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Preprints from April 2004Eisenbud-Green-Hulek-PopescuDavid Eisenbud, Mark Green, Klaus Hulek, and Sorin Popescu have posted two new preprints to the arXiv. "Restricting linear syzygies: algebra and geometry" Math Subject Class: 14N05, 13D02, 14M17 Comments: 26 pages, Plain TeX + diagrams.tex Abstract: In this paper we derive geometric consequences from the presence of a long strand of linear syzygies in the minimal free resolution of a closed scheme in projective space whose homogeneous ideal is generated by quadrics. These consequences are given in terms of intersections with arbitrary linear subspaces. We use our results to bound homological invariants of some well-known projective varieties, to give a combinatorial characterization of quadratic monomial ideals with a long strand of linear syzygies, etc "The geometry of 2-regular algebraic sets" Math Subject Class: 14N05, 14N25, 13D02, 14M20 Comments: 26 pages, Plain TeX + diagrams.tex Abstract: A celebrated Theorem of Del Pezzo and Bertini classifies the nondegenerate irreducible projective varieties X of minimal degree (deg X=1+codim X). There is also a cohomological characterization: X has minimal degree in its linear span if and only if X is 2-regular in the sense of Castelnuovo and Mumford. In this paper we extend these theorems to the reducible case. We prove that any 2-regular projective algebraic set (i.e. reduced subscheme) can be constructed inductively from varieties of minimal degree in a simple way, and we give a geometric criterion similar to minimal degree: a reduced subscheme is 2-regular if and only if it is "small", which means that if L is any linear subspace, then the geometric degree of L\cap X is at most 1 more than the codimension of L\cap X in L. Posted on April 29, 2004
Buchstaber-LazarevV. Buchstaber and A. Lazarev have posted a new preprint, "The discrete Gelfand transform and its dual", to the arXiv. Abstract: We consider the transformation $\ev$ which associates to any element in a K-algebra A a function on the the set of its K-points. This is the analogue of the fundamental Gelfand transform. Both $\ev$ and its dual $\ev^*$ are the maps from a discrete K-module to a topological K-module and we investigate in which case the image of each map is dense. The answer is nontrivial for various choices of K and A already for A=K[x], the polynomial ring in one variable. Applications to the structure of algebras of cohomology operations are given. Posted on April 27, 2004
GuoLi Guo has posted a new preprint, "Ascending Chain Conditions in Free Baxter Algebras", to the arXiv. Math Subject Class: 13E05, 16W99, 47C05 Journal reference: International Journal of Algebra and Computation. 12 (2002), no. 4, 601-622 Comments: 22 pages. No figures
Posted on April 27, 2004
DuncanBenton L. Duncan has posted a new preprint, "Universal operator algebras of directed graphs", to the arXiv. Math Subject Class: 47L40 Comments: 19 pages, uses xypic, submitted to Houston Journal of Mathematics Abstract: We define and investigate properties of universal operator algebras of directed graphs. Results include free products decomposition and continuity of the construction with respect to direct limits. Lastly we prove some K-theoretic results about our algebras. Posted on April 27, 2004
Frankild-Sather-WagstaffAnders Frankild and Sean Sather-Wagstaff have posted a new preprint, "The set of semidualizing complexes is a metric space", to the arXiv. Math Subject Class: 13B40, 13C05, 13C13, 13D05, 13D25, 13D40, 13H10, 05C12, 54E35 Comments: 35 pages, amsart, uses xy-pic Abstract: There has been much speculation about the structure of the set of shift-isomorphism classes of semidualizing complexes over a local ring. In this paper we show that this set can be given the structure of a nontrivial metric space. We investigate the interplay between the metric and several standard algebraic operations, and we provide a new characterization of Gorenstein rings that is motivated by this interplay. In the process, we obtain new results describing the behavior of reflexivity over homomorphisms of finite flat dimension. Posted on April 21, 2004
Jiang-VardyTao Jiang and Alexander Vardy have posted a new preprint, "Asymptotic Improvement of the Gilbert-Varshamov Bound on the Size of Binary Codes", to the arXiv. Math Subject Class: 05C90, 94B65 (Primary) 05A16, 05C69 (secondary) Comments: 10 pages, 3 figures; to appear in the IEEE Transactions on Information Theory, submitted August 12, 2003, revised March 28, 2004 Abstract: Given positive integers $n$ and $d$, let $A_2(n,d)$ denote the maximum size of a binary code of length $n$ and minimum distance $d$. The well-known Gilbert-Varshamov bound asserts that $A_2(n,d) \geq 2^n/V(n,d-1)$, where $V(n,d) = \sum_{i=0}^{d} {n \choose i}$ is the volume of a Hamming sphere of radius $d$. We show that, in fact, there exists a positive constant $c$ such that $$ A_2(n,d) \geq c \frac{2^n}{V(n,d-1)} \log_2 V(n,d-1) $$ whenever $d/n \le 0.499$. The result follows by recasting the Gilbert- Varshamov bound into a graph-theoretic framework and using the fact that the corresponding graph is locally sparse. Generalizations and extensions of this result are briefly discussed. Posted on April 21, 2004
KatoKiriko Kato has posted a new preprint, "Morphisms represented by monomorphisms", to the arXiv. Math Subject Class: 13D02,13D25,16D90 Comments: 22 pages Abstract: Every homomorphism of modules is projective-stably equivalent to an epimorphism but is not always to a monomorphism. We prove that a map is projective-stably equivalent to a monomorphism if and only if its kernel is torsionless, that is, a first syzygy. If it occurs although, there can be various monomorphisms that are projective-stably equivalent to a given map. But in this case there uniquely exists a "perfect" monomorphism to which a given map is projective-stably equivalent. Posted on April 14, 2004
Huneke-LeuschkeCraig Huneke and Graham J. Leuschke have posted a new preprint, "Two theorems about maximal Cohen--Macaulay modules", to the arXiv. Math Subject Class: 13C14; 13A35 Journal reference: Mathematische Annalen, 324 (2002) 391-404 Comments: 14 pages Abstract: This paper contains two theorems concerning the theory of maximal Cohen--Macaulay modules. The first theorem proves that certain Ext groups between maximal Cohen--Macaulay modules $M$ and $N$ must have finite length, provided only finitely many isomorphism classes of maximal Cohen--Macaulay modules exist having ranks up to the sum of the ranks of $M$ and $N$. This has several corollaries. In particular it proves that a Cohen--Macaulay local ring of finite Cohen--Macaulay type has an isolated singularity. A well-known theorem of Auslander gives the same conclusion but requires that the ring be Henselian. Other corollaries of our result include statements concerning when a ring is Gorenstein or a complete intersection on the punctured spectrum, and the recent theorem of Leuschke and Wiegand that the completion of an excellent Cohen--Macaulay local ring of finite Cohen--Macaulay type is again of finite Cohen--Macaulay type. The second theorem proves that a complete local Gorenstein domain of positive characteristic $p$ and dimension $d$ is $F$-rational if and only if the number of copies of $R$ splitting out of $R^{1/p^e}$ divided by $p^{de}$ has a positive limit. This result generalizes work of Smith and Van den Bergh. We call this limit the $F$-signature of the ring and give some of its properties. Posted on April 13, 2004
Huneke-McDermott-MonskyCraig Huneke, Moira A. McDermott, and Paul Monsky have posted a new preprint, "Hilbert-Kunz Functions for Normal Rings", to the arXiv. Math Subject Class: 13D40 Comments: 11 pages Abstract: Let (R,m,k) be an excellent, local, normal ring of characteristic p with a perfect residue field and dim R=d. Let M be a finitely generated R-module. We show that there exists a real number beta(M) such that lambda(M/I^[q]M) = e_{HK}(M) q^d + beta(M) q^{d-1} + O(q^{d-2}). Posted on April 9, 2004
Sharif-YassemiTirdad Sharif and Siamak Yassemi have posted a new preprint, "Special homological dimensions and Intersection Theorem", to the arXiv. Math Subject Class: 13D05; 13D25 Comments: 10 pages Abstract: Let $(R,\fm)$ be commutative Noetherian local ring. It is shown that $R$ is Cohen--Macaulay ring if there exists a Cohen--Macaulay finite (i.e. finitely generated) $R$--module with finite upper Gorenstein dimension. In addition, we show that, in the Intersection Theorem, projective dimension can be replaced by quasi--projective dimension. Posted on April 9, 2004
BrunoFabre Bruno has posted a new preprint, "Hilbert functions and geometry", to the arXiv. Math Subject Class: 14C20, 13A02 Comments: 26 pages, in French Abstract: This note is devoted to the study of the links between the Hilbert function of a subscheme X of the projective space, and its geometric properties. We will assume that X is arithmetically Cohen-Macaulay, which allows us to characterize its Hilbert function by a d integers, where d is the degree of X. We study in this context the geometric description of special linear systems of dimension maximal with respect to their degree on projective Gorenstein curves. Posted on April 9, 2004
Dwyer-Greenlees-IyengarW. Dwyer, J. P. C. Greenlees, and S. Iyengar have posted a new preprint, "Finiteness in derived categories of local rings", to the arXiv. Math Subject Class: 13D05, 13D99 (primary). 18E30, 18G99 (secondary) Comments: 40 pages Abstract: New homotopy invariant finiteness conditions on modules over commutative rings are introduced, and their properties are studied systematically. A number of finiteness results for classical homological invariants like flat dimension, injective dimension, and Gorenstein dimension, are established. It is proved that these specialize to give results concerning modules over complete intersection local rings. A noteworthy feature is the use of techniques based on thick subcategories of derived categories. Posted on April 6, 2004
D'Cruz-Kodiyalam-VermaClare D'Cruz, Vijay Kodiyalam, and Jugal. K. Verma have posted a new preprint, "Bound on the a-invariant and reduction numbers of ideals", to the arXiv.
Comments: 8 pages. to appear in Journal of algebra 274(2004) 594-601 Abstract: Let $R$ be a $d$-dimensional standard graded ring over an Artin local ring. Let $M$ be the unique maximal homogeneous ideal of $R.$ Let $h^i(R)_n$ denote the length of $H^i_M(R)_n$, i.e. the nth graded component of the ith local cohomology module of R with respect to M. Define the Eisenbud-Goto invariant of $R$ to be the number $$EG(R)= \sum_{q=0}^{d-1} \binom{d-1}{q} h^q(R)_{1-q}.$$ We prove that the $a$-invariant of $R$ satisfies $$ a(R) \leq e(R)-length(R_1)+(d-1)(length(R_0)-1)+ EG(R).$$ Using this bound we get upper bounds for the reduction number of an $m$-primary ideal of a Cohen-Macaulay local ring $(R,m)$ whose associated graded ring $G(m)$ has almost maximal depth. Posted on April 6, 2004
ZhengXinxian Zheng has posted a new preprint, "Monomial ideals arising from distributive lattices", to the arXiv. Math Subject Class: 16D25; 16E05; 06D50,;06D99 Abstract: The free resolution and the Alexander dual of squarefree monomial ideals associated with certain subsets of distributive lattices are studied. Posted on April 1, 2004
Herzog-HibiJuergen Herzog and Takayuki Hibi have posted a new preprint, "Level rings arising from meet-distributive meet-semilattices", to the arXiv. Math Subject Class: 13D02; 13H10; 06A12; 06D99 Abstract: The Alexander dual of an arbitrary meet-semilattice is described explicitly. Meet-distributive meet-semilattices whose Alexander dual is level are characterized. Posted on April 1, 2004
OdaSusumu Oda has posted a new preprint, "On unramified finitely generaed extensions of polynomial rings over a field", to the arXiv. Comments: 12 pages Abstract: The Jacobian Conjecture is established (as a corollary of a more general result): If $f_1, ..., f_n$ be elements in a polynomial ring $k[X_1, ..., X_n]$ over a field $k$ of characteristic zero such that $ \det(\partial f_i/ \partial X_j) $ is a nonzero constant, then $k[f_1, ..., f_n] = k[X_1, >..., X_n]$. Posted on April 1, 2004
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