Higher Syzygies Research Articles

Homological dimensions of the Jacobson radical

This work presents results on the finiteness, and on the symmetry properties, of various homological dimensions associated to the Jacobson radical and its higher syzygies, of a semiperfect ring.

Higher syzygies on general polarized Abelian varieties of type (1,⋯,1,d)$(1,\dots ,1,d)$

AbstractIn this paper, we show that a general polarized abelian variety of type and dimension g satisfies property if . In particular, the case affirmatively solves a conjecture by Fuentes García on projective normality.

Betti numbers of fat forests and their Alexander dual

Let k be a field and R=k[x_1,ldots ,x_n]/I=S/I a graded ring. Then R has a t-linear resolution if I is generated by homogeneous elements of degree t, and all higher syzygies are linear. Thus, R has a t-linear resolution if mathrm{Tor}^S_{i,j}(S/I,k)=0 if jne i+t-1. For a graph G on {1,ldots ,n}, the edge algebra is k[x_1,ldots ,x_n]/I, where I is generated by those x_ix_j for which { i,j} is an edge in G. We want to determine the Betti numbers of edge rings with 2-linear resolution. But we want to do that by looking at the edge ring as a Stanley–Reisner ring. For a simplicial complex Delta on [mathbf{n}]={1,ldots ,n} and a field k, the Stanley–Reisner ring k[Delta ] is k[x_1,ldots ,x_n]/I, where I is generated by the squarefree monomials x_{i_1}ldots x_{i_k} for which { i_1,ldots ,i_k} does not belong to Delta . Which Stanley–Reisner rings that are edge rings with 2-linear resolution are known. Their associated complexes has had different names in the literature. We call them fat forests here. We determine the Betti numbers of many fat forests and compare our result with what is known. We also consider Betti numbers of Alexander duals of fat forests.

Tor as a module over an exterior algebra

Let S be a regular local ring with residue field k and let M be a finitely generated S -module. Suppose that f_1,\dots ,f_c \in S is a regular sequence that annihilates M , and let E be an exterior algebra over k generated by c elements. The homotopies for the f_i on a free resolution of M induce a natural structure of graded E -module on Tor ^{S}(M,k) . In the case where M is a high syzygy over the complete intersection R:=S/(f_1,\dots,f_c) we describe this E -module structure in detail, including its minimal free resolution over E . Turning to Ext _R(M, k) we show that, when M is a high syzygy over R , the minimal free resolution of Ext _R(M, k) as a module over the ring of CI operators is the Bernstein–Gel’fand–Gel’fand dual of the E -module Tor ^S(M, k) . For the proof we introduce higher CI operators , and give a construction of a (generally non-minimal) resolution of M over S starting from a resolution of M over R and its higher CI operators.

Minimal Resolutions Over Codimension 2 Complete Intersections

We construct an explicit free resolution T for a maximal Cohen-Macaulay module M over a local complete intersection of codimension 2 with infinite residue field. The resolution is minimal when the module M is a sufficiently high syzygy. Our starting point is a layered free resolution L, described in [7], of length 2 over a regular local ring. We provide explicit formulas for the differential in T in terms of the differential and homotopies on the finite resolution L. One application of our construction is to describe Ulrich modules over a codimension 2 quadratic complete intersection.

A remark on higher syzygies on abelian surfaces

ABSTRACTIn this note, we give a slight improvement of a result of A. Küronya and V. Lozovanu about higher syzygies on abelian surfaces.

The residue field as a high syzygy

ABSTRACTIn this paper, we study commutative local rings whose residue field is a high syzygy. We are able to characterize such rings with embedding dimension at most 3 as those that are Artinian and Gorenstein.

Periodic free resolutions from twisted matrix factorizations

The notion of a matrix factorization was introduced by Eisenbud in the commutative case in his study of bounded (periodic) free resolutions over complete intersections. Since then, matrix factorizations have appeared in a number of applications. In this work, we extend the notion of (homogeneous) matrix factorizations to regular normal elements of connected graded algebras over a field.Next, we relate the category of twisted matrix factorizations to an element over a ring and certain Zhang twists. We also show that in the setting of a quotient of a ring of finite global dimension by a normal regular element, every sufficiently high syzygy module is the cokernel of some twisted matrix factorization. Furthermore, we show that in the noetherian AS-regular setting, there is an equivalence of categories between the homotopy category of twisted matrix factorizations and the singularity category of the hypersurface, following work of Orlov.

Asymptotic behavior of dimensions of syzygies

Let R R be a commutative noetherian local ring and M M be a finitely generated R R -module of infinite projective dimension. It is well-known that the depths of the syzygy modules of M M eventually stabilize to the depth of R R . In this paper, we investigate the conditions under which a similar statement can be made regarding dimension. In particular, we show that if R R is equidimensional and the Betti numbers of M M are eventually non-decreasing, then the dimension of any sufficiently high syzygy module of M M coincides with the dimension of R R .

Projected Canonical Curves and the Clifford Index

We shall work over the complex number field C. Let X be a non-singular projective curve of genus g. We always assume that it is non-hyperelliptic and sometimes identify it with its canonical image in PH(X,KX). By Max Noether’s theorem, the canonical ring of X is generated in degree one and the canonical map is a projectively normal embedding. Furthermore, a well-known theorem of Enriques-Petri states that X is cut out by hyperquadrics if Cliff(X), the Clifford index of X, is bigger than one. Then Mark Green [4] conjectured that the non-vanishing of a certain higher syzygy can be characterized by the Clifford index, which has been verified in many cases. From these, we learn that Cliff(X) reflects the algebraic structure on X better than the gonality gon(X), while two invariants are almost equivalent [2]. For a non-negative integer k, we denote by X the k-th symmetric product of X whose points are considered as effective divisors of degree k on X. When k = 0, we understand that X is one point corresponding to the zero divisor. Let Dk be the open subset of X consisting of effective divisors D which spans scheme theoretically a (k − 1)-plane 〈D〉 in PH(X,KX). For D ∈ Dk, we put KX,D = KX − [D], where [D] denotes the line bundle associated to D. We let ΦKX,D : X → PH(X,KX,D) be the rational map associated to the complete linear system |KX,D|. In this article, we consider two kinds of uniformity questions with respect to Dk:

Regularity on abelian varieties II: Basic results on linear series and defining equations

We apply the theory of M M -regularity developed by the authors [Regularity on abelian varieties, I, J. Amer. Math. Soc. 16 (2003), 285–302] to the study of linear series given by multiples of ample line bundles on abelian varieties. We define an invariant of a line bundle, called M M -regularity index, which governs the higher order properties and (partly conjecturally) the defining equations of such embeddings. We prove a general result on the behavior of the defining equations and higher syzygies in embeddings given by multiples of ample bundles whose base locus has no fixed components, extending a conjecture of Lazarsfeld [proved in Syzygies of abelian varieties, J. Amer. Math. Soc. 13 (2000), 651–664]. This approach also unifies essentially all the previously known results in this area, and is based on Fourier-Mukai techniques rather than representations of theta groups.

Topology of bounded-degree graph complexes

The bounded-degree graph complexes were first introduced by Reiner and Roberts [J. Algebraic Combin. 11 (2000) 135–154]. They arise from the finite free resolution of quadratic Veronese rings and modules. We prove various results about the homotopy types of these complexes, and deduce corresponding characteristic-free results about the quadratic Veronese resolutions. In particular, we characterize the set of multidegrees which support at least one higher syzygy in this resolution. The answer turns out to be independent of the field characteristic.

Higher Syzygies of Elliptic Ruled Surfaces

Let L be a normally generated line bundle on X; we say L satisfies property N_p (notation after Mark Green) if the matrices in the free resolution of R (the homogeneous coordinate ring of X) over S (the homogeneous coordinate ring of the projective space corresponding to the complete linear series |L|) have linear entries until the p-th stage. In this article we prove the following result: Let X be an elliptic ruled surface and let L be a product of p+1 base point free and ample line bundles on X. Then L satisfies property N_p. In particular we prove that numerical classes of all divisors which satisfies property N_p form a convex set. (Recall that Num(X) is generated by the class of a minimal section C_0 and by the class of a fiber f and that C_0 is ample.) As a corollary of the above result we show that the adjoint bundle K_X+(2p+3)A satisfies property N_p, if A is an ample line bundle.

Frobenius Splitting of Schubert Varieties and Linear Syzygies

Introduction. Let G be a connected, simply connected, semi-simple alge? braic group. Let B be a Borel subgroup of G. For an ample line bundle L on G/B, the questions about projective normality of the embedding of G/B in the complete linear system of L and the equations defining the embedding were extensively studied. Ramanathan in [13] proved that the ideal defining the embedding of G/B in the complete linear system of L is generated by quadratic equations. Kempf and Ramanathan proved that the equations defining the embedding of G/B in the complete linear system of L\,... ,Lr are quadratic and that a Schubert variety X is linearly defined with respect to L\,... ,Lr. Ramanathan had conjectured that the method of Frobenius splitting (used in [13]) can be used to study higher syzygies. Let R (BH?(G/B,?Ljj), where the vector (mj) ranges in Nr, be the multigraded coordinate ring of G/B with respect to L\,... ,Lr. In this paper, we use Frobenius splittings to study the higher syzygies of the irrelevant maximal ideal m of R. We prove that m as a /?-module has linear syzygies. Our methods also give simpler proofs of the results obtained in [6] and [13]. Let Z be a projective variety defined over an algebraically closed field. Let L\,... ,L, be ample line bundles on Z and let m denote the irrelevant maximal ideal of the multi-graded ring R = @H?(Z,?Ljj). In the first section we give a criterion for m to have linear syzigies over R. We show that if for every n > 2, the first cohomology of the n-fold product Zn with values in certain ideal sheaves tensored with powers of Lj vanishes then m has linear syzygies (Propositions 1.9 and 1.10). \ye also give a similar criterion for a homogeneous quotient ring of R to have linear syzygies over R (Proposition 1.13). In the second section, we show that any r ample line bundles L\,...,L, on G/B, and their restrictions to a Schubert variety X C G/B over an algebracally closed field of positive characteristic satisfy the above criteria. To prove this result, we use the method of Frobenius splittings. If Z is a Frobenius split variety, and if L is an ample line bundle on Z, then we have Hl(Z,L) = 0 for / > 1. Further if Y is a compatibly split subvariety of Z, then the restriction

Maximal Cohen-Macaulay modules and the quasihomogeneity of isolated Cohen-Macaulay singularities

We conjecture that a complete isolated Cohen-Macaulay singularity of dimension ≥ 2 \geq 2 is graded if and only if sufficiently high syzygy modules of the residue field and of the transpose of the module of Kähler differentials are isomorphic. The "only if" part of the conjecture is proved for hypersurface singularities.