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26. CJM 2008 (vol 60 pp. 721)

Adamus, J.; Bierstone, E.; Milman, P. D.
 Uniform Linear Bound in Chevalley's Lemma We obtain a uniform linear bound for the Chevalley function at a point in the source of an analytic mapping that is regular in the sense of Gabrielov. There is a version of Chevalley's lemma also along a fibre, or at a point of the image of a proper analytic mapping. We get a uniform linear bound for the Chevalley function of a closed Nash (or formally Nash) subanalytic set. Keywords:Chevalley function, regular mapping, Nash subanalytic setCategories:13J07, 32B20, 13J10, 32S10

27. CJM 2008 (vol 60 pp. 556)

Draisma, Jan; Kemper, Gregor; Wehlau, David
 Polarization of Separating Invariants We prove a characteristic free version of Weyl's theorem on polarization. Our result is an exact analogue of Weyl's theorem, the difference being that our statement is about separating invariants rather than generating invariants. For the special case of finite group actions we introduce the concept of \emph{cheap polarization}, and show that it is enough to take cheap polarizations of invariants of just one copy of a representation to obtain separating vector invariants for any number of copies. This leads to upper bounds on the number and degrees of separating vector invariants of finite groups. Keywords:Jan Draisma, Gregor Kemper, David WehlauCategories:13A50, 14L24

28. CJM 2008 (vol 60 pp. 391)

Migliore, Juan C.
 The Geometry of the Weak Lefschetz Property and Level Sets of Points In a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property (WLP), and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its Artinian reduction from having WLP, which in itself is a very algebraic notion. More precisely, we produce level sets of points whose Artinian reductions have socle types 3 and 4 and arbitrary socle degree $\geq 12$ (in the worst case), but fail to have WLP. We also produce a level set of points whose Artinian reduction fails to have unimodal Hilbert function; our example is based on Zanello's example. Finally, we show that a level set of points can have Artinian reduction that has WLP but fails to have the Strong Lefschetz Property. While our constructions are all based on basic double G-linkage, the implementations use very different methods. Keywords:Weak Lefschetz Property, Strong Lefschetz Property, basic double G-linkage, level, arithmetically Gorenstein, arithmetically Cohen--Macaulay, socle type, socle degree, Artinian reductionCategories:13D40, 13D02, 14C20, 13C40, 13C13, 14M05

29. CJM 2007 (vol 59 pp. 880)

van, John E.
 Radical Ideals in Valuation Domains An ideal $I$ of a ring $R$ is called a radical ideal if $I={\mathcalR}(R)$ where ${\mathcal R}$ is a radical in the sense of Kurosh--Amitsur. The main theorem of this paper asserts that if $R$ is a valuation domain, then a proper ideal $I$ of $R$ is a radical ideal if and only if $I$ is a distinguished ideal of $R$ (the latter property means that if $J$ and $K$ are ideals of $R$ such that $J\subset I\subset K$ then we cannot have $I/J\cong K/I$ as rings) and that such an ideal is necessarily prime. Examples are exhibited which show that, unlike prime ideals, distinguished ideals are not characterizable in terms of a property of the underlying value group of the valuation domain. Categories:16N80, 13A18

30. CJM 2007 (vol 59 pp. 109)

Jayanthan, A. V.; Puthenpurakal, Tony J.; Verma, J. K.
 On Fiber Cones of $\m$-Primary Ideals Two formulas for the multiplicity of the fiber cone $F(I)=\bigoplus_{n=0}^{\infty} I^n/\m I^n$ of an $\m$-primary ideal of a $d$-dimensional Cohen--Macaulay local ring $(R,\m)$ are derived in terms of the mixed multiplicity $e_{d-1}(\m | I)$, the multiplicity $e(I)$, and superficial elements. As a consequence, the Cohen--Macaulay property of $F(I)$ when $I$ has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of the reduction number of $I$ and lengths of certain ideals. We also characterize the Cohen--Macaulay and Gorenstein properties of fiber cones of $\m$-primary ideals with a $d$-generated minimal reduction $J$ satisfying $\ell(I^2/JI)=1$ or $\ell(I\m/J\m)=1.$ Keywords:fiber cones, mixed multiplicities, joint reductions, Cohen--Macaulay fiber cones, Gorenstein fiber cones, ideals having minimal and almost minimal mixed multiplicitiesCategories:13H10, 13H15, 13A30, 13C15, 13A02

31. CJM 2005 (vol 57 pp. 1178)

Cutkosky, Steven Dale; Hà, Huy Tài; Srinivasan, Hema; Theodorescu, Emanoil
 Asymptotic Behavior of the Length of Local Cohomology Let $k$ be a field of characteristic 0, $R=k[x_1, \ldots, x_d]$ be a polynomial ring, and $\mm$ its maximal homogeneous ideal. Let $I \subset R$ be a homogeneous ideal in $R$. Let $\lambda(M)$ denote the length of an $R$-module $M$. In this paper, we show that $$\lim_{n \to \infty} \frac{\l\bigl(H^0_{\mathfrak{m}}(R/I^n)\bigr)}{n^d} =\lim_{n \to \infty} \frac{\l\bigl(\Ext^d_R\bigl(R/I^n,R(-d)\bigr)\bigr)}{n^d}$$ always exists. This limit has been shown to be ${e(I)}/{d!}$ for $m$-primary ideals $I$ in a local Cohen--Macaulay ring, where $e(I)$ denotes the multiplicity of $I$. But we find that this limit may not be rational in general. We give an example for which the limit is an irrational number thereby showing that the lengths of these extention modules may not have polynomial growth. Keywords:powers of ideals, local cohomology, Hilbert function, linear growthCategories:13D40, 14B15, 13D45

32. CJM 2005 (vol 57 pp. 724)

Purnaprajna, B. P.
 Some Results on Surfaces of General Type In this article we prove some new results on projective normality, normal presentation and higher syzygies for surfaces of general type, not necessarily smooth, embedded by adjoint linear series. Some of the corollaries of more general results include: results on property $N_p$ associated to $K_S \otimes B^{\otimes n}$ where $B$ is base-point free and ample divisor with $B\otimes K^*$ {\it nef}, results for pluricanonical linear systems and results giving effective bounds for adjoint linear series associated to ample bundles. Examples in the last section show that the results are optimal. Categories:13D02, 14C20, 14J29

33. CJM 2005 (vol 57 pp. 400)

Sabourin, Sindi
 Generalized $k$-Configurations In this paper, we find configurations of points in $n$-dimensional projective space ($\proj ^n$) which simultaneously generalize both $k$-configurations and reduced 0-dimensional complete intersections. Recall that $k$-configurations in $\proj ^2$ are disjoint unions of distinct points on lines and in $\proj ^n$ are inductively disjoint unions of $k$-configurations on hyperplanes, subject to certain conditions. Furthermore, the Hilbert function of a $k$-configuration is determined from those of the smaller $k$-configurations. We call our generalized constructions $k_D$-configurations, where $D=\{ d_1, \ldots ,d_r\}$ (a set of $r$ positive integers with repetition allowed) is the type of a given complete intersection in $\proj ^n$. We show that the Hilbert function of any $k_D$-configuration can be obtained from those of smaller $k_D$-configurations. We then provide applications of this result in two different directions, both of which are motivated by corresponding results about $k$-configurations. Categories:13D40, 14M10

34. CJM 2004 (vol 56 pp. 716)

 Fat Points in $\mathbb{P}^1 \times \mathbb{P}^1$ and Their Hilbert Functions We study the Hilbert functions of fat points in $\popo$. If $Z \subseteq \popo$ is an arbitrary fat point scheme, then it can be shown that for every $i$ and $j$ the values of the Hilbert function $_{Z}(l,j)$ and $H_{Z}(i,l)$ eventually become constant for $l \gg 0$. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in $\popo$. This enables us to compute all but a finite number values of $H_{Z}$ without using the coordinates of points. We also characterize the ACM fat point schemes sing our description of the eventual behaviour. In fact, n the case that $Z \subseteq \popo$ is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function. Keywords:Hilbert function, points, fat points, Cohen-Macaulay, multi-projective spaceCategories:13D40, 13D02, 13H10, 14A15

35. CJM 2004 (vol 56 pp. 742)

Jiang, Chunlan
 Similarity Classification of Cowen-Douglas Operators Let $\cal H$ be a complex separable Hilbert space and ${\cal L}({\cal H})$ denote the collection of bounded linear operators on ${\cal H}$. An operator $A$ in ${\cal L}({\cal H})$ is said to be strongly irreducible, if ${\cal A}^{\prime}(T)$, the commutant of $A$, has no non-trivial idempotent. An operator $A$ in ${\cal L}({\cal H})$ is said to a Cowen-Douglas operator, if there exists $\Omega$, a connected open subset of $C$, and $n$, a positive integer, such that (a) ${\Omega}{\subset}{\sigma}(A)=\{z{\in}C; A-z {\text {not invertible}}\};$ (b) $\ran(A-z)={\cal H}$, for $z$ in $\Omega$; (c) $\bigvee_{z{\in}{\Omega}}$\ker$(A-z)={\cal H}$ and (d) $\dim \ker(A-z)=n$ for $z$ in $\Omega$. In the paper, we give a similarity classification of strongly irreducible Cowen-Douglas operators by using the $K_0$-group of the commutant algebra as an invariant. Categories:47A15, 47C15, 13E05, 13F05

36. CJM 2003 (vol 55 pp. 1019)

Handelman, David
 More Eventual Positivity for Analytic Functions Eventual positivity problems for real convergent Maclaurin series lead to density questions for sets of harmonic functions. These are solved for large classes of series, and in so doing, asymptotic estimates are obtained for the values of the series near the radius of convergence and for the coefficients of convolution powers. Categories:30B10, 30D15, 30C50, 13A99, 41A58, 42A16

37. CJM 2003 (vol 55 pp. 711)

Broughan, Kevin A.
 Adic Topologies for the Rational Integers A topology on $\mathbb{Z}$, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to $\mathbb{Q}$, with a compact completion homeomorphic to the Cantor set. It has a natural place in a family of topologies on $\mathbb{Z}$, which includes the $p$-adics, and one in which the set of rational primes $\mathbb{P}$ is dense. Examples from number theory are given, including the primes and squares, Fermat numbers, Fibonacci numbers and $k$-free numbers. Keywords:$p$-adic, metrizable, quasi-valuation, topological ring,, completion, inverse limit, diophantine equation, prime integers,, Fermat numbers, Fibonacci numbersCategories:11B05, 11B25, 11B50, 13J10, 13B35

38. CJM 2003 (vol 55 pp. 750)

Göbel, Rüdiger; Shelah, Saharon; Strüngmann, Lutz
 Almost-Free $E$-Rings of Cardinality $\aleph_1$ An $E$-ring is a unital ring $R$ such that every endomorphism of the underlying abelian group $R^+$ is multiplication by some ring element. The existence of almost-free $E$-rings of cardinality greater than $2^{\aleph_0}$ is undecidable in $\ZFC$. While they exist in G\"odel's universe, they do not exist in other models of set theory. For a regular cardinal $\aleph_1 \leq \lambda \leq 2^{\aleph_0}$ we construct $E$-rings of cardinality $\lambda$ in $\ZFC$ which have $\aleph_1$-free additive structure. For $\lambda=\aleph_1$ we therefore obtain the existence of almost-free $E$-rings of cardinality $\aleph_1$ in $\ZFC$. Keywords:$E$-rings, almost-free modulesCategories:20K20, 20K30, 13B10, 13B25

39. CJM 2002 (vol 54 pp. 1319)

Yekutieli, Amnon
 The Continuous Hochschild Cochain Complex of a Scheme Let $X$ be a separated finite type scheme over a noetherian base ring $\mathbb{K}$. There is a complex $\widehat{\mathcal{C}}^{\cdot} (X)$ of topological $\mathcal{O}_X$-modules, called the complete Hochschild chain complex of $X$. To any $\mathcal{O}_X$-module $\mathcal{M}$---not necessarily quasi-coherent---we assign the complex $\mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl( \widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr)$ of continuous Hochschild cochains with values in $\mathcal{M}$. Our first main result is that when $X$ is smooth over $\mathbb{K}$ there is a functorial isomorphism $$\mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl( \widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr) \cong \R \mathcal{H}om_{\mathcal{O}_{X^2}} (\mathcal{O}_X, \mathcal{M})$$ in the derived category $\mathsf{D} (\Mod \mathcal{O}_{X^2})$, where $X^2 := X \times_{\mathbb{K}} X$. The second main result is that if $X$ is smooth of relative dimension $n$ and $n!$ is invertible in $\mathbb{K}$, then the standard maps $\pi \colon \widehat{\mathcal{C}}^{-q} (X) \to \Omega^q_{X/ \mathbb{K}}$ induce a quasi-isomorphism $$\mathcal{H}om_{\mathcal{O}_X} \Bigl( \bigoplus_q \Omega^q_{X/ \mathbb{K}} [q], \mathcal{M} \Bigr) \to \mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl( \widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr).$$ When $\mathcal{M} = \mathcal{O}_X$ this is the quasi-isomorphism underlying the Kontsevich Formality Theorem. Combining the two results above we deduce a decomposition of the global Hochschild cohomology $$\Ext^i_{\mathcal{O}_{X^2}} (\mathcal{O}_X, \mathcal{M}) \cong \bigoplus_q \H^{i-q} \Bigl( X, \bigl( \bigwedge^q_{\mathcal{O}_X} \mathcal{T}_{X/\mathbb{K}} \bigr) \otimes_{\mathcal{O}_X} \mathcal{M} \Bigr),$$ where $\mathcal{T}_{X/\mathbb{K}}$ is the relative tangent sheaf. Keywords:Hochschild cohomology, schemes, derived categoriesCategories:16E40, 14F10, 18G10, 13H10

40. CJM 2002 (vol 54 pp. 1100)

Wood, Peter J.
 The Operator Biprojectivity of the Fourier Algebra In this paper, we investigate projectivity in the category of operator spaces. In particular, we show that the Fourier algebra of a locally compact group $G$ is operator biprojective if and only if $G$ is discrete. Keywords:locally compact group, Fourier algebra, operator space, projectiveCategories:13D03, 18G25, 43A95, 46L07, 22D99

41. CJM 2002 (vol 54 pp. 897)

Fortuny Ayuso, Pedro
 The Valuative Theory of Foliations This paper gives a characterization of valuations that follow the singular infinitely near points of plane vector fields, using the notion of L'H\^opital valuation, which generalizes a well known classical condition. With that tool, we give a valuative description of vector fields with infinite solutions, singularities with rational quotient of eigenvalues in its linear part, and polynomial vector fields with transcendental solutions, among other results. Categories:12J20, 13F30, 16W60, 37F75, 34M25

42. CJM 2001 (vol 53 pp. 923)

Geramita, Anthony V.; Harima, Tadahito; Shin, Yong Su
 Decompositions of the Hilbert Function of a Set of Points in $\P^n$ Let $\H$ be the Hilbert function of some set of distinct points in $\P^n$ and let $\alpha = \alpha (\H)$ be the least degree of a hypersurface of $\P^n$ containing these points. Write $\alpha = d_s + d_{s-1} + \cdots + d_1$ (where $d_i > 0$). We canonically decompose $\H$ into $s$ other Hilbert functions $\H \leftrightarrow (\H_s^\prime, \dots, \H_1^\prime)$ and show how to find sets of distinct points $\Y_s, \dots, \Y_1$, lying on reduced hypersurfaces of degrees $d_s, \dots, d_1$ (respectively) such that the Hilbert function of $\Y_i$ is $\H_i^\prime$ and the Hilbert function of $\Y = \bigcup_{i=1}^s \Y_i$ is $\H$. Some extremal properties of this canonical decomposition are also explored. Categories:13D40, 14M10

43. CJM 2000 (vol 52 pp. 123)

Harbourne, Brian
 An Algorithm for Fat Points on $\mathbf{P}^2 Let$F$be a divisor on the blow-up$X$of$\pr^2$at$r$general points$p_1, \dots, p_r$and let$L$be the total transform of a line on$\pr^2$. An approach is presented for reducing the computation of the dimension of the cokernel of the natural map$\mu_F \colon \Gamma \bigl( \CO_X(F) \bigr) \otimes \Gamma \bigl( \CO_X(L) \bigr) \to \Gamma \bigl( \CO_X(F) \otimes \CO_X(L) \bigr)$to the case that$F$is ample. As an application, a formula for the dimension of the cokernel of$\mu_F$is obtained when$r = 7$, completely solving the problem of determining the modules in minimal free resolutions of fat point subschemes\break$m_1 p_1 + \cdots + m_7 p_7 \subset \pr^2$. All results hold for an arbitrary algebraically closed ground field~$k$. Keywords:Generators, syzygies, resolution, fat points, maximal rank, plane, Weyl groupCategories:13P10, 14C99, 13D02, 13H15 44. CJM 1999 (vol 51 pp. 616) Panyushev, Dmitri I.  Parabolic Subgroups with Abelian Unipotent Radical as a Testing Site for Invariant Theory Let$L$be a simple algebraic group and$P$a parabolic subgroup with Abelian unipotent radical$P^u$. Many familiar varieties (determinantal varieties, their symmetric and skew-symmetric analogues) arise as closures of$P$-orbits in$P^u$. We give a unified invariant-theoretic treatment of various properties of these orbit closures. We also describe the closures of the conormal bundles of these orbits as the irreducible components of some commuting variety and show that the polynomial algebra$k[P^u]$is a free module over the algebra of covariants. Categories:14L30, 13A50 45. CJM 1999 (vol 51 pp. 3) Allday, C.; Puppe, V.  On a Conjecture of Goresky, Kottwitz and MacPherson We settle a conjecture of Goresky, Kottwitz and MacPherson related to Koszul duality, \ie, to the correspondence between differential graded modules over the exterior algebra and those over the symmetric algebra. Keywords:Koszul duality, Hirsch-Brown modelCategories:13D25, 18E30, 18G35, 55U15 46. CJM 1998 (vol 50 pp. 719) Göbel, Rüdiger; Shelah, Saharon  Indecomposable almost free modules---the local case Let$R$be a countable, principal ideal domain which is not a field and$A$be a countable$R$-algebra which is free as an$R$-module. Then we will construct an$\aleph_1$-free$R$-module$G$of rank$\aleph_1$with endomorphism algebra End$_RG = A$. Clearly the result does not hold for fields. Recall that an$R$-module is$\aleph_1$-free if all its countable submodules are free, a condition closely related to Pontryagin's theorem. This result has many consequences, depending on the algebra$A$in use. For instance, if we choose$A = R$, then clearly$G$is an indecomposable `almost free' module. The existence of such modules was unknown for rings with only finitely many primes like$R = \hbox{\Bbbvii Z}_{(p)}$, the integers localized at some prime$p$. The result complements a classical realization theorem of Corner's showing that any such algebra is an endomorphism algebra of some torsion-free, reduced$R$-module$G$of countable rank. Its proof is based on new combinatorial-algebraic techniques related with what we call {\it rigid tree-elements\/} coming from a module generated over a forest of trees. Keywords:indecomposable modules of local rings,$\aleph_1$-free modules of rank$\aleph_1\$, realizing rings as endomorphism ringsCategories:20K20, 20K26, 20K30, 13C10

47. CJM 1997 (vol 49 pp. 499)

Fitzgerald, Robert W.
 Gorenstein Witt rings II The abstract Witt rings which are Gorenstein have been classified when the dimension is one and the classification problem for those of dimension zero has been reduced to the case of socle degree three. Here we classifiy the Gorenstein Witt rings of fields with dimension zero and socle degree three. They are of elementary type. Categories:11E81, 13H10
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