Expand all Collapse all | Results 26 - 39 of 39 |
26. CJM 2002 (vol 54 pp. 1319)
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 categories Categories:16E40, 14F10, 18G10, 13H10 |
27. CJM 2002 (vol 54 pp. 897)
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 |
28. CJM 2002 (vol 54 pp. 595)
Lie Algebras of Pro-Affine Algebraic Groups We extend the basic theory of Lie algebras of affine algebraic groups
to the case of pro-affine algebraic groups over an algebraically
closed field $K$ of characteristic 0. However, some modifications
are needed in some extensions. So we introduce the pro-discrete
topology on the Lie algebra $\mathcal{L}(G)$ of the pro-affine
algebraic group $G$ over $K$, which is discrete in the
finite-dimensional case and linearly compact in general. As an
example, if $L$ is any sub Lie algebra of $\mathcal{L}(G)$, we show
that the closure of $[L,L]$ in $\mathcal{L}(G)$ is algebraic in
$\mathcal{L}(G)$.
We also discuss the Hopf algebra of representative functions $H(L)$ of
a residually finite dimensional Lie algebra $L$. As an example, we
show that if $L$ is a sub Lie algebra of $\mathcal{L}(G)$ and $G$
is connected, then the canonical Hopf algebra morphism from $K[G]$
into $H(L)$ is injective if and only if $L$ is algebraically dense
in $\mathcal{L}(G)$.
Categories:14L, 16W, 17B45 |
29. CJM 1999 (vol 51 pp. 881)
The Representation Ring and the Centre of a Hopf Algebra When $H$ is a finite dimensional, semisimple, almost cocommutative
Hopf algebra, we examine a table of characters which extends the
notion of the character table for a finite group. We obtain a
formula for the structure constants of the representation ring in
terms of values in the character table, and give the example of the
quantum double of a finite group. We give a basis of the centre of
$H$ which generalizes the conjugacy class sums of a finite group,
and express the class equation of $H$ in terms of this basis. We
show that the representation ring and the centre of $H$ are dual
character algebras (or signed hypergroups).
Categories:16W30, 20N20 |
30. CJM 1999 (vol 51 pp. 488)
Homological Aspects of Semigroup Gradings on Rings and Algebras This article studies algebras $R$ over a simple artinian ring $A$,
presented by a quiver and relations and graded by a semigroup $\Sigma$.
Suitable semigroups often arise from a presentation of $R$.
Throughout, the algebras need not be finite dimensional. The graded
$K_0$, along with the $\Sigma$-graded Cartan endomorphisms and Cartan
matrices, is examined. It is used to study homological properties.
A test is found for finiteness of the global dimension of a
monomial algebra in terms of the invertibility of the Hilbert
$\Sigma$-series in the associated path incidence ring.
The rationality of the $\Sigma$-Euler characteristic, the Hilbert
$\Sigma$-series and the Poincar\'e-Betti $\Sigma$-series is studied
when $\Sigma$ is torsion-free commutative and $A$ is a division ring.
These results are then applied to the classical series. Finally, we
find new finite dimensional algebras for which the strong no loops
conjecture holds.
Categories:16W50, 16E20, 16G20 |
31. CJM 1999 (vol 51 pp. 294)
A Homotopy of Quiver Morphisms with Applications to Representations It is shown that a morphism of quivers having a certain path
lifting property has a decomposition that mimics the decomposition
of maps of topological spaces into homotopy equivalences composed
with fibrations. Such a decomposition enables one to describe the
right adjoint of the restriction of the representation functor
along a morphism of quivers having this path lifting property.
These right adjoint functors are used to construct injective
representations of quivers. As an application, the injective
representations of the cyclic quivers are classified when the base
ring is left noetherian. In particular, the indecomposable
injective representations are described in terms of the injective
indecomposable $R$-modules and the injective indecomposable
$R[x,x^{-1}]$-modules.
Categories:18A40, 16599 |
32. CJM 1999 (vol 51 pp. 69)
On a Theorem of Hermite and Joubert A classical theorem of Hermite and Joubert asserts that any field
extension of degree $n=5$ or $6$ is generated by an element whose
minimal polynomial is of the form $\lambda^n + c_1 \lambda^{n-1} +
\cdots + c_{n-1} \lambda + c_n$ with $c_1=c_3=0$. We show that this
theorem fails for $n=3^m$ or $3^m + 3^l$ (and more generally, for $n =
p^m$ or $p^m + p^l$, if 3 is replaced by another prime $p$), where $m
> l \geq 0$. We also prove a similar result for division algebras and
use it to study the structure of the universal division algebra $\UD
(n)$.
We also prove a similar result for division algebras and use it to
study the structure of the universal division algebra $\UD(n)$.
Categories:12E05, 16K20 |
33. CJM 1998 (vol 50 pp. 356)
Some norms on universal enveloping algebras The universal enveloping algebra, $U(\frak g)$, of a Lie algebra $\frak g$
supports some norms and seminorms that have arisen naturally in the
context of heat kernel analysis on Lie groups. These norms and seminorms
are investigated here from an algebraic viewpoint. It is shown
that the norms corresponding to heat kernels on the associated Lie
groups decompose as product norms under the natural isomorphism
$U(\frak g_1 \oplus \frak g_2) \cong U(\frak g_1) \otimes U(\frak
g_2)$. The seminorms corresponding to Green's functions are
examined at a purely Lie algebra level for $\rmsl(2,\Bbb C)$. It
is also shown that the algebraic dual space $U'$ is spanned by its
finite rank elements if and only if $\frak g$ is nilpotent.
Categories:17B35, 16S30, 22E30 |
34. CJM 1998 (vol 50 pp. 401)
The hypercentre and the $n$-centre of the unit group of an integral group ring In this paper, we first show that the central height of the unit group of
the integral group ring of a periodic group is at most $2$. We then
give a complete characterization of the $n$-centre of that unit group.
The $n$-centre of the unit group is either the centre or the second
centre (for $n \geq 2$).
Categories:16U60, 20C05 |
35. CJM 1998 (vol 50 pp. 312)
Units in group rings of free products of prime cyclic groups Let $G$ be a free product of cyclic groups of prime order. The
structure of the unit group ${\cal U}(\Q G)$ of the rational group
ring $\Q G$ is given in terms of free products and amalgamated free
products of groups. As an application, all finite subgroups of
${\cal U}(\Q G)$, up to conjugacy, are described and the
Zassenhaus Conjecture for finite subgroups in $\Z G$ is proved. A
strong version of the Tits Alternative for ${\cal U}(\Q G)$ is
obtained as a corollary of the structural result.
Keywords:Free Products, Units in group rings, Zassenhaus Conjecture Categories:20C07, 16S34, 16U60, 20E06 |
36. CJM 1998 (vol 50 pp. 3)
Subgroups of the adjoint group of a radical ring It is shown that the adjoint group $R^\circ$ of an arbitrary
radical ring $R$ has a series with abelian factors and that its finite
subgroups are nilpotent. Moreover, some criteria for subgroups of
$R^\circ$ to be locally nilpotent are given.
Categories:16N20, 20F19 |
37. CJM 1997 (vol 49 pp. 1265)
Hecke algebras and class-group invariant Let $G$ be a finite group. To a set of subgroups of order two we associate
a $\mod 2$ Hecke algebra and construct a homomorphism, $\psi$, from its
units to the class-group of ${\bf Z}[G]$. We show that this homomorphism
takes values in the subgroup, $D({\bf Z}[G])$. Alternative constructions of
Chinburg invariants arising from the Galois module structure of
higher-dimensional algebraic $K$-groups of rings of algebraic integers
often differ by elements in the image of $\psi$. As an application we show
that two such constructions coincide.
Categories:16S34, 19A99, 11R65 |
38. CJM 1997 (vol 49 pp. 788)
Trace functions in the ring of fractions of polycyclic group rings, II We prove the existence of trace functions in the rings of fractions of
polycyclic-by-finite group rings or their homomorphic images. In
particular a trace function exists in the ring of fractions of $KH$,
where $H$ is a polycyclic-by-finite group and $\char K > N$, where
$N$ is a constant depending on $H$.
Categories:20C07, 16A08, 16A39 |
39. CJM 1997 (vol 49 pp. 772)
Finite dimensional representations of $U_t\bigl(\rmsl (2)\bigr)$ at roots of unity All finite dimensional indecomposable representations of
$U_t (\rmsl (2))$ at roots of $1$ are determined.
Categories:16G10, 16G70, 17B37 |