A Note on the Associated Primes of Local Cohomology Modules

In this note we give a simple proof of the following result: Let R be a commutative Noetherian ring,  an ideal of R and M a finite R-module, if H_ ⁱ (M) has finite support for all i < n, then Ass(H_ ⁿ (M)) is finite.     

Fields interpretable in rosy theories

We are working in a monster model ℭ of a rosy theory T. We prove the following theorems, generalizing the appropriate results from the finite Morley rank case and o-minimal structures. If R is a ⋁-definable integral domain of positive, finite U^t-rank, then its field of fractions is interpretable in ℭ. If A and M are infinite, definable, abelian groups such that A acts definably and faithfully on M as a group of automorphisms, M is A-minimal and U^t(M) is finite, then there is an infinite field interpretable in ℭ. If G is an infinite, solvable but non nilpotent-by-finite, definable group of finite U^t-rank and T has NIP, then there is an infinite field interpretable in 〈G, ·〉.     

On a class of finite rings

In [7], Corbas determined all finite rings in which the product of any two zero-divisors is zero, and showed that they are of two types, one of characteristic p ²and the other of characteristic p².

The purpose of this paper is to address the problem of the classification of finite rings such that.

(i)the set of all zero-divisors form an ideal M.

(ii)M ³=(0); and.

(iii)M ³≠(0).

Because of (i), these rings are called completely primary and we shall call a finite completely primary ring R which satisfies conditions (i), (ii) and (iii), a ring with property(T). These rings are of three types, namely, of characteristic p p ² and p ³. The characteristic p ² case is subdivided into cases in which p?M ² p?ann(M)?M ² and p?M ?ann(M), where ann(M) denotes the two-sided annihilator of where M in R.     

The residual finiteness of negatively curved polygons of finite groups

A subgroup M⊂G is almost malnormal provided that for each g∈G−M, the intersection M ^g ∩M is finite. It is proven that the free product of two virtually free groups amalgamating a finitely generated almost malnormal subgroup, is residually finite. A consequence of a generalization of this result is that an acute-angled n-gon of finite groups is residually finite if n≥4. Another consequence is that if G acts properly discontinuously and cocompactly on a 2-dimensional hyperbolic building whose chambers have acute angles and at least 4 sides, then G is residually finite. Oblatum 17-VII-2000 & 13-II-2002?Published online: 29 April 2002     

Finite jet determination of local analytic CR automorphisms and their parametrization by 2-jets in the finite type case

We show that germs of local real-analytic CR automorphisms of a real-analytic hypersurface M in $\mathbb{C}$² at a point p M are uniquelydetermined by their jets of some finite order at p if and only if M is not Levi-flat near p. This seems to be the first necessary and sufficient result on finite jet determination and the first result of this kind in the infinite type case.If M is of finite type at p, we prove a stronger assertion: the local real-analytic CR automorphisms of M fixing p are analytically parametrized (and hence uniquely determined) by their 2-jets at p. This result is optimal since the automorphisms of the unit sphere are not determined by their 1-jets at a point of the sphere. The finite type condition is necessary since otherwise the needed jet order can be arbitrarily high [Kow1,2], [Z2]. Moreover, we show, by an example, that determination by 2-jets fails for finite type hypersurfaces already in $\mathbb{C)$³.We also give an application to the dynamics of germs of local biholomorphisms of $\mathbb{C)$².     

Optimal boundary control of the wave equation with pointwise control constraints

In optimal control problems frequently pointwise control constraints appear. We consider a finite string that is fixed at one end and controlled via Dirichlet conditions at the other end with a given upper bound M for the L ^∞-norm of the control. The problem is to control the string to the zero state in a given finite time. If M is too small, no feasible control exists. If M is large enough, the optimal control problem to find an admissible control with minimal L ²-norm has a solution that we present in this paper.     

On Finite Complete Presentations and Exact Decompositions of Semigroups

We prove that given a finite (zero) exact right decomposition (M, T) of a semigroup S, if M is defined by a finite complete presentation, then S is also defined by a finite complete presentation. Exact right decompositions are natural generalizations to semigroups of coset decompositions in groups. As a consequence, we deduce that any Zappa–Szép extension of a monoid defined by a finite complete presentation, by a finite monoid, is also defined by such a presentation.

It is also proved that a semigroup M ⁰[A; I, J; P], where A and P satisfy some very general conditions, is also defined by a finite complete presentation.     

Some remarks on dualisability and endodualisability

The fundamental problem of dualisability and the particular problem of endodualisability are discussed. It is proved tha every finite generating algebra of a quasi-variety generated by a finite dualisable algebra D is also dualisable. The corresponding result for endodualisability is true when D is subdirectly irreducible. Under special conditions, it is also proved that a finite algebra M is endodualisable if and only if any finite power M ⁿ of M is endodualisable. Received January 27, 1999; accepted in final form September 17, 1999.     

Groups acting on finite dimensional spaces with finite stabilizers

It is shown that every H -group G of type admits a finite dimensional G-CW-complex X with finite stabilizers and with the additional property that for each finite subgroup H, the fixed point subspace X ^H is contractible. This establishes conjecture (5.1.2) of [9]. The construction of X involves joining a family of spaces parametrized by the poset of non-trivial finite subgroups of G and ultimately relies on the theorem of Cornick and Kropholler that if M is a -module which is projective as a -module for all finite then M has finite projective dimension. Received: April 30, 1997     

Amenability of the Second Dual of Hypergroup Algebras

The amenability of the Banach algebra L ¹(G), the measure algebra M(G) and their second duals of a locally compact group have been considered by a number of authors. During these investigations it has been shown that L ¹(G)^** is amenable if and only if G is finite. If LUC (G)^*, the dual of the space of left uniformly continuous functions on G, is amenable, then G is compact and M(G) is amenable. Finally, if M(G)^** is amenable, then G is finite. The aim of this paper is to generalize all of the above results to the locally compact hypergroups.     

Minimal surfaces in R3 with one end¶and bounded curvature

We study complete minimal surfaces M immersed in R ³, with finite topology and one end. We give conditions which oblige M to be conformally a compact Riemann surface punctured in one point, and we show that M can be parametrized by meromorphic data on this compact Riemann surface. The goal is to prove that when M is also embedded, then the end of M is asymptotic to an end of a helicoid (or M is a plane). Received: 13 January 1997 / Revised version: 15 September 1997     

Finiteness properties of minimax and coatomic local cohomology modules

Let R be a noetherian ring, \mathfraka{\mathfrak{a}} an ideal of R, and M an R-module. We prove that for a finite module M, if Hⁱ_\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is minimax for all i ≥ r ≥ 1, then Hⁱ_\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is artinian for i ≥ r. A local–global principle for minimax local cohomology modules is shown. If Hⁱ_\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is coatomic for i ≤ r (M finite) then Hⁱ_\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is finite for i ≤ r. We give conditions for a module which is locally minimax to be a minimax module. A non-vanishing theorem and some vanishing theorems are proved for local cohomology modules.     

Complete hypersurfaces with constant mean curvature and finite L p norm curvature in Euclidean spaces

In this paper, we consider complete hypersurfaces in R ⁿ⁺¹ with constant mean curvature H and prove that M ⁿ is a hyperplane if the L ² norm curvature of M ⁿ satisfies some growth condition and M ⁿ is stable. It is an improvement of a theorem proved by H. Alencar and M. do Carmo in 1994. In addition, we obtain that M ⁿ is a hyperplane (or a round sphere) under the condition that M ⁿ is strongly stable (or weakly stable) and has some finite L ^p norm curvature. Received: 14 July 2007     

Some results on the cofiniteness of local cohomology modules

Let R be a commutative Noetherian ring, a an ideal of R, M an R-module and t a non-negative integer. In this paper we show that the class of minimax modules includes the class of AF modules. The main result is that if the R-module Ext _R ^t (R/a,M) is finite (finitely generated), H _a ⁱ (M) is a-cofinite for all i < t and H _a ^t (M) is minimax then H _a ^t (M) is a-cofinite. As a consequence we show that if M and N are finite R-modules and H _a ⁱ (N) is minimax for all i < t then the set of associated prime ideals of the generalized local cohomology module H _a ^t (M,N) is finite.     

Relative invariants for representations of finite dimensional algebras

 Let M be a finite dimensional module over a finite dimensional basic K-algebra Λ, where K is an algebraically closed field. We associate with M a weight θ ^M (i.e. an element of the dual of the Grothendieck group of mod-Λ) in module theoretic terms. Let β be a dimension vector with θ ^M (β)=0. We generalize a construction of relative invariants of quivers due to Schofield [S] and define a relative invariant polynomial function d ^M _β on the variety of modules of dimension vector β, such that d ^M _β (N) = 0 for some module N if and only if there is a nonzero morphism from M to N. Assuming char (K) = 0, we conclude from the main result of Schofield-Van den Bergh [SV] that relative invariants of this form span all the spaces of relative invariants. To get algebra generators of the algebra of semi-invariants it is sufficient to take the d ^M _β with M indecomposable. Received: 31 July 2001     

The associated primes of local cohomology modules over rings of small dimension

Let R be a commutative Noetherian local ring of dimension d, I an ideal of R, and M a finitely generated R-module. We prove that the set of associated primes of the local cohomology module H ⁱ _I (M) is finite for all i≥ 0 in the following cases: (1) d≤ 3; (2) d= 4 and $R$ is regular on the punctured spectrum; (3) d= 5, R is an unramified regular local ring, and M is torsion-free. In addition, if $d>0$ then H ^{d − 1} _I (M) has finite support for arbitrary R, I, and M. Received: 31 October 2000 / Revised version: 8 January 2001     

Classification of Finite Groups with a CC-Subgroup

Abstract

A proper subgroup M of a group G is called a CC-subgroup of G if the centralizer C _G (m) of every m ∈ M ^# = M ? {1} is contained in M. In this paper we classify all finite groups containing a CC-subgroup, extending work of many authors.     

Torsion-Free Space Groups, 2-Cohomology, and Scott Modules

For a subgroupCof orderpof a finite groupG, we find the summandMof thep-adic permutation module ind_C^GZ_psuch thatH²(G, M)≠0, and determine whenMis the Scott module. This is applied to the study of torsion-free space groups.     

Multiplication operators on the Bergman space via analytic continuation

In this paper, using the group-like property of local inverses of a finite Blaschke product ?, we will show that the largest C^?-algebra in the commutant of the multiplication operator M? by ? on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of ?−1°? over the unit disk. If the order of the Blaschke product ? is less than or equal to eight, then every C^?-algebra contained in the commutant of M? is abelian and hence the number of minimal reducing subspaces of M? equals the number of connected components of the Riemann surface of ?−1°? over the unit disk.     

Characterization of modules of finite projective dimension via Frobenius functors

Let M be a finitely generated module over a local ring R of characteristic p > 0. If depth(R) = s, then the property that M has finite projective dimension can be characterized by the vanishing of the functor Extⁱ_R(M, ^fnR){{\rm Ext}^i_R(M, ^{f^n}R)} for s + 1 consecutive values i > 0 and for infinitely many n. In addition, if R is a d-dimensional complete intersection, then M has finite projective dimension can be characterized by the vanishing of the functor Extⁱ_R(M, ^fnR){{\rm Ext}^i_R(M, ^{f^n}R)} for some i ≥ d and some n > 0.