcal{O}}_Y}\mathcal{G})$ is exact (which one will express by saying that $\mathcal{F}$ is cohomologically flat over $Y$ in dimension $n$). Another important property for the components ${\mathcal{T}}_i$ of ${\mathcal{T}}_{\bullet }(\mathcal{G})={\mathcal{H}}_i({\mathcal{L}}_{\bullet }{\otimes }_Y\mathcal{G})$ is the semi-continuity property of the function $y\mapsto {\dim }_{\kappa (y)}({\mathcal{T}}_i(\kappa (y)))$; when ${\mathcal{T}}_i$ is exact, this property is replaced by a continuity property, the converse being moreover true according to Grauert when $Y$ is reduced (7.8.4).

6.1.3.

In §§ 6 and 7, we have systematically made use of hypercohomology, taking everywhere as arguments complexes of sheaves instead of sheaves, although the necessity of this point of view will appear only in later chapters. The cohomological formalism developed on this occasion will moreover become more transparent in the chapter of this Treatise that will be devoted to the elaboration of an algebra of cohomological functors of coherent sheaves, including the duality formalism. But that will require developments which fall outside the framework of the present chapter.

6.1.4.

To abbreviate, given two complexes $K^{\bullet }$, $K^{\prime \bullet }$ in an abelian category $\mathcal{C}$, we shall say that a morphism of complexes $f:K^{\bullet }\to K^{\prime \bullet }$ is a homotopism if there exists a morphism $g:K^{\prime \bullet }\to K^{\bullet }$ such that the composite morphisms $f\circ g$ and $g\circ f$ are both homotopic to the identity (by abuse of language, when such a homotopism exists, one also says that $K^{\bullet }$ and $K^{\prime \bullet }$ are homotopic). When one can define the hypercohomology of a covariant additive functor $T$ from $\mathcal{C}$ into an abelian category ${\mathcal{C}}^{\prime }$ with respect to a complex of $\mathcal{C}$ $(0_{III},\mathrm{11.4.3})$, it is immediate that a homotopism $K^{\bullet }\to K^{\prime \bullet }$ of complexes of $\mathcal{C}$ canonically defines an isomorphism $R^{\bullet }T(K^{\bullet })\to R^{\bullet }T(K^{\prime \bullet })$ for hypercohomology (loc. cit.).

6.2. Hypercohomology of complexes of modules on a prescheme

6.2.1.

Let $X$ be a prescheme and ${\mathcal{K}}^{\bullet }=({\mathcal{K}}^i{)}_{i\in \mathbb{Z}}$ a complex of ${\mathcal{O}}_X$-modules whose differential is of degree +1. Recall that for every morphism $f:X\to Y$ of preschemes, one has defined $(0_{III},\mathrm{12.4.1})$ the ${\mathcal{O}}_Y$-modules of hypercohomology ${\mathcal{H}}^n(f,{\mathcal{K}}^{\bullet })$ (also denoted ${\mathcal{H}}^n({\mathcal{K}}^{\bullet })$ or $R^n{f}_{\ast }({\mathcal{K}}^{\bullet })$) for every $n\in \mathbb{Z}$; the hypercohomology ${\mathcal{H}}^{\bullet }(f,{\mathcal{K}}^{\bullet })$ is the abutment of the two spectral functors ${}^{\prime }\mathcal{E}(f,{\mathcal{K}}^{\bullet })$ and ${}^{\prime \prime }\mathcal{E}(f,{\mathcal{K}}^{\bullet })$, whose E_2 terms are given by

  'E_2^{p,q} = ℋ^p(ℋ^q(f, 𝒦^•))                                                (6.2.1.1)
  ''E_2^{p,q} = ℋ^p(f, ℋ^q(𝒦^•)) = R^p f_*(ℋ^q(𝒦^•))                            (6.2.1.2)

where ${\mathcal{H}}^{\bullet }(f,{\mathcal{K}}^{\bullet })$ is the complex whose component of degree $i$ is ${\mathcal{H}}^i(f,{\mathcal{K}}^{\bullet })=R^i{f}_{\ast }({\mathcal{K}}^i)$ (loc. cit.). Recall also that when $Y$ is reduced to a point, one denotes the corresponding hypercohomology by $H^{\bullet }(X,{\mathcal{K}}^{\bullet })$ (which is formed of modules over $\Gamma (X,{\mathcal{O}}_X)$ independent of the punctual prescheme $Y$ considered); when $Y=X$ and $f=1_X$, one has ${\mathcal{H}}^n(f,{\mathcal{K}}^{\bullet })={\mathcal{H}}^n({\mathcal{K}}^{\bullet })$ (cohomology of the complex ${\mathcal{K}}^{\bullet }$); when ${\mathcal{K}}^i=0$ except for $i=i_0$, one has

  ℋ^n(f, 𝒦^•) = R^{n − i_0} f_*(𝒦^{i_0}).

The spectral sequence ${}^{\prime }\mathcal{E}(f,{\mathcal{K}}^{\bullet })$ is always regular; the two spectral sequences are biregular when ${\mathcal{K}}^{\bullet }$ is bounded below $(0_{III},\mathrm{12.4.1})$. Every homotopism $h:{\mathcal{K}}^{\bullet }\to {\mathcal{K}}^{\prime \bullet }$ of complexes of ${\mathcal{O}}_X$-modules (6.1.4) gives an isomorphism ${\mathcal{H}}^{\bullet }(f,{\mathcal{K}}^{\bullet })\stackrel{\sim }{\to }{\mathcal{H}}^{\bullet }(f,{\mathcal{K}}^{\prime \bullet })$ for hypercohomology. The same holds when one assumes only that ${\mathcal{H}}^{\bullet }(h):{\mathcal{H}}^{\bullet }({\mathcal{K}}^{\bullet })\to {\mathcal{H}}^{\bullet }({\mathcal{K}}^{\prime \bullet })$ is an isomorphism and that ${\mathcal{K}}^{\bullet }$ and ${\mathcal{K}}^{\prime \bullet }$ are bounded below, as follows at once from $(0_{III},\mathrm{11.1.5})$ applied to the spectral sequence (6.2.1.2) and to the analogous one for ${\mathcal{K}}^{\prime \bullet }$. Finally, for every open cover $\mathfrak{U}=(U_{\alpha })$ of $X$, one has also defined $(0_{III},\mathrm{12.4.5})$ the hypercohomology $H^{\bullet }(\mathfrak{U},{\mathcal{K}}^{\bullet })$ as the cohomology of the bicomplex $C^{\bullet }(\mathfrak{U},{\mathcal{K}}^{\bullet })$ (whose component of bi-indices $(i,j)$ is by definition $C^i(\mathfrak{U},{\mathcal{K}}^j)$); the $H^n(\mathfrak{U},{\mathcal{K}}^{\bullet })$ are again modules over $\Gamma (X,{\mathcal{O}}_X)$.

Proposition (6.2.2).

Let $X$ be a scheme and $\mathfrak{U}=(U_{\alpha })$ a cover of $X$ by affine open sets. For every complex ${\mathcal{K}}^{\bullet }$ of quasi-coherent ${\mathcal{O}}_X$-modules, the hypercohomology modules $H^{\bullet }(X,{\mathcal{K}}^{\bullet })$ and $H^{\bullet }(\mathfrak{U},{\mathcal{K}}^{\bullet })$ are canonically isomorphic.

Proof. Indeed, every finite intersection $V$ of open sets of the cover $\mathfrak{U}$ is affine (I, 5.5.6), so $H^q(V,{\mathcal{K}}^i)=0$ for every $i$ and every $q>0$ (1.3.1); the proposition is therefore a particular case of $(0_{III},\mathrm{12.4.7})$.

Proposition (6.2.3).

Let $f:X\to Y$ be a quasi-compact and separated morphism of preschemes. For every complex ${\mathcal{K}}^{\bullet }$ of quasi-coherent ${\mathcal{O}}_X$-modules, the ${\mathcal{O}}_Y$-modules ${\mathcal{H}}^n(f,{\mathcal{K}}^{\bullet })$ are quasi-coherent.

Proof. Since the ${\mathcal{H}}^q(f,{\mathcal{K}}^i)=R^q{f}_{\ast }({\mathcal{K}}^i)$ are quasi-coherent ${\mathcal{O}}_Y$-modules (1.4.10), the same holds for ${}^{\prime }{E}_2^{p,q}$, which, by (6.2.1.1), is a quotient of a kernel of a homomorphism of quasi-coherent modules by an image of such a homomorphism (I, 4.1.1). For the same reason, all the ${\mathcal{O}}_Y$-modules ${}^{\prime }{E}_r^{p,q}$, $B_r{(}^{\prime }{E}_r^{p,q})$, $Z_r{(}^{\prime }{E}_r^{p,q})$ of the first spectral sequence are quasi-coherent. The regularity of the spectral sequence ${}^{\prime }\mathcal{E}(f,{\mathcal{K}}^{\bullet })$ implies that $Z_{\infty }{(}^{\prime }{E}_2^{p,q})$ is equal to one of the $Z_k{(}^{\prime }{E}_2^{p,q})$, hence is quasi-coherent, and the same holds for $B_{\infty }{(}^{\prime }{E}_2^{p,q})=\underrightarrow{\lim }{B}_k{(}^{\prime }{E}_2^{p,q})$ $(0_{III},\mathrm{11.2.4})$ and (I, 4.1.1); the ${}^{\prime }{E}_{\infty }^{p,q}$ are therefore also quasi-coherent. The preceding spectral sequence being regular, the filtration of the $F^p({\mathcal{H}}^n(f,{\mathcal{K}}^{\bullet }))$ is discrete and exhaustive; in other words, the ${\mathcal{O}}_Y$-module ${\mathcal{H}}^n(f,{\mathcal{K}}^{\bullet })$ is the union of an increasing sequence $({\mathcal{G}}_k{)}_{k\ge 0}$ of ${\mathcal{O}}_Y$-modules such that ${\mathcal{G}}_0=0$ and such that each ${\mathcal{G}}_k/{\mathcal{G}}_{k-1}$ is equal to one of the ${\mathcal{O}}_Y$-modules ${}^{\prime }{E}_{\infty }^{p,q}$, hence is quasi-coherent. By induction on $k$, one deduces that the ${\mathcal{G}}_k$ are quasi-coherent (I, 4.1.17), and since ${\mathcal{H}}^n(f,{\mathcal{K}}^{\bullet })=\underrightarrow{\lim }{\mathcal{G}}_k$, the proposition is proved (I, 4.1.1).

Corollary (6.2.4).

Under the hypotheses of (6.2.3), for every affine open $V$ of $Y$, the canonical homomorphism

  H^n(f⁻¹(V), 𝒦^•) → Γ(V, ℋ^n(f, 𝒦^•))                                          (6.2.4.1)

is bijective for every $n\in \mathbb{Z}$.

Proof. The proof is the same as that of (1.4.11), using (6.2.2), replacing $\mathcal{F}$ by ${\mathcal{K}}^{\bullet }$, ${\mathcal{K}}^{\bullet }$ by $f_{\ast }({\mathcal{C}}^{\bullet }(\mathfrak{U},{\mathcal{K}}^{\bullet }))$, ${\mathcal{H}}^{\bullet }({\mathcal{K}}^{\bullet })$ by ${\mathcal{H}}^{\bullet }(f,{\mathcal{K}}^{\bullet })$, and noting that the latter is a quasi-coherent ${\mathcal{O}}_Y$-module by (6.2.3).

Proposition (6.2.5).

Let $Y$ be a locally noetherian prescheme, $f:X\to Y$ a proper morphism, ${\mathcal{K}}^{\bullet }$ a complex of ${\mathcal{O}}_X$-modules such that the ${\mathcal{O}}_X$-modules ${\mathcal{H}}^q({\mathcal{K}}^{\bullet })$ are coherent. Then the ${\mathcal{O}}_Y$-modules ${\mathcal{H}}^n(f,{\mathcal{K}}^{\bullet })$ are coherent.

Proof. The question being local on $Y$, one may restrict to the case where $Y$ is noetherian and affine, and it is therefore, by (6.2.4), a matter of proving that the $H^n(X,{\mathcal{K}}^{\bullet })$ are $\Gamma (Y,{\mathcal{O}}_Y)$-modules of finite type. One has then $H^{\bullet }(X,{\mathcal{K}}^{\bullet })=H^{\bullet }(\mathfrak{U},{\mathcal{K}}^{\bullet })$ (6.2.2), where one may assume that $\mathfrak{U}$ is finite, since $X$ is quasi-compact. The cochains of each complex $C^{\bullet }(\mathfrak{U},{\mathcal{K}}^i)$ being alternating by definition, there is an integer $r>0$ such that $C^i(\mathfrak{U},{\mathcal{K}}^j)=0$ for $i<0$ and $i>r$; one concludes $(0_{III},\mathrm{11.3.3})$ that the two spectral sequences of the bicomplex $C^{\bullet }(\mathfrak{U},{\mathcal{K}}^{\bullet })$ are biregular. Since the intersections of the sets of $\mathfrak{U}$ are affine open sets (I, 5.5.6), each functor $\mathcal{F}\mapsto C^i(\mathfrak{U},\mathcal{F})$ is exact in the category of quasi-coherent ${\mathcal{O}}_X$-modules; hence $H_q^i(C^{\bullet }(\mathfrak{U},{\mathcal{K}}^{\bullet }))=C^i(\mathfrak{U},{\mathcal{H}}^q({\mathcal{K}}^{\bullet }))$, and the E_2 terms of the second spectral sequence of $C^{\bullet }(\mathfrak{U},{\mathcal{K}}^{\bullet })$ are given $(0_{III},\mathrm{11.3.2})$ by

  ''E_2^{p,q} = H^p(C^•(𝔘, ℋ^q(𝒦^•))) = H^p(𝔘, ℋ^q(𝒦^•)) = H^p(X, ℋ^q(𝒦^•))

by (1.4.1); since $f$ is proper, these are $\Gamma (Y,{\mathcal{O}}_Y)$-modules of finite type (3.2.1). The spectral sequence ${}^{\prime \prime }\mathcal{E}(C^{\bullet }(\mathfrak{U},{\mathcal{K}}^{\bullet }))$ being biregular, one deduces that the $H^n(X,{\mathcal{K}}^{\bullet })$ are $\Gamma (Y,{\mathcal{O}}_Y)$-modules of finite type $(0_{III},\mathrm{11.1.8})$.

A fortiori, if ${\mathcal{K}}^{\bullet }$ is a complex of coherent ${\mathcal{O}}_X$-modules, the ${\mathcal{O}}_Y$-modules ${\mathcal{H}}^n(f,{\mathcal{K}}^{\bullet })$ are coherent under the hypotheses of (6.2.5) relative to $Y$ and $f$ $(0_I,\mathrm{5.3.4})$.

6.2.6.

The hypercohomology ${\mathcal{H}}^{\bullet }(f,{\mathcal{K}}^{\bullet })$ is a cohomological functor in the category of complexes of ${\mathcal{O}}_X$-modules bounded below $(0_{III},\mathrm{12.4.4})$. It is a cohomological functor in the category of all complexes of ${\mathcal{O}}_X$-modules when the morphism $f$ is quasi-compact and the space underlying $X$ is locally noetherian: indeed, it then follows from (G, II, 3.10.1) that $f_{\ast }$ commutes with inductive limits (the question being local on $Y$), and one may apply $(0_{III},\mathrm{11.5.2})$.

Finally, if $f$ is separated, ${\mathcal{H}}^{\bullet }(f,{\mathcal{K}}^{\bullet })$ is a cohomological functor in the category of complexes of quasi-coherent ${\mathcal{O}}_X$-modules. This is immediate when $Y$ is affine, since $X$ is then a scheme, and so, by virtue of the canonical isomorphism (6.2.2), one is reduced to seeing that ${\mathcal{K}}^{\bullet }\mapsto H^{\bullet }(\mathfrak{U},{\mathcal{K}}^{\bullet })$ is a cohomological functor in the category of complexes of quasi-coherent ${\mathcal{O}}_X$-modules, which is immediate since the functor ${\mathcal{K}}^{\bullet }\mapsto C^{\bullet }(\mathfrak{U},{\mathcal{K}}^{\bullet })$ is exact in this category (I, 1.3.7). In the general case, for every affine open $V$

of $Y$, $f^{-1}(V)$ is a scheme, and to apply what precedes, it suffices to verify that for an exact sequence $0\to {\mathcal{K}}^{\bullet }\to {\mathcal{K}}^{\prime \bullet }\to {\mathcal{K}}^{\prime \prime \bullet }\to 0$ of complexes of quasi-coherent ${\mathcal{O}}_X$-modules, the homomorphism $\partial :{\mathcal{H}}^n(f,{\mathcal{K}}^{\prime \prime \bullet })|V\to {\mathcal{H}}^{n+1}(f,{\mathcal{K}}^{\bullet })|V$ does not depend on the affine open cover $\mathfrak{U}$ of $f^{-1}(V)$ used to define it. But this follows from the fact that, if ${\mathfrak{U}}^{\prime }$ is an affine open cover finer than $\mathfrak{U}$, the diagram

                       ↗  H^•(f⁻¹(V), 𝒦^•)
  H^•(𝔘,  𝒦^•)  ≀
                  ↓
                       ↘
  H^•(𝔘', 𝒦^•)

of canonical isomorphisms is commutative, as is the diagram

  H^n(𝔘,  𝒦''^•)   →^∂   H^{n+1}(𝔘,  𝒦^•)
       ↓ ≀                  ↓ ≀
  H^n(𝔘', 𝒦''^•)   →^∂   H^{n+1}(𝔘', 𝒦^•).

When one of the preceding conditions is satisfied and ${\mathcal{K}}^{\bullet }\to {\mathcal{K}}^{\prime \bullet }$ is a homotopism (6.1.4), the corresponding isomorphism ${\mathcal{H}}^{\bullet }(f,{\mathcal{K}}^{\bullet })\stackrel{\sim }{\to }{\mathcal{H}}^{\bullet }(f,{\mathcal{K}}^{\prime \bullet })$ is then an isomorphism of $\partial$-functors $(0_{III},\mathrm{11.4.4})$.

6.2.7.

All the preceding applies naturally without change (apart from notation) to a complex ${\mathcal{K}}_{\bullet }$ of quasi-coherent ${\mathcal{O}}_X$-modules whose differential is of degree $-1$; it suffices to consider the complex ${\mathcal{K}}^{\bullet }=({\mathcal{K}}^i)$ where ${\mathcal{K}}^i={\mathcal{K}}_{-i}$ for every $i\in \mathbb{Z}$.

6.3. Hypertor of two complexes of modules

6.3.1.

Let $A$ be a commutative ring and $P_{\bullet }$, $Q_{\bullet }$ two complexes of $A$-modules whose differentials are of degree $-1$; let $L_{\bullet ,\bullet }$ (resp. $M_{\bullet ,\bullet }$) be a Cartan–Eilenberg projective resolution of $P_{\bullet }$ (resp. $Q_{\bullet }$) $(0_{III},\mathrm{11.6.1})$; $L_{\bullet ,\bullet }{\otimes }_A{M}_{\bullet ,\bullet }$ is then (for the sum of the first degrees and the sum of the second degrees) a bicomplex (with differentials of degree $-1$), whose homology $H_{\bullet }(L_{\bullet ,\bullet }{\otimes }_A{M}_{\bullet ,\bullet })$ does not depend on the chosen Cartan–Eilenberg resolutions $L_{\bullet ,\bullet }$, $M_{\bullet ,\bullet }$, and is by definition the hyperhomology of the bifunctor $P_{\bullet }{\otimes }_A{Q}_{\bullet }$ in $P_{\bullet }$ and $Q_{\bullet }$ $(0_{III},\mathrm{11.6.5})$. We shall set by definition

  Tor_n^A(P_•, Q_•) = H_n(L_{•,•} ⊗_A M_{•,•})                                  (6.3.1.1)

and we shall say that this $A$-module is the hypertor of index $n$ of the two complexes $P_{\bullet }$, $Q_{\bullet }$. One knows that in the category of complexes of $A$-modules bounded below, the $To{r}_n^A(P_{\bullet },Q_{\bullet })$ form a homological bifunctor in $P_{\bullet }$, $Q_{\bullet }$ $(0_{III},\mathrm{11.6.5})$. Moreover:

Proposition (6.3.2).

The bifunctor $To{r}_n^A(P_{\bullet },Q_{\bullet })$ is the common abutment of two spectral bifunctors ${}^{\prime }E(P_{\bullet },Q_{\bullet })$, ${}^{\prime \prime }E(P_{\bullet },Q_{\bullet })$, whose E_2 terms are

  'E_{p,q}^2 = H_p(Tor_q^A(P_•, Q_•))                                           (6.3.2.1)
  ''E_{p,q}^2 = ⊕_{q'+q''=q} Tor_p^A(H_{q'}(P_•), H_{q''}(Q_•))                  (6.3.2.2)

where, in (6.3.2.1), $To{r}_q^A(P_{\bullet },Q_{\bullet })$ denotes the bicomplex formed by the $A$-modules $To{r}_q^A(P_i,Q_j)$. The spectral sequence (6.3.2.2) is always regular; if $P_{\bullet }$ and $Q_{\bullet }$ are bounded below, or if $A$ is of finite cohomological dimension, the two spectral sequences (6.3.2.1) and (6.3.2.2) are biregular.

Proof. This follows from $(0_{III},\mathrm{11.6.5})$, since when $A$ is of finite cohomological dimension $n$, every $A$-module admits a projective resolution of length $n$ (M, VI, 2.1).

Corollary (6.3.3).

Let $P_{\bullet }^{\prime }$, $Q_{\bullet }^{\prime }$ be two complexes of $A$-modules, $u:P_{\bullet }\to P_{\bullet }^{\prime }$, $v:Q_{\bullet }\to Q_{\bullet }^{\prime }$ two homomorphisms of complexes. If the homomorphisms $H_{\bullet }(u):H_{\bullet }(P_{\bullet })\to H_{\bullet }(P_{\bullet }^{\prime })$, $H_{\bullet }(v):H_{\bullet }(Q_{\bullet })\to H_{\bullet }(Q_{\bullet }^{\prime })$ deduced respectively from $u$ and $v$ are bijective, then the homomorphism $To{r}_{\bullet }^A(P_{\bullet },Q_{\bullet })\to To{r}_{\bullet }^A(P_{\bullet }^{\prime },Q_{\bullet }^{\prime })$ deduced from $u$ and $v$ is bijective.

Proof. Indeed, the homomorphism of spectral sequences ${}^{\prime \prime }E(P_{\bullet },Q_{\bullet }){\to }^{\prime \prime }E(P_{\bullet }^{\prime },Q_{\bullet }^{\prime })$ deduced from $u$ and $v$ is then an isomorphism on the E_2 terms, and the conclusion follows from the fact that these spectral sequences are regular by (6.3.2) $(0_{III},\mathrm{11.1.5})$.

Proposition (6.3.4).

Let $P_{\bullet }$, $Q_{\bullet }$ be two complexes of $A$-modules, bounded below. Let $L_{\bullet ,\bullet }$ (resp. $M_{\bullet ,\bullet }$) be a bicomplex formed by flat $A$-modules such that for every $i$, $L_{i,\bullet }$ (resp. $M_{i,\bullet }$) is a resolution of $P_i$ (resp. $Q_i$). Then one has canonical isomorphisms

  Tor_•^A(P_•, Q_•) ⥲ H_•(L_{•,•} ⊗_A Q_•) ⥲ H_•(P_• ⊗_A M_{•,•}) ⥲ H_•(L_{•,•} ⊗_A M_{•,•}).
                                                                                (6.3.4.1)

Proof. This follows from $(0_{III},\mathrm{11.6.5},(ii)and(iii))$ and from the definition of flat $A$-modules.

Remarks (6.3.5).

(i) With the notations of (6.3.1), the bicomplexes $L_{\bullet ,\bullet }{\otimes }_A{M}_{\bullet ,\bullet }$ and $M_{\bullet ,\bullet }{\otimes }_A{L}_{\bullet ,\bullet }$ are canonically isomorphic, whence a canonical isomorphism $To{r}_{\bullet }^A(P_{\bullet },Q_{\bullet })\stackrel{\sim }{\to }To{r}_{\bullet }^A(Q_{\bullet },P_{\bullet })$.

(ii) If $F$ and $G$ are two $A$-modules, $P_{\bullet }$ and $Q_{\bullet }$ the complexes of $A$-modules reduced to $F$ and $G$ respectively in degree 0 and zero in the other degrees, then two projective resolutions $L_{\bullet }$, $M_{\bullet }$ of $F$ and $G$ respectively may be considered as Cartan–Eilenberg resolutions of $P_{\bullet }$ and $Q_{\bullet }$ by completing them by zeros. One has consequently in this case $To{r}_{\bullet }^A(P_{\bullet },Q_{\bullet })\stackrel{\sim }{\to }To{r}_{\bullet }^A(F,G)$.

Proposition (6.3.6).

Let $(P_{\bullet }^{\lambda })$, $(Q_{\bullet }^{\mu })$ be two filtered inductive systems of complexes of $A$-modules; one has a canonical isomorphism

  lim_→  Tor_•^A(P_•^λ, Q_•^μ) ⥲ Tor_•^A(lim_→ P_•^λ, lim_→ Q_•^μ).             (6.3.6.1)
        λ,μ                              λ          μ

Proof. Set $P_{\bullet }=\underrightarrow{\lim }{P}_{\bullet }^{\lambda }$, $Q_{\bullet }=\underrightarrow{\lim }{Q}_{\bullet }^{\mu }$; by functoriality, it is clear that the $To{r}_{\bullet }^A(P_{\bullet }^{\lambda },Q_{\bullet }^{\mu })$ form an inductive system and that the applications $To{r}_{\bullet }^A(P_{\bullet }^{\lambda },Q_{\bullet }^{\mu })\to To{r}_{\bullet }^A(P_{\bullet },Q_{\bullet })$ deduced

from the canonical applications $P_{\bullet }^{\lambda }\to P_{\bullet }$, $Q_{\bullet }^{\mu }\to Q_{\bullet }$ form an inductive system of homomorphisms, whence a canonical homomorphism (6.3.6.1), and more generally a canonical homomorphism ${\underrightarrow{\lim }}^{\prime \prime }E(P_{\bullet }^{\lambda },Q_{\bullet }^{\mu }){\to }^{\prime \prime }E(P_{\bullet },Q_{\bullet })$ of which (6.3.6.1) is the homomorphism of abutments. In addition, the spectral sequence ${}^{\prime \prime }E(P_{\bullet },Q_{\bullet })$ is regular (6.3.2), and the same holds for the spectral sequence ${\underrightarrow{\lim }}^{\prime \prime }E(P_{\bullet }^{\lambda },Q_{\bullet }^{\mu })$, as follows from the definitions $(0_{III},\mathrm{11.1.7})$ and from the proof of $(0_{III},\mathrm{11.3.3})$; to show that (6.3.6.1) is bijective, it therefore suffices $(0_{III},\mathrm{11.1.5})$ to prove that the homomorphism

  lim_→ ''E(P_•^λ, Q_•^μ) → ''E(P_•, Q_•)                                       (6.3.6.2)

is bijective on the E_2 terms. Since the functor $H_{\bullet }$ commutes with the inductive limit of complexes of modules, one is finally reduced to proving that for two filtered inductive systems $(F^{\lambda })$, $(G^{\mu })$ of $A$-modules, the canonical homomorphism

  lim_→ (Tor_•^A(F^λ, G^μ)) → Tor_•^A(lim_→ F^λ, lim_→ G^μ)
       λ,μ                              λ          μ

is bijective. For that, consider for each $F^{\lambda }$ the canonical free resolution

  L_•^λ : ⋯ → L_{i+1}^λ → L_i^λ → ⋯ → L_1^λ → L_0^λ → 0

where $L_0^{\lambda }$ is the $A$-module of formal linear combinations of elements of $F^{\lambda }$ and $L_{i+1}^{\lambda }$ is the $A$-module of formal linear combinations of elements of $Ker(L_i^{\lambda }\to L_{i-1}^{\lambda })$; one verifies immediately that the $L_{\bullet }^{\lambda }$ form an inductive system of complexes, and if one sets $F=\underrightarrow{\lim }{F}^{\lambda }$, $L_i=\underrightarrow{\lim }{L}_i^{\lambda }$, the $L_i$ form a resolution $L_{\bullet }$ of $F$, since the functor $\underrightarrow{\lim }$ is exact; in addition, the $L_i$, inductive limits of free $A$-modules, are flat $(0_I,\mathrm{6.1.2})$. One considers similarly for each $\mu$ the canonical free resolution $M_{\bullet }^{\mu }$ of $G^{\mu }$, and $M_{\bullet }=\underrightarrow{\lim }{M}_{\bullet }^{\mu }$ is a flat resolution of $G=\underrightarrow{\lim }{G}^{\mu }$. One then has Tor_•^A(lim_→ F^λ, lim_→ G^μ) = H_•(L_• ⊗_A M_•) by virtue of (6.3.5) and (6.3.4); but $H_{\bullet }(L_{\bullet }{\otimes }_A{M}_{\bullet })=\underrightarrow{\lim }{H}_{\bullet }(L_{\bullet }^{\lambda }{\otimes }_A{M}_{\bullet }^{\mu })$ since $H_{\bullet }$ commutes with inductive limits of complexes of modules; since $H_{\bullet }(L_{\bullet }^{\lambda }{\otimes }_A{M}_{\bullet }^{\mu })=To{r}_{\bullet }^A(F^{\lambda },G^{\mu })$, this completes the proof.

When one assumes that there exists $i_0$ such that $P_i^{\lambda }=Q_i^{\mu }=0$ for $i<i_0$ for all $\lambda$ and $\mu$, one proves in the same way that the canonical homomorphism

  lim_→ 'E(P_•^λ, Q_•^μ) → 'E(P_•, Q_•)                                         (6.3.6.3)

is bijective.

Proposition (6.3.7).

Assume $P_{\bullet }$ and $Q_{\bullet }$ bounded below. If the complex $P_{\bullet }$ is formed of flat $A$-modules, one has a canonical $A$-isomorphism of $\partial$-functors in $Q_{\bullet }$

  Tor_•^A(P_•, Q_•) ⥲ H_•(P_• ⊗_A Q_•).                                         (6.3.7.1)

Proof. Indeed, the spectral sequence (6.3.2.1) is biregular and degenerate, and the existence of the isomorphism (6.3.7.1) follows from $(0_{III},\mathrm{11.1.6})$. In addition, by computing the hypertor from a Cartan–Eilenberg projective resolution of $P_{\bullet }$ (6.3.4), one sees at once that the isomorphism so defined is an isomorphism of $\partial$-functors in $Q_{\bullet }$.

6.3.8.

Let $\rho :A\to A^{\prime }$ be a homomorphism of rings. We propose to define an $A$-homomorphism of degree 0 functorial in $P_{\bullet }$, $Q_{\bullet }$, canonically associated with $\rho$:

  ρ_{P_•, Q_•} : Tor_•^A(P_•, Q_•) → Tor_•^{A'}(P_• ⊗_A A', Q_• ⊗_A A').        (6.3.8.1)

For this, consider a Cartan–Eilenberg projective resolution $L_{\bullet ,\bullet }$ of $P_{\bullet }$; consider on the other hand a Cartan–Eilenberg projective resolution $L_{\bullet ,\bullet }^{\prime }$ of $P_{\bullet }{\otimes }_A{A}^{\prime }$. We shall see that one can define an $A^{\prime }$-homomorphism of complexes $L_{\bullet ,\bullet }{\otimes }_A{A}^{\prime }\to L_{\bullet ,\bullet }^{\prime }$, determined up to homotopy. Indeed, the construction of $L_{\bullet ,\bullet }$ is entirely determined when one is given (arbitrarily), for each $i$, a projective resolution $(X_{ij}^B{)}_{j\ge 0}$ of $B_i(P_{\bullet })$ and a projective resolution $(X_{ij}^H{)}_{j\ge 0}$ of $H_i(P_{\bullet })$, which are respectively equal to $B_i^I(L_{\bullet ,\bullet })$ and $H_i^I(L_{\bullet ,\bullet })$; one deduces successively $Z_i^I(L_{\bullet ,\bullet })=H_i^I(L_{\bullet ,\bullet })\oplus B_i^I(L_{\bullet ,\bullet })$, then $L_{i,\bullet }=Z_i^I(L_{\bullet ,\bullet })\oplus B_{i-1}^I(L_{\bullet ,\bullet })$. That being so, $X_{i,\bullet }^B{\otimes }_A{A}^{\prime }$ is in general no longer a resolution of $B_i(P_{\bullet }){\otimes }_A{A}^{\prime }$, but it is still a complex formed of projective $A^{\prime }$-modules, and there is therefore an $A^{\prime }$-homomorphism $X_{i,\bullet }^B{\otimes }_A{A}^{\prime }\to B_i^I(L_{\bullet ,\bullet }^{\prime })$ compatible with the augmentations, and determined up to homotopy (M, V, 1.1). One has similarly an $A^{\prime }$-homomorphism $X_{i,\bullet }^H{\otimes }_A{A}^{\prime }\to H_i^I(L_{\bullet ,\bullet }^{\prime })$ determined up to homotopy, from which one deduces, by the construction recalled above, an $A^{\prime }$-homomorphism $L_{i,\bullet }{\otimes }_A{A}^{\prime }\to L_{i,\bullet }^{\prime }$ for every $i$; these homomorphisms (for $i\in \mathbb{Z}$) are compatible with the differentials $L_{i,\bullet }\to L_{i-1,\bullet }$ and the analogous ones for $L_{i,\bullet }^{\prime }$, by virtue of the same construction, and they therefore constitute the desired $A^{\prime }$-homomorphism $L_{\bullet ,\bullet }{\otimes }_A{A}^{\prime }\to L_{\bullet ,\bullet }^{\prime }$.

To define (6.3.8.1), it then suffices to consider similarly a Cartan–Eilenberg projective resolution $M_{\bullet ,\bullet }$ (resp. $M_{\bullet ,\bullet }^{\prime }$) of $Q_{\bullet }$ (resp. $Q_{\bullet }{\otimes }_A{A}^{\prime }$), and an $A^{\prime }$-homomorphism $M_{\bullet ,\bullet }{\otimes }_A{A}^{\prime }\to M_{\bullet ,\bullet }^{\prime }$. From these homomorphisms one deduces an $A^{\prime }$-homomorphism $(L_{\bullet ,\bullet }{\otimes }_A{A}^{\prime }){\otimes }_{A^{\prime }}(M_{\bullet ,\bullet }{\otimes }_A{A}^{\prime })\to L_{\bullet ,\bullet }^{\prime }{\otimes }_{A^{\prime }}{M}_{\bullet ,\bullet }^{\prime }$, then by composition an $A$-homomorphism of bicomplexes $L_{\bullet ,\bullet }{\otimes }_A{M}_{\bullet ,\bullet }\to L_{\bullet ,\bullet }^{\prime }{\otimes }_{A^{\prime }}{M}_{\bullet ,\bullet }^{\prime }$, and on passing to homology one obtains (6.3.8.1), which is well defined since it comes from a morphism of complexes defined up to homotopy.

If ${\rho }^{\prime }:A^{\prime }\to A^{\prime \prime }$ is a second homomorphism of rings, and ${\rho }^{\prime \prime }:A\to A^{\prime \prime }$ the composite homomorphism ${\rho }^{\prime }\circ \rho$, it is clear that ${\rho }_{P_{\bullet },Q_{\bullet }}^{\prime \prime }={\rho }_{P_{\bullet }^{\prime },Q_{\bullet }^{\prime }}^{\prime }\circ {\rho }_{P_{\bullet },Q_{\bullet }}$, where $P_{\bullet }^{\prime }=P_{\bullet }{\otimes }_A{A}^{\prime }$, $Q_{\bullet }^{\prime }=Q_{\bullet }{\otimes }_A{A}^{\prime }$.

Note further that the morphism of bicomplexes $L_{\bullet ,\bullet }{\otimes }_A{M}_{\bullet ,\bullet }\to L_{\bullet ,\bullet }^{\prime }{\otimes }_{A^{\prime }}{M}_{\bullet ,\bullet }^{\prime }$ considered above defines functorial morphisms (in $P_{\bullet }$ and $Q_{\bullet }$) of spectral sequences

  'E_{pq}^r(P_•, Q_•) → 'E_{pq}^r(P_• ⊗_A A', Q_• ⊗_A A')
  ''E_{pq}^r(P_•, Q_•) → ''E_{pq}^r(P_• ⊗_A A', Q_• ⊗_A A'),

independent of the Cartan–Eilenberg resolutions considered, and having also the preceding transitivity property.

Proposition (6.3.9).

Let $\rho :A\to A^{\prime }$ be a homomorphism of rings such that $A^{\prime }$ is a flat $A$-module. Then one has functorial canonical isomorphisms

  Tor_•^{A'}(P_• ⊗_A A', Q_• ⊗_A A') ⥲ Tor_•^A(P_•, Q_•) ⊗_A A'                 (6.3.9.1)
  'E(P_• ⊗_A A', Q_• ⊗_A A') ⥲ 'E(P_•, Q_•) ⊗_A A'                              (6.3.9.2)
  ''E(P_• ⊗_A A', Q_• ⊗_A A') ⥲ ''E(P_•, Q_•) ⊗_A A'.

Proof. Indeed, given the exactness of the functor $M{\otimes }_A{A}^{\prime }$ in $M$, $L_{\bullet ,\bullet }{\otimes }_A{A}^{\prime }$ and $M_{\bullet ,\bullet }{\otimes }_A{A}^{\prime }$ are then Cartan–Eilenberg projective resolutions of $P_{\bullet }{\otimes }_A{A}^{\prime }$ and $Q_{\bullet }{\otimes }_A{A}^{\prime }$ respectively, whence the conclusion.

6.3.10.

Let $\rho :A\to A^{\prime }$ be a homomorphism of rings; for every complex $P_{\bullet }^{\prime }$ of $A^{\prime }$-modules, $P_{\bullet ,[\rho ]}^{\prime }$ is a complex of $A$-modules; moreover, the identity application $P_{\bullet ,[\rho ]}^{\prime }\to P_{\bullet }^{\prime }$ can be considered as composed of the canonical applications

  P'_{•,[ρ]} → P'_{•,[ρ]} ⊗_A A' →^μ P'_•,

where $\mu$ is the $A^{\prime }$-homomorphism $\mu (x\otimes a^{\prime })=a^{\prime }x$. If $Q_{\bullet }^{\prime }$ is a second complex of $A^{\prime }$-modules, one has therefore canonical functorial homomorphisms of degree 0

  Tor_•^A(P'_{•,[ρ]}, Q'_{•,[ρ]}) → Tor_•^{A'}(P'_{•,[ρ]} ⊗_A A', Q'_{•,[ρ]} ⊗_A A')
                                  → Tor_•^{A'}(P'_•, Q'_•)                      (6.3.10.1)

where the first arrow is the $A$-homomorphism defined in (6.3.8) and the second is deduced from the $A^{\prime }$-homomorphisms $P_{\bullet ,[\rho ]}^{\prime }{\otimes }_A{A}^{\prime }\to P_{\bullet }^{\prime }$ and $Q_{\bullet ,[\rho ]}^{\prime }{\otimes }_A{A}^{\prime }\to Q_{\bullet }^{\prime }$ by functoriality. One has analogous homomorphisms for the spectral sequences of (6.3.2), and obvious transitivity properties, which we leave to the reader to state.

Proposition (6.3.11).

Let $\rho :A\to A^{\prime }$ be a homomorphism of rings making $A^{\prime }$ a flat $A$-module. For every complex $P_{\bullet }^{\prime }$ of $A^{\prime }$-modules and every complex $Q_{\bullet }$ of $A$-modules bounded below, one has a functorial canonical isomorphism

  Tor_•^A(P'_{•,[ρ]}, Q_•) ⥲ Tor_•^{A'}(P'_•, Q_• ⊗_A A').                      (6.3.11.1)

Proof. Indeed, if $M_{\bullet ,\bullet }$ is a Cartan–Eilenberg projective resolution of $Q_{\bullet }$, $M_{\bullet ,\bullet }{\otimes }_A{A}^{\prime }$ is a Cartan–Eilenberg projective resolution of $Q_{\bullet }{\otimes }_A{A}^{\prime }$, and one has, up to a canonical isomorphism, $P_{\bullet ,[\rho ]}^{\prime }{\otimes }_A{M}_{\bullet ,\bullet }=P_{\bullet }^{\prime }{\otimes }_{A^{\prime }}(M_{\bullet ,\bullet }{\otimes }_A{A}^{\prime })$; the conclusion follows from (6.3.4).

Remark (6.3.12).

Let $(A^{\lambda })$ be a filtered inductive system of rings, and let $(P_{\bullet }^{\lambda })$, $(Q_{\bullet }^{\lambda })$ be two inductive systems of complexes of $(A^{\lambda })$-modules; one then has a canonical isomorphism generalizing (6.3.6.1)

  lim_→ Tor_•^{A^λ}(P_•^λ, Q_•^λ) ⥲ Tor_•^A(P_•, Q_•)                            (6.3.12.1)

where $A=\underrightarrow{\lim }{A}^{\lambda }$, $P_{\bullet }=\underrightarrow{\lim }{P}_{\bullet }^{\lambda }$, $Q_{\bullet }=\underrightarrow{\lim }{Q}_{\bullet }^{\lambda }$. Once the homomorphisms

  Tor_•^{A^λ}(P_•^λ, Q_•^λ) → Tor_•^{A^μ}(P_•^μ, Q_•^μ)

for $\lambda \le \mu$ are defined, with the help of (6.3.10), the proof is that of (6.3.6).

Proposition (6.3.13).

Let $S$ be a multiplicative subset of $A$, $P_{\bullet }$ and $Q_{\bullet }$ two complexes of $A$-modules in which the homotheties defined by the elements of $S$ are bijective, so that, if $A^{\prime }=S^{-1}A$, $P_{\bullet }$ and $Q_{\bullet }$ are formed of $A^{\prime }$-modules. Then one has a canonical isomorphism $To{r}_{\bullet }^A(P_{\bullet },Q_{\bullet })\stackrel{\sim }{\to }To{r}_{\bullet }^{A^{\prime }}(P_{\bullet },Q_{\bullet })$.

Proof. Indeed, the hypothesis implies that the canonical homomorphisms $P_{\bullet }\to P_{\bullet }{\otimes }_A{A}^{\prime }$, $Q_{\bullet }\to Q_{\bullet }{\otimes }_A{A}^{\prime }$ are bijective. On the other hand, the functoriality of the hypertor shows that every $s\in S$ defines a bijective homothety in $To{r}_{\bullet }^A(P_{\bullet },Q_{\bullet })$, and consequently

  Tor_•^A(P_•, Q_•) → Tor_•^A(P_•, Q_•) ⊗_A A'

is also a bijective homomorphism. Since $A^{\prime }$ is a flat $A$-module, the conclusion follows from (6.3.9), and one has similarly canonical isomorphisms for the spectral sequences.

6.4. Local hypertor functors of complexes of quasi-coherent modules: case of affine schemes

6.4.1.

Let $S$ be an affine scheme with ring $A$, and $X$, $Y$ two affine $S$-schemes with rings $B$, $C$ respectively, so that $B$ and $C$ are algebras over $A$. Every complex ${\mathcal{P}}_{\bullet }$ (resp. ${\mathcal{Q}}_{\bullet }$) of quasi-coherent ${\mathcal{O}}_X$-modules (resp. ${\mathcal{O}}_Y$-modules) is of the form $P_{\bullet }$ (resp. $Q_{\bullet }$), where $P_{\bullet }$ (resp. $Q_{\bullet }$) is a complex of $B$-modules (resp. $C$-modules) (I, 1.3.7 and 1.3.8). One may obviously consider $P_{\bullet }$ and $Q_{\bullet }$ as complexes of $A$-modules and form the $To{r}_n^A(P_{\bullet },Q_{\bullet })$; furthermore, by virtue of the bifunctorial character of $To{r}_n^A(P_{\bullet },Q_{\bullet })$, the $A$-algebras $B$ and $C$ operate on this $A$-module, and these operations make it a $(B,C)$-bimodule, or, equivalently, a module over $B{\otimes }_{A}C=A(X{\times }_{S}Y)$. One has thereby defined a quasi-coherent ${\mathcal{O}}_{X{\times }_{S}Y}$-module

  𝒯or_n^{𝒪_S}(𝒫_•, 𝒬_•) = (Tor_n^A(P_•, Q_•))~                                   (6.4.1.1)

which is called the local hypertor of index $n$ of the complexes ${\mathcal{P}}_{\bullet }$ and ${\mathcal{Q}}_{\bullet }$, and which is also denoted $\mathcal{T}o{r}_n^S({\mathcal{P}}_{\bullet },{\mathcal{Q}}_{\bullet })$.

Lemma (6.4.2).

With the notations of (6.4.1), suppose that the ring $A$ is of the form $R^{-1}{A}^{\prime }$, where $A^{\prime }$ is a ring and $R$ a multiplicative subset of $A^{\prime }$. Let $S^{\prime }=\mathrm{Spec}(A^{\prime })$, so that $X$ and $Y$ may be considered as $S^{\prime }$-preschemes and one has $X{\times }_{S^{\prime }}Y=X{\times }_{S}Y$ (I, 1.6.2 and 3.2.4). Then one has $\mathcal{T}o{r}_{\bullet }^{S^{\prime }}({\mathcal{P}}_{\bullet },{\mathcal{Q}}_{\bullet })=\mathcal{T}o{r}_{\bullet }^S({\mathcal{P}}_{\bullet },{\mathcal{Q}}_{\bullet })$.

Proof. This follows from formula (6.4.1.1) and from (6.3.13).

6.4.3.

With the notations and hypotheses of (6.4.1), let $\mathcal{F}=F$ be a quasi-coherent ${\mathcal{O}}_X$-module, $\mathcal{G}=G$ a quasi-coherent ${\mathcal{O}}_Y$-module; considering $\mathcal{F}$ and $\mathcal{G}$ as complexes of modules, one will denote by $\mathcal{T}o{r}_n^S(\mathcal{F},\mathcal{G})$ or $\mathcal{T}o{r}_n^{{\mathcal{O}}_S}(\mathcal{F},\mathcal{G})$ their hypertor of index $n$; it follows from (6.3.5, (ii)) that one has

  𝒯or_n^{𝒪_S}(ℱ, 𝒢) = (Tor_n^A(F, G))~.                                          (6.4.3.1)

Returning now to the general case of two complexes of quasi-coherent modules ${\mathcal{P}}_{\bullet }$, ${\mathcal{Q}}_{\bullet }$, formulas (6.4.1.1) and (6.4.3.1) show, taking into account Prop. (6.3.2), that $\mathcal{T}o{r}_{\bullet }^S({\mathcal{P}}_{\bullet },{\mathcal{Q}}_{\bullet })$ is the abutment of two spectral sequences ${}^{\prime }\mathcal{E}({\mathcal{P}}_{\bullet },{\mathcal{Q}}_{\bullet })$, ${}^{\prime \prime }\mathcal{E}({\mathcal{P}}_{\bullet },{\mathcal{Q}}_{\bullet })$, whose E_2 terms are given by

  'ℰ_{pq}^2 = ℋ_p(𝒯or_q^S(𝒫_•, 𝒬_•))                                            (6.4.3.2)
  ''ℰ_{pq}^2 = ⊕_{q'+q''=q} 𝒯or_p^S(ℋ_{q'}(𝒫_•), ℋ_{q''}(𝒬_•))                   (6.4.3.3)

where $\mathcal{T}o{r}_q^S({\mathcal{P}}_{\bullet },{\mathcal{Q}}_{\bullet })$ is the bicomplex of quasi-coherent ${\mathcal{O}}_{X{\times }_{S}Y}$-modules $\mathcal{T}o{r}_q^S({\mathcal{P}}_i,{\mathcal{Q}}_j)$.

6.4.4.

Consider now two further affine schemes $X^{(1)}=\mathrm{Spec}(B^{(1)})$, $Y^{(1)}=\mathrm{Spec}(C^{(1)})$, where $B^{(1)}$ and $C^{(1)}$ are $A$-algebras, and suppose given two

$S$-morphisms $u:X^{(1)}\to X$, $v:Y^{(1)}\to Y$, corresponding to $A$-homomorphisms $\varphi :B\to B^{(1)}$, $\psi :C\to C^{(1)}$. Consider the complexes $u^{\ast }({\mathcal{P}}_{\bullet })=(P_{\bullet }{\otimes }_B{B}^{(1)})$ of ${\mathcal{O}}_{X^{(1)}}$-modules, $v^{\ast }({\mathcal{Q}}_{\bullet })=(Q_{\bullet }{\otimes }_C{C}^{(1)})$ of ${\mathcal{O}}_{Y^{(1)}}$-modules. The canonical $A$-homomorphisms

  P_• → P_• ⊗_B B^{(1)},   Q_• → Q_• ⊗_C C^{(1)}

give by functoriality an $A$-homomorphism

  Tor_•^A(P_•, Q_•) → Tor_•^A(P_• ⊗_B B^{(1)}, Q_• ⊗_C C^{(1)});

moreover, again by functoriality, this homomorphism is in fact a homomorphism of $(B{\otimes }_{A}C)$-modules. From this one concludes that one has thereby defined a $(u{\times }_{S}v)$-morphism

  θ : 𝒯or_•^S(𝒫_•, 𝒬_•) → 𝒯or_•^S(u^*(𝒫_•), v^*(𝒬_•))                            (6.4.4.1)

and consequently, a homomorphism of ${\mathcal{O}}_{X^{(1)}{\times }_S{Y}^{(1)}}$-modules

  θ^♯ : (u ×_S v)^*(𝒯or_•^S(𝒫_•, 𝒬_•)) → 𝒯or_•^S(u^*(𝒫_•), v^*(𝒬_•))             (6.4.4.2)

which is evidently a morphism of bi-$\partial$-functors in the categories of quasi-coherent modules bounded below.

The homomorphism (6.4.4.2) is not necessarily bijective; however:

Lemma (6.4.5).

With the notations of (6.4.4), suppose that $u$ and $v$ are open immersions; then the homomorphism (6.4.4.2) is bijective.

Proof. Identify $X^{(1)}$ (resp. $Y^{(1)}$) with an open subset of $X$ (resp. $Y$); $X^{(1)}$ (resp. $Y^{(1)}$) is then a union of open sets of the form $D(f)$ (resp. $D(g)$), where $f\in B$ (resp. $g\in C$), and the induced preschemes $D(\varphi (f))$ and $D(f)$ (resp. $D(\psi (g))$ and $D(g)$) are isomorphic. It will suffice to prove the lemma when $X^{(1)}$ (resp. $Y^{(1)}$) is of the form $D(f)$ (resp. $D(g)$); indeed, if this point is established, and if one returns to the general case, it suffices to prove that the restriction of ${\theta }^{♯}$ to each open set $D(f){\times }_{S}D(g)$ is an isomorphism; now, if $u_1:D(f)\to X^{(1)}$, $v_1:D(g)\to Y^{(1)}$ are the canonical injections, the preceding restriction is none other than $(u_1{\times }_S{v}_1{)}^{\ast }({\theta }^{♯})$; but it is immediate, by virtue of definitions (6.4.4) and $(0_I,\mathrm{4.4.8})$, that on composing it with the canonical homomorphism

  (u_1 ×_S v_1)^*(𝒯or_•^S(u^*(𝒫_•), v^*(𝒬_•))) → 𝒯or_•^S(u'^*(𝒫_•), v'^*(𝒬_•))   (6.4.5.1)

where $u^{\prime }=u\circ u_1$ and $v^{\prime }=v\circ v_1$, one obtains the canonical homomorphism

  (u' ×_S v')^*(𝒯or_•^S(𝒫_•, 𝒬_•)) → 𝒯or_•^S(u'^*(𝒫_•), v'^*(𝒬_•))               (6.4.5.2)

and if one knows that (6.4.5.1) and (6.4.5.2) are isomorphisms, it will follow that the same holds for $(u_1{\times }_S{v}_1{)}^{\ast }({\theta }^{♯})$.

Suppose then that $X^{(1)}=D(f)$ and $Y^{(1)}=D(g)$, so that $B^{(1)}=B_f$ and $C^{(1)}=C_g$; $u^{\ast }({\mathcal{P}}_{\bullet })$ (resp. $v^{\ast }({\mathcal{Q}}_{\bullet })$) is then identified with $(P_{\bullet }){}_f$ (resp. $(Q_{\bullet }){}_g$); on the other hand, $X^{(1)}{\times }_S{Y}^{(1)}$ is identified with the open subscheme $D(f\otimes g)$ of $X{\times }_{S}Y=\mathrm{Spec}(B{\otimes }_{A}C)$ (II, 4.3.2.4); it is a matter of proving that the homomorphism

  (Tor_•^A(P_•, Q_•))_{f ⊗ g} → Tor_•^A((P_•)_f, (Q_•)_g)                       (6.4.5.3)

deduced by functoriality from the canonical homomorphisms $P_{\bullet }\to (P_{\bullet }{)}_f$, $Q_{\bullet }\to (Q_{\bullet }{)}_g$, is bijective. Now $(0_I,\mathrm{1.6.1})$, one may write $(P_{\bullet }{)}_f=\underrightarrow{\lim }{P}_{\bullet }^{(n)}$, where the $P_{\bullet }^{(n)}$ are all complexes of $B$-modules identical to $P_{\bullet }$, the application $P_{\bullet }^{(m)}\to P_{\bullet }^{(n)}$ for $m\le n$ being multiplication by $f^{n-m}$; one has an analogous result for $Q_{\bullet }$ replacing $f$ by $g$; on the other hand, it is clear that the homomorphism

  Tor_•^A(P_•^{(m)}, Q_•^{(m)}) → Tor_•^A(P_•^{(n)}, Q_•^{(n)})

corresponding to the homomorphisms $P_{\bullet }^{(m)}\to P_{\bullet }^{(n)}$ and $Q_{\bullet }^{(m)}\to Q_{\bullet }^{(n)}$ is by definition multiplication by $(f\otimes g{)}^{n-m}$. The conclusion follows from $(0_I,\mathrm{1.6.1})$ applied to the first member of (6.4.5.3) and from (6.3.6).

6.4.6.

With the notations of (6.4.4), one defines similarly canonical homomorphisms of spectral functors

  (u ×_S v)^*('ℰ(𝒫_•, 𝒬_•)) → 'ℰ(u^*(𝒫_•), v^*(𝒬_•))                              (6.4.6.1)
  (u ×_S v)^*(''ℰ(𝒫_•, 𝒬_•)) → ''ℰ(u^*(𝒫_•), v^*(𝒬_•))

and the reasoning of (6.4.5) shows that when $u$ and $v$ are open immersions, the homomorphisms (6.4.6.1) are bijective: indeed, taking into account (6.3.6.2) and (6.3.6.3), it proves that it is an isomorphism on the E_2 terms, and (6.4.5) shows that it is an isomorphism on the abutments; one concludes therefore with the help of $(0_{III},\mathrm{11.1.2})$ and $(0_{III},\mathrm{11.2.4})$.

6.5. Local hypertor functors of complexes of quasi-coherent modules: general case

6.5.1.

Consider now an arbitrary prescheme $S$ and two arbitrary $S$-preschemes $X$, $Y$; let ${\mathcal{P}}_{\bullet }$ (resp. ${\mathcal{Q}}_{\bullet }$) be a complex of quasi-coherent ${\mathcal{O}}_X$-modules (resp. ${\mathcal{O}}_Y$-modules). Set $Z=X{\times }_{S}Y$; we are going to define quasi-coherent ${\mathcal{O}}_Z$-modules $\mathcal{T}o{r}_n^S({\mathcal{P}}_{\bullet },{\mathcal{Q}}_{\bullet })$, called the local hypertor of ${\mathcal{P}}_{\bullet }$ and ${\mathcal{Q}}_{\bullet }$, which reduce to those already defined in (6.4) when $S$, $X$, and $Y$ are affine.

When ${\mathcal{P}}_{\bullet }$ and ${\mathcal{Q}}_{\bullet }$ reduce respectively to their terms of degree 0, $\mathcal{F}$ and $\mathcal{G}$ (the others being zero), one will write $\mathcal{T}o{r}_n^S(\mathcal{F},\mathcal{G})$ instead of $\mathcal{T}o{r}_n^S({\mathcal{P}}_{\bullet },{\mathcal{Q}}_{\bullet })$.

6.5.2.

Suppose first that $S$ is affine, and let $(X_{\lambda })$, $(Y_{\mu })$ be covers of $X$ and $Y$ respectively by affine open sets; then the $Z_{\lambda \mu }=X_{\lambda }{\times }_S{Y}_{\mu }$ form an affine open cover of $Z$. Set ${\mathcal{P}}_{\lambda ,\bullet }={\mathcal{P}}_{\bullet }|X_{\lambda }$, ${\mathcal{Q}}_{\mu ,\bullet }={\mathcal{Q}}_{\bullet }|Y_{\mu }$; we have therefore for every pair $(\lambda ,\mu )$ a quasi-coherent ${\mathcal{O}}_{Z_{\lambda \mu }}$-module ${\mathcal{F}}_{\lambda \mu }=\mathcal{T}o{r}_n^S({\mathcal{P}}_{\lambda ,\bullet },{\mathcal{Q}}_{\mu ,\bullet })$, and it must be shown that the ${\mathcal{F}}_{\lambda \mu }$ satisfy the gluing condition $(0_I,\mathrm{3.3.1})$. For this, it suffices to verify that for every affine open set $U\subset X_{\lambda }\cap X_{{\lambda }^{\prime }}$ (resp. $V\subset Y_{\mu }\cap Y_{{\mu }^{\prime }}$), the restrictions of ${\mathcal{F}}_{\lambda \mu }$ and ${\mathcal{F}}_{{\lambda }^{\prime }{\mu }^{\prime }}$ to $U{\times }_{S}V$ are canonically isomorphic; but this follows at once from the existence of canonical isomorphisms of these restrictions onto $\mathcal{T}o{r}_n^S({\mathcal{P}}_{\bullet }|U,{\mathcal{Q}}_{\bullet }|V)$ (6.4.5). Moreover, it follows at once from this definition and from (6.4.5) that the ${\mathcal{O}}_Z$-module so defined does not depend (up to isomorphism) on the open covers $(X_{\lambda })$, $(Y_{\mu })$ considered;