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Preprints from June 2004Nagel-RoemerUwe Nagel and Tim Roemer have posted a new preprint, "Extended degree functions and monomial modules", to the arXiv. Math Subject Class: 13C99; 13D99; 13P10 Comments: 20 pages Abstract: The arithmetic degree, the smallest extended degree, and the homological degree are invariants that have been proposed as alternatives of the degree of a module if this module is not Cohen-Macaulay. We compare these degree functions and study their behavior when passing to the generic initial or the lexicographic submodule. This leads to various bounds and to counterexamples to a conjecture of Gunston and Vasconcelos, respectively. Particular attention is given to the class of sequentially Cohen-Macaulay modules. The results in this case lead to an algorithm that computes the smallest extended degree. Posted on June 22, 2004
KnafHagen Knaf has posted a new preprint, "Regular local algebras over a Pruefer domain: weak dimension and regular sequences", to the arXiv. Math Subject Class: 13D05,13H05 Comments: 25 pages, 1 figure Abstract: A not necessarily noetherian local ring O is called regular if every finitely generated ideal I of O possesses finite projective dimension. In the article localizations O of a finitely presented, flat algebra A over a Pruefer domain R at a prime q are investigated with respect to regularity: this property of O is shown to be equivalent to the finiteness of the weak homological dimension wdim(O). A formula to compute wdim(O) is provided. Furthermore regular sequences within the maximal ideal M of O are studied: it is shown that regularity of O implies the existence of a maximal regular sequence of length wdim(O). If height(p) is finite, where p is the intersection of q with R, then this sequence can be choosen such that the radical of the ideal generated by the members of the sequence equals M. As a consequence it is proved that if O is regular, then the (noetherian) factor ring O/pO is Cohen-Macaulay. If pR_p is not finitely generated, then O/pO itself is regular. Posted on June 22, 2004
D'Andrea-SombraCarlos D'Andrea and Martin Sombra have posted a new preprint, "The Cayley-Menger determinant is irreducible for $n\geq 3$", to the arXiv. Math Subject Class: Primary 12E05; Secondary 52A38 Comments: 7 pages, 4 figures Abstract: We prove that the Cayley-Menger determinant of an $n$-dimensional simplex is an absolutely irreducible polynomial for $n\geq3.$ We also study the irreducibility of polynomials associated to related geometric constructions. Posted on June 22, 2004
Singh-SwansonAnurag K. Singh and Irena Swanson have posted a new preprint, "Associated primes of local cohomology modules and of Frobenius powers", to the arXiv. Journal reference: International Mathematics Research Notices 33 (2004) 1703-1733 Abstract: We construct normal hypersurfaces whose local cohomology modules have infinitely many associated primes. These include unique factorization domains of characteristic zero with rational singularities, as well as F-regular unique factorization domains of positive characteristic. As a consequence, we answer a question on the associated primes of Frobenius powers of ideals, which arose from the localization problem in tight closure theory. Posted on June 22, 2004
SinghAnurag Singh has posted a few preprints to the arXiv. "p-torsion elements in local cohomology modules" Journal reference: Mathematical Research Letters 7 (2000) 165-176 Abstract: We construct an example where a local cohomology module of a hypersurface has p-torsion elements for every prime integer p, and consequently has infinitely many associated prime ideals. We also answer a related question of Hochster. "p-torsion elements in local cohomology modules. II" Journal reference: Local Cohomology and Its Applications, Marcel Dekker (2002) 155-167 Abstract: Lyubeznik conjectured that local cohomology modules of regular rings have finitely many associated primes. We examine this conjecture for polynomial rings over the integers, and record some equational identities that arise from studying special cases. "Associated primes of local cohomology modules" Journal reference: Proceedings of the 48th Algebra Symposium at Nagoya University, (2003) 133-145 Abstract: This is an expanded version of a lecture on finiteness properties of local cohomology modules given at the 48th Algebra Symposium held at Nagoya University in August 2003. Posted on June 22, 2004
Chan-HuangC-Y. Jean Chan and I-Chiau Huang have posted a new preprint, "Module structure of an injective resolution", to the arXiv. Math Subject Class: 13C11; 13D25 Comments: 30 pages Abstract: Let A be the ring obtained by localizing the polynomial ring k[X,Y,Z,W] over a field k at the maximal ideal (X,Y,Z,W) and modulo the ideal (XW-YZ). Let p be the ideal of A generated by X and Y. We study the module structure of a minimal injective resolution of A/p in details using local cohomology. Applications include the description of Ext^i(M,A/p), where M is a module constructed by Dutta, Hochster and McLaughlin, and the Yoneda product of Ext^*(A/p,A/p). Posted on June 22, 2004
Fontana-Houston-LucasMarco Fontana, Evan Houston, and Thomas Lucas have posted a new preprint, "Integral Domains whose Simple Overrings are Intersections of Localizations", to the arXiv. Math Subject Class: 13A15, 13F05 Abstract: Call a domain $R$ an sQQR-domain if each simple overring of $R$, i.e., each ring of the form $R[u]$ with $u$ in the quotient field of $R$, is an intersection of localizations of $R$. We characterize Prüfer domains as integrally closed sQQR-domains. In the presence of certain finiteness conditions, we show that the sQQR-property is very strong; for instance, a Mori sQQR-domain must be a Dedekind domain. We also show how to construct sQQR-domains which have (non-simple) overrings which are not intersections of localizations. Posted on June 22, 2004
HashimotoMitsuyasu Hashimoto has posted a new preprint, "A pure subalgebra of a finitely generated algebra is finitely generated", to the arXiv. Math Subject Class: 13E15 Comments: 4 pages Abstract: We prove the following. Let $R$ be a Noetherian ring, $B$ a finitely generated $R$-algebra, and $A$ a pure $R$-subalgebra of $B$. Then $A$ is finitely generated over $R$. Posted on June 22, 2004
Matusevich-Miller-WaltherLaura Felicia Matusevich, Ezra Miller, and Uli Walther have posted a new preprint, "Homological Methods for Hypergeometric Families", to the arXiv. Abstract: We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for hypergeometric systems H_A(\beta) arising from a d x n integer matrix A and a parameter \beta \in \CC^d. To do so we introduce an Euler-Koszul functor for hypergeometric families over \CC^d, whose homology generalizes the notion of a hypergeometric system, and we prove a homology isomorphism with our general homological construction above. We show that a parameter \beta is rank-jumping for H_A(\beta) if and only if \beta lies in the Zariski closure of the set of \ZZ^d-graded degrees \alpha where the local cohomology \bigoplus_{i < d}H^i_\frakm(\CC[\NN A])_\alpha of the semigroup ring \CC[\NN A] supported at its maximal graded ideal \frakm is nonzero. Consequently, H_A(\beta) has no rank-jumps over \CC^d if and only if \CC[\NN A] is Cohen-Macaulay of dimension d. Posted on June 22, 2004
Baker-RichterAndrew Baker and Birgit Richter have posted a new preprint, "Realizability of algebraic Galois extensions by strictly commutative ring spectra", to the arXiv. Math Subject Class: 55P42, 55P43, 55S35, 55P91, 55P92, 13B05 Abstract: We discuss some of the basic ideas of Galois theory for commutative Salgebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups.We describe the general framework developed by Rognes. Central roles are played by the notion of strong duality and a trace or norm mapping constructed by Greenlees and May in the context of generalized Tate cohomology. We give some examples where algebraic data on coefficient rings ensures strong topological consequences. We consider the issue of passage from algebraic Galois extensions to topological ones applying obstruction theories of Robinson and Goerss-Hopkins to produce topological models for algebraic Galois extensions and the necessary morphisms of commutative S-algebras. Examples such as the Real K-theory spectrum as a KO-algebra indicate that more exotic phenomena occur in the topological setting. We show how in certain cases topological abelian Galois extensions are classified by the same Harrison groups as algebraic ones and this leads to computable Harrison groups for such spectra. We consider the Tate spectrum associated to a G-Galois extension and show that it is trivial, thus generalising an analogous result for algebraic Galois extensions. We end by proving an analogue of Hilbert's theorem 90 for the units associated with a Galois extension. Posted on June 22, 2004
MoodyJohn Atwell Moody has posted a new preprint, "Graded rings and the Nash process", to the arXiv. Math Subject Class: 13A30 Abstract: Let V be an irreducible variety over a field k and let ...-> V_3 -> V_1 -> V_0=V be the sequence of iterated Nash blowups. Beyond the work of Hironaka, Gonzales-Sprinberg, and Spivakovsky in the case of surfaces, it is not know whether the inverse limit lim V_i is a variety, or whether it has the structre of Proj(R) for some graded ring R. In this paper we show it is nevertheless possible to control lim V_i from below. There is a graded ring R (or sheaf if V is not affine) and a natural map lim V_i -> Proj(R) if the limit is a variety. Although the map goes in the wrong direction, whenever $R$ is finite type the sequence of Nash blowups does terminate successfully in finitely many steps. For projective varieties, convergence of the Nash process is essentially controlled by a Hilbert series. Calculations are described for toric varieties and the technique is illustrated by the Hilbert series calculation which verifies the Camacho, Neto and Sad result that a dicritical vector field cannot be resolved. Posted on June 15, 2004
HellusMichael Hellus has posted a new preprint, "On the associated primes of Matlis duals of top local cohomology modules", to the arXiv. Math Subject Class: 13D45; 14B15 Comments: 10 pages Abstract: After motivating the question we prove various results about the set of associated primes of Matlis duals of top local cohomology modules. In some cases we can calculate this set, for the general situation we present a conjecture. An easy application of this theory is the well-known fact that Krull dimension can be expressed by the vanishing of local cohomology modules. Posted on June 15, 2004
Heinzer-Kim-UlrichWilliam Heinzer, Mee-Kyoung Kim, and Bernd Ulrich have posted a new preprint, "The Gorenstein and complete intersection properties of associated graded rings", to the arXiv. Math Subject Class: 13A30 Abstract: Let I be an m-primary ideal of a Noetherian local ring (R,m). We consider the Gorenstein and complete intersection properties of the associated graded ring G(I) and the fiber cone F(I) of I as reflected in their defining ideals as homomorphic images of polynomial rings over R/I and R/m respectively. In case all the higher conormal modules of I are free over R/I, we observe that: (i) G(I) is Cohen-Macaulay iff F(I) is Cohen- Macaulay, (ii) G(I) is Gorenstein iff both F(I) and R/I are Gorenstein, and (iii) G(I) is a relative complete intersection iff F(I) is a relative complete intersection. In case R/I is Gorenstein, we give a necessary and sufficient condition for G(I) to be Gorenstein in terms of residuation of powers of I with respect to a reduction J of I with \mu(J) = dim R and the reduction number r of I with respect to J. We prove that G(I) is Gorenstein iff J:I^{r-i} = J + I^{i+1}, for i = 0, ...,r-1. If (R,m) is a Gorenstein local ring and I \subseteq m is an ideal having a reduction J with reduction number r such that \mu(J) = ht(I) = g > 0, we prove that the extended Rees algebra R[It, t^-1}] is quasi-Gorenstein with \a-invariant a if and only if J^n:I^r = I^{n+a-r+g-1} for every integer n. Posted on June 15, 2004
Blickle-BonduManuel Blickle and Raphael Bondu have posted a new preprint, "Local cohomology multiplicities in positive characterisitic", to the arXiv. Math Subject Class: 14B15; 13D45 Comments: 15 pages Abstract: We give a description of certain local cohomology invariants (introduced by Lyubeznik) for a class of mildly singular varieties in positive characteristic. The novelty of our approuch lies in using the new Riemann-Hilbert type correspondenc of Emerton and Kisin to express these invariants in terms of 'etale cohomology with ZZ/pZZ coefficients. Thus we are able to obtain results in positive characteristic whereas earlier approaches were restricted (due to the use of de Rham theory) to the analytic case. Posted on June 15, 2004
ZingerAleksey Zinger has posted a new preprint, "On the Structure of Certain Natural Cones over Moduli Spaces of Genus-One Holomorphic Maps", to the arXiv. Math Subject Class: 53D99, 14N35 Comments: 38 pages Abstract: We show that certain naturally arising cones over the main component of a moduli space of $J_0$-holomorphic maps into $P^n$ have a well-defined euler class. We also prove that this is the case if the standard complex structure $J_0$ on $P^n$ is replaced by a nearby almost complex structure $J$. The genus-zero analogue of the cone considered in this paper is always a vector bundle. The genus-zero Gromov-Witten invariant of a projective hypersurface is the euler class of such a vector bundle. As shown in a separate paper, the "genus-one part" of the genus-one GW-invariant of a threefold in $P^4$ is the euler class of the corresponding cone considered below. The remaining part is a multiple of the genus-zero GW-invariant. This should be the case for any complete intersection in a projective space. Posted on June 15, 2004
PlotkinBoris Plotkin has posted a new preprint, "Notes on Engel groups and Engel elements in groups. Some generalizations", to the arXiv. Math Subject Class: 20F19; 16R99 Comments: 17pages Abstract: Engel groups and Engel elements became popular in 50s. We consider in the paper the more general nil-groups and nil-elements in groups. All these notions are related to nilpotent groups and nilpotent radicals in groups. These notions generate problems which are parallel to Burnside problems for periodic groups. The first three theorems of the paper are devoted to nil-groups and Engel groups, while the other results are connected with the further generalizations. These generalizations extend the theory to solvable groups and solvable radicals in groups. The paper has two parts. The first one (sections 2-4) deals with old ideas, while the second one (sections 5-9) is devoted to generalizations. Posted on June 15, 2004
VelicheOana Veliche has posted a new preprint, "Gorenstein projective dimension for complexes", to the arXiv. Math Subject Class: 18Gxx Comments: Accepted for publication in Transactions AMS, submitted October 8, 2003 Abstract: We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology groups have a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension. Posted on June 6, 2004
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