Gorchinskii, S. O.; Parshin, A. N. Adelic Lefschetz formula for the action of a one-dimensional torus. (English) Zbl 1152.14017 Uraltseva, N.N.(ed.), Proceedings of the St. Petersburg Mathematical Society. Vol. XI. Transl. from the Russian. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-4204-8/hbk). Transl., Ser. 2, Am. Math. Soc. 218, 31-48 (2006); translation from Tr. St-Petersb. Mat. Obshch. 11, 37-57 (2005). The authors generalize the classical Lefschetz fixed point theorem in algebraic topology and the Lefschetz formula for the cohomology of coherent sheaf \(\mathcal{F}\) on a projective algebraic variety \(X\). Instead of a single endomorphism they consider a family of automorphisms of \(X\) parametrized by a one-dimensional torus \({\mathbb{G}}_m\). In the case when \(\mathcal{F}\) is a sheaf corresponding to a vector bundle \(F\) and the set of fixed points \(Z\) is finite, each term in the Lefschetz formula can be regarded as the trace of the action of \(f\in{\mathbb{G}}_m\) on the infinite dimensional space \({\widehat{\mathcal{O}}}_{X,x}\otimes F_x\) where \({\widehat{\mathcal{O}}}_{X,x}\) is the local ring at the point \(x\in Z\), and \(F_x\) is the fiber of the vector bundle over \(x\). The authors prove the Lefschetz formula using the cohomology theory of the adelic complexes \({\mathbb{A}}_X(\mathcal{F})^{\bullet}\) of coherent sheaves \(\mathcal{F}\) and the notion of traces of the action of the group \({\mathbb{G}}_m\) on the components of the adelic complex. The adelic Lefschetz formula for a locally free sheaf on a nonsingular projective variety of arbitrary dimension has the form \[ Tr( {\mathbb{G}}_m,H^{\bullet}(X,\mathcal{F}))=Tr({\mathbb{G}}_m, {\mathbb{A}}^{fix}_X(\mathcal{F}) ), \] where \({\mathbb{A}}^{fix}_X(\mathcal{F}) \) is the “fixed” part of the adelic complex connected with the set of fixed points \(X^{{\mathbb{G}}_m}\). Since the complex \({\mathbb{A}}^{fix}_X(\mathcal{F}) \) is defined for any scheme, they conjecture that the adelic Lefschetz formula holds for an arbitrary proper scheme over a field \(k\).For the entire collection see [Zbl 1101.11001]. Reviewer: V. Uma (Chennai) Cited in 1 ReviewCited in 1 Document MSC: 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14L30 Group actions on varieties or schemes (quotients) 14C99 Cycles and subschemes 14J99 Surfaces and higher-dimensional varieties Keywords:Lefschetz formula; Lefschetz number; Adelic complexes; one-dimensional torus PDFBibTeX XMLCite \textit{S. O. Gorchinskii} and \textit{A. N. Parshin}, Transl., Ser. 2, Am. Math. Soc. 218, 31--48 (2006; Zbl 1152.14017); translation from Tr. St-Petersb. Mat. Obshch. 11, 37--57 (2005) Full Text: arXiv