Bifunctor cohomology and Cohomological finite generation for reductive groups

Antoine Touzé (1, 2) , Wilberd van Der Kallen (3)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 LMJL - Laboratoire de Mathématiques Jean Leray
3 Utrecht Mathematical Institute

Let G be a reductive linear algebraic group over afield k. Let A be a finitely generated commutative k-algebra on which G acts rationally by k-algebra automorphisms. Invariant theory states that the ring of invariants A(G) = H-0(G, A) is finitely generated. We show that in fact the full cohomology ring H*(G, A) is finitely generated. The proof is based on the strict polynomial bifunctor cohomology classes constructed in [22]. We also continue the study of bifunctor cohomology of Gamma*(gl((1))).

Domaines: K-théorie et homologie [math.KT]; Théorie des représentations [math.RT]

Antoine Touzé, Wilberd van Der Kallen. Bifunctor cohomology and Cohomological finite generation for reductive groups. Duke Mathematical Journal, 2010, 151 (2), pp.251-278. ⟨10.1215/00127094-2009-065⟩. ⟨hal-00638188⟩