Classifying spaces, group cohomology and Chow rings

Il corso si terrà ogni venerdì dalle 11 alle 14. Le lezioni saranno in aula G di matematica e online al link Google Meet meet.google.com/gve-otgc-hoh.

Per ricevere comunicazioni sul corso iscrivetevi su google classroom: link invito.

Testi:

Il corso è basato sul libro Group homology and algebraic cycles di Burt Totaro. Per i prerequisiti (che comunque ripasseremo/rifaremo in classe) suggerisco:

• Hatcher, Algebraic topology e May, a concise course in algebraic topology per la topologia algebrica.

• Hartshorne, Algebraic geometry o Liu, Algebraic geometry and arithmetic curves (consigliato per chi ha interessi più aritmetici) per la geometria algebrica.

• Fulton, Intersection theory per la teoria dell'intersezione.

• Edidin-Graham, Equivariant intersection theory , Totaro, The Chow ring of a classifying space per la teoria dell'intersezione equivariante. I calcoli per ricavare gli anelli di Chow dei classificanti dei gruppi classici si può trovare in Molina-Vistoli, On the Chow rings of classifying spaces for classical groups.

Ricevimento:

Ricevimento su appuntamento.

Sinossi:

Group cohomology draws a powerful connection between algebra and geometry: a group G naturally determines a topological space BG, the classifying space of G, and we can define the cohomology ring of G as the cohomology of the space BG. The challenge is then to determine how the properties of G are connected to those of the cohomology ring H*(BG).

On the algebro-geometric side, one can construct a different classifying object BG, which can be seen either as an algebraic stack or as a limit of algebraic varieties. A natural analog of the singular cohomology ring in this setting is given by the Chow ring CH*(BG). Over the complex numbers the two are tied by the cycle map CH^i(BG) -> H^2i(BG), which is an isomorphism after tensoring by the rational numbers.

In this short course we will introduce these two invariants, carry over some basic computations and prove the comparison theorem between the two. Time allowing, we will delve into deeper questions on the depth and regularity of these rings.

The course is based on the book "Group cohomology and algebraic cycles", by Burt Totaro.

Appunti