Lemma 20.31.7 (0FP6)—The Stacks project

Table of contents
Part 1: Preliminaries
Chapter 20: Cohomology of Sheaves
Section 20.31: Cup product
Lemma 20.31.7 (cite)

previous lemma

Lemma 20.31.7. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ and $g : (Y, \mathcal{O}_ Y) \to (Z, \mathcal{O}_ Z)$ be morphisms of ringed spaces. The relative cup product of Remark 20.28.7 is compatible with compositions in the sense that the diagram

\[ \xymatrix{ R(g \circ f)_*K \otimes _{\mathcal{O}_ Z}^\mathbf {L} R(g \circ f)_*L \ar@{=}[rr] \ar[d] & & Rg_*Rf_*K \otimes _{\mathcal{O}_ Z}^\mathbf {L} Rg_*Rf_*L \ar[d] \\ R(g \circ f)_*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \ar@{=}[r] & Rg_*Rf_*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} L) & Rg_*(Rf_*K \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L) \ar[l] } \]

is commutative in $D(\mathcal{O}_ Z)$ for all $K, L$ in $D(\mathcal{O}_ X)$.

Proof. This is true because going around the diagram either way we obtain the map adjoint to the map

\begin{align*} & L(g \circ f)^*\left(R(g \circ f)_*K \otimes _{\mathcal{O}_ Z}^\mathbf {L} R(g \circ f)_*L\right) \\ & = L(g \circ f)^*R(g \circ f)_*K \otimes _{\mathcal{O}_ X}^\mathbf {L} L(g \circ f)^*R(g \circ f)_*L) \\ & \to K \otimes _{\mathcal{O}_ X}^\mathbf {L} L \end{align*}

in $D(\mathcal{O}_ X)$. To see this one uses that the composition of the counits like so

\[ L(g \circ f)^*R(g \circ f)_* = Lf^* Lg^* Rg_* Rf_* \to Lf^* Rf_* \to \text{id} \]

is the counit for $L(g \circ f)^*$ and $R(g \circ f)_*$. See Categories, Lemma 4.24.9. $\square$

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