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Given a sequence of related modules M n defined over a sequence of related polynomial rings, one may ask how to simultaneously compute a finite Gröbner basis for each M n. Furthermore, one may ask how to simultaneously compute the module of syzygies of each M n. In this paper we address both questions. Working in the setting of OI-modules over a Noetherian polynomial OI-algebra, we provide OI-analogues of Buchberger's Criterion, Buchberger's Algorithm for computing Gröbner bases, and Schreyer's Theorem for computing syzygies. We also establish a stabilization result for Gröbner bases. [ABSTRACT FROM AUTHOR]

We invoke the Bernstein–Gel $'$ fand–Gel $'$ fand (BGG) correspondence to study subcomplexes of free resolutions given by two well-known complexes, the Koszul and the Eagon–Northcott. This approach provides a complete characterization of the ranks of free modules in a subcomplex in the Koszul case and imposes numerical restrictions in the Eagon–Northcott case. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to size up the minimal graded free resolution of a homogeneous ideal in terms of its generating degrees, by itself an ambitious objective. As understood, sizing up means looking closely at the two available parameters: the shifts and the Betti numbers. Since, in general, bounds for the shifts can behave quite steeply, we filter the difficulty by the subadditivity of the syzygies. The method we applied is hopefully new and sheds additional light on the structure of the minimal free resolution. For the Betti numbers, we apply the Boij–Söderberg techniques in order to get polynomial upper bounds for them. It is expected that the landscape of hypersurface singularities already contains most of the difficult corners of the arbitrary case. In this regard, we treat some facets of this case, including an improvement of one of the basic results of a recent work by Busé–Dimca–Schenck–Sticlaru. [ABSTRACT FROM AUTHOR]

The characteristic-free resolution of K_((7,5,3) ) is applied to the Lascoux resolution of K_((7,5,3) ) F (characteristic zero resolution) in this paper. This research teaches us about the relationship between the resolution of the weyl module K_((7,5,3) ) F in the characteristic-free mode and the Lascoux mode. In this work, let R be a commutative ring with 1, Ӻ be a free R-module and Ḍ_į be "the divided power algebra" of degree į. Ṃ is a left-graded module with for Ѡ = Ƶ_21^ҝ⋵ Å and Ѵ ⋵〖 Ḍ〗_(ᴯ_1 )⊗ Ḍ_(ᴯ_2 ) . We have Ѡ(Ѵ)= Ƶ_(21 )^ҝ (Ѵ)=〖 ∂〗_(21 )^ҝ (Ѵ). where the separator ẋ vanishes between Ƶ_ḁᵬ^((ț))and ∂_ḁᵬ^((ț)). Also by using Capelli identities we prove the sequences and the subsequences of the terms of characteristic zero satisfy the commutative for each diagram in these sequences and subsequences. Finally we get the reduction of the terms of the resolution of the Weyl module for characteristic free to the terms of the resolution of the Weyl module for characteristic 0.

We prove that in the polynomial ring Q = k [ x , y , z , w ] , with k an algebraically closed field of characteristic zero, all Gorenstein homogeneous ideals I satisfying the inclusions (x , y , z , w) 4 ⊆ I ⊆ (x , y , z , w) 2 can be obtained by doubling from a grade three perfect ideal J ⊂ I such that Q / J is a locally Gorenstein ring. Moreover, a graded minimal free resolution of the Q -module Q / I can be completely described in terms of a graded minimal free resolution of the Q -module Q / J and a homogeneous embedding of a shift of the canonical module ω Q / J into Q / J. [ABSTRACT FROM AUTHOR]

A net in P 2 is a configuration of lines A and points X satisfying certain incidence properties. Nets appear in a variety of settings, ranging from quasigroups to combinatorial design to classification of Kac–Moody algebras to cohomology jump loci of hyperplane arrangements. For a matroid M and rank r, we associate a monomial ideal (a monomial variant of the Orlik–Solomon ideal) to the set of flats of M of rank ≤ r . In the context of line arrangements in P 2 , applying Alexander duality to the resulting ideal yields insight into the combinatorial structure of nets. [ABSTRACT FROM AUTHOR]

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This paper shows that Cohen-Macaulay algebras can be algebraically approximated in such a way that their Cohen-Macaulayness and minimal Betti numbers are preserved. This is achieved by showing that finitely generated modules over power series rings can be algebraically approximated in a manner that preserves their diagrams of initial exponents and their minimal Betti numbers. These results are also applied to obtain an approximation result for flat homomorphisms from rings of power series to Cohen-Macaulay algebras. [ABSTRACT FROM AUTHOR]

A generalization of the plane de Jonquières transformation to arbitrary dimension is studied, with an eye for the ideal theoretic side. In particular, one considers structural properties of the corresponding base ideal and of its defining relations. Useful throughout is the idea of downgraded sequences of forms, a tool considered in many sources for the rounding-up of ideals of defining relations. The emphasis here is on the case where the supporting Cremona transformation of the de Jonquières transformation is the identity map. In this case we give some results on the homological behavior of the graph of the transformation. [ABSTRACT FROM AUTHOR]

We give a new proof of Hilbert's Syzygy Theorem for monomial ideals. In addition, we prove the following. If S = k[xi,..., xn] is a polynomial ring over a field, M is a squarefree monomial ideal in S, and each minimal generator of M has degree larger than i, then pd(S/M) ≤ n -- i. [ABSTRACT FROM AUTHOR]

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 ≡ J mod m N and with the same Hilbert function as I , then the Betti numbers β i R (R / I) and β i R (R / J) coincide for 0 ≤ i ≤ p. Moreover, we present several cases in which an ideal J such that I ≡ J mod m N is forced to have the same Hilbert function as I , and therefore the same Betti numbers. [ABSTRACT FROM AUTHOR]

Let k be a field of characteristic zero, n < N be positive integers, P be the polynomial ring k [ x , y , z , w ] , F be the homogeneous polynomial x n + y n + z n + w n , K be the ideal (x N , y N , z N , w N) , and P ‾ be the hypersurface ring P ‾ = P / (F). We describe the minimal multi-homogeneous resolution of P ‾ / K P ‾ by free P ‾ -modules, the socle degrees of P ‾ / K P ‾ , and the minimal multi-homogeneous resolution of the Gorenstein ring P / (K : F) by free P -modules. Our arguments use Stanley's theorem that every Artinian monomial complete intersection over a polynomial ring with coefficients from a field of characteristic zero has the strong Lefschetz property as well as a multi-grading on P for which both ideals K and (F) are homogeneous. The resolution of P ‾ / K P ‾ by free P ‾ -modules is obtained from a Differential Graded Algebra resolution of P / (K : F) by free P -modules, together with one homotopy map. The multi-grading is used to prove that the resulting resolution is minimal. [ABSTRACT FROM AUTHOR]

We generalize a result of Eisenbud, Huneke and Ulrich on the maximal graded shifts of a module with prescribed annihilator and prove a linear regularity bound for ideals in a polynomial ring depending only on the first p − c steps in the resolution, where p = pd (S / I) and c = codim (I). [ABSTRACT FROM AUTHOR]

Given a monomial ideal in a polynomial ring over a field, we define the generalized Newton complementary dual of the given ideal. We show good properties of such duals including linear quotients and isomorphism between the special fiber rings. We construct the cellular free resolutions of duals of strongly stable ideals generated in the same degree. When the base ideal is generated in degree two, we provide an explicit description of cellular free resolution of the dual of a compatible generalized stable ideal. [ABSTRACT FROM AUTHOR]

Let k be a field of characteristic 0. Using the method of idealization, we show that there is a non-Koszul, quadratic, Artinian, Gorenstein, standard graded k-algebra of regularity 3 and codimension 8, answering a question of Mastroeni, Schenck, and Stillman. We also show that this example is minimal in the sense that no other idealization that is non-Koszul, quadratic, Artinian, Gorenstein algebra, with regularity 3 has smaller codimension. We also construct an infinite family of graded, quadratic, Artinian, Gorenstein algebras A m , indexed by an integer m ≥ 2 , with the following properties: (1) there are minimal first syzygies of the defining ideal in degree m + 2 , (2) for m ≥ 3 , A m is not Koszul, (3) for m ≥ 7 , the Hilbert function of A m is not unimodal, and thus (4) for m ≥ 7 , A m does not satisfy the weak or strong Lefschetz properties. In particular, the subadditivity property fails for quadratic Gorenstein ideals. Finally, we show that the idealization of a construction of Roos yields non-Koszul quadratic Gorenstein algebras such that the residue field k has a linear resolution for precisely α steps for any integer α ≥ 2 . Thus there is no finite test for the Koszul property even for quadratic Gorenstein algebras. [ABSTRACT FROM AUTHOR]

Let k be a field of arbitrary characteristic, A a Noetherian k -algebra and consider the polynomial ring A [ x ] = A [ x 0 , ... , x n ]. We consider graded submodules of A [ x ] m having a special set of generators, a marked basis over a quasi-stable module. Such a marked basis shares many interesting properties with a Gröbner basis, including the existence of a Noetherian reduction relation. The set of submodules of A [ x ] m having a marked basis over a given quasi-stable module possesses a natural affine scheme structure which we will exhibit. Furthermore, the syzygies of a module with such a marked basis are generated by a marked basis, too (over a suitable quasi-stable module in ⊕ i = 1 m ′ A [ x ] (− d i)). We apply marked bases and related properties to the investigation of Quot functors (and schemes). More precisely, for a given Hilbert polynomial, we explicitly construct (up to the action of a general linear group) an open cover of the corresponding Quot functor, made up of open subfunctors represented by affine schemes. This provides a new proof that the Quot functor is the functor of points of a scheme. We also exhibit a procedure to obtain the equations defining a given Quot scheme as a subscheme of a suitable Grassmannian. Thanks to the good behaviour of marked bases with respect to Castelnuovo-Mumford regularity, we can adapt our methods in order to study the locus of the Quot scheme given by an upper bound on the regularity of its points. [ABSTRACT FROM AUTHOR]

The question we address in this paper is: which monomial ideals have minimal cellular resolutions, that is, minimal resolutions obtained from homogenizing the chain maps of CW-complexes? Velasco gave families of examples of monomial ideals that do not have minimal cellular resolutions, but those examples have large minimal generating sets. In this paper, we show that if a monomial ideal has at most four generators, then the ideal and its (monomial) Artinian reductions have minimal cellular resolutions. When the ideal is generated by two monomials, we can give a precise description of the CW-complex supporting minimal free resolution of the ideal and its Artinian reduction. Also, in this case, we compute the multigraded Betti numbers, Cohen-Macaulay type and determine when the corresponding algebra is a level algebra. [ABSTRACT FROM AUTHOR]

Given a representation of a finite group G over some commutative base ring k , the cofixed space is the largest quotient of the representation on which the group acts trivially. If G acts by k -algebra automorphisms, then the cofixed space is a module over the ring of G -invariants. When the order of G is not invertible in the base ring, little is known about this module structure. We study the cofixed space in the case that G is the symmetric group on n letters acting on a polynomial ring by permuting its variables. When k has characteristic 0, the cofixed space is isomorphic to an ideal of the ring of symmetric polynomials in n variables. Localizing k at a prime integer p while letting n vary reveals striking behavior in these ideals. As n grows, the ideals stay stable in a sense, then jump in complexity each time n reaches a multiple of p. [ABSTRACT FROM AUTHOR]

We provide constructive versions of Hilbert's syzygy theorem for Z and Z/NZ following Schreyer's method. Moreover, we extend these results to arbitrary coherent strict Bézout rings with a divisibility test for the case of finitely generated modules whose module of leading terms is finitely generated. [ABSTRACT FROM AUTHOR]

Using discrete Morse theory, we give an algorithm that prunes the excess of information in the Taylor resolution and constructs a new cellular free resolution for an arbitrary monomial ideal. The pruned resolution is not simplicial in general, but we can slightly modify our algorithm in order to obtain a simplicial resolution. We also show that the Lyubeznik resolution fits into our pruning strategy. The pruned resolution is not always minimal but it is a lot closer to the minimal resolution than the Taylor and the Lyubeznik resolutions as we will see in some examples. We finally use our methods to give a different approach to the theory of splitting of monomial ideals. We deduce from this splitting strategy that the pruned resolution is always minimal in the case of edge ideals of paths and cycles. [ABSTRACT FROM AUTHOR]

In this paper we study those submonoids of N d with a nontrivial pseudo-Frobenius set. In the affine case, we prove that they are the affine semigroups whose associated algebra over a field has maximal projective dimension possible. We prove that these semigroups are a natural generalization of numerical semigroups and, consequently, most of their invariants can be generalized. In the last section we introduce a new family of submonoids of N d and using its pseudo-Frobenius elements we prove that the elements in the family are direct limits of affine semigroups. [ABSTRACT FROM AUTHOR]

We investigate whether the property of having linear quotients is inherited by ideals generated by multigraded shifts of a Borel ideal and a squarefree Borel ideal. We show that the ideal generated by the first multigraded shifts of a Borel ideal has linear quotients, as do the ideal generated by the $k$ th multigraded shifts of a principal Borel ideal and an equigenerated squarefree Borel ideal for each $k$. Furthermore, we show that equigenerated squarefree Borel ideals share the property of being squarefree Borel with the ideals generated by multigraded shifts. [ABSTRACT FROM AUTHOR]