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The Stacks project

Lemma 63.3.13. Let f : X \to Y and g : Y \to Z be composable morphisms of schemes which are separated and locally of finite type. Let \mathcal{F} be an abelian sheaf on X_{\acute{e}tale}. Then g_!f_!\mathcal{F} = (g \circ f)_!\mathcal{F} as subsheaves of (g \circ f)_*\mathcal{F}.

Proof. We strongly urge the reader to prove this for themselves. Let W \in Z_{\acute{e}tale} and s \in (g \circ f)_*\mathcal{F}(W) = \mathcal{F}(X_ W). Denote T \subset X_ W the support of s; this is a closed subset. Observe that s is a section of (g \circ f)_!\mathcal{F} if and only if T is proper over W. We have f_!\mathcal{F} \subset f_*\mathcal{F} and hence g_!f_!\mathcal{F} \subset g_!f_*\mathcal{F} \subset g_*f_*\mathcal{F}. On the other hand, s is a section of g_!f_!\mathcal{F} if and only if (a) T is proper over Y_ W and (b) the support T' of s viewed as section of f_!\mathcal{F} is proper over W. If (a) holds, then the image of T in Y_ W is closed and since f_!\mathcal{F} \subset f_*\mathcal{F} we see that T' \subset Y_ W is the image of T (details omitted; look at stalks).

The conclusion is that we have to show a closed subset T \subset X_ W is proper over W if and only if T is proper over Y_ W and the image of T in Y_ W is proper over W. Let us endow T with the reduced induced closed subscheme structure. If T is proper over W, then T \to Y_ W is proper by Morphisms, Lemma 29.41.7 and the image of T in Y_ W is proper over W by Cohomology of Schemes, Lemma 30.26.5. Conversely, if T is proper over Y_ W and the image of T in Y_ W is proper over W, then the morphism T \to W is proper as a composition of proper morphisms (here we endow the closed image of T in Y_ W with its reduced induced scheme structure to turn the question into one about morphisms of schemes), see Morphisms, Lemma 29.41.4. \square


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