Resolutions over strict complete intersections

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.

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

Almost complete intersections and factorial rings

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.

Surgery, as developed by Browder, Kervaire, Milnor, Novikov, Sullivan, Wall and others is a method for comparing homotopy types of topological spaces with difieomorphism or homeomorphism types of manifolds of dimension‚ 5. In this paper, a modiflcation of this theory is presented, where instead of flxing a homotopy type one considers a weaker information. Roughly speaking, one compares n-dimensional compact manifolds with topological spaces whose k-skeletons are flxed, where k is at least [n=2]. A particularly attractive example which illustrates the concept is given by complete intersections. By the Lefschetz hyperplane theorem, a complete intersection of complex dimension n has the same n-skeleton as n and one can use the modifled theory to obtain information about their difieomorphism type although the homotopy classiflcation is not known. The theory reduces this classiflcation result to the determination of complete intersections in a certain bordism group. This was under certain restrictions carried out in [Tr]. The restrictions are: If d = d1¢:::¢dr is the total degree of a complete intersection X n1;:::;dr of complex dimension n, then the assumption is, that for all primes p with p(pi1)• n+1, the total degree d is divisible by p [(2n+1)=(2pi1)]+1 . Theorem A. Two complete intersections X n

Almost complete intersections and the Lex-Plus-Powers Conjecture

We propose a question on the annihilators of positive Koszul homologies of a system of parameters of an almost complete intersection R. The question can be stated in terms of the acyclicity of certain (finite) residual approximation complexes whose zeroth homologies are the residue field of R. We show that our question has an affirmative answer for the first Koszul homology of any almost complete intersection, as well as for all positive Koszul homologies of certain system of parameters which exist in some almost complete intersection rings with small multiplicities. The statement about the first Koszul homology is shown to be equivalent to the monomial conjecture and thus follows from its validity.

In 1997 Batyrev, Ciocan-Fontanine, Kim, and van Straten suggested a construction of Landau–Ginzburg models for Fano complete intersections in Grassmannians similar to Givental’s construction for complete intersections in smooth toric varieties. We show that for a Fano complete intersection in a Grassmannian the result of the above construction is birational to a complex torus. In other words, the complete intersections under consideration have very weak Landau–Ginzburg models.

Finite projective algebras B over arbitrary commutative rings A are discussed with respect to dualizing B-modules over B and over A. This leads to quasi-Frobenius algebras in a natural way. Several change of rings characterizations of quasi-Frobenius algebras are given. Secondly, locally complete intersections and complete intersections are considered unter the point of view of quasi-Frobenius algebras resp. Frobenius algebras. Characterizations of complete intersections are obtained by using algebraic K-theory and differential modules. Some applications refer to representations of affine curves as idealtheoretic complete intersections.

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.

We study a one-parameter family of gauged linear sigma models (GLSMs) naturally associated to a complete intersection in weighted projective space. In the positive phase of the family we recover Gromov-Witten theory of the complete intersection, while in the negative phase we obtain a Landau--Ginzburg-type theory. Focusing on the negative phase, we develop foundational properties which allow us to state and prove a genus-zero comparison theorem that generalizes the multiple log-canonical correspondence and should be viewed as analogous to quantum Serre duality in the positive phase. Using this comparison result, along with the crepant transformation conjecture and quantum Serre duality, we prove a genus-zero correspondence between the GLSMs which arise at the two phases, thereby generalizing the Landau-Ginzburg/Calabi-Yau correspondence to complete intersections.

Let K be a field and I a monomial ideal of the polynomial ring S = K[x1, ..., xn]. We show that if either: 1) I is almost complete intersection, 2) I can be generated by less than four monomials; or 3) I is the Stanley-Reisner ideal of a locally complete intersection simplicial complex on [n], then Stanley's conjecture holds for S/I.

Tame quivers, semi-invariants, and complete intersections

We study subcanonical codimension 2 subvarieties ofPn, n ⩾ 4, using as our main tool the rank 2 vector bundle canonically associated to them. With this method we prove first that every smooth canonical surface inP4 is a complete intersection. Next we study smooth varieties of codimension 2inPn,n⩾6; it is well known that all of them are subcanonical and R. Hartshorne conjectured that they are always complete intersections, if n⩾7. We prove this conjecture in the particular case of a variety X for which the integer e such that Ωx=θx(e) is 0 or negative. This result, togheter with a strong result by Z. Ran, provides a quadratic bound for the degree of a non-complete intersection variety of codimension 2inPn, n⩾6.

Syzygies in Hilbert schemes of complete intersections