Almost complete intersections and the Lex-Plus-Powers Conjecture

On the Hilbert function of Artinian local complete intersections of codimension three

Given a 0-dimensional affine K-algebra R=K[x1,…,xn]∕I, where I is an ideal in a polynomial ring K[x1,…,xn] over a field K, or, equivalently, given a 0-dimensional affine scheme, we construct effective algorithms for checking whether R is a complete intersection at a maximal ideal, whether R is locally a complete intersection, and whether R is a strict complete intersection. These algorithms are based on Wiebe’s characterization of 0-dimensional local complete intersections via the 0-th Fitting ideal of the maximal ideal. They allow us to detect which generators of I form a regular sequence resp. a strict regular sequence, and they work over an arbitrary base field K. Using degree filtered border bases, we can detect strict complete intersections in certain families of 0-dimensional ideals.

Let 𝕏 be a set of K-rational points in ℙ1×ℙ1 over a field K of characteristic zero, let 𝕐 be a fat point scheme supported at 𝕏, and let R𝕐 be the bihomogeneous coordinate ring of 𝕐. In this paper we investigate the module of Kahler differentials ΩR𝕐∕K1. We describe this bigraded R𝕐-module explicitly via a homogeneous short exact sequence and compute its Hilbert function in a number of special cases, in particular when the support 𝕏 is a complete intersection or an almost complete intersection in ℙ1×ℙ1. Moreover, we introduce a Kahler different for 𝕐 and use it to characterize ACM reduced schemes in ℙ1×ℙ1 having the Cayley–Bacharach property.

Random monomial ideals

A criterion is established, in terms of the Massey products structure carried by the homology of partial resolutions, for the Yoneda cohomology algebra ExtA(k, k) to be a free module over the universal envelope of a free graded Lie subalgebra.It is shown that several conjectures on the (co)homology of local rings, in particular on the asymptotic behaviour of the Betti numbers, follow from such a structure.For all rings with edim A -depths * 3, and for Gorenstein rings with edim A -depth A = 4, the following dichotomy is proved: Either A is a complete intersection, or xiA(k, k) contains a nonabelian free graded Lie subalgebra.0. Ten conjectures on the homology of local rings (in place of Introduction).In this paper (A, rrt, k) will denote a noetherian local ring A with maximal ideal m and residue field k.A well-known homological invariant of A is its Poincar series 00 ; = 0 where b (= b(A)) is the /'th Betti number of A, i.e. the rank of the /'th free module in the unique, up to noncanonical isomorphism, yl-free resolution X of k.Much of the work on the homology of local rings has been motivated by the following famous conjecture:C0. PA(t) represents a rational function.Recently this has been settled in the negative by D. Anick,1 who constructed for each k a A>algebra A requiring 5 generators for m and having m3 = 0, with transcendental Poincar series.On the other hand, more than 20 years of research have produced different kinds of conditions, which ensure the rationality of PA(t).Before collecting for further reference several of them in a theorem, we introduce some notation, to be kept for the rest of the paper.edim A = embedding dimension of A := dimt m/m2 (= bx(A)); dim A -Krull dimension of A ;depth A = maximal length of an ^-regular sequence in m; d(A) = complete intersection defect of A '-dim^ a/no.-(dim R -dim A), where (R, n) is a regular local ring such that R/a is isomorphic to the m-adic

Let \((S, \mathfrak {n})\) be a regular local ring and let \(I \subseteq \mathfrak {n}^{2} \) be a perfect ideal of S. Sharp upper bounds on the minimal number of generators of I are known in terms of the Hilbert function of R = S/I. Starting from information on the ideal I, for instance the minimal number of generators, a difficult task is to determine good bounds on the minimal number of generators of the leading ideal I∗ which defines the tangent cone of R or to give information on its graded structure. Motivated by papers of S. C. Kothari and S. Goto et al. concerning the leading ideal of a complete intersection I = (f, g) in a regular local ring, we present results provided ht (I) = 2. If I is a complete intersection, we prove that the Hilbert function of R determines the graded Betti numbers of the leading ideal and, as a consequence, we recover most of the results of the previously quoted papers. The description is more complicated if ν(I) > 2 and a careful investigation can be provided when ν(I) = 3. Several examples illustrating our results are given.

Recently, nearly complete intersection ideals were defined by Boocher and Seiner to establish lower bounds on Betti numbers for monomial ideals (arXiv:1706.09866). Stone and Miller then characterized nearly complete intersections using the theory of edge ideals (arXiv:2101.07901). We extend their work to fully characterize nearly complete intersections of arbitrary generating degrees and use this characterization to compute minimal free resolutions of nearly complete intersections from their degree 2 part.

LetP be a positively graded polynomial ring over a fieldk of characteristic zero, letI be a homogeneous ideal ofP, and setR=P/I. The paper investigates the homological properties of someR-modules canonically associated withR, among them the module Ω R/k of Kahler differentials and the conormal moduleI/I 2. It is shown that a subexponential bound on the Betti numbers of any of these modules implies thatI is generated by aP-regular sequence. In particular, the finiteness of the projective dimension of the conormal module impliesR is a complete intersection. Similarly, the finiteness of the projective dimension of the differential module impliesR is a reduced complete intersection. This provides strong converses to some well-known properties of complete intersections, and establishes special cases of conjectures of Vasconcelos. The proofs of these results make extensive use of differential graded homological algebra. The crucial step is to show that any homomorphism of complexes from the minimal cotangent complexL R/k of Andre and Quillen into the minimal free resolution of the irrelevant maximal ideal m ofR, which extends the Euler map Ω R/k →, is a split embedding of gradedR-modules.

This note is meant to be an introduction to cohomological methods and their use in the theory of error-correcting codes. In particular we consider evaluation codes on a complete intersection. The dimension of the code is determined by the Koszul complex for X⊂ℙ2 and a lower bound for the minimal distance is obtained through linkage. By way of example our result fits the well-known formula for the minimal distance of the Generalized Reed-Muller code.

We investigate the minimal free resolutions of cyclic modules $R/I$, where $I$ is an almost complete intersection in the local ring $R$. Our results concern various binomial lower bounds for the Betti numbers of the resolution. For example, we show that the sum of the Betti numbers is at least $2^{d}$ where $d$ is the dimension of $R$.

We describe a practical and effective method for reconstructing the\ndeformation class of a Fano manifold X from a Laurent polynomial f that\ncorresponds to X under Mirror Symmetry. We explore connections to nef\npartitions, the smoothing of singular toric varieties, and the construction of\nembeddings of one (possibly-singular) toric variety in another. In particular,\nwe construct degenerations from Fano manifolds to singular toric varieties; in\nthe toric complete intersection case, these degenerations were constructed\npreviously by Doran--Harder. We use our method to find models of orbifold del\nPezzo surfaces as complete intersections and degeneracy loci, and to construct\na new four-dimensional Fano manifold.\n

Castelnuovo Mumford regularity with respect to multigraded ideals

A set of n homogeneous polynomials in n variables is a regular sequence if the associated polynomial system has only the obvious solution (0,0,...,0) . Denote by p_k(n) the power sum symmetric polynomial in n variables x_1^k+x_2^k+...+x_n^k . The interpretation of the q -analogue of the binomial coefficient as Hilbert function leads us to discover that n consecutive power sums in n variables form a regular sequence. We consider then the following problem: describe the subsets A \subset \mathbb N^* of cardinality n such that the set of polynomials p_a(n) with a\in A is a regular sequence. We prove that a necessary condition is that n! divides the product of the degrees of the elements of A . To find an easily verifiable sufficient condition turns out to be surprisingly difficult already for n = 3 . Given positive integers a < b < c with gcd (a,b,c) = 1 , we conjecture that p_a(3), p_b(3), p_c(3) is a regular sequence if and only if abc ≡ 0\ (\mathrm{mod}\ 6) . We provide evidence for the conjecture by proving it in several special instances.

We study how Betti numbers of ideals in a local ring change under small perturbations. Given p∈N and given an ideal I of a Noetherian local ring (R,m), our main result states that there exists N>0 such that if J is an ideal with I≡JmodmN and with the same Hilbert function as I, then the Betti numbers βiR(R/I) and βiR(R/J) coincide for 0≤i≤p. Moreover, we present several cases in which an ideal J such that I≡JmodmN is forced to have the same Hilbert function as I, and therefore the same Betti numbers.

Forcing the strong Lefschetz and the maximal rank properties