arXiv subjects

Richard F. Bartels

Publications and source records attributed to Richard F. Bartels.

3 recordsLinked to original sources

arXiv2026

Cohen-Macaulay approximations over generically Gorenstein rings

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.

math.AC↗

arXiv2026

Cohen-Macaulay approximations and the $\text{SC}_r$-condition

We study the relation between MCM approximations and FID hulls of modules over a Cohen-Macaulay local ring $R$ with canonical module, specifically when $R$ is generically Gorenstein. We then generalize a result of Kato, who proved that a Gorenstein complete local ring $R$ satisfies the $\text{SC}_{2}$-condition if and only if $R$ is a UFD. For $r \geq 3$, we prove a criterion for when an MCM $R$-module $M$ satisfies the $\text{SC}_{r}$-condition, assuming that its first syzygy $Ω_{R}^{1}(M)$ satisfies the $\text{SC}_{r-1}$-condition.

math.AC↗

arXiv2024

Some properties of ideals in Cohen-Macaulay local rings

For a Cohen-Macaulay local ring $(R,\mathfrak{m})$ with canonical module, we study how relations between $\text{index}(R)$ and $\text{g}\ell\ell(R)$ and between $\text{index}(R)$ and $e(R)$ are preserved when factoring out regular sequences and localizing at prime ideals. We then give conditions for when ideals in a one-dimensional Cohen-Macaulay local ring are Elias and Burch, and use these conditions to study the relationship between Elias, Burch, and Ulrich ideals.

math.AC↗