5,492 research outputs found

    Generic Syzygy Schemes

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    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 result

    Syzygy modules with semidualizing or G-projective summands

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    Let R be a commutative Noetherian local ring with residue class field k. In this paper, we mainly investigate direct summands of the syzygy modules of k. We prove that R is regular if and only if some syzygy module of k has a semidualizing summand. After that, we consider whether R is Gorenstein if and only if some syzygy module of k has a G-projective summand.Comment: 13 pages, to appear in Journal of Algebr

    Looking out for stable syzygy bundles

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    We study (slope-)stability properties of syzygy bundles on a projective space P^N given by ideal generators of a homogeneous primary ideal. In particular we give a combinatorial criterion for a monomial ideal to have a semistable syzygy bundle. Restriction theorems for semistable bundles yield the same stability results on the generic complete intersection curve. From this we deduce a numerical formula for the tight closure of an ideal generated by monomials or by generic homogeneous elements in a generic two-dimensional complete intersection ring.Comment: This paper contains an appendix by Georg Hein: Semistability of the general syzygy bundle. The new version is quite ne

    Syzygy Modules and Injective Cogenerators for Noether Rings

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    In this paper, we focus on nn-syzygy modules and the injective cogenerator determined by the minimal injective resolution of a noether ring. We study the properties of nn-syzygy modules and a category Rn(mod  R)R_n(\mod R) which includes the category consisting of all nn-syzygy modules and their applications on Auslander-type rings. Then, we investigate the injective cogenerators determined by the minimal injective resolution of RR. We show that RR is Gorenstein with finite self-injective dimension at most nn if and only if \id R\leq n and \fd \bigoplus_{i=0}^n I_i(R)< \infty. Some known results can be our corollaries

    Characterizations of regular local rings via syzygy modules of the residue field

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    Let RR be a commutative Noetherian local ring with residue field kk. We show that if a finite direct sum of syzygy modules of kk surjects onto `a semidualizing module' or `a non-zero maximal Cohen-Macaulay module of finite injective dimension', then RR is regular. We also prove that RR is regular if and only if some syzygy module of kk has a non-zero direct summand of finite injective dimension.Comment: 7 page

    Stability and Unobstructedness of Syzygy Bundles

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    It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle Ed1,...,dnE_{d_1,..., d_n} on \PP^N defined as the kernel of a general epimorphism \xymatrix{\phi:\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 Ed,nE_{d,n} on \PP^N associated to nn generic forms f1,...,fn∈K[X0,X1,...,XN]f_1,...,f_n\in K[X_0,X_1,..., X_N] of the same degree dd. Our first goal is to prove that Ed,nE_{d,n} is stable if N+1≤n≤(d+22)+N−2N+1\le n\le\tbinom{d+2}{2}+N-2. This bound improves, in general, the bound n≤d(N+1)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−1n-1 vector bundles on \PP^N containing syzygy bundles. We prove that if N+1≤n≤(d+22)+N−2N+1\le n\le\tbinom{d+2}{2}+N-2 and N≠3N\ne 3, then the syzygy bundle Ed,nE_{d,n} is unobstructed and it belongs to a generically smooth irreducible component of dimension n(d+NN)−n2n\tbinom{d+N}{N}-n^2, if N≥4N \geq 4, and n(d+22)+n(d−12)−n2n\tbinom{d+2}{2}+n\tbinom{d-1}{2}-n^2, if N=2.Comment: 32 pages, minor change
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