Institut für Mathematik       Universität Zürich

Algebra and Topology

course by Lorenzo Mantovani and Christian Dahlhausen in spring term 2020 with exercise class by Emil Jacobsen.
Tuesday, 15.15-17.00, and Thursday, 15.15-17.00, online via BigBlueButton.


Lecture notes

Here you find lecture notes for the second half of the course: version: 28th May 2020

Board diary (for the second half of the course)

07.04.Motivation for cohomology, abelian categories, injective objects, sheaves have enough injectives
21.04.Existence and properties of right-derived functors
23.04.Derived functors via F-acyclic resolutions, injective abelian groups
28.04.Projective objects, left-derived functors, enough projective sheaves on finitely generated spaces, sheaf cohomology
30.04.Flasque sheaves are acyclic, cohomology on open/closed subspaces, Mayer-Vietoris sequences
05.05.Godement resolution, Čech cohomology
07.05.Čech cohomology equals sheaf cohomology on paracompact spaces, first cohomology on good covers
12.05.an example, first cohomology in terms of torsors
14.05.Higher direct images, homotopy invariance of sheaf cohomology
19.05.Sheaves in terms of étalé spaces, étalé spaces of locally constant sheaves
21.05.complements on (locally) constant sheaves, fundamental groupoid and its monodromy representation
26.05.equivalence of the monodromy functor, classification of locally constant sheaves on spheres and on the torus
28.05.alternative proof of homotopy invariance of sheaf cohomology for locally constant sheaves, outlook