## Algebraic K-Theory Seminar

Mark Walker

University of Nebraska

The total rank and toral rank conjectures

**Abstract:**Let R be local ring of dimension d and F a complex of free R-modules whose homology modules are of finite length. It has been conjectured that the sum of the ranks of the free modules occurring in F (i.e., the total rank of F) must be at least 2^d. When F is the minimal resolution of a module of finite length, this conjecture is a weak form of the well-known Buchsbaum-Eisenbud-Horrocks Conjecture. Assume X is a compact CW complex that admits a free action by a d-dimensional torus. The Toral Rank Conjecture, due to Halperin, predicts that the sum of the ranks of the rational homology groups of X must be at least 2^d. There is a variant of this conjecture, due to Carlsson, concerning spaces that admit a free action by an elementary abelian p-group. In this talk I will discuss the relationship between these conjectures and recent progress toward settling them.

Wednesday September 13, 2017 at 10:30 AM in SEO 1227