Tate resolution and coherent sheaf cohomology
Vortrag von Prof. Dr. Frank-Olaf Schreyer
Sprecher eingeladen von: Prof. Dr. Joseph Ayoub
Datum: 04.11.13 Zeit: 13.15 - 14.45 Raum: Y27H25
The Tate resolution of a coherent sheaf $\mathcal F$ on a projective space $\mathbb P^n =\mathbf P(W)$ is a minimal complex $T(\mathcal F)$ of graded free $E=\Lambda W^*$-module over the exterior algebra, which encodes the cohomology of $\mathcal F$: $$T^d(\mathcal F) = \sum_{i=0}^n H^i(\matchcal F(d-i) \otimes \omega_E(i-d),$$ where $\omega_E=Hom_K(E,K)=\Lambda W$ denotes the dualizing module of the exterior algebra. In the talk I will review this concept and its generalization to product of projective spaces. In the case of products, the Tate resolution is more difficult to define. It is no longer a complex of finitely generated modules, but it still governs the cohomology, the Beilinson monads and image complexes $R\pi_* \mathcal F(a)$ for any partial projection $\pi$ onto factors. Application include the computation of direct image complexes for morphisms between projective varieties, in terms of Beilinson monads, suitable for Computer algebra packages.