The main theme of this paper is to study different “Gorenstein defect categories” and their connections. This is done by studying rings for which ${\mathbb{𝕂}}_{ac}\left(\text{ Prj-}R\right)={\mathbb{𝕂}}_{tac}\left(\text{ Prj-}R\right)$, that is, rings enjoying the property that every acyclic complex of projectives is totally acyclic. Such studies have been started by Iyengar and Krause over commutative Noetherian rings with a dualizing complex. We show that a virtually Gorenstein Artin algebra is Gorenstein if and only if it satisfies the above mentioned property. Then, we introduce recollements connecting several categories which help in providing categorical characterizations of Gorenstein rings. Finally, we study relative singularity categories that lead us to some more “Gorenstein defect categories”.

Let $D$ be an integral domain with quotient field $K$, $X$ be an indeterminate over $D$, $D\left[\left[X\right]\right]$ be the power series ring over $D$, and $c\left(f\right)$ be the ideal of $D$ generated by the coefficients of $f\in D\left[\left[X\right]\right]$. We will say that a star operation $\ast$ on $D$ is a c-star operation if (i) $c{\left(fg\right)}^{\ast }={\left(c\left(f\right)c\left(g\right)\right)}^{\ast }$ for all $0\ne f,g\in D\left[\left[X\right]\right]$ and (ii) ${\left(AB\right)}^{\ast }\subseteq {\left(AC\right)}^{\ast }$ implies $B^{\ast }\subseteq C^{\ast }$ for all nonzero fractional ideals $A,B,C$ of $D$. Assume that $D$ admits a c-star operation $\ast$, and let $Kr\left(\left(D,\ast \right)\right)=\left\{f∕g\mid f,g\in D\left[\left[X\right]\right],g\ne 0,\text{and }c\left(f\right)\subseteq c{\left(g\right)}^{\ast }\right\}$. Among other things, we show that $Kr\left(\left(D,\ast \right)\right)$ is a Bézout domain, $D$ is completely integrally closed, the $v$-operation on $D$ is a c-star operation, and $Kr\left(\left(D,v\right)\right)$ is a completely integrally closed Bézout domain. We also show that if $V$ is a rank-one valuation domain, then the $v$-operation on $V$ is a c-star operation, $Kr\left(\left(V,v\right)\right)$ is a rank-one valuation domain, and $Kr\left(\left(V,v\right)\right)$ is a DVR if and only if $V$ is a DVR. Using this result, we show that if $D$ is a generalized Krull domain, then $Kr\left(\left(D,v\right)\right)$ is a one-dimensional generalized Krull domain.

Let $S$ be a polynomial ring in $n$ variables over a field $K$ of any characteristic. Let $M$ be a strongly stable submodule of a finitely generated graded free $S$-module $F$, with all basis elements of $F$ of the same degree. The existence of a general strongly stable submodule $\tilde{M}$ of a finitely generated graded free $S$-module $\tilde{F}$, $rank\tilde{F}\ge rankF$, which preserves values and positions of the extremal Betti numbers of $M$, is proved.

It is proved that a module $M$ over a Noetherian ring $R$ of positive characteristic $p$ has finite flat dimension if there exists an integer $t\ge 0$ such that ${Tor}_i^R\left(M{,}^{f^e}R\right)=0$ for $t\le i\le t+dimR$ and infinitely many $e$. This extends results of Herzog, who proved it when $M$ is finitely generated. It is also proved that when $R$ is a Cohen–Macaulay local ring, it suffices that the $Tor$ vanishing holds for one $e\ge {log}_{p}e\left(R\right)$, where $e\left(R\right)$ is the multiplicity of $R$.

Given the complement of a hyperplane arrangement, let $\mathrm{\Gamma }$ be the closure of the graph of the map inverting each of its defining linear forms. The characteristic polynomial manifests itself in the Hilbert series of $\mathrm{\Gamma }$ in two different-seeming ways, one due to Terao and the other to Huh and Katz. We define an extension of the no broken circuit complex of a matroid and use it to give a direct Gröbner basis argument that the polynomials extracted from the Hilbert series in these two ways agree.

Let $R$ be an integral domain, and let $A,A_1,A_2,\dots ,A_s$ be overrings of $R$, where $A$ is of the form $S^{-1}R$, where $S=R\setminus {\mathfrak{𝔭}}_1\cup \cdots \cup {\mathfrak{𝔭}}_n$ for some prime ideals ${\mathfrak{𝔭}}_i$, and where each $A_i$, $i\ge 2$, is of the form $S_i^{-1}R$ for some multiplicatively closed subset $S_i$ of $R$. It is shown that if $A\subseteq A_1\cup \cdots \cup A_s$, then $A\subseteq A_i$ for some $i$.

We explore the relationship between secant ideals and initial ideals of $I_{2,n}$, the ideal of the Grassmannian, $Gr\left(2,{ℂ}^n\right)$. The $\left(r-1\right)$-secant of $I_{2,n}$ is the ideal generated by the $2r\times 2r$ subpfaffians of a generic $n\times n$ skew-symmetric matrix. It has been conjectured that for a weight vector $\omega$ in the tropical Grassmannian, the secant of the initial ideal of $I_{2,n}$ with respect to $\omega$ is equal to the initial ideal of the secant. We show that this conjecture is not true in general. Using the correspondence between weight vectors in the tropical Grassmannian and binary leaf-labeled trees, we also give necessary and sufficient conditions for the conjecture to hold in terms of the topology of the tree associated to $\omega$. In the course of proving this result, we show that the ideal ${in}_{\omega }\left(I_{2,n}^{\left\{r\right\}}\right)$ is always prime, thus giving a new class of prime initial ideals of the Pfaffian ideals.

Let $R$ be a graded ring. We introduce a class of graded $R$-modules called Gröbner-coherent modules. Roughly, these are graded $R$-modules that are coherent as ungraded modules because they admit an adequate theory of Gröbner bases. The class of Gröbner-coherent modules is formally similar to the class of coherent modules: for instance, it is an abelian category closed under extension. However, Gröbner-coherent modules come with tools for effective computation that are not present for coherent modules.

We study properties of the Stanley–Reisner rings of simplicial complexes with isolated singularities modulo two generic linear forms. Miller, Novik, and Swartz proved that if a complex has homologically isolated singularities, then its Stanley–Reisner ring modulo one generic linear form is Buchsbaum. Here we examine the case of nonhomologically isolated singularities, providing many examples in which the Stanley–Reisner ring modulo two generic linear forms is a quasi-Buchsbaum but not Buchsbaum ring.

Given a local noetherian ring $R$ whose formal completion is integral, we introduce and study the $p$-radical closure $R^{prc}$. Roughly speaking, this is the largest purely inseparable $R$-subalgebra inside the formal completion $\widehat{R}$. It turns out that the finitely generated intermediate rings $R\subset A\subset R^{prc}$ have rather peculiar properties. They can be used in a systematic way to provide examples of integral local rings whose normalization is nonfinite, that do not admit a resolution of singularities, and whose formal completion is nonreduced.