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Let $(R,m)$ be a Gorenstein local ring of characteristic $p$. Let $^f R$ denote the $R-R$-bimodule with additve group $R$ and left and right scalar multiplication given by $a\circ r\bullet b = a^p rb$. Then $\text{Hom}_R(^fR,R)\cong\ ^fR$.

Since $^fR\cong R$ as right $R$-module, the above isomorphism is true as right $R$-modules. How can I show the isomorphism for left $R$-modules using the Gorenstein condition?

Bromelain
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  • It may be helpful to be clear about what is meant by $\operatorname{Hom}_R(\mbox{}^f R,R)$ . Do you intend this to be the collection of right $R$-linear maps, left $R$-linear maps, or $R$-bimodule maps? – metalspringpro Oct 12 '23 at 13:41
  • @metalspringpro Sorry for causing the confusion. Here I mean the left $R$-linear map, and should only focus on the left $R$-module isomorphism instead. – Bromelain Oct 12 '23 at 16:02
  • With that in mind, I recommend taking a look at Theorem 3.3.7 (b) in "Cohen-Macaulay Rings" by Bruns and Herzog. – metalspringpro Oct 12 '23 at 16:08
  • @metalspringpro But this theorem requires $^fR$ to be module finite. I don’t think it’s a finite $R$-module – Bromelain Oct 12 '23 at 16:15
  • There is a standard approach in this theory for reducing to the $F$-finite case that one can apply first. – metalspringpro Oct 12 '23 at 16:54
  • @metalspringpro Could you provide more detail or references about the standard approach? – Bromelain Oct 12 '23 at 17:13
  • The construction is known as the $\Gamma$-construction and can be found in Section 6 of this paper https://www.jstor.org/stable/2154942. See also Ch. 10, S. 3 of Cohen-Macaulay Representations by Leuschke-Wiegand. The point is to complete and then extend the residue field of the completion so the resulting extension is faithfully flat and $F$-finite, the idea being that a complete local ring with $F$-finite residue field is itself $F$-finite. – metalspringpro Oct 12 '23 at 17:29
  • @metalspringpro Thanks！ – Bromelain Oct 12 '23 at 18:09

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