Stable cohomology over local rings

Abstract

For a commutative noetherian ring R with residue field k stable cohomology modules Ext ˆ R n ( k , k ) have been defined for each n ∈ Z , but their meaning has remained elusive. It is proved that the k-rank of any Ext ˆ R n ( k , k ) characterizes important properties of R, such as being regular, complete intersection, or Gorenstein. These numerical characterizations are based on results concerning the structure of Z -graded k-algebra carried by stable cohomology. It is shown that in many cases it is determined by absolute cohomology through a canonical homomorphism of algebras Ext R ( k , k ) → Ext ˆ R ( k , k ) . Some techniques developed in the paper are applicable to the study of stable cohomology functors over general associative rings.