nLab Gorenstein ring spectrum

Contents

Contents

1. 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.

2. Examples

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

3. References

Last revised on April 4, 2017 at 10:05:52. See the history of this page for a list of all contributions to it.