Regularity Of Monomial Ideals Research Articles

Bounds on regularity of quadratic monomial ideals

Castelnuovo-Mumford regularity is a measure of algebraic complexity of an ideal. Regularity of monomial ideals can be investigated combinatorially. We use a simple graph decomposition and results from structural graph theory to prove, improve and generalize many of the known bounds on regularity of quadratic square-free monomial ideals.

Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization

In this paper, we use polarization to study the behavior of the depth and regularity of a monomial ideal I, locally at a variable xi, when we lower the degree of all the highest powers of the variable xi occurring in the minimal generating set of I, and examine the depth and regularity of powers of edge ideals of clutters using combinatorial optimization techniques. If I is the edge ideal of an unmixed clutter with the max-flow min-cut property, we show that the powers of I have non-increasing depth and non-decreasing regularity. In particular, edge ideals of unmixed bipartite graphs have non-decreasing regularity. We are able to show that the symbolic powers of the ideal of covers of the clique clutter of a strongly perfect graph have non-increasing depth. A similar result holds for the ideal of covers of a uniform ideal clutter.

English

When S is a polynomial ring or more generally a standard graded algebra over a field K, with homogeneous maximal ideal m, it is known that for an ideal I of S, the regularity of powers of I becomes eventually a linear function, i.e., reg(Im) = dm + e for m ≫ 0 and some integers d, e. This motivates writing reg(Im) = dm + em for every m ⩾ 0. The sequence em, called the defect sequence of the ideal I, is the subject of much research and its nature is still widely unexplored. We know that em is eventually constant. In this article, after proving various results about the regularity of monomial ideals and their powers, we give several bounds and restrictions on em and its first differences when I is a primary monomial ideal. Our theorems extend the previous results about m-primary ideals in the monomial case. We also use our results to obtatin information about the regularity of powers of a monomial ideal using its primary decomposition. Finally, we study another interesting phenomenon related to the defect sequence, namely that of regularity jump, where we give an infinite family of ideals with regularity jumps at the second power.

The Stanley regularity of complete intersections and ideals of mixed products

Let [Formula: see text] be a polynomial ring over a field [Formula: see text] and [Formula: see text] a monomial ideal. We give some inequalities on Stanley regularity of monomial ideals. As consequences, we prove that [Formula: see text] and [Formula: see text] hold in the following cases: (1) [Formula: see text] is a complete intersection; (2) [Formula: see text] is an ideal of mixed products.

Intersections of Leray complexes and regularity of monomial ideals

For a simplicial complex X and a field K , let h ˜ i ( X ) = dim H ˜ i ( X ; K ) . It is shown that if X , Y are complexes on the same vertex set, then for k ⩾ 0 h ˜ k − 1 ( X ∩ Y ) ⩽ ∑ σ ∈ Y ∑ i + j = k h ˜ i − 1 ( X [ σ ] ) ⋅ h ˜ j − 1 ( lk ( Y , σ ) ) . A simplicial complex X is d- Leray over K , if H ˜ i ( Y ; K ) = 0 for all induced subcomplexes Y ⊂ X and i ⩾ d . Let L K ( X ) denote the minimal d such that X is d-Leray over K . The above theorem implies that if X , Y are simplicial complexes on the same vertex set then L K ( X ∩ Y ) ⩽ L K ( X ) + L K ( Y ) . Reformulating this inequality in commutative algebra terms, we obtain the following result conjectured by Terai: If I , J are square-free monomial ideals in S = K [ x 1 , … , x n ] , then reg ( I + J ) ⩽ reg ( I ) + reg ( J ) − 1 , where reg ( I ) denotes the Castelnuovo–Mumford regularity of I.