Semi-infinite cohomology of associative algebras and bar duality

Abstract

A similar approach for associative algebra semiinfinite cohomology was presented in [Ar]. Let k be a base field. Consider a “right semiregular A-module” SA := Ind A N Coind N k (k) and its endomorphism algebra EndA(SA) =: A . Suppose A is augmented, i. e. there is a two-sided ideal A ♯ ∈ A such that the quotient algebra is k, let k denote the trivial left A-module. Then semiinfinite cohomology with coefficients in a graded A-module can be viewed as a “two sided derived functor” of the functor HomA♯(k, SA ⊗A ∗) (see [V], [Ar]).