The Stacks project

36.22 Künneth formula

For the case where the base is a field, please see Varieties, Section 33.29. Consider a cartesian diagram of schemes

\[ \xymatrix{ & X \times _ S Y \ar[ld]^ p \ar[rd]_ q \ar[dd]^ f \\ X \ar[rd]_ a & & Y \ar[ld]^ b \\ & S } \]

Let $K \in D(\mathcal{O}_ X)$ and $M \in D(\mathcal{O}_ Y)$. There is a canonical map

36.22.0.1
\begin{equation} \label{perfect-equation-kunneth} Ra_*K \otimes _{\mathcal{O}_ S}^\mathbf {L} Rb_*M \longrightarrow Rf_*(Lp^*K \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} Lq^*M) \end{equation}

Namely, we can use the maps $Ra_*K \to Ra_*Rp_* Lp^*K = Rf_*Lp^*K$ and $Rb_*M \to Rb_*Rq_* Lq^*M = Rf_*Lq^*M$ and then we can use the relative cup product (Cohomology, Remark 20.28.7).

Lemma 36.22.1. In the situation above, if $a$ and $b$ are quasi-compact and quasi-separated and $X$ and $Y$ are tor-independent over $S$, then (36.22.0.1) is an isomorphism for $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ and $M \in D_\mathit{QCoh}(\mathcal{O}_ Y)$.

Proof. This follows from the following sequence of isomorphisms

\begin{align*} Rf_*(Lp^*K \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} Lq^*M) & = Ra_*Rp_*(Lp^*K \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} Lq^*M) \\ & = Ra_*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} Rp_*Lq^*M) \\ & = Ra_*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} La^*Rb_*M) \\ & = Ra_*K \otimes _{\mathcal{O}_ S}^\mathbf {L} Rb_*M \end{align*}

The first equality holds because $f = a \circ p$. The second equality by Lemma 36.21.1. The third equality by Lemma 36.21.5. The fourth equality by Lemma 36.21.1. We omit the verification that the composition of these isomorphisms is the same as the map (36.22.0.1). $\square$

Remark 36.22.2. With $X, Y, S, a, b, p, q, f$ as in the introduction to this section suppose we have an $\mathcal{O}_ X$-module $\mathcal{F}$ and an $\mathcal{O}_ Y$-module $\mathcal{G}$. Then we have

\begin{align*} & p^{-1}\mathcal{F} \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G} \\ & = p^{-1}(\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{O}_ X) \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}(\mathcal{O}_ Y \otimes _{\mathcal{O}_ Y} \mathcal{G}) \\ & = p^{-1}\mathcal{F} \otimes _{p^{-1}\mathcal{O}_ X} p^{-1}\mathcal{O}_ X \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{O}_ Y \otimes _{q^{-1}\mathcal{O}_ Y} q^{-1}\mathcal{G} \\ & = p^{-1}\mathcal{F} \otimes _{q^{-1}\mathcal{O}_ X} \mathcal{O}_{X \times _ S Y} \otimes _{q^{-1}\mathcal{O}_ Y} q^{-1}\mathcal{G} \\ & = p^{-1}\mathcal{F} \otimes _{q^{-1}\mathcal{O}_ X} \mathcal{O}_{X \times _ S Y} \otimes _{\mathcal{O}_{X \times _ S Y}} \mathcal{O}_{X \times _ S Y} \otimes _{q^{-1}\mathcal{O}_ Y} q^{-1}\mathcal{G} \\ & = p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G} \end{align*}

This is occasionally useful.

Lemma 36.22.3. Let $a : X \to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{F}^\bullet $ be a locally bounded complex of $a^{-1}\mathcal{O}_ S$-modules. Assume for all $n \in \mathbf{Z}$ the sheaf $\mathcal{F}^ n$ is a flat $a^{-1}\mathcal{O}_ S$-module and $\mathcal{F}^ n$ has the structure of a quasi-coherent $\mathcal{O}_ X$-module compatible with the given $a^{-1}\mathcal{O}_ S$-module structure (but the differentials in the complex $\mathcal{F}^\bullet $ need not be $\mathcal{O}_ X$-linear). Then the following hold

  1. $Ra_*\mathcal{F}^\bullet $ is locally bounded,

  2. $Ra_*\mathcal{F}^\bullet $ is in $D_\mathit{QCoh}(\mathcal{O}_ S)$,

  3. $Ra_*\mathcal{F}^\bullet $ locally has finite tor dimension,

  4. $\mathcal{G} \otimes _{\mathcal{O}_ S}^\mathbf {L} Ra_*\mathcal{F}^\bullet = Ra_*(a^{-1}\mathcal{G} \otimes _{a^{-1}\mathcal{O}_ S} \mathcal{F}^\bullet )$ for $\mathcal{G} \in \mathit{QCoh}(\mathcal{O}_ S)$, and

  5. $K \otimes _{\mathcal{O}_ S}^\mathbf {L} Ra_*\mathcal{F}^\bullet = Ra_*(a^{-1}K \otimes _{a^{-1}\mathcal{O}_ S}^\mathbf {L} \mathcal{F}^\bullet )$ for $K \in D_\mathit{QCoh}(\mathcal{O}_ S)$.

Proof. Parts (1), (2), (3) are local on $S$ hence we may and do assume $S$ is affine. Since $a$ is quasi-compact we conclude that $X$ is quasi-compact. Since $\mathcal{F}^\bullet $ is locally bounded, we conclude that $\mathcal{F}^\bullet $ is bounded.

For (1) and (2) we can use the first spectral sequence $R^ pa_*\mathcal{F}^ q \Rightarrow R^{p + q}a_*\mathcal{F}^\bullet $ of Derived Categories, Lemma 13.21.3. Combining Cohomology of Schemes, Lemma 30.4.5 and Homology, Lemma 12.24.11 we conclude.

Let us prove (3) by the induction principle of Cohomology of Schemes, Lemma 30.4.1. Namely, for a quasi-compact open of $U$ of $X$ consider the condition that $R(a|_ U)_*(\mathcal{F}^\bullet |_ U)$ has finite tor dimension. If $U, V$ are quasi-compact open in $X$, then we have a relative Mayer-Vietoris distinguished triangle

\[ R(a|_{U \cup V})_*\mathcal{F}^\bullet |_{U \cup V} \to R(a|_ U)_*\mathcal{F}^\bullet |_ U \oplus R(a|_ V)_*\mathcal{F}^\bullet |_ V \to R(a|_{U \cap V})_*\mathcal{F}^\bullet |_{U \cap V} \to \]

by Cohomology, Lemma 20.33.5. By the behaviour of tor amplitude in distinguished triangles (see Cohomology, Lemma 20.44.6) we see that if we know the result for $U$, $V$, $U \cap V$, then the result holds for $U \cup V$. This reduces us to the case where $X$ is affine. In this case we have

\[ Ra_*\mathcal{F}^\bullet = a_*\mathcal{F}^\bullet \]

by Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) and the vanishing of higher direct images of quasi-coherent modules under an affine morphism (Cohomology of Schemes, Lemma 30.2.3). Since $\mathcal{F}^ n$ is $S$-flat by assumption and $X$ affine, the modules $a_*\mathcal{F}^ n$ are flat for all $n$. Hence $a_*\mathcal{F}^\bullet $ is a bounded complex of flat $\mathcal{O}_ S$-modules and hence has finite tor dimension.

Proof of part (5). Denote $a' : (X, a^{-1}\mathcal{O}_ S) \to (S, \mathcal{O}_ S)$ the obvious flat morphism of ringed spaces. Part (5) says that

\[ K \otimes _{\mathcal{O}_ S}^\mathbf {L} Ra'_*\mathcal{F}^\bullet = Ra'_*(L(a')^*K \otimes _{a^{-1}\mathcal{O}_ S}^\mathbf {L} \mathcal{F}^\bullet ) \]

Thus Cohomology, Equation (20.49.2.1) gives a functorial map from the left to the right and we want to show this map is an isomorphism. This question is local on $S$ hence we may and do assume $S$ is affine. The rest of the proof is exactly the same as the proof of Lemma 36.21.1 except that we have to show that the functor $K \mapsto Ra'_*(L(a')^*K \otimes _{a^{-1}\mathcal{O}_ S}^\mathbf {L} \mathcal{F}^\bullet )$ commutes with direct sums. This is where we will use $\mathcal{F}^ n$ has the structure of a quasi-coherent $\mathcal{O}_ X$-module. Namely, observe that $K \mapsto L(a')^*K \otimes _{a^{-1}\mathcal{O}_ S}^\mathbf {L} \mathcal{F}^\bullet $ commutes with arbitrary direct sums. Next, if $\mathcal{F}^\bullet $ consists of a single quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}^\bullet = \mathcal{F}^ n[-n]$ then we have $L(a')^*G \otimes _{a^{-1}\mathcal{O}_ S}^\mathbf {L} \mathcal{F}^\bullet = La^*K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{F}^ n[-n]$, see Cohomology, Lemma 20.27.4. Hence in this case the commutation with direct sums follows from Lemma 36.4.2. Now, in general, since $S$ is affine (hence $X$ quasi-compact) and $\mathcal{F}^\bullet $ is locally bounded, we see that

\[ \mathcal{F}^\bullet = (\mathcal{F}^ a \to \ldots \to \mathcal{F}^ b) \]

is bounded. Arguing by induction on $b - a$ and considering the distinguished triangle

\[ \mathcal{F}^ b[-b] \to (\mathcal{F}^ a \to \ldots \to \mathcal{F}^ b) \to (\mathcal{F}^ a \to \ldots \to \mathcal{F}^{b - 1}) \to \mathcal{F}^ b[-b + 1] \]

the proof of this part is finished. Some details omitted.

Proof of part (4). Let $a' : (X, a^{-1}\mathcal{O}_ S) \to (S, \mathcal{O}_ S)$ be as above. Since $\mathcal{F}^\bullet $ is a locally bounded complex of flat $a^{-1}\mathcal{O}_ S$-modules we see the complex $a^{-1}\mathcal{G} \otimes _{a^{-1}\mathcal{O}_ S} \mathcal{F}^\bullet $ represents $L(a')^*\mathcal{G} \otimes _{a^{-1}\mathcal{O}_ S}^\mathbf {L} \mathcal{F}^\bullet $ in $D(a^{-1}\mathcal{O}_ S)$. Hence (4) follows from (5). $\square$

Lemma 36.22.4. Let $f : X \to Y$ be a morphism of schemes with $Y = \mathop{\mathrm{Spec}}(A)$ affine. Let $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ be a finite affine open covering such that all the finite intersections $U_{i_0 \ldots i_ p} = U_{i_0} \cap \ldots \cap U_{i_ p}$ are affine. Let $\mathcal{F}^\bullet $ be a bounded complex of $f^{-1}\mathcal{O}_ Y$-modules. Assume for all $n \in \mathbf{Z}$ the sheaf $\mathcal{F}^ n$ is a flat $f^{-1}\mathcal{O}_ Y$-module and $\mathcal{F}^ n$ has the structure of a quasi-coherent $\mathcal{O}_ X$-module compatible with the given $p^{-1}\mathcal{O}_ Y$-module structure (but the differentials in the complex $\mathcal{F}^\bullet $ need not be $\mathcal{O}_ X$-linear). Then the complex $\text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}^\bullet ))$ is K-flat as a complex of $A$-modules.

Proof. We may write

\[ \mathcal{F}^\bullet = (\mathcal{F}^ a \to \ldots \to \mathcal{F}^ b) \]

Arguing by induction on $b - a$ and considering the distinguished triangle

\[ \mathcal{F}^ b[-b] \to (\mathcal{F}^ a \to \ldots \to \mathcal{F}^ b) \to (\mathcal{F}^ a \to \ldots \to \mathcal{F}^{b - 1}) \to \mathcal{F}^ b[-b + 1] \]

and using More on Algebra, Lemma 15.57.7 we reduce to the case where $\mathcal{F}^\bullet $ consists of a single quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ placed in degree $0$. In this case the Čech complex for $\mathcal{F}$ and $\mathcal{U}$ is homotopy equivalent to the alternating Čech complex, see Cohomology, Lemma 20.23.6. Since $U_{i_0 \ldots i_ p}$ is always affine, we see that $\mathcal{F}(U_{i_0 \ldots i_ p})$ is $A$-flat. Hence $\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F})$ is a bounded complex of flat $A$-modules and hence K-flat by More on Algebra, Lemma 15.57.9. $\square$

Let $X, Y, S, a, b, p, q, f$ be as in the introduction to this section. Assume $S = \mathop{\mathrm{Spec}}(A)$ affine. On $X$ let $\mathcal{F}^\bullet $ be a complex of $a^{-1}\mathcal{O}_ S$-modules. On $Y$ let $\mathcal{G}^\bullet $ be a complex of $b^{-1}\mathcal{O}_ S$-modules. Then there exists a “cup product” map

36.22.4.1
\begin{equation} \label{perfect-equation-de-rham-kunneth} R\Gamma (X, \mathcal{F}^\bullet ) \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{G}^\bullet ) \longrightarrow R\Gamma (X \times _ S Y, \text{Tot}(p^{-1}\mathcal{F}^\bullet \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}^\bullet )) \end{equation}

in $D(A)$. This is constructed using the pullback maps $R\Gamma (X, \mathcal{F}^\bullet ) \to R\Gamma (X \times _ S Y, p^{-1}\mathcal{F}^\bullet )$ and $R\Gamma (Y, \mathcal{G}^\bullet ) \to R\Gamma (X \times _ S Y, q^{-1}\mathcal{G}^\bullet )$ and the cup product constructed in Cohomology, Section 20.31.

Lemma 36.22.5. The cup product (36.22.4.1) is an isomorphism if $\mathcal{F}^\bullet $, resp. $\mathcal{G}^\bullet $ consist of a single $S$-flat quasi-coherent $\mathcal{O}_ X$, resp. $\mathcal{O}_ Y$-module sitting in degree zero and $X$ and $Y$ are quasi-compact with affine diagonal.

This lemma is true without the assumption on the affineness of the diagonals of $X$ and $Y$ replaced by $X$ and $Y$ are quasi-separated. To see this one replaces open coverings in the proof below by hypercoverings.

Proof. Assume $\mathcal{F}^\bullet = \mathcal{F}$ and $\mathcal{G}^\bullet = \mathcal{G}$ are both reduced to a single quasi-coherent module placed in degree $0$ and both $\mathcal{F}$ and $\mathcal{G}$ are flat over $S$. Choose finite affine open coverings $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ and $\mathcal{V} : Y = \bigcup _{j \in J} V_ j$. This determines an affine open covering $\mathcal{W} : X \times _ S Y = \bigcup _{(i, j) \in I \times J} U_ i \times _ S V_ j$. Note that $\mathcal{W}$ is a refinement of $\text{pr}_1^{-1}\mathcal{U}$ and of $\text{pr}_2^{-1}\mathcal{V}$. Thus by the discussion in Cohomology, Section 20.25 we obtain maps

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet (\mathcal{W}, p^*\mathcal{F}) \quad \text{and}\quad \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G}) \to \check{\mathcal{C}}^\bullet (\mathcal{W}, q^*\mathcal{G}) \]

well defined up to homotopy and compatible with pullback maps on cohomology. In Cohomology, Equation (20.25.3.2) we have constructed a map of complexes

\[ \text{Tot}( \check{\mathcal{C}}^\bullet (\mathcal{W}, p^*\mathcal{F}) \otimes _ A \check{\mathcal{C}}^\bullet (\mathcal{W}, q^*\mathcal{G})) \longrightarrow \check{\mathcal{C}}^\bullet (\mathcal{W}, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G}) \]

which is compatible with the cup product on cohomology by Cohomology, Lemma 20.31.3. Combining the above we obtain a map of complexes

36.22.5.1
\begin{equation} \label{perfect-equation-kunneth-on-cech} \text{Tot}( \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes _ A \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G})) \to \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{W}, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G})) \end{equation}

We claim this is the map in the statement of the lemma, i.e., the source and target of this arrow are the same as the source and target of (36.22.4.1). Namely, by Cohomology of Schemes, Lemma 30.2.2 and Cohomology, Lemma 20.25.2 the canonical maps

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \to R\Gamma (X, \mathcal{F}), \quad \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G}) \to R\Gamma (Y, \mathcal{G}) \]

and

\[ \check{\mathcal{C}}^\bullet (\mathcal{W}, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G}) \to R\Gamma (X \times _ S Y, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G}) \]

are isomorphisms. On the other hand, the complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ is homotopy equivalent to the alternating Čech complex $\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F})$ by Cohomology, Lemma 20.23.6. As the modules $\mathcal{F}^ n(U_{i_0\ldots i_ s})$ are $A$-flat, we see that the complex $\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F})$ is K-flat (More on Algebra, Lemma 15.57.9). Hence $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ is a K-flat complex of $A$-modules too and we conclude that $\text{Tot}( \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes _ A \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G}))$ represents the derived tensor product $R\Gamma (X, \mathcal{F}) \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{G})$ as claimed. Finally, we have $p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G} = p^{-1}\mathcal{F} \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}$ by Remark 36.22.2.

We still have to show that (36.22.5.1) is a quasi-isomorphism. We will do this using dimension shifting. Set $d(\mathcal{F}) = \max \{ d \mid H^ d(X, \mathcal{F}) \not= 0\} $. Assume $d(\mathcal{F}) > 0$. Set $U = \coprod \nolimits _{i \in I} U_ i$. This is an affine scheme as $I$ is finite. Denote $j : U \to X$ the morphism which is the inclusion $U_ i \to X$ on each $U_ i$. Since the diagonal of $X$ is affine, the morphism $j$ is affine, see Morphisms, Lemma 29.11.11. It follows that $\mathcal{F}' = j_*j^*\mathcal{F}$ is $S$-flat, see Morphisms, Lemma 29.25.4. It also follows that $d(\mathcal{F}') = 0$ by combining Cohomology of Schemes, Lemmas 30.2.4 and 30.2.2. For all $x \in X$ we have $\mathcal{F}_ x \to \mathcal{F}'_ x$ is the inclusion of a direct summand: if $x \in U_ i$, then $\mathcal{F}' \to (U_ i \to X)_*\mathcal{F}|_{U_ i}$ gives a splitting. We conclude that $\mathcal{F} \to \mathcal{F}'$ is injective and $\mathcal{F}'' = \mathcal{F}'/\mathcal{F}$ is $S$-flat as well. Also $d(\mathcal{F}'') < d(\mathcal{F})$. In this way we reduce to the case $d(\mathcal{F}) = 0$.

Arguing in the same fashion for $\mathcal{G}$ we find that we may assume that both $\mathcal{F}$ and $\mathcal{G}$ have nonzero cohomology only in degree $0$. Observe that this means that $\Gamma (X, \mathcal{F})$ is quasi-isomorphic to the finite complex $\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F})$ of flat $A$-modules sitting in degrees $0, \ldots , |I|$. It follows that $\Gamma (X, \mathcal{F})$ is a flat $A$-module. Let $V \subset Y$ be an affine open. Consider the affine open covering $\mathcal{U}_ V : X \times _ S V = \bigcup _{i \in I} U_ i \times _ S V$. It is immediate that

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes _ A \mathcal{G}(V) = \check{\mathcal{C}}^\bullet (\mathcal{U}_ V, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} q^*\mathcal{G}) \]

(equality of complexes). By the flatness of $\mathcal{G}(V)$ over $A$ we see that $\Gamma (X, \mathcal{F}) \otimes _ A \mathcal{G}(V) \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes _ A \mathcal{G}(V)$ is a quasi-isomorphism. Since the sheafification of $V \mapsto \check{\mathcal{C}}^\bullet (\mathcal{U}_ V, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} q^*\mathcal{G})$ represents $Rq_*(p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} q^*\mathcal{G})$ by Cohomology of Schemes, Lemma 30.7.1 we conclude that

\[ Rq_*(p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} q^*\mathcal{G}) \cong \Gamma (X, \mathcal{F}) \otimes _ A \mathcal{G} \]

on $Y$ with obvious notation. Using this and Lemma 36.21.1 (which applies as $\Gamma (X, \mathcal{F})$ is $A$-flat) we conclude that $H^ n(X \times _ S Y, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} q^*\mathcal{G})$ is zero for $n > 0$ and isomorphic to $H^0(X, \mathcal{F}) \otimes _ A H^0(Y, \mathcal{G})$ for $n = 0$. This finishes the proof (except that we should check that the isomorphism is indeed given by cup product in degree $0$; we omit the verification). $\square$

Lemma 36.22.6. The cup product (36.22.4.1) is an isomorphism in $D(A)$ if the following assumptions hold

  1. $X$ and $Y$ are quasi-compact with affine diagonal,

  2. $\mathcal{F}^\bullet $ is bounded,

  3. $\mathcal{G}^\bullet $ is bounded below,

  4. $\mathcal{F}^ n$ is a flat $a^{-1}\mathcal{O}_ S$-module and $\mathcal{F}^ n$ has the structure of a quasi-coherent $\mathcal{O}_ X$-module compatible with the given $a^{-1}\mathcal{O}_ S$-module structure (but the differentials in the complex $\mathcal{F}^\bullet $ need not be $\mathcal{O}_ X$-linear),

  5. $\mathcal{G}^ m$ is a flat $b^{-1}\mathcal{O}_ S$-module and $\mathcal{G}^ m$ has the structure of a quasi-coherent $\mathcal{O}_ Y$-module compatible with the given $b^{-1}\mathcal{O}_ S$-module structure (but the differentials in the complex $\mathcal{G}^\bullet $ need not be $\mathcal{O}_ Y$-linear).

Proof. Suppose that we have maps of complexes of $p^{-1}\mathcal{O}_ S$-modules

\[ \mathcal{F}_1^\bullet \to \mathcal{F}_2^\bullet \to \mathcal{F}_3^\bullet \to \mathcal{F}_1^\bullet [1] \]

Then by the functoriality of the cup product we obtain a commutative diagram

\[ \xymatrix{ R\Gamma (X, \mathcal{F}_1^\bullet ) \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{G}^\bullet ) \ar[r] \ar[d] & R\Gamma (X \times _ S Y, \text{Tot}(p^{-1}\mathcal{F}_1^\bullet \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}^\bullet )) \ar[d] \\ R\Gamma (X, \mathcal{F}_2^\bullet ) \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{G}^\bullet ) \ar[r] \ar[d] & R\Gamma (X \times _ S Y, \text{Tot}(p^{-1}\mathcal{F}_2^\bullet \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}^\bullet )) \ar[d] \\ R\Gamma (X, \mathcal{F}_3^\bullet ) \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{G}^\bullet ) \ar[r] \ar[d] & R\Gamma (X \times _ S Y, \text{Tot}(p^{-1}\mathcal{F}_3^\bullet \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}^\bullet )) \ar[d] \\ R\Gamma (X, \mathcal{F}_1^\bullet [1]) \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{G}^\bullet ) \ar[r] & R\Gamma (X \times _ S Y, \text{Tot}(p^{-1}\mathcal{F}_1^\bullet [1] \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}^\bullet )) } \]

If the original maps form a distinguished triangle in the homotopy category of complexes of $p^{-1}\mathcal{O}_ S$-modules, then the columns of this diagram form distinguished triangles in $D(A)$.

Suppose that $\mathcal{F}^ n = 0$ for $n < i$. Then we may consider the termwise split short exact sequence of complexes

\[ 0 \to \sigma _{\geq i + 1}\mathcal{F}^\bullet \to \mathcal{F}^\bullet \to \mathcal{F}^ i[-i] \to 0 \]

where the truncation is as in Homology, Section 12.15. This produces the distinguished triangle

\[ \sigma _{\geq i + 1}\mathcal{F}^\bullet \to \mathcal{F}^\bullet \to \mathcal{F}^ i[-i] \to (\sigma _{\geq i + 1}\mathcal{F}^\bullet )[1] \]

where the final arrow is given by the boundary map $\mathcal{F}^ i \to \mathcal{F}^{i + 1}$. It follows from the discussion above that it suffices to prove the lemma for $\mathcal{F}^ i[-i]$ and $\sigma _{\geq i + 1}\mathcal{F}^\bullet $. Since $\sigma _{\geq i + 1}\mathcal{F}^\bullet $ has fewer nonzero terms, by induction, if we can prove the lemma if $\mathcal{F}^\bullet $ is nonzero only in single degree, then the lemma follows. Thus we may assume $\mathcal{F}^\bullet $ is nonzero only in one degree.

Assume $\mathcal{F}^\bullet $ is the complex which has an $S$-flat quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ sitting in degree $0$ and is zero in other degrees. Observe that $R\Gamma (X, \mathcal{F})$ has finite tor dimension by Lemma 36.22.3 for example. Say it has tor amplitude in $[i, j]$. Pick $N \gg 0$ and consider the distinguished triangle

\[ \sigma _{\geq N + 1}\mathcal{G}^\bullet \to \mathcal{G}^\bullet \to \sigma _{\leq N}\mathcal{G}^\bullet \to (\sigma _{\geq N + 1}\mathcal{G}^\bullet )[1] \]

in the homotopy category of complexes of $b^{-1}\mathcal{O}_ S$-modules on $Y$. Now observe that both

\[ R\Gamma (X, \mathcal{F}) \otimes _ A^\mathbf {L} R\Gamma (Y, \sigma _{\geq N + 1}\mathcal{G}^\bullet ) \quad \text{and}\quad R\Gamma (X \times _ S Y, \mathcal{F} \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\sigma _{\geq N + 1}\mathcal{G}^\bullet )) \]

have vanishing cohomology in degrees $\leq N + i$. Thus, using the arguments given above, if we want to prove our statement in a given degree, then we may assume $\mathcal{G}^\bullet $ is bounded. Repeating the arguments above one more time we may also assume $\mathcal{G}^\bullet $ is nonzero only in one degree. This case is handled by Lemma 36.22.5. $\square$


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