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Gorenstein ring spectrum

Contents

Idea

The definition of a Gorenstein ring spectrum is motivated by the Gorenstein condition for rings. A Gorenstein ring, RR, is a commutative Noetherian local ring such that the Ext-group Ext R *(k,R)Ext^{\ast}_R(k, R) is one dimensional as a kk-vector space, where kk is the residue field of RR. This last condition may be restated as the property that the homology of the (right derived) Hom complex Hom R(k,R)Hom_R(k, R) is equivalent to a suspension of kk.

Thus, a ring spectrum Rk\mathbf{R} \to \mathbf{k} is said to be Gorenstein if there is an equivalence of R\mathbf{R}-module spectra Hom R(k,R)Σ akHom_{\mathbf{R}}(\mathbf{k}, \mathbf{R}) \simeq \Sigma^a\mathbf{k} for some integer aa.

A ring, RkR \to k, is Gorenstein if and only if the ring spectrum HRHkH R \to H k is Gorenstein.

Examples

Examples from representation theory, from chromatic stable homotopy theory and from rational homotopy theory are given in (Greenlees16, Sec. 23).

References

Last revised on April 4, 2017 at 06:05:52. See the history of this page for a list of all contributions to it.