The Stacks project

Lemma 48.8.4. Let $f : X \to Y$ and $g : Y \to Z$ be composable morphisms of quasi-compact and quasi-separated schemes and set $h = g \circ f$. Let $a, b, c$ be the adjoints of Lemma 48.3.1 for $f, g, h$. For any $K \in D_\mathit{QCoh}(\mathcal{O}_ Z)$ the diagram

\[ \xymatrix{ Lf^*(Lg^*K \otimes _{\mathcal{O}_ Y}^\mathbf {L} b(\mathcal{O}_ Z)) \otimes _{\mathcal{O}_ X}^\mathbf {L} a(\mathcal{O}_ Y) \ar@{=}[d] \ar[r] & a(Lg^*K \otimes _{\mathcal{O}_ Y}^\mathbf {L} b(\mathcal{O}_ Z)) \ar[r] & a(b(K)) \ar@{=}[d] \\ Lh^*K \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*b(\mathcal{O}_ Z) \otimes _{\mathcal{O}_ X}^\mathbf {L} a(\mathcal{O}_ Y) \ar[r] & Lh^*K \otimes _{\mathcal{O}_ X}^\mathbf {L} c(\mathcal{O}_ Z) \ar[r] & c(K) } \]

is commutative where the arrows are ( and we have used $Lh^* = Lf^* \circ Lg^*$ and $c = a \circ b$.

Proof. In this proof we will write $f_*$ for $Rf_*$ and $f^*$ for $Lf^*$, etc, and we will write $\otimes $ for $\otimes ^\mathbf {L}_{\mathcal{O}_ X}$, etc. The composition of the top arrows is adjoint to a map

\[ g_*f_*(f^*(g^*K \otimes b(\mathcal{O}_ Z)) \otimes a(\mathcal{O}_ Y)) \to K \]

The left hand side is equal to $K \otimes g_*f_*(f^*b(\mathcal{O}_ Z) \otimes a(\mathcal{O}_ Y))$ by Derived Categories of Schemes, Lemma 36.22.1 and inspection of the definitions shows the map comes from the map

\[ g_*f_*(f^*b(\mathcal{O}_ Z) \otimes a(\mathcal{O}_ Y)) \xleftarrow {g_*\epsilon } g_*(b(\mathcal{O}_ Z) \otimes f_*a(\mathcal{O}_ Y)) \xrightarrow {g_*\alpha } g_*(b(\mathcal{O}_ Z)) \xrightarrow {\beta } \mathcal{O}_ Z \]

tensored with $\text{id}_ K$. Here $\epsilon $ is the isomorphism from Derived Categories of Schemes, Lemma 36.22.1 and $\beta $ comes from the counit map $g_*b \to \text{id}$. Similarly, the composition of the lower horizontal arrows is adjoint to $\text{id}_ K$ tensored with the composition

\[ g_*f_*(f^*b(\mathcal{O}_ Z) \otimes a(\mathcal{O}_ Y)) \xrightarrow {g_*f_*\delta } g_*f_*(ab(\mathcal{O}_ Z)) \xrightarrow {g_*\gamma } g_*(b(\mathcal{O}_ Z)) \xrightarrow {\beta } \mathcal{O}_ Z \]

where $\gamma $ comes from the counit map $f_*a \to \text{id}$ and $\delta $ is the map whose adjoint is the composition

\[ f_*(f^*b(\mathcal{O}_ Z) \otimes a(\mathcal{O}_ Y)) \xleftarrow {\epsilon } b(\mathcal{O}_ Z) \otimes f_*a(\mathcal{O}_ Y) \xrightarrow {\alpha } b(\mathcal{O}_ Z) \]

By general properties of adjoint functors, adjoint maps, and counits (see Categories, Section 4.24) we have $\gamma \circ f_*\delta = \alpha \circ \epsilon ^{-1}$ as desired. $\square$

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