Stable cohomology over local rings

AbstractFor a commutative noetherian ring R with residue field k stable cohomology modules ExtˆRn(k,k) have been defined for each n∈Z, but their meaning has remained elusive. It is proved that the k-rank of any ExtˆRn(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 ExtR(k,k)→ExtˆR(k,k). Some techniques developed in the paper are applicable to the study of stable cohomology functors over general associative rings