The Stacks project

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. $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 20.31: Cup product

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FP6. Beware of the difference between the letter 'O' and the digit '0'.