|
|
|
|
|
Preprints from May 2004Holm-JorgensenHenrik Holm and Peter Jorgensen have posted two new preprints to the arXiv. "Cohen-Macaulay injective, projective, and flat dimension" Math Subject Class: 13D05, 13D25 Comments: 18 pages Abstract: We define three new homological dimensions - Cohen-Macaulay injective, projective, and flat dimension - which inhabit a theory similar to that of classical injective, projective, and flat dimension. Finiteness of the new dimensions characterizes Cohen-Macaulay rings with dualizing modules. "Semi-dualizing modules and related Gorenstein homological dimensions" Math Subject Class: 13D05, 13D07, 13D25, 18G10, 18G25 Comments: 25 pages Abstract: A semi-dualizing module over a commutative noetherian ring A is a finitely generated module C with RHom_A(C,C) \simeq A in the derived category D(A). We show how each such module gives rise to three new homological dimensions which we call C-Gorenstein projective, C-Gorenstein injective, and C-Gorenstein flat dimension, and investigate the properties of these dimensions. Posted on May 28, 2004
Baker-RichterAndrew Baker and Birgit Richter have posted a new preprint, "Invertible modules for commutative $\mathbb{S}$-algebras with residue fields", to the arXiv. Math Subject Class: 55P15; 55P42; 55P60 Abstract: The aim of this note is to understand invertible modules over a commutative $\mathbb{S}$-algebra in the sense of Elmendorf, Kriz, Mandell & May in some very well-behaved cases. Our main result shows that as long as the commutative $\mathbb{S}$-algebra $R$ has `reductions mod $\mathfrak{m}$' for all maximal ideals $\mathfrak{m}\ideal R_*$, and Noetherian localisations $(R_*)_{\mathfrak{m}}$, then for every invertible $R$-module $U$, $U_*=\pi_*U$ is an invertible $R_*$-module. Posted on May 28, 2004
Dibaei-YassemiMohammad T. Dibaei and Siamak Yassemi have posted a new preprint, "Associated primes and cofiniteness of local cohomology modules", to the arXiv. Math Subject Class: 13D45 Comments: 8 pages Abstract: Let $\mathfrak{a}$ be an ideal of Noetherian ring $R$ and let $M$ be an $R$-module such that $\mathrm{Ext}^i_R(R/\mathfrak{a},M)$ is finite $R$-module for every $i$. If $s$ is the first integer such that the local cohomology module $\mathrm{H}^s_\mathfrak{a}(M)$ is non $\mathfrak{a}$-cofinite, then we show that $\mathrm{Hom}_{R}(R/\mathfrak{a}, \mathrm{H}^s_\mathfrak{a}(M))$ is finite. Specially, the set of associated primes of $\mathrm{H}^s_\mathfrak{a}(M)$ is finite. Posted on May 27, 2004
WeimannMartin Weimann has posted a new preprint, "La trace via le calcul residuel: une nouvelle version du theoreme d'Abel-inverse, formes abeliennes", to the arXiv. Math Subject Class: 32B10;32C30 Comments: 33 pages Abstract: We show here how residue calculus (residue currents, Grothendick (sic) residues, duality theorem) can be used to obtain an algebraic characterization of the Abel-transform of a meromorphic form on germs of analytic sets. We prove by this way a stronger version of Abel-inverse theorem with an "algebraic" approach and we show the link with Wood's theorem. Furthermore, we obtain a new method to bound easily the dimension of the vector space of abelian forms on an algebraic projective hypersurface. Posted on May 27, 2004
VaccarinoF. Vaccarino has posted a new preprint, "The ring of multisymmetric functions", to the arXiv. Math Subject Class: 05E05, 13A50, 20C30 Comments: 12 pages, 0 figures, amsppt Abstract: Let R be a commutative ring and let n,m be two positive integers. The symmetric group on n letters acts diagonally on the ring of polynomials in nxm variables with coefficients in R. The subrings of invariants for this action is called the ring of multisymmetric functions since these are the usual symmetric functions when m=1. In this paper we will give a presentation in terms of generators and relations that holds for any R and any n,m answering in this way to a classical question. I would like to thank M.Brion, C.De Concini and C.Procesi, in alphabetical order, for useful discussions. Posted on May 27, 2004
BrennerHolger Brenner has posted a new preprint, "A linear bound for Frobenius powers and an inclusion bound for tight closure", to the arXiv. Math Subject Class: 13A35; 13D02; 14J60 Abstract: Let I denote an R_+ -primary homogeneous ideal in a normal standard-graded Cohen-Macaulay domain over a field of positive characteristic p. We give a linear degree bound for the Frobenius powers I^[q] of I, q=p^e in terms of the minimal slope of the top-dimensional syzygy bundle on the projective variety Proj R. This provides an inclusion bound for tight closure. In the same manner we give a linear bound for the Castelnuovo-Mumford regularity of the Frobenius powers I^[q]. Posted on May 26, 2004
Brenner-HeinHolger Brenner and Georg Hein have posted a new preprint, "Restriction of the cotangent bundle to elliptic curves and Hilbert-Kunz fnctions", to the arXiv. Math Subject Class: 13A35; 13D02; 13D40; 14H60 Abstract: We describe the possible restrictions of the cotangent bundle \Omega_{\PP^N} to an elliptic curve C \subset \PP^N. We apply this in positive characteristic to the computation of the Hilbert-Kunz function of a homogeneous R_+-primary ideal I \subset R in the graded section ring R = \bigoplus_{n \in \NN} \Gamma(C, Ĝ(n)). Posted on May 26, 2004
DrenskyVesselin Drensky has posted a new preprint, "Invariants of unipotent transformations acting on noetherian relatively free algebras", to the arXiv. Math Subject Class: 16R10 (primary); 16R30 (secondary) Comments: 8 pages Abstract: The classical theorem of Weitzenboeck states that the algebra of invariants of a single unipotent transformation $g$ in $GL_m(K)$ acting on the polynomial algebra $K[x_1,...,x_m]$ over a field $K$ of characteristic 0 is finitely generated. Recently the author and C.K. Gupta have started the study of the algebra of $g$-invariants of relatively free algebras of rank $m$ in varieties of associative algebras. They have shown that the algebra of invariants is not finitely generated if the variety contains the algebra $UT_2(K)$ of $2\times 2$ upper triangular matrices. The main result of the present paper is that the algebra of invariants is finitely generated if and only if the variety does not contain the algebra $UT_2(K)$. As a by-product of the proof we have established also the finite generation of the algebra of $g$-invariants of the mixed trace algebra generated by $m$ generic $n\times n$ matrices and the traces of their products. Posted on May 25, 2004
Aslaksen-Drensky-SadikovaHelmer Aslaksen, Vesselin Drensky, and Liliya Sadikova have posted a new preprint, "Defining relations of invariants of two 3 \times 3 matrices", to the arXiv. Math Subject Class: 16R30 Comments: 14 pages Abstract: Over a field of characteristic 0, the algebra of invariants of several $n\times n$ matrices under simultameous conjugation by $GL_n$ is generated by traces of products of generic matrices. Teranishi, 1986, found a minimal system of eleven generators of the algebra of invariants of two $3\times 3$ matrices. Nakamoto, 2002, obtained an explicit, but very complicated, defining relation for a similar system of generators over $\mathbb Z$. In this paper we have found another natural set of eleven generators of this algebra of invariants over a field of characteristic 0 and have given the defining relation with respect to this set. Our defining relation is much simpler than that of Nakamoto. The proof is based on easy computer calculations with standard functions of Maple but the explicit form of the relation has been found with methods of representation theory of general linear groups. Posted on May 23, 2004
Eisenbud-Huneke-UlrichDavid Eisenbud, Craig Huneke, and Bernd Ulrich have posted a new preprint, "The Regularity of Tor and Graded Betti Numbers", to the arXiv. Abstract: Let S=K[x_1,..., x_n], let A,B be finitely generated graded S-modules, and let m=(x_1,...,x_n). We give bounds for the Castelnuovo-Mumford regularity of the local cohomology of Tor_i(A,B) under the assumption that the Krull dimension of Tor_1(A,B) is at most 1. We apply the results to syzygies, Groebner bases, products and powers of ideals, and to the relationship of the Rees and Symmetric algebras. For example we show that any homogeneous linearly presented m-primary ideal has some power equal to a power of m; and if the first (roughly) (n-1)/2 steps of the resolution of I are linear, then I^2 is a power of m. Posted on May 20, 2004
Elias-Colomé-NinJuan Elias and Gemma Colomé-Nin have posted a new preprint, "Bigraded structures and the depth of blow-up algebras", to the arXiv. Math Subject Class: 13A30; 13C14; 13D40 Abstract: Let $R$ be a Cohen-Macaulay local ring, and let $I\subset R$ be an ideal with minimal reduction $J$. In this paper we attach to the pair $I$, $J$ a non-standard bigraded module $\Sigma^{I,J}$. The study of the bigraded Hilbert function of $\SIJ$ allows us to prove a improved version of Wang's conjecture and a weak version of Sally's conjecture, both on the depth of the associated graded ring $gr_I(R)$. The module $\SIJ$ can be considered as a refinement of the Sally's module previously introduced by W. Vasconcelos. Posted on May 20, 2004
Heinzer-Rotthaus-WiegandWilliam Heinzer, Christel Rotthaus, and Sylvia Wiegand have posted a new preprint, "Integral closures of ideals in completions of regular local domains", to the arXiv. Math Subject Class: 13H05 Comments: 9 pages Abstract: In this paper we exhibit an example of a three-dimensional regular local domain (A, n) having a height-two prime ideal P with the property that the extension PA^ of P to the n-adic completion A^ of A is not integrally closed. We use a construction we have studied in earlier papers: For R=k[x,y,z], where k is a field of characteristic zero and x,y,z are indeterminates over k, the example A is an intersection of the localization of the power series ring k[y,z][[x]] at the maximal ideal (x,y,z) with the field k(x,y,z,f, g) where f, g are elements of (x,y,z)k[y,z][[x]] that are algebraically independent over k(x,y,z). The elements f, g are chosen in such a way that using results from our earlier papers A is Noetherian and it is possible to describe A as a nested union of rings associated to A that are localized polynomial rings over k in five variables. Posted on May 20, 2004
Boutin-KemperMireille Boutin and Gregor Kemper have posted a new preprint, "On Reconstructing Configurations of Points in ${\mathbb P}^2$ from a Joint Distribution of Invariants", to the arXiv. Math Subject Class: 68U;14L Abstract: Consider the diagonal action of the projective group $\PGL_3$ on $n$ copies of ${\mathbb P}^2$. In addition, consider the action of the symmetric group $\Sigma_n$ by permuting the copies. In this paper we find a set of generators for the invariant field of the combined group $\Sigma_n \times \PGL_3$. As the main application, we obtain a reconstruction principle for point configurations in ${\mathbb P}^2$ from their sub-configurations of five points. Finally, we address the question of how such reconstruction principles pass down to subgroups. Posted on May 20, 2004
Sidman-Van TuylJessica Sidman and Adam Van Tuyl have posted a new preprint, "Multigraded regularity: syzygies and fat points", to the arXiv. Math Subject Class: 13D02, 13D40, 14F17 Comments: AMS LaTex, 18 pages Abstract: The Castelnuovo-Mumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations. We study connections between two recent definitions of multigraded regularity with a view towards a better understanding of the multigraded Hilbert function of fat point schemes in P^{n_1} x ... x P^{n_k}. Posted on May 17, 2004
Ha-TrungHuy Tai Ha and Ngo Viet Trung have posted a new preprint, "Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups", to the arXiv. Math Subject Class: 14M05; 13A30; 13H10 Comments: 21 pages, to appear in Trans. Amer. Math. Soc Abstract: This paper addresses problems related to the existence of arithmetic Macaulayfications of projective schemes. Let Y be the blow-up of a projective scheme X = Proj R along the ideal sheaf of I \subset R. It is known that there are embeddings Y \cong Proj k[(I^e)_c] for c \ge d(I)e + 1, where d(I) denotes the maximal generating degree of I, and that there exists a Cohen-Macaulay ring of the form k[(I^e)_c] if and only if H^0(Y,O_Y) = k, H^i(Y,O_Y) = 0 for i = 1,...,dim Y-1, Y is equidimensional and Cohen-Macaulay. Cutkosky and Herzog asked when there is a linear bound on c and e ensuring that k[(I^e)_c] is a Cohen-Macaulay ring. We obtain a surprising compelte answer to this question, namely, that under the above conditions, there are well determined invariants a and b such that k[(I^e)_c] is Cohen-Macaulay for all c > d(I)e + a and e > b. Our approach is based on recent results on the asymptotic linearity of the Castelnuovo-Mumford regularity of ideal powers. We also investigate the existence of a Cohen-Macaulay Rees algebra of the form R[(I^e)_ct] (which provides an arithmetic Macaulayfication for X). If R has negative a*-invariant, we prove that such a Cohen-Macaulay Rees algebra exists if and only if f_*O_Y = O_X, R^i f*O_Y = 0 for i > 0, Y is equidimensional and Cohen-Macaulay. Especially, these conditions imply the Cohen-Macaulayness of R[(I^e)_ct] for all c > d(I)e + a and e > b. The above results can be applied to obtain several new classes of Cohen-Macaulay algebras. Posted on May 17, 2004
Huneke-TrungCraig Huneke and Ngo Viet Trung have posted a new preprint, "On the core of ideals", to the arXiv. Math Subject Class: 13A02 Comments: 20 pages, to appear in Compositio Math Abstract: Our focus in this paper is in effective computation of the core core(I) of an ideal I which is defined to be the intersection of all minimal reductions of I. The first main result is a closed formula for the graded core(m) of the maximal graded ideal m of an arbitrary standard graded algebra A over a field k. This formula allows us to study basic properties of the graded core and to construct counter-examples to some open questions on the core of ideals in a local ring. For instance, we can show that in general, core(m \otimes E) \neq core(m)\otimes E, where E is a field extension of k. From this it follows that the equation core(I R') = core(I)R' does not hold for an arbitrary flat local homomorphism R \to R' of Cohen-Macaulay local rings. The second main result proves the formulae core(I)= (J^r:I^r)I = (J^r:I^r)J = J^{r+1}:I^r for any equimultiple ideal I in a Cohen-Macaulay ring R with with characteristic zero residue field, where J is a minimal reduction of I and r is its reduction number. This result has been obtained independently by Polini-Ulrich and Hyry-Smith in the one-dimensional case or when R is a Gorenstein ring. Moreover, we can prove that core(I) = IK, where K is the conductor of R in the blowing-up ring at I. Posted on May 17, 2004
BrennerHolger Brenner has posted a new preprint, "The Hilbert-Kunz function in graded dimension two", to the arXiv. Math Subject Class: 13A35; 13D40; 14G15; 14H60 Abstract: Let R denote a two-dimensional normal standard-graded domain over the algebraic closure K of a finite field of characteristic p, and let I denote a homogeneous primary ideal. We prove that the Hilbert-Kunz function of I has the form Posted on May 17, 2004
Shpilrain-YuVladimir Shpilrain and Jie-Tai Yu have posted a new preprint, "Test polynomials, retracts, and the Jacobian conjecture", to the arXiv. Math Subject Class: 14R10, 14R15 Comments: 7 pages Abstract: Let K[x,y] be the algebra of two-variable polynomials over a field K. A polynomial p=p(x, y) is called a test polynomial (for automorphisms) if, whenever \phi(p)=p for a mapping \phi of K[x,y], this \phi must be an automorphism. Here we show that p \in C[x,y] is a test polynomial if and only if p does not belong to any proper retract of C[x,y]. This has the following corollary that may have application to the Jacobian conjecture: if a mapping \phi of C[x,y] with invertible Jacobian matrix is ``invertible on one particular polynomial", then it is an automorphism. More formally: if there is a non-constant polynomial p and an injective mapping \psi of C[x,y] such that \psi(\phi(p)) =p, then \phi is an automorphism. Posted on May 17, 2004
EpsteinNeil Epstein has posted a new preprint, "A tight closure analogue of analytic spread", to his website and also to the arXiv. MSC: 13A35; 13B22 Abstract. An analogue of the theory of integral closure and reductions is developed for a more general class of closures, called Nakayama closures. It is shown that tight closure is a Nakayama closure by proving a ``Nakayama lemma for tight closure''. Then, after strengthening A. Vraciu's theory of $*$-independence and the special part of tight closure, it is shown that all minimal $*$-reductions of an ideal in an analytically irreducible excellent local ring of positive characteristic have the same minimal number of generators. This number is called the $*$-spread of the ideal, by analogy with the notion of analytic spread. Posted on May 17, 2004
Baghdadi-Fontana-PicozzaSaid El Baghdadi, Marco Fontana, and Giampaolo Picozza have posted a new preprint, "Semistar Dedekind Domains", to the arXiv. Math Subject Class: 13A15; 13G05; 13E99 Abstract: Let $D$ be an integral domain and $\star$ a semistar operation on $D$. As a generalization of the notion of Noetherian domains to the semistar setting, we say that $D$ is a $\star$--Noetherian domain if it has the ascending chain condition on the set of its quasi--$\star$--ideals. On the other hand, as an extension the notion of Prüfer domain (and of Prüfer $v$--multiplication domain), we say that $D$ is a Prüfer $\star$--multiplication domain (P$\star$MD, for short) if $D_M$ is a valuation domain, for each quasi--$\star_{_{f}}$--maximal ideal $M$ of $D$. Finally, recalling that a Dedekind domain is a Noetherian Prüfer domain, we define a $\star$--Dedekind domain to be an integral domain which is $\star$--Noetherian and a P$\star$MD. In the present paper, after a preliminary study of $\star$--Noetherian domains, we investigate the $\star$--Dedekind domains. We extend to the $\star$--Dedekind domains the main classical results and several characterizations proven for Dedekind domains. In particular, we obtain a characterization of a $\star$--Dedekind domain by a property of decomposition of any semistar ideal into a ``semistar product'' of prime ideals. Moreover, we show that an integral domain $D$ is a $\star$--Dedekind domain if and only if the Nagata semistar domain Na$(D, \star)$ is a Dedekind domain. Several applications of the general results are given for special cases of the semistar operation $\star$. Posted on May 11, 2004
Hartmann-SheplerJulia Hartmann and Anne V. Shepler have posted a new preprint, "Jacobians of Reflection Groups", to the arXiv. Math Subject Class: 13A50, 20C, 52C35 Comments: 11 pages Abstract: Steinberg showed that when a finite reflection group acts on a real or complex vector space of finite dimension, the Jacobian determinant of a set of basic invariants factors into linear forms which define the reflecting hyperplanes. This result generalizes verbatim to fields whose characteristic is prime to the order of the group. Our main theorem gives a generalization of Steinberg's result for arbitrary fields using a ramification formula of Benson and Crawley-Boevey. As an intermediate result, we show that every finite group which fixes a hyperplane pointwise has a polynomial ring of invariants. Posted on May 11, 2004
Diaconis-ErikssonPersi Diaconis and Nicholas Eriksson have posted a new preprint, "Markov bases for noncommutative Fourier analysis of ranked data", to the arXiv. Comments: 16 pages, 2 figures. Submitted to the Journal of Symbolic Computation, special issue on Computational Algebraic Statistics Abstract: To calibrate Fourier analysis of $S_5$ ranking data by Markov chain Monte-Carlo techniques, a set of moves (Markov basis) is needed. We calculate this Markov basis, and use it to provide a new statistical analysis of two datasets. The calculation involves a large Gröbner basis computation (45825 generators in 120 indeterminates), but reduction to a minimal basis and by natural symmetries leads to a remarkably small basis (14 elements). This symmetry is exploited to calculate a Markov basis for $S_6$. Finally, we improve a bound on the degree of the generators for a Markov basis for $S_n$ and provide evidence that this ideal is actually generated in degree 3. Posted on May 7, 2004
GanWee Liang Gan has posted a new preprint, "Chevalley restriction theorem for the cyclic quiver", to the arXiv. Comments: 5pp, LaTeX Abstract: We prove a Chevalley restriction theorem and its double analogue for the cyclic quiver. Posted on May 7, 2004
Schlomiuk-VulpeDana Schlomiuk and Nicolae Vulpe have posted a new preprint, "Planar quadratic vector fields with invariant lines of total multiplicity at least five", to the arXiv. Math Subject Class: 34C05; 13A50 Report number: CRM-2922, Departement de Mathematiques et de Statistiques, Universite de Montreal, May, 2003 Comments: 50 pages, 4 Postscript figures, Latex Abstract: In this article we consider the action of affine group and time rescaling on planar quadratic differential systems. We construct a system of representatives of the orbits of systems with at least five invariant lines, including the line at infinity and including multiplicities. For each orbit we exhibit its configuration. We characterize in terms of algebraic invariants and comitants and also geometrically, using divisors of the complex projective plane, the class of quadratic differential systems with at least five invariant lines. These conditions are such that no matter how a system may be presented, one can verify by using them whether the system has or does not have at least five invariant lines and to check to which orbit (or family of orbits) it belongs. Posted on May 7, 2004
Mueller-RitzenthalerJuergen Mueller and Christophe Ritzenthaler have posted a new preprint, "On the ring of invariants of ordinary quartic curves in characteristic 2", to the arXiv. MSC-class: 14L24 Abstract. In this article a complete set of invariants for ordinary quartic curves in Posted on May 7, 2004
SullivantSeth Sullivant has posted a new preprint, "Small contingency tables with large gaps", to the arXiv. Comments: 6 pages Abstract: We construct examples of contingency tables on $n$ binary random variables where the gap between the linear programming lower/upper bound and the true integer lower/upper bounds on cell entries is exponentially large. These examples provide evidence that linear programming may not be an effective heuristic for detecting disclosures when releasing margins of multi-way tables. Posted on May 4, 2004
Bollobas-NikiforovBela Bollobas and Vladimir Nikiforov have posted a new preprint, "Graphs and Hermitian matrices: discrepancy and singular values", to the arXiv. Math Subject Class: 05C50; 15A18 Comments: 22 pages, accepted in Discrete Math Abstract: We introduce a measure of discrepancy of Hermitian matrices and establish an inequality between the second singular value of a Hermitian matrix and its discrepancy. These results are applied to answer two questions of Fan Chung about graph eigenvalues. Posted on May 4, 2004
|
|
|
home | photos | conferences
Juan Migliore has posted some pictures from the recent Hartshorne conference. Go see! More recent news from the commalg community.
Upcoming: All recent and upcoming conferences.
search provided by
|
|
|
|
|
commalg home | faq | people | photos | conferences | links |movable type design and original content © 2001-2008 |
