Research Article
Ring homomorphisms and finite Gorenstein dimension
- LL Avramov, H-B Foxby
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- Published online by Cambridge University Press:
- 01 September 1997, pp. 241-270
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The local structure of homomorphisms of commutative noetherian rings is investigated from the point of view of dualizing complexes. A concept of finite Gorenstein dimension, which substantially weakens the notion of finite flat dimension, is introduced for homomorphisms. It is shown to impose structural constraints, due to a remarkable equivalence of subcategories of the derived category of all modules.
An essential part of this study is the development of relative notions of dualizing complexes and Bass numbers. It is proved that the Bass numbers of local homomorphisms are rigid, extending a known result for local rings. Quasi-Gorenstein homomorphisms are introduced as local homomorphisms that base-change a dualizing complex for the source ring into one for the target. They are shown to have the stability properties of the Gorenstein homomorphism that they generalize.
1991 Mathematics Subject Classification: primary 13H10, 13D23, 14E40; secondary 13C15.
S-Unit equations, binary forms and curves of genus 2
- NP Smart
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- 01 September 1997, pp. 271-307
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In this paper a practical algorithm is given to find all binary forms with rational coefficients of given degree with discriminant divisible by a given finite set of rational primes, up to an obvious equivalence relation.
This is done by adapting the finiteness result of Evertse and Gy\H{o}ry. A technical assumption that all fields used have class number $1$ is made to aid the exposition.
All binary forms of degree less than or equal to $6$ with $2$-power discriminant are then calculated. This is then used to complete the table of curves of genus $2$ which had previously been computed by Merriman and the author. Some ranks of the associated Jacobian varieties are also computed.
1991 Mathematics Subject Classification: primary 11E76, 11G30; secondary 11Y40, 11D61.
Finiteness properties of certain metabelian arithmetic groups in the function field case
- K-U Bux
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- Published online by Cambridge University Press:
- 01 September 1997, pp. 308-322
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Consider the group scheme
$$ R \mapsto G(R) := R \rtimes R^* = \left( \begin{array}{cc} 1 & R \\ 0 & R^* \end{array} \right) = \left( \begin{array}{cc} 1 & *\\ 0 & * \end{array} \right) \cap \mathrm{GL}_2(R) $$
where $R$ is an arbitrary commutative ring with $1 \neq 0$ and a unit $x \in R^*$ acts on $R$ by multiplication.
We will study the finiteness properties of subgroups of $G(\Oka_S)$ where $\Oka_S$ is an $S$-arithmetic subring of a global function field. The subgroups we are interested in are of the form $\Oka_S \rtimes Q$ where $Q$ is a subgroup of $\Oka_S^*$. The finiteness properties of these metabelian groups can be expressed in terms of the $\Sigma$-invariant due to R. Bieri and R. Strebel.
{\sc Theorem A.} {\it Let $S$ be a finite set of places of a global function field (regarded as normalized discrete valuations) and $\Oka_S$ the corresponding $S$-arithmetic ring. Let $Q$ be a subgroup of $\Oka_S^*$. Then $Q$ is finitely generated and for all integers $n \geq 1$ the following are equivalent:
\begin{enumerate} \item[\rm (1)] $\Oka_S \rtimes Q$ is of type FP$_n$;
\item[\rm (2)] $\Oka_S$ is $n$-tame as a $\ZZZ Q$-module;
\item[\rm (3)] each $p \in S$ restricts to a non-trivial homomorphism $p|_{Q} : Q \rightarrow \RRR$ and the set $\{ [p|_Q] \mid p \in S \} \subseteq \SSS(Q)$ is $n$-tame. \end{enumerate}
If these conditions hold for at least one $n \geq 1$ then the identity
$$ \Sigma_{\Oka_S}^c(Q,\ZZZ) = \{ [p|_Q] \mid p \in S \} $$
holds.} {\sc Theorem B.} {\it Let $r$ denote the rank of $Q$. Then the following hold: \begin{enumerate} \item[\rm (1)] the group $\Oka_S \rtimes Q$ is not of type FP$_{r+1}$;
\item[\rm (2)] if $Q$ has maximum rank $r=|S|-1$ then the group $\Oka_S \rtimes Q$ is of type FP$_r$.
\end{enumerate} In particular, $G(\Oka_S) = \Oka_S \rtimes \Oka_S^*$ is of type FP$_{|S|-1}$ but not of type FP$_{|S|}$.}
1991 Mathematics Subject Classification: 20E08, 20F16, 20G30, 52A20.
Analytical functional models and local spectral theory
- E Albrecht, J Eschmeier
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- 01 September 1997, pp. 323-348
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In 1959 E. Bishop used a Banach-space version of the analytic duality principle established by e Silva, K\"{o}the, Grothendieck and others to study connections between spectral decomposition properties of a Banach-space operator and its adjoint. According to Bishop a continuous linear operator $T \in L(X)$ on a Banach space $X$ satisfies property $(\beta)$ if the multiplication operator ${\cal O}(U,X) \rightarrow {\cal O}(U,X),$ $f \mapsto (z-T)f,$ is injective with closed range for each open set $U$ in the complex plane. In the present article the analytic duality principle in its original locally convex form is used to develop a complete duality theory for property $(\beta)$. At the same time it is shown that, up to similarity, property $(\beta)$ characterizes those operators occurring as restrictions of operators decomposable in the sense of C. Foias, and that its dual property, formulated as a spectral decomposition property for the spectral subspaces of the given operator, characterizes those operators occurring as quotients of decomposable operators. It is proved that, unlike the situation for commuting subnormal operators, each finite commuting system of operators with property $(\beta)$ can be extended to a finite commuting system of decomposable operators. Meanwhile the results of this paper have been used to prove the existence of invariant subspaces for subdecomposable operators with sufficiently rich spectrum.
1991 Mathematics Subject Classification: 47A11, 47B40.
Weak covering properties of weak topologies
- A Dow, H Junnila, J Pelant
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- 01 September 1997, pp. 349-368
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We consider covering properties of weak topologies of Banach spaces, especially of weak or point-wise topologies of function spaces $C(K)$, for compact spaces $K$. We answer questions posed by A. V. Arkhangel'skii, S. P. Gul'ko, and R. W. Hansell. Our main results are the following. A Banach space of density at most $\omega_1$ is hereditarily metaLindel\" of in its weak topology. If the weight of a compact space $K$ is at most $\omega_1$, then the spaces $C_w(K)$ and $C_p(K)$ are hereditarily metaLindel\" of. Let $\bar T$ be the one-point compactification of a tree $T$. Then the space $C_p(\bar T)$ is hereditarily $\sigma$-metacompact. If $T$ is an infinitely branching full tree of uncountable height and of cardinality bigger than ${\bf c}$, then the weak topology of the unit sphere of $C(\bar T)$ is not $\sigma$-fragmented by any metric. The space $C_p(\beta\omega_1)$ is neither metaLindel\" of nor $\sigma$-relatively metacompact. The space $C_p(\beta\omega_2)$ is not $\sigma$-relatively metaLindel\" of. Under the set-theoretic axiom $\diamondsuit$, there exists a scattered compact space $K_1$ such that the space $C_p(K_1)$ is not $\sigma$-relatively metacompact, and under a related axiom \boxd, there exists a scattered compact space $K_2$ such that the space $C_p(K_2)$ is not $\sigma$-relatively metaLindel\" of.
1991 Mathematics Subject Classification: 54C35, 46B20, 54E20, 54D30.
On finite and infinite Ck-[Ascr ]-determinacy
- H Brodersen
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- 01 September 1997, pp. 369-435
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Let $f:({\mathbb R}^n,0)\rightarrow(\Bbb R^p,0)$ be a $C^\infty$ map-germ. We define $f$ to be finitely, or $\infty$-, $\mathcal A$-determined, if there exists an integer $m$ such that all germs $g$ with $j^mg(0)=j^mf(0)$, or if all germs $g$ with the same infinite Taylor series as $f$, respectively, are $\mathcal A$-equivalent to $f$. For any integer $k$, $0\le k<\infty$, we can consider $\mathcal A$\rq s $C^k$ counterpart (consisting of $C^k$ diffeomorphisms) $\mathcal A^{(k)}$, and we can define the notion of finite, or $\infty$-,$\mathcal A^{(k)}$-determinacy in a similar manner. Consider the following conditions for a $C^{\infty}$ germ $f$: (a$_k$) $f$ is $\infty$-$\mathcal A^{(k)}$-determined, (b$_k$) $f$ is finitely $\mathcal A^{(k)}$-determined, (t) $m_n^{\infty}\theta(f)\subset tf(m_n\theta(n))+\omega f(m_p\theta(p))$, (g) there exists a representative $ f:U\rightarrow\Bbb R^p$ defined on some neighbourhood $U$ of 0 in $\Bbb R^n$ such that the multigerm of $f$ is stable at every finite set $S\subset U-\lbrace0\rbrace$, and (g$^\prime$) every $f^\prime $ with $j^\infty f^\prime(0)=j^\infty f(0)$ satisfies condition (g). We also define a technical condition which will imply condition $(\mbox{g})$ above. This condition is a collection of $p+1$ \Lojasiewicz inequalities which express that the multigerm of $f$ is stable at any finite set of points outside 0 and only becomes unstable at a finite rate when we approach 0. We will denote this condition by (e). With this notation we prove the following. For any $ C^\infty$ map germ $f:(\Bbb R^n,0)\rightarrow(\Bbb R^p,0)$ the conditions (e), (t), (g$'$) and ($\mbox{a}_{\infty}$) are equivalent conditions. Moreover, each of these conditions is equivalent to any of ($\mbox{a}_k$) $(p+1\le k<\infty)$, ($\mbox{b}_k$) $(p+1\le k<\infty)$.
1991 Mathematics Subject Classification: 58C27.
ℤ4-Kerdock codes, orthogonal spreads, and extremal euclidean line-sets
- AR Calderbank, PJ Cameron, WM Kantor, JJ Seidel
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- Published online by Cambridge University Press:
- 01 September 1997, pp. 436-480
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When $m$ is odd, spreads in an orthogonal vector space of type $\Omega^+ (2m+2,2)$ are related to binary Kerdock codes and extremal line-sets in $\RR^{2^{m+1}}$ with prescribed angles. Spreads in a $2m$-dimensional binary symplectic vector space are related to Kerdock codes over $\ZZ_4$ and extremal line-sets in $\CC^{2^m}$ with prescribed angles. These connections involve binary, real and complex geometry associated with extraspecial 2-groups. A geometric map from symplectic to orthogonal spreads is shown to induce the Gray map from a corresponding $\ZZ_4$-Kerdock code to its binary image. These geometric considerations lead to the construction, for any odd composite $m$, of large numbers of $\ZZ_4$-Kerdock codes. They also produce new $\ZZ_4$-linear Kerdock and Preparata codes.
1991 Mathematics Subject Classification: primary 94B60; secondary 51M15, 20C99.