nLab Gorenstein ring spectrum

Idea

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

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

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

Examples

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

References

• William Dwyer, John Greenlees, Srikanth Iyengar, Duality in algebra and topology, Advances in Maths

  200 (2006) 357-402, (arXiv:math/0510247)

• John Greenlees, Homotopy Invariant Commutative Algebra over fields, (arXiv:1601.02473)