Some problems in algebraic geometry - Donovan, P (1968)
Let X be a protective non-singular algebraic variety over an algebraically closed field k. Let f: X ⟶ X be an endomorphism of X and let F be a locally free sheaf on X. Then the vector spaces Hq [X;F] are known to be of finite rank and to be zero except for finitely many integers q. Let Φ: f*F → F be a morphism of sheaves. Then endomorphisms, eq, of each Hq [X;F] may be constructed as the composite: Hq[X;F] f*⟶ Hq[X;f*F] Φ*⟶ Hq[X;F]. Set L(f,F,Φ) = ∑ (-)qtrace(eq) . If f has a non-singular fixed point set and subject to a slight restriction on the nature of the endomorphism induced by f on the normal bundle of each fixed component, Y, formulae are obtained of the form: L(f,F,Φ) = ∑Y v'(Y,Φ), with the sum being taken over the set of fixed components, in the two special cases: (α) f is periodic with period prime to the characteristic of k. (β) each fixed component is a point or a curve. The definition of v'(Y;Φ) involves Chern characters and the like of the restriction of F and Φ to Y, as well as the induced action on the normal bundle. If k has characteristic p ≠ 0, in case (α) a stronger p-adic lifting of the above formula is obtained. The methods used are setting up an appropriate formalism and using the theory of derived categories in homological algebra. The algebraic geometry used is essentially that of the Borel-Serre paper (1958) vintage.