Expand all Collapse all | Results 26 - 50 of 52 |
26. CMB 2010 (vol 53 pp. 667)
On the Endomorphism Rings of Local Cohomology Modules Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ a proper ideal
of $R$. We show that if $n:=\operatorname{grade}_R\mathfrak{a}$, then
$\operatorname{End}_R(H^n_\mathfrak{a}(R))\cong \operatorname{Ext}_R^n(H^n_\mathfrak{a}(R),R)$. We also
prove that, for a nonnegative integer $n$ such that
$H^i_\mathfrak{a}(R)=0$ for every $i\neq n$, if $\operatorname{Ext}_R^i(R_z,R)=0$ for
all $i >0$ and $z \in \mathfrak{a}$, then
$\operatorname{End}_R(H^n_\mathfrak{a}(R))$ is a homomorphic
image of $R$, where $R_z$ is the ring of fractions of $R$ with
respect to a multiplicatively closed subset $\{z^j \mid j \geqslant
0 \}$ of $R$. Moreover, if $\operatorname{Hom}_R(R_z,R)=0$ for all $z
\in \mathfrak{a}$,
then $\mu_{H^n_\mathfrak{a}(R)}$ is an isomorphism, where $\mu_{H^n_\mathfrak{a}(R)}$
is the canonical ring homomorphism $R \rightarrow \operatorname{End}_R(H^n_\mathfrak{a}(R))$.
Keywords:local cohomology module, endomorphism ring, Matlis dual functor, filter regular sequence Categories:13D45, 13D07, 13D25 |
27. CMB 2010 (vol 53 pp. 577)
A Unified Approach to Local Cohomology Modules using Serre Classes This paper
discusses the connection between the local cohomology modules and
the Serre classes of $R$-modules. This connection has provided a common
language for expressing some results regarding the local cohomology
$R$-modules that have appeared in different papers.
Keywords:associated prime ideals, local cohomology modules, Serre class Category:13D45 |
28. CMB 2010 (vol 53 pp. 602)
Notes on Diagonal Coinvariants of the Dihedral Group
The bigraded Hilbert function and the minimal free resolutions for the
diagonal coinvariants of the dihedral groups are exhibited, as well as for
all their bigraded invariant Gorenstein quotients.
Categories:13D02, 20C33, 20F55 |
29. CMB 2010 (vol 53 pp. 404)
Invariant Theory of Abelian Transvection Groups Let $G$ be a finite group acting linearly on the vector space $V$ over a field of arbitrary characteristic. The action is called \emph{coregular} if the invariant ring is generated by algebraically independent homogeneous invariants, and the \emph{direct summand property} holds if there is a surjective $k[V]^G$-linear map $\pi\colon k[V]\to k[V]^G$. The following Chevalley--Shephard--Todd type theorem is proved. Suppose $G$ is abelian. Then the action is coregular if and only if $G$ is generated by pseudo-reflections and the direct summand property holds.
Category:13A50 |
30. CMB 2009 (vol 53 pp. 77)
Constructing (Almost) Rigid Rings and a UFD Having Infinitely Generated Derksen and Makar-Limanov Invariants |
Constructing (Almost) Rigid Rings and a UFD Having Infinitely Generated Derksen and Makar-Limanov Invariants An example is given of a UFD which has an infinitely generated Derksen invariant. The ring is "almost rigid" meaning that the Derksen invariant is equal to the Makar-Limanov invariant. Techniques to show that a ring is (almost) rigid are discussed, among which is a generalization of Mason's abc-theorem.
Categories:14R20, 13A50, 13N15 |
31. CMB 2009 (vol 53 pp. 247)
Root Extensions and Factorization in Affine Domains An integral domain R is IDPF (Irreducible Divisors of Powers Finite) if, for every non-zero element a in R, the ascending chain of non-associate irreducible divisors in R of $a^{n}$ stabilizes on a finite set as n ranges over the positive integers, while R is atomic if every non-zero element that is not a unit is a product of a finite number of irreducible elements (atoms). A ring extension S of R is a \emph{root extension} or \emph{radical extension} if for each s in S, there exists a natural number $n(s)$ with $s^{n(s)}$ in R. In this paper it is shown that the ascent and descent of the IDPF property and atomicity for the pair of integral domains $(R,S)$ is governed by the relative sizes of the unit groups $\operatorname{U}(R)$ and $\operatorname{U}(S)$ and whether S is a root extension of R. The following results are deduced from these considerations. An atomic IDPF domain containing a field of characteristic zero is completely integrally closed. An affine domain over a field of characteristic zero is IDPF if and only if it is completely integrally closed. Let R be a Noetherian domain with integral closure S. Suppose the conductor of S into R is non-zero. Then R is IDPF if and only if S is a root extension of R and $\operatorname{U}(S)/\operatorname{U}(R)$ is finite.
Categories:13F15, 14A25 |
32. CMB 2009 (vol 52 pp. 535)
A Note on Locally Nilpotent Derivations\\ and Variables of $k[X,Y,Z]$ We strengthen certain results
concerning actions of $(\Comp,+)$ on $\Comp^{3}$
and embeddings of $\Comp^{2}$ in $\Comp^{3}$,
and show that these results are in fact valid
over any field of characteristic zero.
Keywords:locally nilpotent derivations, group actions, polynomial automorphisms, variable, affine space Categories:14R10, 14R20, 14R25, 13N15 |
33. CMB 2009 (vol 52 pp. 72)
A SAGBI Basis For $\mathbb F[V_2\oplus V_2\oplus V_3]^{C_p}$ Let $C_p$ denote the cyclic group of order $p$, where $p \geq 3$ is
prime. We denote by $V_n$ the indecomposable $n$ dimensional
representation of $C_p$ over a field $\mathbb F$ of characteristic
$p$. We compute a set of generators, in fact a SAGBI basis, for
the ring of invariants $\mathbb F[V_2 \oplus V_2 \oplus V_3]^{C_p}$.
Category:13A50 |
34. CMB 2008 (vol 51 pp. 406)
Condensed and Strongly Condensed Domains This paper deals with the concepts of condensed and strongly condensed
domains. By definition, an integral domain $R$ is condensed
(resp. strongly condensed) if each pair of ideals $I$ and $J$ of $R$,
$IJ=\{ab/a \in I, b \in J\}$ (resp. $IJ=aJ$ for some $a \in I$ or
$IJ=Ib$ for some $b \in J$). More precisely, we investigate the
ideal theory of condensed and strongly condensed domains in
Noetherian-like settings, especially Mori and strong Mori domains and
the transfer of these concepts to pullbacks.
Categories:13G05, 13A15, 13F05, 13E05 |
35. CMB 2008 (vol 51 pp. 439)
On the Maximal Spectrum of Semiprimitive Multiplication Modules An $R$-module $M$ is called a multiplication module if for each
submodule $N$ of $M$, $N=IM$ for some ideal $I$ of $R$. As
defined for a commutative ring $R$, an $R$-module $M$ is said to be
semiprimitive if the intersection of maximal submodules of $M$ is
zero. The maximal spectra of a semiprimitive multiplication
module $M$ are studied. The isolated points of $\Max(M)$ are
characterized algebraically. The relationships among the maximal
spectra of $M$, $\Soc(M)$ and $\Ass(M)$ are studied. It is shown
that $\Soc(M)$ is exactly the set of all elements of $M$ which
belongs to every maximal submodule of $M$ except for a finite
number. If $\Max(M)$ is infinite, $\Max(M)$ is a one-point
compactification of a discrete space if and only if $M$ is Gelfand and for
some maximal submodule $K$, $\Soc(M)$ is the intersection of all
prime submodules of $M$ contained in $K$. When $M$ is a
semiprimitive Gelfand module, we prove that every intersection
of essential submodules of $M$ is an essential submodule if and only if
$\Max(M)$ is an almost discrete space. The set of uniform
submodules of $M$ and the set of minimal submodules of $M$
coincide. $\Ann(\Soc(M))M$ is a summand submodule of $M$ if and only if
$\Max(M)$ is the union of two disjoint open subspaces $A$ and
$N$, where $A$ is almost discrete and $N$ is dense in itself. In
particular, $\Ann(\Soc(M))=\Ann(M)$ if and only if $\Max(M)$ is almost
discrete.
Keywords:multiplication module, semiprimitive module, Gelfand module, Zariski topolog Category:13C13 |
36. CMB 2007 (vol 50 pp. 598)
Artinian Local Cohomology Modules Let $R$ be a commutative Noetherian ring, $\fa$ an ideal
of $R$ and $M$ a finitely generated $R$-module. Let $t$ be a
non-negative integer. It is known that if the local cohomology
module $\H^i_\fa(M)$ is finitely generated for all $i Keywords:local cohomology module, Artinian module, reflexive module Categories:13D45, 13E10, 13C05 |
37. CMB 2005 (vol 48 pp. 275)
Krull Dimension of Injective Modules Over Commutative Noetherian Rings Let $R$ be a commutative Noetherian
integral domain with field of fractions $Q$. Generalizing a
forty-year-old theorem of E. Matlis, we prove that the $R$-module
$Q/R$ (or $Q$) has Krull dimension if and only if $R$ is semilocal
and one-dimensional. Moreover, if $X$ is an injective module over
a commutative Noetherian ring such that $X$ has Krull dimension,
then the Krull dimension of $X$ is at most $1$.
Categories:13E05, 16D50, 16P60 |
38. CMB 2003 (vol 46 pp. 304)
Localization of the Hasse-Schmidt Algebra The behaviour of the Hasse-Schmidt algebra of higher derivations under
localization is studied using Andr\'e cohomology. Elementary
techniques are used to describe the Hasse-Schmidt derivations on
certain monomial rings in the nonmodular case. The localization
conjecture is then verified for all monomial rings.
Categories:13D03, 13N10 |
39. CMB 2003 (vol 46 pp. 3)
Condensed Domains An integral domain $D$ with identity is condensed (resp., strongly
condensed) if for each pair of ideals $I$, $J$ of $D$, $IJ=\{ij; i\in I,
j\in J\}$ (resp., $IJ=iJ$ for some $i\in I$ or $IJ =Ij$ for some
$j\in J$). We show that for a Noetherian domain $D$, $D$ is condensed
if and only if $\Pic(D)=0$ and $D$ is locally condensed, while a local
domain is strongly condensed if and only if it has the two-generator
property. An integrally closed domain $D$ is strongly condensed if and
only if $D$ is a B\'{e}zout generalized Dedekind domain with at most one
maximal ideal of height greater than one. We give a number of
equivalencies for a local domain with finite integral closure to be
strongly condensed. Finally, we show that for a field extension
$k\subseteq K$, the domain $D=k+XK[[X]]$ is condensed if and only if
$[K:k]\leq 2$ or $[K:k]=3$ and each degree-two polynomial in $k[X]$
splits over $k$, while $D$ is strongly condensed if and only if $[K:k]
\leq 2$.
Categories:13A15, 13B22 |
40. CMB 2002 (vol 45 pp. 272)
The Transfer in the Invariant Theory of Modular Permutation Representations II In this note we show that the image of the transfer for permutation
representations of finite groups is generated by the transfers of
special monomials. This leads to a description of the image of the
transfer of the alternating groups. We also determine the height of
these ideals.
Keywords:polynomial invariants of finite groups, permutation representation, transfer Category:13A50 |
41. CMB 2002 (vol 45 pp. 119)
The Grade Conjecture and the $S_{2}$ Condition Sufficient conditions are given in order to prove the lowest unknown case
of the grade conjecture. The proof combines vanishing results of local
cohomology and the $S_{2}$ condition.
Categories:13D22, 13D45, 13D25, 13C15 |
42. CMB 2000 (vol 43 pp. 362)
Examples of Half-Factorial Domains In this paper, we determine some sufficient conditions for an $A +
XB[X]$ domain to be an HFD. As a consequence we give new examples
of HFDs of the type $A + XB[X]$.
Keywords:atomic domain, HFD Categories:13A05, 13B30, 13F15, 13G05 |
43. CMB 2000 (vol 43 pp. 312)
On the Prime Ideals in a Commutative Ring If $n$ and $m$ are positive integers, necessary and sufficient
conditions are given for the existence of a finite commutative ring $R$
with exactly $n$ elements and exactly $m$ prime ideals. Next,
assuming the Axiom of Choice, it is proved that if $R$ is a
commutative ring and $T$ is a commutative $R$-algebra which is
generated by a set $I$, then each chain of prime ideals of $T$ lying
over the same prime ideal of $R$ has at most $2^{|I|}$ elements. A
polynomial ring example shows that the preceding result is
best-possible.
Categories:13C15, 13B25, 04A10, 14A05, 13M05 |
44. CMB 2000 (vol 43 pp. 126)
Sur l'annulation de certains modules de cohomologie d'AndrÃ©-Quillen Soient $A$ un anneau noeth\'erien, $B$ un anneau r\'egulier
essentiellement de type fini sur $A$. Si la cohomologie
d'Andr\'e-Quillen $H^q (A,B,B) = 0$ pour tout $q \geq 2$ alors $A$
est un anneau r\'egulier.
Category:13D03 |
45. CMB 2000 (vol 43 pp. 100)
A Gorenstein Ring with Larger Dilworth Number than Sperner Number A counterexample is given to a conjecture of Ikeda by finding a class of
Gorenstein rings of embedding dimension $3$ with larger Dilworth number than
Sperner number. The Dilworth number of $A[Z/pZ\oplus Z/pZ]$ is computed
when $A$ is an unramified principal Artin local ring.
Categories:13E15, 16S34 |
46. CMB 1999 (vol 42 pp. 231)
Generating Ideals in Rings of Integer-Valued Polynomials Let $R$ be a one-dimensional locally analytically irreducible
Noetherian domain with finite residue fields. In this note it is
shown that if $I$ is a finitely generated ideal of the ring
$\Int(R)$ of integer-valued polynomials such that for each $x \in
R$ the ideal $I(x) =\{f(x) \mid f \in I\}$ is strongly
$n$-generated, $n \geq 2$, then $I$ is $n$-generated, and some
variations of this result.
Categories:13B25, 13F20, 13F05 |
47. CMB 1999 (vol 42 pp. 155)
Non--Cohen-Macaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants This paper contains two essentially independent results in the
invariant theory of finite groups. First we prove that, for any
faithful representation of a non-trivial $p$-group over a field of
characteristic $p$, the ring of vector invariants of $m$ copies of
that representation is not \comac\ for $m\geq 3$. In the second
section of the paper we use Poincar\'e series methods to produce upper
bounds for the degrees of the generators for the ring of invariants as
long as that ring is Gorenstein. We prove that, for a finite
non-trivial group $G$ and a faithful representation of dimension $n$
with $n>1$, if the ring of invariants is Gorenstein then the ring is
generated in degrees less than or equal to $n(|G|-1)$. If the ring of
invariants is a hypersurface, the upper bound can be improved to $|G|$.
Category:13A50 |
48. CMB 1999 (vol 42 pp. 125)
Modular Vector Invariants of Cyclic Permutation Representations Vector invariants of finite groups (see the introduction for an
explanation of the terminology) have often been used to illustrate the
difficulties of invariant theory in the modular case: see,
\eg., \cite{Ber}, \cite{norway}, \cite{fossum}, \cite{MmeB},
\cite{poly} and \cite{survey}. It is therefore all the more
surprising that the {\it unpleasant} properties of these invariants
may be derived from two unexpected, and remarkable, {\it nice}
properties: namely for vector permutation invariants of the cyclic
group $\mathbb{Z}/p$ of prime order in characteristic $p$ the
image of the transfer homomorphism $\Tr^{\mathbb{Z}/p} \colon
\mathbb{F}[V] \lra \mathbb{F}[V]^{\mathbb{Z}/p}$ is a prime ideal,
and the quotient algebra $\mathbb{F}[V]^{\mathbb{Z}/p}/ \Im
(\Tr^{\mathbb{Z}/p})$ is a polynomial algebra on the top Chern
classes of the action.
Keywords:polynomial invariants of finite groups Category:13A50 |
49. CMB 1998 (vol 41 pp. 359)
Embedding the Hopf automorphism group into the Brauer group Let $H$ be a faithfully projective Hopf algebra over a commutative
ring $k$. In \cite{CVZ1, CVZ2} we defined the Brauer group
$\BQ(k,H)$ of $H$ and an homomorphism $\pi$ from Hopf automorphism
group $\Aut_{\Hopf}(H)$ to $\BQ(k,H)$. In this paper, we show that
the morphism $\pi$ can be embedded into an exact sequence.
Categories:16W30, 13A20 |
50. CMB 1998 (vol 41 pp. 3)
Root closure in Integral Domains, III {If A is a subring of a commutative ring B and if n
is a positive integer, a number of sufficient conditions are given for
``A[[X]]is n-root closed in B[[X]]'' to be equivalent to ``A is n-root
closed in B.'' In addition, it is shown that if S is a multiplicative
submonoid of the positive integers ${\bbd P}$ which is generated by
primes, then there exists a one-dimensional quasilocal integral domain
A (resp., a von Neumann regular ring A) such that $S = \{ n \in {\bbd P}\mid
A$ is $n$-root closed$\}$ (resp., $S = \{n \in {\bbd P}\mid A[[X]]$
is $n$-rootclosed$\}$).
Categories:13G05, 13F25, 13C15, 13F45, 13B99, 12D99 |