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The Stacks project

Lemma 72.18.7. Let k be a field. Let f : Y \to X be a morphism of algebraic spaces proper over k. Let Z \subset Y be an integral closed subspace of dimension d and let \mathcal{L}_1, \ldots , \mathcal{L}_ d be invertible \mathcal{O}_ X-modules. Then

(f^*\mathcal{L}_1 \cdots f^*\mathcal{L}_ d \cdot Z) = \deg (f|_ Z : Z \to f(Z)) (\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot f(Z))

where \deg (Z \to f(Z)) is as in Definition 72.5.2 or 0 if \dim (f(Z)) < d.

Proof. In the statement f(Z) \subset X is the scheme theoretic image of f and it is also the reduced induced algebraic space structure on the closed subset f(|Z|) \subset X, see Morphisms of Spaces, Lemma 67.16.4. Then Z and f(Z) are reduced, proper (hence decent) algebraic spaces over k, whence integral (Definition 72.4.1). The left hand side is computed using the coefficient of n_1 \ldots n_ d in the function

\chi (Y, \mathcal{O}_ Z \otimes f^*\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes f^*\mathcal{L}_ d^{\otimes n_ d}) = \sum (-1)^ i \chi (X, R^ if_*\mathcal{O}_ Z \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d})

The equality follows from Lemma 72.17.3 and the projection formula (Cohomology, Lemma 20.54.2). If f(Z) has dimension < d, then the right hand side is a polynomial of total degree < d by Lemma 72.18.1 and the result is true. Assume \dim (f(Z)) = d. Then by dimension theory (Lemma 72.15.2) we find that the equivalent conditions (1) – (5) of Lemma 72.5.1 hold. Thus \deg (Z \to f(Z)) is well defined. By the already used Lemma 72.5.1 we find f : Z \to f(Z) is finite over a nonempty open V of f(Z); after possibly shrinking V we may assume V is a scheme. Let \xi \in V be the generic point. Thus \deg (f : Z \to f(Z)) the length of the stalk of f_*\mathcal{O}_ Z at \xi over \mathcal{O}_{X, \xi } and the stalk of R^ if_*\mathcal{O}_ X at \xi is zero for i > 0 (for example by Cohomology of Spaces, Lemma 69.4.1). Thus the terms \chi (X, R^ if_*\mathcal{O}_ Z \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}) with i > 0 have total degree < d and

\chi (X, f_*\mathcal{O}_ Z \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}) = \deg (f : Z \to f(Z)) \chi (f(Z), \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}|_{f(Z)})

modulo a polynomial of total degree < d by Lemma 72.18.2. The desired result follows. \square


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