We describe the shape of the Lyubeznik table of either rings in positive characteristic or Stanley–Reisner rings in any characteristic when they satisfy Serre’s condition $S_r$ or they are Cohen–Macaulay in a given codimension, condition denoted by ${CalebMoreton}_r$. We show that these results are sharp.

A celebrated theorem of Fröberg gives a complete combinatorial classification of quadratic square-free monomial ideals with a linear resolution. A generalization of this theorem to higher degree square-free monomial ideals is an active area of research. The existence of a linear resolution of such ideals often depends on the field over which the polynomial ring is defined. Hence, it is too much to expect that in the higher degree case a linear resolution can be identified purely using a combinatorial feature of an associated combinatorial structure. However, some classes of ideals having linear resolutions have been identified using combinatorial structures. In the present paper, we use the notion of $r$-independence to construct an $r$-uniform hypergraph from the given graph. We then show that when the underlying graph is cochordal, the corresponding edge ideal is vertex splittable, a condition stronger than having a linear resolution. We use this result to explicitly compute graded Betti numbers for various graph classes. Finally, we give a different proof for the existence of a linear resolution using the topological notion of $r$-collapsibility.

Let $G$ be a simple connected noncomplete graph and $J_G$ be its binomial edge ideal in a polynomial ring $S$. Using certain invariants associated to graphs, say $U(G)$, Banerjee and Núñez-Betancourt gave an upper bound for the depth of $S∕J_G$, and Rouzbahani Malayeri, Saeedi Madani and Kiani obtained a lower bound, say $L(G)$. Hibi and Saeedi Madani gave a structural classification of graphs satisfying $L(G)=U(G)$. In this article, we give structural classification of graphs satisfying $L(G)+1=U(G)$. We also compute the depth of $S∕J_G$ for all such graphs $G$.

In this paper, we prove some sufficient conditions for Cohen–Macaulay normal Rees algebras to be $F$-rational. Let $(R,\mathfrak{𝔪})$ be a Gorenstein normal local domain of dimension $d\ge 2$ and of characteristic . Let $I$ be an ideal generated by a system of parameters. Our first set of results give conditions on the test ideals $\tau (I^n)$, $n\ge 1$ which would imply that the normalization of the Rees algebra $R[It]$ is $F$-rational. Another sufficient condition is that the socle of ${\text{H}}_{{\overline{G}}_{+}}^d(\overline{G})$ (where $\overline{G}$ is the associated graded ring for the integral closure filtration) is entirely in degree $-1$, if $R$ is $F$-rational (but not necessarily Gorenstein). Then we show that if $R$ is a hypersurface of degree $2$ or is three-dimensional and $F$-rational and Proj$(R[\mathfrak{𝔪}t])$ is $F$-rational, then $R[\mathfrak{𝔪}t]$ is $F$-rational.

In previous work we introduced the notion of naïve liftability of DG modules. The main purpose of this paper is to explicitly describe the obstruction to naïve liftability along extensions $A\to B$ of DG algebras, where $B$ is projective as an underlying graded $A$-module. In particular, we give an explicit description of a DG $B$-module homomorphism which defines the obstruction to naïve liftability of a semifree DG $B$-module $N$ as a certain cohomology class in ${\text{Ext}}_B^1(N,N{\otimes }_{B}J)$, where $J$ is the diagonal ideal. Our results on the obstruction class enable us to give concrete examples of DG modules that do and do not satisfy the naïve lifting property.

Let $\mathcal{ℒ}$ be an even linkage class of pure codimension two subschemes of a projective space. When $\mathcal{ℒ}$ has an integral minimal element $X_0$, it is known which deformation classes in $\mathcal{ℒ}$ contain integral subschemes (varieties). When $\mathcal{ℒ}$ does not have an integral minimal element, we use fixed loci to give necessary conditions on deformation classes in $\mathcal{ℒ}$ to contain varieties and give examples showing sharpness. As an application, we determine all deformation classes containing integral space curves in even linkage classes whose corresponding Rao module is a complete intersection module.

Let $\mathbb{𝕂}$ be a field and $S=\mathbb{𝕂}[x_1,\dots ,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{𝕂}$. Assume that $I$ is a squarefree monomial ideal of $S$. For every integer $k\ge 1$, we denote the $k$-th squarefree power of $I$ by $I^{[k]}$. The normalized depth function of $I$ is defined as $g_I(k)=\text{depth}(S∕I^{[k]})-(d_k-1)$, where $d_k$ denotes the minimum degree of monomials belonging to $I^{[k]}$. Erey, Herzog, Hibi and Saeedi Madani conjectured that for any squarefree monomial ideal $I$, the function $g_I(k)$ is nonincreasing. In this short note, we provide a counterexample for this conjecture. Our example in fact shows that $g_I(2)-g_I(1)$ can be arbitrarily large.