Cohen-Macaulay approximations over generically Gorenstein rings
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910065026400256 |
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| author | Bartels, Richard F. |
| author_facet | Bartels, Richard F. |
| contents | Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring with canonical module that is generically Gorenstein. In this paper, I prove isomorphisms relating the minimal MCM approximations and minimal FID hulls of modules constructed from a canonical ideal $\,ω\subset R$, including $\,ω/xR$, with $\,x \in ω\,$ a nonzerodivisor, $\,(ω/xR)^{\vee}:=\text{Ext}^1_R(ω/xR,ω)$, $\,R/ω^2$, and $\,ω/ω^2$. I also prove that if $R$ is not Gorenstein, then $δ_{R}\left(ω/xR \right)=δ_{R}\left(\left(ω/xR \right)^{\vee} \right)=0\,$ and $\,γ_{R}\left(Ω^{1}_{R}\left(ω/xR \right) \right)=γ_{R}\left(Ω^{1}_{R}\left(\left(ω/xR\right)^{\vee}\right) \right)=0$, where $δ_R$ is Auslander's $\,δ$-invariant and $γ_R$ is the dual $γ$-invariant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21587 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cohen-Macaulay approximations over generically Gorenstein rings Bartels, Richard F. Commutative Algebra Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring with canonical module that is generically Gorenstein. In this paper, I prove isomorphisms relating the minimal MCM approximations and minimal FID hulls of modules constructed from a canonical ideal $\,ω\subset R$, including $\,ω/xR$, with $\,x \in ω\,$ a nonzerodivisor, $\,(ω/xR)^{\vee}:=\text{Ext}^1_R(ω/xR,ω)$, $\,R/ω^2$, and $\,ω/ω^2$. I also prove that if $R$ is not Gorenstein, then $δ_{R}\left(ω/xR \right)=δ_{R}\left(\left(ω/xR \right)^{\vee} \right)=0\,$ and $\,γ_{R}\left(Ω^{1}_{R}\left(ω/xR \right) \right)=γ_{R}\left(Ω^{1}_{R}\left(\left(ω/xR\right)^{\vee}\right) \right)=0$, where $δ_R$ is Auslander's $\,δ$-invariant and $γ_R$ is the dual $γ$-invariant. |
| title | Cohen-Macaulay approximations over generically Gorenstein rings |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2603.21587 |