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Preprints from March 2003EnescuFlorian Enescu has posted a new preprint, "Strong test modules and multiplier ideals", to the arXiv and his web page. MSC: 13A35 (primary);14B05 (secondary) Abstract: We introduce the notion of strong test module and show that a large number of such modules appear in the tight closure theory of complete domains: the test ideal (this has already been known), the parameter test module, and the module of relative test elements. They also appear as certain multiplier ideals, a concept of interest in algebraic geometry. Posted on March 28, 2003
Brun-RoemerMorten Brun and Tim Roemer have posted a new preprint, "Betti numbers of Z^n-graded modules", to the arXiv. MSC: 13D07 13D02 18G15 Abstract: Let S=K[X_1,...,X_n] be the polynomial ring over a field K. For bounded below Z^n-graded S-modules M and N we show that if Tor^S_p(M,N) is nonzero, then for every i between 0 and p, the dimension of the K-vector space Tor^S_i(M,N) is at least as big as the binomial coefficient (p,i). In particular, we get lower bounds for the total Betti numbers. These results are related to a conjecture of Buchsbaum and Eisenbud. Posted on March 28, 2003
LittleJohn B. Little hasposted a new preprint, "A key equation and the computation of error values for codes from order domains", to the arXiv. MSC: 94B27; 94B35; 13P10 Abstract: We study the computation of error values in the decoding of codes constructed from order domains. Our approach is based on a sort of analog of the key equation for decoding Reed-Solomon and BCH codes. We identify a key equation for all codes from order domains which have finitely-generated value semigroups; the field of fractions of the order domain may have arbitrary transcendence degree, however. We provide a natural interpretation of the construction using the theory of Macaulay's inverse systems and duality. O'Sullivan's generalized Berlekamp-Massey-Sakata (BMS) decoding algorithm applies to the duals of suitable evaluation codes from these order domains. When the BMS algorithm does apply, we will show how it can be understood as a process for constructing a collection of solutions of our key equation. Posted on March 27, 2003
Iyengar-Sather-WagstaffSrikanth Iyengar and Sean Sather-Wagstaff have posted a new preprint, "G-dimension over local homomorphisms. Applications to the Frobenius endomorphism", to the arXiv. MSC: 13D05, 13D25, 13B10, 13H10, 14B25 Abstract: We develop a theory of G-dimension for modules over local homomorphisms which encompasses the classical theory of G-dimension for finite modules over local rings. As an application, we prove that a local ring R of characteristic p is Gorenstein if and only if it possesses a nonzero finite module of finite projective dimension that has finite G-dimension when considered as an R-module via some power of the Frobenius endomorphism of R. We also prove results that track the behavior of Gorenstein properties of local homomorphisms under (de)composition. Posted on March 24, 2003
BensonDave Benson has posted a new preprint, "Commutative algebra in the cohomology of groups", to the Groups, Representations and Cohomology Preprint Archive. These are the official notes from Dave's lectures in the Introductory Workshop in Commutative Algebra at MSRI. Abstract: Commutative algebra is used extensively in the cohomology of groups. In this series of lectures, I concentrate on finite groups, but I also discuss the cohomology of finite group schemes, compact Lie groups, $p$-compact groups, infinite discrete groups and profinite groups. I describe the role of various concepts from commutative algebra, including finite generation, Krull dimension, depth, associated primes, the Cohen-Macaulay and Gorenstein conditions, local cohomology, Grothendieck's local duality, and Castelnuovo-Mumford regularity. Contents: Posted on March 24, 2003
NevoEran Nevo has posted a new preprint, "Algebraic Shifting and Basic Constructions on Simplicial Complexes", to the arXiv. MSC: 05E99, 55Nxx, 13Dxx Abstract: We try to understand the behavior of algebraic shifting with respect to basic constructions on simplicial complexes, like union and join. In particular we give a complete combinatortial description of the shifting of disjoint union in terms of the shifting of its components. As a corollary, we prove the following, conjectured by Kalai: $\Delta(K \cup L) = \Delta (\Delta(K) \cup \Delta(L))$, where $K,L$ are complexes, $\cup$ means disjoint union, and $\Delta$ is the shifting operator. We give an example showing that replacing the operation 'union' with the operation 'join' in the above equation is wrong, disproving a conjecture made by Kalai. We adopt a homological point of view on the algebraic shifting operator, which is used throughout this work. Posted on March 21, 2003
TeissierBernard Teissier has posted a new preprint, "Valuations, Deformations, and Toric Geometry", to the arXiv. MSC: 13A18; 14B05 Abstract: A study of the relation between a noetherian local domain with a given valuation and its associated graded ring with respect to the valuation, which in some cases is an esentially toric variety, possibly of infinite embedding dimension, but of finite Krull dimension. After extension of the valuation to a suitable completion (whose existence is not established in the paper) the relation becomes much more precise, and suggests a possible way to establish local uniformization for an excellent equicharacteristic local domain with an algebraically closed residue field, by deforming a partial toric embedded resolution of the essentially toric spectrum of the associated graded ring. Posted on March 18, 2003
HowaldJason Howald has posted a new preprint, "Multiplier Ideals of Sufficiently General Polynomials", to the arXiv. MSC: 14M25 14Q99 Abstract: It is well known that the multiplier ideal $\multr{I}$ of an ideal $I$ determines in a straightforward way the multiplier ideal $\multr{f}$ of a sufficiently general element $f$ of $I$. We give an explicit condition on a polynomial $f \in \CC[x_1,...,x_n]$ which guarantees that it is a sufficiently general element of the most natural associated monomial ideal, the ideal generated by its terms. This allows us to directly calculate the multiplier ideal $\multr{f}$ (for all $r$) of ``most'' polynomials $f$. Posted on March 18, 2003
SchoutensHans Schoutens has posted a new preprint, "Log-terminal singularities and vanishing theorems", to the arXiv and his web page. MSC: 14F17, 14B05, 13H10 Abstract: Generalizing work of Smith and Hara, we give a new characterization of log-terminal singularities for finitely generated algebras over $\mathbb C$, in terms of purity properties of ultraproducts of characteristic $p$ Frobenii. The first application is a Boutôt-type theorem for log-terminal singularities: given a pure morphism $Y\to X$ between affine $\mathbb Q$-Gorenstein varieties of finite type over $\mathbb C$, if $Y$ has at most a log-terminal singularities, then so does $X$. The second application is the Vanishing for Maps of Tor for log-terminal singularities: if $A\subset R$ is a Noether Normalization of a finitely generated $\mathbb C$-algebra $R$ and $S$ is a finitely generated $R$-algebra with log-terminal singularities, then the natural morphism $\operatorname{Tor}^A_i(M,R) \to \operatorname{Tor}^A_i(M,S)$ is zero, for every $A$-module $M$ and every $i\geq 1$. The final application is the Kawamata-Viehweg Vanishing Theorem for a connected projective variety $X$ of finite type over $\mathbb C$ whose affine cone has a log-terminal vertex (for some choice of polarization). As a smooth Fano variety has this latter property, we obtain a proof of the following conjecture of Smith for quotients of smooth Fano varieties: if $G$ is the complexification of a real Lie group acting algebraically on a projective smooth Fano variety $X$, then for any numerically effective line bundle $\mathcal L$ on any GIT quotient $Y:=X//G$, each cohomology module $H^i(Y,\mathcal L)$ vanishes for $i>0$, and, if $\mathcal L$ is moreover big, then $H^i(Y,\mathcal L^{-1})$ vanishes for $i<\operatorname{dim}Y$. Posted on March 18, 2003
BensonDavid Benson has posted a new preprint, "Dickson invariants, regularity and computation in group cohomology", to the arXiv. MSC: 20J06; 20C20; 13D45; 13A50 Abstract: In this paper, we investigate the commutative algebra of the cohomology ring $H^*(G,k)$ of a finite group $G$ over a field $k$. We relate the concept of quasi-regular sequence, introduced by Benson and Carlson, to the local cohomology of the cohomology ring. We give some slightly strengthened versions of quasi-regularity, and relate one of them to Castelnuovo--Mumford regularity. We prove that the existence of a quasi-regular sequence in either the original sense or the strengthened ones is true if and only if the Dickson invariants form a quasi-regular sequence in the same sense. The proof involves the notion of virtual projectivity, introduced by Carlson, Peng and Wheeler. As a by-product of this investigation, we give a new proof of the Bourguiba--Zarati theorem on depth and Dickson invariants, in the context of finite group cohomology, without using the machinery of unstable modules over the Steenrod algebra. Finally, we describe an improvement of Carlson's algorithm for computing the cohomology of a finite group using a finite initial segment of a projective resolution of the trivial module. In contrast to Carlson's algorithm, ours does not depend on verifying any conjectures during the course of the calculation, and is always guaranteed to work. Posted on March 18, 2003
PanovTaras E. Panov has posted a new preprint, "Stanley-Reisner rings and torus actions", to the arXiv. MSC: 13F55; 55T20; 57N65; 57R19; 57R91 Abstract: We review a class of problems on the borders of topology of torus actions, commutative homological algebra and combinatorial geometry, which is currently being investigated by Victor Buchstaber and the author. The text builds on the lectures delivered on the transformation group courses in Osaka City University and Universitat Autonoma de Barcelona. We start with discussing several well-known results and problems on combinatorial geometry of polytopes and simplicial complexes, and then move gradually towards investigating the combinatorial structures associated with spaces acted on by the torus. Parallelly, we set up the required commutative algebra apparatus, including Cohen-Macaulay/Gorenstein rings and Stanley-Reisner face rings of simplicial complexes. Posted on March 17, 2003
KuranoKazuhiko Kurano has posted a new preprint, "Numerical equivalence defined on Chow groups of Noetherian local rings", to his webpage. Posted on March 13, 2003
Watanabe-YoshidaKei-ichi Watanabe and Ken-ichi Yoshida have posted a new preprint, "Minimal Hilbert-Kunz multiplicity", to the arXiv. MSC: 13A35; 13H15 Abstract: In this paper, we ask the following question: what is the minimal value of the difference $e_{HK}(I) - e_{HK}(I')$ for ideals $I' \supseteq I$ with $l_A(I'/I) =1$? In order to answer to this question, we define the notion of minimal Hilbert-Kunz multiplicity for strongly F-regular rings. Moreover, we calculate this invariant for quotient singularities and for the coordinate ring of the Segre embedding: $P^{r-1} \times P^{s-1} \hookrightarrow P^{rs-1}$, respectively. Posted on March 13, 2003
HashimotoMitsuyasu Hashimoto has posted a new preprint, "'Geometric quotients are algebraic schemes' based on Fogarty's idea", to the arXiv and his web page. MSC: 13E15; 14L24 (Primary), 13C15 (Secondary) Abstract: Let S be a Noetherian scheme, f:X->Y a surjective S-morphism of S-schemes, with X of finite type over S. We discuss what makes Y of finite type. First, we prove that if S is excellent, Y is reduced, and f is universally open, then Y is of finite type. We apply this to understand Fogarty's theorem in "Geometric quotients are algebraic schemes, Adv. Math. 48 (1983), 166--171" for the special case that the group scheme G is flat over the Noetherian base scheme S. Namely, we prove that if G is a flat S-group scheme of finite type acting on X and f is its strict orbit space, then Y is of finite type. Utilizing the technique used there, we also prove that Y is of finite type if f is flat. The same is true if S is excellent, f is proper, and Y is Noetherian. Posted on March 13, 2003
BrennerHolger Brenner has posted a new preprint, "Computing the tight closure in dimension two", to the arXiv. MSC: 13A35, 14H60 Abstract: We study computational aspects of the tight closure of a homogeneous primary ideal in a two-dimensional normal standard-graded domain. We show how to use slope criteria for the sheaf of relations for generators of the ideal to compute the tight closure. Our method gives in particular an algorithm to compute the tight closure of three elements under the condition that we are able to compute the Harder-Narasimhan filtration. We apply this to the computation of (x^a,y^a,z^a)^* in K[x,y,z]/(F), where F is a homogeneous polynomial. Posted on March 13, 2003
YanagawaKohji Yanagawa has posted a new preprint, "Derived Category of Squarefree Modules and Local Cohomology with Monomial Ideal Support", to the arXiv. MSC: 13D02, 13D45, 13F55, 18E30 Abstract: A "squarefree module" over a polynomial ring $S = k[x_1, .., x_n]$ is a generalization of a Stanley-Reisner ring, and allows us to apply homological methods to the study of monomial ideals systematically. Let $Sq$ be the category of squarefree modules. Then the derived category $D^b(Sq)$ of $Sq$ has three duality functors which act on $D^b(Sq)$ just like three transpositions of the symmetric group $S_3$ (up to translation). This phenomenon is closely related to the Koszul dulaity (in particular, the Bernstein-Gel'fand-Gel'fand correspondence). We also study the local cohomology module $H_{I_\Delta}^i(S)$ at a Stanley-Reisner ideal $I_\Delta$ using squarefree modules. Among other things, we see that Hochster's formula on the Hilbert function of $H_m^i(S/I_\Delta)$ is also a formula on the characteristic cycle of $H_{I_\Delta}^{n-i}(S)$ as a module over the Weyl algebra $S<\partial_1, ..., \partial_n >$ (if $chara(k)=0$). Posted on March 13, 2003
JorgensenPeter Jorgensen has posted a new preprint, "Recognizing dualizing complexes", to the arXiv. MSC: 13D25; 16E45 Abstract: Let A be a noetherian local commutative ring and let M be a suitable complex of A-modules. This paper proves that M is a dualizing complex for A if and only if the trivial extension A \ltimes M is a Gorenstein Differential Graded Algebra. As a corollary follows that A has a dualizing complex if and only if it is a quotient of a Gorenstein local Differential Graded Algebra. Posted on March 13, 2003
YoshinoYuji Yoshino has posted a new preprint, "Modules of G-dimension zero over local rings with the cube of maximal ideal being zero", to the arXiv. MSC: 13C14, 13D05, 16G50 Abstract: Let $(R, \m)$ be a commutative Noetherian local ring with $\m^3 =(0)$. We give a condition for $R$ to have a non-free module of G-dimension zero. We shall also construct a family of non-isomorphic indecomposable modules of G-dimension zero with parameters in an open subset of projective space. We shall finally show that the subcategory consisting of modules of G-dimension zero over $R$ is not necessarily a contravariantly finite subcategory in the category of finitely generated $R$-modules. Posted on March 13, 2003
KaruKalle Karu has posted a new preprint, "Local Strong Factorization of birational maps", to the arXiv. MSC: 14E05, 13H05 Abstract: The strong factorization conjecture states that a proper birational map between smooth algebraic varieties over a field of characteristic zero can be factored as a sequence of smooth blowups followed by a sequence of smooth blowdowns. We prove a local version of the strong factorization conjecture for toric varieties. Combining this result with the monomialization theorem of S.D.Cutkosky, we obtain a strong factorization theorem for local rings dominated by a valuation. Posted on March 13, 2003
SandersJan A. Sanders has posted a new preprint, "Normal form theory and spectral sequences", to the arXiv. MSC: 34C20; 18G40 Abstract: The concept of unique normal form is formulated in terms of a spectral sequence. As an illustration of this technique some results of Baider and Churchill concerning the normal form of the anharmonic oscillator are reproduced. The aim of this paper is to show that spectral sequences give us a natural framework in which to formulate normal form theory. Posted on March 13, 2003
Domokos-FrenkelM. Domokos and P. E. Frenkel have posted a new preprint, "On orthogonal invariants in characteristic 2", to the arXiv. MSC: 13A50, 15A72, 20G05 Abstract: Working over an algebraically closed base field k of characteristic 2, the ring of invariants R^G is studied, where G is the orthogonal group O(n) or the special orthogonal group SO(n), acting naturally on the coordinate ring R of the m-fold direct sum $k^n \oplus...\oplus k^n$ of the standard vector representation. It is proved for O(2), O(3)=SO(3), SO(4), and O(4), that there exists an m-linear invariant with m arbitrarily large, which is not expressible as a polynomial of invariants of lower degree. This is in sharp contrast with the uniform description of the ring of invariants valid in all other characteristics, and supports the conjecture that the same phenpmena occur for all n. For general even n, new O(n)-invariants are constructed, which are not expressible as polynomials of the quadratic invariants. In contrast with these results, it is shown that rational invariants have a uniform description valid in all characteristics. Similarly, if $m \leq n$, then R^O(n) is generated by the obvious invariants. For all n, the algebra R^G is a finitely generated module over the subalgebra generated by the quadratic invariants. Finally we mention that for even n, an n-linear SO(n)-invariant is given, which distinguishes between SO(n) and O(n) (just like the determinant in all characteristics different from 2). Posted on March 13, 2003
Eto-YoshidaKazufumi Eto and Ken-ichi Yoshida have posted a new preprint, "Notes on Hilbert-Kunz multiplicity of Rees algebras", to the arXiv. MSC: 13H15; 13A30; 13H35 Abstract: In this paper, we estimate the Hilbert-Kunz multiplicity of the (extended) Rees algebras in terms of some invariants of the base ring. Also, we give an explicit formula for the Hilbert-Kunz multiplicities of Rees algebras over Veronese subrings. Posted on March 5, 2003
Ein-Lazarsfeld-Smith-VarolinLawrence Ein, Robert Lazarsfeld, Karen E. Smith and Dror Varolin have posted a new preprint, "Jumping Coefficients of Multiplier Ideals", to the arXiv. Abstract: We study in this paper some local invariants attached via multiplier ideals to an effective divisor or ideal sheaf on a smooth complex variety. First considered (at least implicitly) by Libgober and by Loeser and Vaquie, these jumping coefficients consist of an increasing sequence of rational numbers beginning with the log canonical threshold of the divisor or ideal in question. They encode interesting geometric and algebraic information, and we show that they arise naturally in several different contexts. Given a polynomial f having only isolated singularities, results of Varchenko, Loeser and Vaquie imply that if \xi is a jumping number of f = 0 lying in the interval (0, 1], then -\xi is a root of the Bernstein-Sato polynomial of f. We adapt an argument of Kollar to show prove that this holds also when the singular locus of f has positive dimension. In a more algebraic direction, we show that the number of such jumping coefficients bounds the uniform Artin-Rees number of the principal ideal (f) in the sense of Huneke: in the case of isolated singularities, this in turn leads to bounds involving the Milnor and Tyurina numbers of f . Along the way, we establish a general result relating multiplier to Jacobian ideals. We also explore the extension of these ideas to the setting of graded families of ideals. The paper contains many concrete examples. Posted on March 5, 2003
Ebrahimi-FardKurusch Ebrahimi-Fard has posted a new preprint, "On the associative Nijenhuis Relation", to the arXiv. MSC: 08B20, 83C47, 35Q15, 17A30 Abstract: In this brief note we would like to give the construction of a free commutative unital associative Nijenhuis algebra on a commutative unital associative algebra based on an augmented modified quasi-shuffle product. Posted on March 3, 2003
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