Regularity and multiplicity of Veronese type algebras

Let [Formula: see text] be a finite simple graph and [Formula: see text] denote its vertex cover ideal in a polynomial ring over a field [Formula: see text]. In this paper, we show that all symbolic powers of vertex cover ideals of certain vertex-decomposable graphs have linear quotients. Using these results, we give various conditions on a subset [Formula: see text] of the vertices of [Formula: see text] so that all symbolic powers of vertex cover ideals of [Formula: see text], obtained from [Formula: see text] by adding a whisker to each vertex in [Formula: see text], have linear quotients. For instance, if [Formula: see text] is a vertex cover of [Formula: see text], then all symbolic powers of [Formula: see text] have linear quotients. Moreover, we compute the Castelnuovo–Mumford regularity of symbolic powers of certain vertex cover ideals.

For a monomial ideal I, we consider the ith homological shift ideal of I, denoted by $${\text {HS}}_i(I)$$ , that is, the ideal generated by the ith multigraded shifts of I. Some algebraic properties of this ideal are studied. It is shown that for any monomial ideal I and any monomial prime ideal P, $${\text {HS}}_i(I(P))\subseteq {\text {HS}}_i(I)(P)$$ for all i, where I(P) is the monomial localization of I. In particular, we consider the homological shift ideal of some families of monomial ideals with linear quotients. For any $${{\mathbf {c}}}$$ -bounded principal Borel ideal I and for the edge ideal of complement of any path graph, it is proved that $${\text {HS}}_i(I)$$ has linear quotients for all i. As an example of $${{\mathbf {c}}}$$ -bounded principal Borel ideals, Veronese type ideals are considered and it is shown that the homological shift ideal of these ideals are polymatroidal. This implies that for any polymatroidal ideal which satisfies the strong exchange property, $${\text {HS}}_j(I)$$ is again a polymatroidal ideal for all j. Moreover, for any edge ideal with linear resolution, the ideal $${\text {HS}}_j(I)$$ is characterized and it is shown that $${\text {HS}}_1(I)$$ has linear quotients.

In this paper, we introduce the concept of equivariant linear quotients in [Formula: see text], extending the notion of linear quotients for monomial ideals in the finite polynomial ring [Formula: see text] over a field [Formula: see text]. We also define the notions of equivariant stability and equivariant strong stability. Our main results show that if [Formula: see text] is a square-free monomial ideal with equivariant linear quotients, then the ideal [Formula: see text] also possesses equivariant linear quotients (see Theorem 3.12). We demonstrate that if [Formula: see text] is a square-free equivariant stable ideal, then [Formula: see text] remains a square-free equivariant stable ideal (see Theorem 4.7). Moreover, we provide examples illustrating that these results do not generally extend to non-squarefree ideals. Alongside, we examine the asymptotic behavior of the Castelnuovo–Mumford regularity and the projective dimension of these chains of ideals.

Powers of ideals associated to (C4,2K2)-free graphs

Castelnuovo–Mumford regularity: examples of curves and surfaces

Let be a standard graded ring, be the irrelevant ideal of R and 𝔞0 be an ideal of R 0. In this paper, as a generalization of the concept of Castelnouvo–Mumford regularity reg(M) of a finitely generated graded R-module M, we define the regularity of M with respect to 𝔞0 + R +, say reg𝔞0 + R + (M). We also study some relations of this new invariant with the classic one. To this end, we need to consider the cohomological dimension of some finitely generated R 0-modules. Also, we will express reg𝔞0 + R + (M) in terms of some invariants of the minimal graded free resolution of M and see that in a special case this invariant is independent of the choice of 𝔞0.

Non-commutative Castelnuovo–Mumford regularity and AS-regular algebras

The paper begins by overviewing the basic facts on geometric exceptional collections. Then we derive, for any coherent sheaf $\cF$ on a smooth projective variety with a geometric collection, two spectral sequences: the first one abuts to $\cF$ and the second one to its cohomology. The main goal of the paper is to generalize Castelnuovo–Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on smooth projective varieties X with a geometric collection σ. We define the notion of regularity of a coherent sheaf $\cF$ on X with respect to σ. We show that the basic formal properties of the Castelnuovo–Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we show that in case of coherent sheaves on $\PP^n$ and for a suitable geometric collection of coherent sheaves on $\PP^n$ both notions of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a smooth quadric hypersurface $Q_n \subset \PP^{n+1}$ (n odd) with respect to a suitable geometric collection and we compare it with the Castelnuovo–Mumford regularity of their extension by zero in $\PP^{n+1}$.

Castelnuovo Mumford regularity with respect to multigraded ideals

Castelnuovo–Mumford regularity and Ratliff–Rush closure

Generalized Castelnuovo–Mumford regularity for affine Kac–Moody algebras

McCullough and Peeva found sequences of counterexamples to the Eisenbud–Goto conjecture on the Castelnuovo–Mumford regularity by using Rees-like algebras, where entries of each sequence have increasing dimensions and codimensions. In this paper we suggest another method to construct counterexamples to the conjecture with any fixed dimension $n\geq 3$ and any fixed codimension $e\geq 2$. Our strategy is an unprojection process and utilizes the possible complexity of homogeneous ideals with three generators. Furthermore, our counterexamples exhibit how singularities affect the Castelnuovo–Mumford regularity.

New upper and lower bounds on the Castelnuovo–Mumford regularity are given in terms of the Hilbert coefficients. Examples are provided to show that these bounds are in some sense nearly sharp.

A famous theorem of Kalai and Meshulam is that r e g ( I + J ) ≤ r e g ( I ) + r e g ( J ) − 1 reg(I + J) \leq reg(I) + reg(J) -1 for any squarefree monomial ideals I I and J J . This result was subsequently extended by Herzog to the case where I I and J J are any monomial ideals. In this paper we conjecture that the Castelnuovo–Mumford regularity is subadditive on binomial edge ideals. Specifically, we propose that r e g ( J G ) ≤ r e g ( J H 1 ) + r e g ( J H 2 ) − 1 reg(J_{G}) \leq reg(J_{H_{1}}) + reg(J_{H_{2}}) -1 whenever G G , H 1 H_{1} , and H 2 H_{2} are graphs satisfying E ( G ) = E ( H 1 ) ∪ E ( H 2 ) E(G) = E(H_{1}) \cup E(H_{2}) and J ∗ J_{\ast } is the associated binomial edge ideal. We prove a special case of this conjecture which strengthens the celebrated theorem of Malayeri–Madani–Kiani that r e g ( J G ) reg(J_{G}) is bounded above by the minimal number of maximal cliques covering the edges of the graph G G . From this special case we obtain a new upper bound for r e g ( J G ) reg(J_{G}) , namely that r e g ( J G ) ≤ h t ( J G ) + 1 reg(J_{G}) \leq ht(J_{G}) +1 . Our upper bound gives an analogue of the well-known result that r e g ( I ( G ) ) ≤ h t ( I ( G ) ) + 1 reg(I(G)) \leq ht(I(G)) +1 where I ( G ) I(G) is the edge ideal of the graph G G . We additionally prove that this conjecture holds for graphs admitting a combinatorial description for its Castelnuovo–Mumford regularity, that is for closed graphs, bipartite graphs with J G J_{G} Cohen–Macaulay, and block graphs. Finally, we give examples to show that our new upper bound is incomparable with Malayeri–Madani–Kiani’s upper bound for r e g ( J G ) reg(J_{G}) given by the size of a maximal clique disjoint set of edges.

This note has two goals. The first is to give a short and self contained introduction to the Castelnuovo–Mumford regularity for standard graded rings \(R=\bigoplus _{i\in {\mathbb N}} R_i\) over general base rings R0. The second is to present a simple and concise proof of a classical result due to Cutkosky, Herzog and Trung and, independently, to Kodiyalam asserting that the regularity of powers Iv of an homogeneous ideal I of R is eventually a linear function in v. Finally we show how the flexibility of the definition of the Castelnuovo–Mumford regularity over general base rings can be used to give a simple proof of a result proved by the authors in “Maximal minors and linear powers”.