5,492 research outputs found
Generic Syzygy Schemes
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
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
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
In this paper, we focus on -syzygy modules and the injective cogenerator
determined by the minimal injective resolution of a noether ring. We study the
properties of -syzygy modules and a category which includes
the category consisting of all -syzygy modules and their applications on
Auslander-type rings. Then, we investigate the injective cogenerators
determined by the minimal injective resolution of . We show that is
Gorenstein with finite self-injective dimension at most 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
Let be a commutative Noetherian local ring with residue field . We
show that if a finite direct sum of syzygy modules of surjects onto `a
semidualizing module' or `a non-zero maximal Cohen-Macaulay module of finite
injective dimension', then is regular. We also prove that is regular if
and only if some syzygy module of has a non-zero direct summand of finite
injective dimension.Comment: 7 page
Stability and Unobstructedness of Syzygy Bundles
It is a longstanding problem in Algebraic Geometry to determine whether the
syzygy bundle 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 on \PP^N associated to generic forms of the same degree . Our first goal is to prove that
is stable if . This bound improves,
in general, the bound given by G. Hein in \cite{B}, Appendix A.
In the last part of the paper, we study moduli spaces of stable rank
vector bundles on \PP^N containing syzygy bundles. We prove that if and , then the syzygy bundle is
unobstructed and it belongs to a generically smooth irreducible component of
dimension , if , and
, if N=2.Comment: 32 pages, minor change
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