Stable cohomology over local rings
Let \(R\) be a noetherian local ring with residue field \(k\). In the present paper, the authors study the absolute cohomology module \(\mathcal E = \text{Ext}_R(k, k)\) and the stable cohomology module \(\mathcal S = \widehat{\text{Ext}}_R(k, k)\). This paper consists of ten sections excluding the introduction. In the first five sections, the authors develop general properties of stable cohomology theory. In Section 6, they give a criterion for \(R\) to be regular, complete intersection, or Gorenstein by using a homogeneous component \(\widehat{\text{Ext}}^n_R(k, k)\) of \(\mathcal S\). In Sections 7--9, they study the structure of \(\mathcal S\). Assume that \(R\) is not regular. It is known that the natural map \(\iota \: \mathcal E \to \mathcal S\) of graded \(k\)-algebras is injective. The authors compute the cokernel of \(\iota\) and show that \(\iota\) splits if \(\text{depth\,} \mathcal E \geq 2\) (in general \(\text{depth\,} \mathcal E \geq 1\)). The condition \(\text{depth\,} \mathcal E \geq 2\) is not strong. They give several classes of rings such that \(\text{depth} \,\mathcal E \geq 2\). On the other hand, \(\text{depth} \,\mathcal E = 1\) if \(R\) is a Golod ring. In Section 10, the authors studied the structure of \(\mathcal S\) when \(R\) is Golod.
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