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© 2019 Elsevier B.V. In a first part of this paper, we prove constancy of the canonical graded Betti table among the smooth curves in linear systems on Gorenstein weak Fano toric surfaces. In a second part, we show that Green's canonical syzygy conjecture holds for all smooth curves of genus at most 32 or Clifford index at most 6 on arbitrary toric surfaces. Conversely we use known results on Green's conjecture (due to Lelli-Chiesa)to obtain new facts about graded Betti tables of projectively embedded toric surfaces. ispartof: JOURNAL OF PURE AND APPLIED ALGEBRA vol:224 issue:2 pages:507-527 status: published

A closed subscheme of codimension two T ⊂ P 2 is a quasi complete intersection (q.c.i.) of type ( a , b , c ) if there exists a surjective morphism O ( − a ) ⊕ O ( − b ) ⊕ O ( − c ) → I T . We give bounds on deg ⁡ ( T ) in function of a , b , c and r, the least degree of a syzygy between the three polynomials defining the q.c.i. (see Theorem 6 ). As a by-product we recover a theorem of du Plessis-Wall on the global Tjurina number of plane curves (see Theorem 20 ) and some other related results.

For an abelian category A , we define the category PEx( A ) of pullback diagrams of short exact sequences in A , as a subcategory of the functor category Fun( Δ , A ) for a fixed diagram category Δ. For any object M in PEx ( A ) , we prove the existence of a short exact sequence 0 → K → P → M → 0 of functors, where the objects are in PEx( A ) and P ( i ) ∈ Proj ( A ) for any i ∈ Δ . As an application, we prove that if ( C , D , E ) is a triple of syzygy finite classes of objects in mod Λ satisfying some special conditions, then Λ is an Igusa–Todorov algebra. Finally, we study lower triangular matrix Artin algebras and determine in terms of their components, under reasonable hypothesis, when these algebras are syzygy finite or Igusa–Todorov.

We show that if R is a two dimensional standard graded domain (with the graded maximal ideal m) of characteristic p > 0 and I ⊂ R is a graded ideal with l ( R / I ) ∞ , then the F-threshold c I ( m ) can be expressed in terms of a strong HN (Harder-Narasimhan) slope of the canonical syzygy bundle on Proj R . Thus c I ( m ) is a rational number. This gives us a well defined notion, of the F-threshold c I ( m ) in characteristic 0, in terms of a HN slope of the syzygy bundle on Proj R . This generalizes our earlier result (in [13] ) where we have shown that if I has homogeneous generators of the same degree, then the F-threshold c I ( m ) is expressed in terms of the minimal strong HN slope (in char p) and in terms of the minimal HN slope (in char 0), respectively, of the canonical syzygy bundle on Proj R . In the present more general setting, the relevant slope may not be the minimal one. Here we also prove that, for a given pair ( R , I ) over a field of characteristic 0, if ( m p , I p ) is a reduction mod p of ( m , I ) then c I p ( m p ) ≠ c ∞ I ( m ) implies c I p ( m p ) has p in the denominator, for almost all p.

Directed graphs called “syzygy quivers” are used to study the asymptotic growth rates of the dimensions of the syzygies of representations of finite dimensional algebras. For any finite dimensional syzygy-finite representation (e.g. any representation of a monomial algebra), we show that this growth rate is poly-exponential, i.e. the product of a polynomial and an exponential function, and give a procedure for computing the corresponding degree and base from a syzygy quiver. We characterize the growth rates arising in this context. The bases of the occurring exponential functions are the Perron numbers: real, nonnegative algebraic integers b whose irreducible polynomial over Q has no root with modulus larger than b. Modulo this restriction, arbitrary degrees and bases occur. Moreover, we show that these growth rates are invariant under stable derived equivalences.

An A-hypergeometric system is not irreducible, if its parameter vector is resonant. In this paper, we present a way of computing a finite system of generators of the first syzygy module of an irreducible A-hypergeometric quotient. In particular, if the semigroup generated by A is simplicial and scored, then an explicit system of generators is given.

It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle $E_{d_1,..., d_n}$ on $\PP^N$ defined as the kernel of a general epimorphism $\xymatrix{��:\cO(-d_1)\oplus...\oplus\cO(-d_n)\ar[r] &\cO}$ is (semi)stable. In this note, we restrict our attention to the case of syzygy bundles $E_{d,n}$ on $\PP^N$ associated to $n$ generic forms $f_1,...,f_n\in K[X_0,X_1,..., X_N]$ of the same degree $d$. Our first goal is to prove that $E_{d,n}$ is stable if $N+1\le n\le\tbinom{d+2}{2}+N-2$. This bound improves, in general, the bound $n\le d(N+1)$ given by G. Hein in \cite{B}, Appendix A. In the last part of the paper, we study moduli spaces of stable rank $n-1$ vector bundles on $\PP^N$ containing syzygy bundles. We prove that if $N+1\le n\le\tbinom{d+2}{2}+N-2$ and $N\ne 3$, then the syzygy bundle $E_{d,n}$ is unobstructed and it belongs to a generically smooth irreducible component of dimension $n\tbinom{d+N}{N}-n^2$, if $N \geq 4$, and $n\tbinom{d+2}{2}+n\tbinom{d-1}{2}-n^2$, if N=2., 32 pages, minor changes

For a finite dimensional vector space G we define the k-th generic syzygy scheme Gensyz_k(G) by explicit equations. We show that the syzygy scheme Syz(f) of any syzygy in the linear strand of a projective variety X which is cut out by quadrics is a cone over a linear section of a corresponding generic syzygy scheme. We also give a geometric description of Gensyz_k(G) for k=0,1,2. In particular Gensyz_2(G) is the union of a Pl"ucker embedded Grassmannian and a linear space. From this we deduce that every smooth, non-degenerate projective curve C which is cut out by quadrics and has a p-th linear syzygy of rank p+3 admits a rank 2 vector bundle E with det E = O_C(1) and h^0(E) at least p+4., Comment: 12 Pages. This paper is a completely rewritten version of the first part of math.AG/0108078. It also contains several new results

This paper presents characterizations of border bases of zero-dimensional polyno- mial ideals that are analogous to the known characterizations of Grobner bases. Based on a Border Division Algorithm, a variant of the usual Division Algorithm, we characterize border bases as border prebases with one of the following equivalent properties: special generation, generation of the border form ideal, conuence of the corresponding rewrite relation, reduction of S-polynomials to zero, and lifting of syzygies. The last characterization relies on a detailed study of the relative position of the border terms and their syzygy module. In particular, a border prebasis is a border basis if and only if all fundamental syzygies of neighboring border terms lift; these liftings are easy to compute.

Let K be a field, X = {X 1 ,…, X n } and Y = {Y 1 ,…, Y r } sets of indeterminates, and f ∈ K[[X]], g ∈ K[[Y]] two non-zero formal power series with f ∈ (X),g ∈ (Y) . Set Δ g = (∂g/∂Y i : 1≤i≤r) and suppose that Δ g is a ( Y )-primary ideal (e.g., if char K = 0 and g is an isolated singularity). Write R ≔ K[[X,Y]]/(f+g), R 1 ≔ K[[X]]/(f) and R ≔ R/Δ g R . The main aim of this paper is to relate an arbitrary maximal Cohen–Macaulay (MCM for short) R -module N to the higher order syzygy Ω R r (N/Δ g N) , and in this way relate indecomposable MCM R -modules to higher order syzygies of certain indecomposable R -modules. The R -modules in question are deformations of MCM R 1 -modules and are weakly liftable. We find resolutions of the higher order syzygy modules in question which are shown to be minimal in certain situations, and express these in terms of matrix factorizations. The theory is shown to be applicable with almost complete success to singularities of Knorrer-type and, in any case, to give detailed information about MCM R -modules. Our techniques generalise and simplify those of Herzog and Popescu [7] , and we further use a lifting theorem for maps in Koszul complexes and a technique involving iterated mapping cones which may be of independent interest. [3pt]

We prove a characteristic-free plethysm formula of the complex SS2ϑ for a map of finite free modules ϑ. We also study a family of subcomplexes of a Schur complex, the ν-Schur complexes. Using these machineries from characteristic-free representation theory of general linear group, we give an example of determinantal ideal of generic symmetric matrix whose second Betti number depends on the characteristic of the base field. We also give a short proof of Kurano's first syzygy theorem.

This can be regarded as a generalization of Serre’s Theorem that every torsion-free group of finite virtual cohomological dimension has finite cohomological dimension (see [2]). For suppose that G is torsion-free and that H has cohomological dimension n < co. Then the nth syzygy in any projective resolution of Z over ZG is projective as ZH-module, and Moore’s Conjecture implies at once that it is also projective as ZG-module. One of the special cases of the conjecture which we prove here also implies Serre’s Theorem. It is natural to generalize the conjecture, because, in addition to group rings, it makes perfect sense for crossed products and in fact for strongly group-graded rings. Recall that a G-graded ring is a ring A with a direct sum decomposition

We investigate for an artin algebra Λ when categories of dth syzygy modules are closed under extensions. In this case they are functorially finite resolving, and have an associated cotilting module. We show that this holds for all d for algebras Λ which are k-Gorenstein for all k, that is, in a minimal injective resolution 0 → Λ → I0 → I1 → ⋯ → Ij → ⋯ we have pdΛ Ij ⩽ j for all j. from this we get a correspondence between indecomposable projective and indecomposable injective modules over these algebras, which can be applied to prove that if Λ is k-Gorenstein for all k and idΛ Λ < ∞, then id ΛΛ < ∞.

Campanella's refinements of Dubreil's theorem are sharp upper and lower bounds, formulated in postulational terms, for the number of forms of any given degree in the standard bases of 2-codimensional perfect homogeneous polynomial ideals. This note considers the implications of upper-bound-extremality. Its principal discovery is that such extremality implies a splitting of the syzygy module of any standard basis of such an ideal into syzygy modules of the standard bases of ‘component’ ideals explicitly describable in terms of the standard basis in question.

Every finitely generated rank 2 third syzygy module over a regular local ring is known to be self-dual. We show more generally that any finitely generated rank 2 reflexive module is self-dual, and that the isomorphism is skew symmetric. We use this ismorphism to estimate how large k may be if the module is a k th syzygy, and how closely allied rank 3 modules are related to their duals.

Let R be a commutative noetherian local ring with residue field k. We introduce two new invariants of R-modules and compute them for the module k. As a consequence, we show, when R is nonregular, that no direct sum of the syzygy modules of k surjects onto a nonzero module of finite projective dimension. A dual statement is also true: no direct sum of the cosyzygy modules of k (in an injective resolution) contains a nonzero module of finite injective dimension with nonzero scale.

In this paper we construct certain indecomposable syzygy modules over graded hypersurface rings and give a lower bound of their rank. Thereby we show that there is no upper bound for the ranks of the indecomposable maximal Cohen-Macaulay modules over a graded hypersurface ring of dimension at most two and of multiplicity at most three.

Let A be a noetherian AS-regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and grA the category of finitely generated graded left A-modules. Following Jørgensen, we define the Castelnuovo–Mumford regularity reg(M•) of a complex M•∈Db(grA) in terms of the local cohomologies or the minimal projective resolution of M•. Let A! be the quadratic dual ring of A. For the Koszul duality functor G:Db(grA)→Db(grA!), we have reg(M•)=max{i∣Hi(G(M•))≠0}. Using these concepts, we interpret results of Martinez-Villa and Zacharia concerning weakly Koszul modules (also called componentwise linear modules) over A!. As an application, refining a result of Herzog and Römer, we show that if J is a monomial ideal of an exterior algebra E=⋀〈y1,…,yd〉, d≥3, then the (d−2)nd syzygy of E/J is weakly Koszul.

The lifting problem for homogeneous ideals is studied. A relation between a homogeneous ideal J and its liftings is established using a syzygy basis of J . This relation is then used to obtain an algorithm for finding all the liftings of a homogeneous ideal. As an application of the algorithm, we discover the first example of a homogeneous ideal of dimension 0 in four variables which is not liftable to a radical ideal over the field of rational numbers.

Let R be a commutative ring with identity and M be a finitely generated perfect R-module of homological dimension n with a projective resolution F:0→Fn→fnfn−1→·→Fk→fkFk−1→·→F1→1F0→M→0 where rank fk is rk. In this paper, we show that for 1

A different proof of the fact that the first syzygy module of minors of certain size defined by a generic matrix over a commutative ring with identity is generated by linear relations of certain type is given. The proof utilizes the Grobner basis theory and the division algorithm over the polynomial ring Z [ X ] that the author developed in a separate paper.