The Stacks project

Lemma 43.19.1. Let $X$ be a nonsingular variety.

  1. If $\mathcal{F}$ and $\mathcal{G}$ are coherent $\mathcal{O}_ X$-modules, then there are canonical isomorphisms

    \[ \text{Tor}_ i^{\mathcal{O}_{X \times X}}(\mathcal{O}_\Delta , \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times X}} \text{pr}_2^*\mathcal{G}) = \Delta _*\text{Tor}_ i^{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G}) \]
  2. If $K$ and $M$ are in $D_\mathit{QCoh}(\mathcal{O}_ X)$, then there is a canonical isomorphism

    \[ L\Delta ^* \left( L\text{pr}_1^*K \otimes _{\mathcal{O}_{X \times X}}^\mathbf {L} L\text{pr}_2^*M \right) = K \otimes _{\mathcal{O}_ X}^\mathbf {L} M \]

    in $D_\mathit{QCoh}(\mathcal{O}_ X)$ and a canonical isomorphism

    \[ \mathcal{O}_\Delta \otimes _{\mathcal{O}_{X \times X}}^\mathbf {L} L\text{pr}_1^*K \otimes _{\mathcal{O}_{X \times X}}^\mathbf {L} L\text{pr}_2^*M = \Delta _*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} M) \]

    in $D_\mathit{QCoh}(X \times X)$.

Proof. Let us explain how to prove (1) in a more elementary way and part (2) using more general theory. As (2) implies (1) the reader can skip the proof of (1).

Proof of (1). Choose an affine open $\mathop{\mathrm{Spec}}(A) \subset X$. Then $A$ is a Noetherian $\mathbf{C}$-algebra and $\mathcal{F}$, $\mathcal{G}$ correspond to finite $A$-modules $M$ and $N$ (Cohomology of Schemes, Lemma 30.9.1). By Derived Categories of Schemes, Lemma 36.3.9 we may compute $\text{Tor}_ i$ over $\mathcal{O}_ X$ by first computing the $\text{Tor}$'s of $M$ and $N$ over $A$, and then taking the associated $\mathcal{O}_ X$-module. For the $\text{Tor}_ i$ over $\mathcal{O}_{X \times X}$ we compute the tor of $A$ and $M \otimes _\mathbf {C} N$ over $A \otimes _\mathbf {C} A$ and then take the associated $\mathcal{O}_{X \times X}$-module. Hence on this affine patch we have to prove that

\[ \text{Tor}_ i^{A \otimes _\mathbf {C} A}(A, M \otimes _\mathbf {C} N) = \text{Tor}_ i^ A(M, N) \]

To see this choose resolutions $F_\bullet \to M$ and $G_\bullet \to M$ by finite free $A$-modules (Algebra, Lemma 10.71.1). Note that $\text{Tot}(F_\bullet \otimes _\mathbf {C} G_\bullet )$ is a resolution of $M \otimes _\mathbf {C} N$ as it computes Tor groups over $\mathbf{C}$! Of course the terms of $F_\bullet \otimes _\mathbf {C} G_\bullet $ are finite free $A \otimes _\mathbf {C} A$-modules. Hence the left hand side of the displayed equation is the module

\[ H_ i(A \otimes _{A \otimes _\mathbf {C} A} \text{Tot}(F_\bullet \otimes _\mathbf {C} G_\bullet )) \]

and the right hand side is the module

\[ H_ i(\text{Tot}(F_\bullet \otimes _ A G_\bullet )) \]

Since $A \otimes _{A \otimes _\mathbf {C} A} (F_ p \otimes _\mathbf {C} G_ q) = F_ p \otimes _ A G_ q$ we see that these modules are equal. This defines an isomorphism over the affine open $\mathop{\mathrm{Spec}}(A) \times \mathop{\mathrm{Spec}}(A)$ (which is good enough for the application to equality of intersection numbers). We omit the proof that these isomorphisms glue.

Proof of (2). The second statement follows from the first by the projection formula as stated in Derived Categories of Schemes, Lemma 36.22.1. To see the first, represent $K$ and $M$ by K-flat complexes $\mathcal{K}^\bullet $ and $\mathcal{M}^\bullet $. Since pullback and tensor product preserve K-flat complexes (Cohomology, Lemmas 20.26.5 and 20.26.8) we see that it suffices to show

\[ \Delta ^*\text{Tot}( \text{pr}_1^*\mathcal{K}^\bullet \otimes _{\mathcal{O}_{X \times X}} \text{pr}_2^*\mathcal{M}^\bullet ) = \text{Tot}( \mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{M}^\bullet ) \]

Thus it suffices to see that there are canonical isomorphisms

\[ \Delta ^*(\text{pr}_1^*\mathcal{K} \otimes _{\mathcal{O}_{X \times X}} \text{pr}_2^*\mathcal{M}) \longrightarrow \mathcal{K} \otimes _{\mathcal{O}_ X} \mathcal{M} \]

whenever $\mathcal{K}$ and $\mathcal{M}$ are $\mathcal{O}_ X$-modules (not necessarily quasi-coherent or flat). We omit the details. $\square$

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