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26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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$. Keywords:Generators, syzygies, resolution, fat points, maximal rank, plane, Weyl groupCategories:13P10, 14C99, 13D02, 13H15 35. CJM 1999 (vol 51 pp. 616) Panyushev, Dmitri I.  Parabolic Subgroups with Abelian Unipotent Radical as a Testing Site for Invariant Theory Let$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]$is a free module over the algebra of covariants. Categories:14L30, 13A50 36. CJM 1999 (vol 51 pp. 3) Allday, C.; Puppe, V.  On a Conjecture of Goresky, Kottwitz and MacPherson We settle a conjecture of Goresky, Kottwitz and MacPherson related to Koszul duality, \ie, to the correspondence between differential graded modules over the exterior algebra and those over the symmetric algebra. Keywords:Koszul duality, Hirsch-Brown modelCategories:13D25, 18E30, 18G35, 55U15 37. CJM 1998 (vol 50 pp. 719) Göbel, Rüdiger; Shelah, Saharon  Indecomposable almost free modules---the local case Let$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

38. 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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