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We study the variation of the local Langlands correspondence for ${\rm GL}_{n}$ in characteristic-zero families. We establish an existence and uniqueness theorem for a correspondence in families, as well as a recognition theorem for when a given pair of Galois- and reductive-group- representations can be identified as local Langlands correspondents. The results, which can be used to study local-global compatibility questions along eigenvarieties, are largely analogous to those of Emerton, Helm, and Moss on the local Langlands correspondence over local rings of mixed characteristic. We apply the theory to the interpolation of local zeta integrals and of $L$- and $\varepsilon$-factors., 28 pages. Several corrections and improvements (main results unchanged)

We study divided power structures on finitely generated $k$-algebras, where $k$ is a field of positive characteristic $p$. As an application we show examples of $0$-dimensional Gorenstein $k$-schemes that do not lift to a fixed noetherian local ring of non-equal characteristic. We also show that Frobenius neighbourhoods of a singular point of a general hypersurface of large dimension have no liftings to mildly ramified rings of non-equal characteristic.

We give a theory of ideles with coefficients for smooth surfaces over a field. It is an analogue of Beilinson/Huber's theory of higher adeles, but handling cycle module sheaves instead of quasi-coherent ones. We prove that they give a flasque resolution of the cycle module sheaves in the Zariski topology. As a technical ingredient we show the Gersten property for cycle modules on equicharacteristic complete regular local rings, which might be of independent interest.

In a recent paper of Ellenberg, Oberlin, and Tao, the authors asked whether there are Besicovitch phenomena in F_q[[t]]^n. In this paper, we answer their question in the affirmative by explicitly constructing a Kakeya set in F_q[[t]]^n of measure 0. Furthermore, we prove that any Kakeya set in F_q[[t]]^2 or Z_p^2 is of Minkowski dimension 2., Comment: 10 pages

A new method is demonstrated to quantify local ring strain, which is based on the expectation value of orbital angular momentum along the internuclear axis. In contrast to energy based methods which provide overall ring strain, this method is able to identify the local strain in every part of the ring. The formalism is benchmarked on several cycloalkanes in which the presence of ring strain is well understood. The ring strain plays a decisive role in carbon nanotubes (CNTs) properties; for instance, the hydrogen storage capability of CNTs is related to their diameter, which in turn has a close relation to the ring strain in their CC bonds. On this basis, the ring strain in five CNTs with different diameters is analyzed and the results reflected meaningful correlation between the CNTs diameter and ring strain. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Let (R,m) be a local ring and U_R=Spec(R) -{m} be the punctured spectrum of R. Gabber conjectured that if R is a complete intersection of dimension 3, then the abelian group Pic(U_R) is torsion-free. In this note we prove Gabber's statement for the hypersurface case. We also point out certain connections between Gabber's Conjecture, Van den Bergh's notion of non-commutative crepant resolutions and some well-studied questions in homological algebra over local rings., Some statements/typos fixed thanks to corrections from the referees, main results remain the same

For a given saturated fusion system F we define the ring A(S) F of the F-invariants of the Burnside ring functor. We show how this ring is related to the Burnside ring of the fusion system F and how it appears naturally in the analogue of Segal�s conjecture for the classifying spectrum BF. We give an explicit description of A(S) F and we prove it is a local ring.

Let A be a Noetherian local ring and let E be a finitely generated A-module having rank r. In this note one deals with the expected inequality l(∧ r E) ≥ height (Fitt r (E)), where height (Fitt r (E)) is the codimension of the rth Fitting ideal of E, and l(M) stands for the analytic spread of a module M. One establishes cases where the inequality holds as well as where it fails. A special case where the inequality holds implies the celebrated Zak inequality for the dimension of the image of the Gauss map.

We give an elementary theory of Henselian local rings and construct the Henselisation of a local ring. All our theorems have an algorithmic content. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We consider random Hermitian matrices in which distant above-diagonal entries are independent but nearby entries may be correlated. We find the limit of the empirical distribution of eigenvalues by combinatorial methods. We also prove that the limit has an algebraic Stieltjes transform by an argument based on dimension theory of Noetherian local rings. © 2008 Wiley Periodicals, Inc.

An additive functor from the category of flat right R-modules to the category of abelian groups is continuous if it is isomorphic to a functor of the form−⊗ R M, where M is a left R-module. It is shown that for any simple subfunctor A of−⊗ M there is a unique indecomposable flat cotorsion module U R for which A(U)≠0. It is also proved that every subfunctor of a continuous functor contains a simple subfunctor. This implies that every flat right R-module may be purely embedded into a product of indecomposable flat cotorsion modules.If CE(R) is the cotorsion envelope of R R and S= End; R CE(R), then a local ring monomorphism is constructed from R/J(R) to S/J(S). This local morphism of rings is used to associate a semiperfect ring to any semilocal ring. It also proved that if R is a semilocal ring and M a simple left R-module, then the functor−⊗ R M on the category of flat right R-modules is uniform, and therefore contains a unique simple subfunctor.

In this note, it is proved that over a commutative noetherian henselian non-Gorenstein local ring there are infinitely many isomorphism classes of indecomposable totally reflexive modules, if there is a nonfree cyclic totally reflexive module.

Our focus in this paper is in effective computation of the core core(I) of an ideal I which is defined to be the intersection of all minimal reductions of I. The first main result is a closed formula for the graded core(m) of the maximal graded ideal m of an arbitrary standard graded algebra A over a field k. This formula allows us to study basic properties of the graded core and to construct counter-examples to some open questions on the core of ideals in a local ring. For instance, we can show that in general, core(m \otimes E) \neq core(m)\otimes E, where E is a field extension of k. From this it follows that the equation core(I R') = core(I)R' does not hold for an arbitrary flat local homomorphism R \to R' of Cohen-Macaulay local rings. The second main result proves the formulae core(I)= (J^r:I^r)I = (J^r:I^r)J = J^{r+1}:I^r for any equimultiple ideal I in a Cohen-Macaulay ring R with with characteristic zero residue field, where J is a minimal reduction of I and r is its reduction number. This result has been obtained independently by Polini-Ulrich and Hyry-Smith in the one-dimensional case or when R is a Gorenstein ring. Moreover, we can prove that core(I) = IK, where K is the conductor of R in the blowing-up ring at I., 20 pages, to appear in Compositio Math

Let $M$ be a module over the ring $R$ . Extensive use is made of Krull codimension to study further the Artinian-finitary automorphism group $\[ F_1\hbox{\rm Aut}_RM \,{=}\, \{g\,{\in}\hbox{ Aut}_RM \,{:}\, M(g - 1) \hbox{ is } R\hbox{-Artinian}\} \]$ of $M$ over $R$ . Substantial progress is made where either $M$ is residually Noetherian or $R$ is commutative. There are some group-theoretic consequences of the two main structure theorems.

The depth of the associated graded ring of the powers of an ideal $I$ of a local ring $R$ is studied. It is proved that the depth of the associated graded ring of $I^n$ is asymptotically constant when $n$ tends to infinity, and this value is characterized in terms of Valabrega–Valla conditions of $I^m$ for some large integer $m\ge 0$ . As a corollary, a generalization is obtained of the $2$ -dimensional algebraic version of the Grauert–Riemenschneider vanishing theorem (due to Huckaba and Huneke) to ideals satisfying the second Valabrega–Valla condition. The positiveness of Hilbert coefficients is also studied, and Valabrega–Valla conditions are linked to the vanishing of the cohomology groups of the closed fiber of the blowing up of ${\bf Spec}(R)$ along the closed sub-scheme defined by $I$ .

We establish an inequality involving colengths of the tight closure of ideals of systems of parameters in local rings with some mild conditions. As an application, we prove and refine a result by Goto and Nakamura, conjectured by Watanabe and Yoshida, which states that the Hilbert-Samuel multiplicity of a parameter ideal is greater than or equal to the colength of the tight closure of the ideal.

Let K be the quotient field of a 2-dimensional regular local ring (R, m) and let v be a prime divisor of R, i.e., a valuation of K birationally dominating R which is residually transcendental over R. Zariski showed that: such prime divisor v is uniquely associated to a simplem-primary integrally closed ideal I of R, there are only finitely many simple v-ideals including I, and all the other v-ideals can be uniquely factored into products of simple v-ideals. The number of nonmaximal simple v-ideals is called the rank of v or the rank of I as well. It is also known that such anm-primary ideal I is minimally generated by o(I)+1 elements, where o(I) denotes the m-adic order of I. Given a simple valuation ideal of order two associated to a prime divisor v of arbitrary rank, in this paper we find minimal generating sets of all the simple v-ideals and the value semigroup v(R) in terms of its rank and the v-value difference of two elements in a regular system of parameters of R. We also obtain unique factorizations of all the composite v-ideals and describe the complete sequence of v-ideals as explicitly as possible. (© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We study finitely generated modules $M$ over a ring $R$, noetherian on both sides. If $M$ has finite Gorenstein dimension $\mbox{G-dim}_RM$ in the sense of Auslander and Bridger, then it determines two other cohomology theories besides the one given by the absolute cohomology functors ${\rm Ext}^n_R(M,\ )$. Relative cohomology functors ${\rm Ext}^n_{\mathcal G}(M,\ )$ are defined for all non-negative integers $n$; they treat the modules of Gorenstein dimension $0$ as projectives and vanish for $n > \mbox{G-dim}_RM$. Tate cohomology functors $\widehat{\rm Ext}^n_R(M,\ )$ are defined for all integers $n$; all groups $\widehat{\rm Ext}^n_R(M,N)$ vanish if $M$ or $N$ has finite projective dimension. Comparison morphisms $\varepsilon_{\mathcal G}^n \colon {\rm Ext}^n_{\mathcal G}(M,\ ) \to {\rm Ext}^n_R(M,\ )$ and $\varepsilon_R^n \colon {\rm Ext}^n_R(M,\ ) \to \widehat{\rm Ext}^n_R(M,\ )$ link these functors. We give a self-contained treatment of modules of finite G-dimension, establish basic properties of relative and Tate cohomology, and embed the comparison morphisms into a canonical long exact sequence $0 \to {\rm Ext}^1_{\mathcal G}(M,\ ) \to \cdots \to {\rm Ext}^n_{\mathcal G}(M,\ ) \to {\rm Ext}^n_R(M,\ ) \to \widehat{\rm Ext}^n_R(M,\ ) \to {\rm Ext}^{n+1}_{\mathcal G}(M,\ ) \to \cdots$. We show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings. 2000 Mathematical Subject Classification: 16E05, 13H10, 18G25.

Let (R,m) be a local ring with prime ideals p and q such that p+q is an m-primary ideal. If R is regular and contains a field, and dim(R/p)+dim(R/q)=dim(R), we prove that p^{(r)}\cap q^{(n)}\subseteq m^{m+n} for all positive integers r and s. This is proved using a generalization of Serre's Intersection Theorem which we apply to a hypersurface R/fR. The generalization gives conditions that guarantee that Serre's bound on the intersection dimension dim(R/p)+dim(R/q) \leq dim(R) holds when R is nonregular., Comment: 29 pages

Let (A, [mfr ]) be a local ring. For convenience we will assume throughout this paper that the residue field of A is infinite.Let I be an ideal of A. An ideal J ⊆I is called a reduction of I if JIn = In+1 for some integer n. The least number n with this property is denoted by rJ (I). A reduction of I is said to be minimal if it does not contain any other reduction of I. The reduction number r(I) of I is the minimum of rJ(I) for all minimal reductions J of I. A minimal reduction of I usually has better properties than I. It can be viewed as an approximation of I and the reduction number is a measure for how it is different from I. The minimal number of generators of every minimal reduction of I is equal to the dimension of the fibre ring [oplus ]n[ges ]0In/[mfr ]In. This invariant is called the analytic spread of I and denoted by [lscr ](I). All these notions have played an important role in ideal theory since their introduction by Northcott and Rees [16].

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.

This work was motivated in part by the following general question: given an ideal I in a Cohen–Macaulay (abbreviated to CM) local ring R such that dim R / I =0, what information about I and its associated graded ring can be obtained from the Hilbert function and Hilbert polynomial of I ? By the Hilbert (or Hilbert–Samuel) function of I , we mean the function H I ( n ) =λ( R / I n ) for all n [ges ]1, where λ denotes length. Samuel [ 23 ] showed that for large values of n , the function H I ( n ) coincides with a polynomial P I ( n ) of degree d =dim R . This polynomial is referred to as the Hilbert, or Hilbert–Samuel, polynomial of I . The Hilbert polynomial is often written in the form formula here where e 0 ( I ), [ctdot ], e d ( I ) are integers uniquely determined by I . These integers are known as the Hilbert coefficients of I .

In this paper we continue our study of the Canonical Element Conjecture (henceforth C.E.C.) via the dualizing complex. Throughout the work (A, m, k) will denote a noetherian complete local ring A of dimension n, m its maximal ideal and k=A/m. Since A is complete, we can find a complete local Gorenstein ring (R, mR, k) (complete intersection) such that dim R=dim A and A=R/I. Let Ω denote the canonical module of A, that is, Ω=HomR (A, R), which may be identified with the annihilator of I in R, an ideal of R. When A is a domain, we change notation and denote I by P; in this case P is a height 0 prime ideal of R.

Let ( R , [mfr ]) be a local Noetherian ring. We show that if R is complete, then an R -module M satisfies local duality if and only if the Bass numbers μ i ([mfr ], M ) are finite for all i . The class of modules with finite Bass numbers includes all finitely generated, all Artinian, and all Matlis reflexive R -modules. If the ring R is not complete, we show by example that modules with finite Bass numbers need not satisfy local duality. We prove that Matlis reflexive modules satisfy local duality in general, where R is any local ring with a dualizing complex.

This paper is about the local geometry of a real surfaces. It introduces machinery for studying families of subsets which are determined by conditions which are similar to base conditions, but also involve positivity/non-negativity. The methods used are the real spectrum and Zariski's theory of complete ideals.

Let G be a unipotent algebraic group over K (a field of characteristic 0) which acts rationally on an affine scheme X = Spec A over K, where A is a commutative K-algebra. The problem of finding sufficient and manageable conditions to guarantee that the geometric quotient X/G exists is of fundamental interest in the theory of moduli spaces for local objects such as isolated singularities or (Cohen-Macaulay) modules over the local ring of a singularity (cf. [L-P], [G-P2], [G-H-P], [H]).

The in-house information communication network using local area network (LAN) is required to cope flexibly with the expansion/modification of the system and to avoid the system-down in the generation of a fault by detaching the fault domain. This paper considers the star-ring LAN with the ring spurs and a dual ring. It proposes also the structure and control of the wiring concentrator of ring LAN with the dedicated automatic reconfiguration control by the ring spur bypass and the ring wrapback. Since the ring contains both the ordinary terminals and WC, the reconfiguration-dedicated medium access control (MAC) frame is introduced anew and can be sent only by wiring concentration (WC), making unnecessary the specification modification of the MAC layer of the terminal. The WC which handles the localization and detachment of the fault point is determined by this mechanism. The test ring is automatically expanded from the local ring in WC to the ring containing the ring spur and the ring containing the dual ring successively by testing the conductivity. Using WC, a ring LAN system with the automatic ring degeneration/expansion function can be realized, while providing the terminal with the function of connection/detachment/shift on LAN (hot-pluggability). The time required for reconfiguration by the proposed reconfiguration control is evaluated by summing up the times for the major control steps and by experimental examination.

Let A be a commutative Noetherian local ring satisfying the approximation property. This means that any system of polynomial equations over A having a solution in the completion Ǎ has also a solution in A. Then A is proven to admit a dualizing complex.

This paper studies Frobenius maps on injective hulls of residue fields of complete local rings with a view toward providing constructive descriptions of objects originating from the theory of tight closure. Specifically, the paper describes algorithms for computing parameter test ideals, and tight closure of certain submodules of the injective hull of residue fields of a class of well-behaved rings which includes all quasi-Gorenstein complete local rings., Comment: Minor corrections, an example added. To appear in the Journal of the London Mathematical Society

This paper is devoted to the introduction of extension rings S : = R[x]/gR[x] with a suitable polynomial g ϵ R[x] of arbitrary commutative rings R with identity and to the development of a normal basis concept of S over R, which is similar to that of Galois extensions of finite fields. We prove new results for Galois extensions of local rings and apply them together with the Chinese remainder theorem to solve the above task in a constructive way.

A commutative Noetherian local ring of finite global dimension is a regular local ring and the structure of such a ring is well described 1n many texts (see §1.9). It is perhaps surprising that very little seems to be known about the corresponding non-commutat1ve rings and it is our aim 1n this thesis to examine some of the ways in which the commutative theory can, and cannot be extended to a right Noetherian local ring of finite right global dimension. Chapter 1 contains basic definitions and, in Chapter 2, results are obtained on the projective and Injective dimensions of modules over right Noetherian local rings. Commutative regular local rings are domains and we begin Chapter 3 by considering the question of when a right Noetherian local ring R of finite right global dimension is a prime ring. An example of Stafford's shows that R is not necessarily prime; however, by examining the Nilpotent radical of R, we are able to extend results of Ramras [57] and Walker [75] and show that R is indeed prime when certain prime ideals are localisable. In Chapter 4 we consider the lattice of prime ideals of R when R is an AR-r1ng and provide theorems which generalise some of the basic results from the theory of commutative regular local rings. Section 4.3 contains examples which not only illustrate points arising in Chapters 3 and 4 but also show that some of the techniques which are mainstays of the commutative theory, fall dramatically in a non-commutat1ve setting. In Chapter 5, we generalise the concepts of regular sequences and Cohen Macaulay rings enabling us to prove, in §5.2 and chapter 6, that a right Noetherian local ring of finite right global dimension, which 1s integral over its centre, is a prime ring and exhibits many properties similar to those enjoyed by commutative regular local rings. Examples are provided which show that some alternative approaches are not applicable to the rings considered in these chapters. Regular local rings are Gorensteln rings and, as such, are the subject of an elegant structure theorem due to Bass [5]. In Chapter 7, we generalise his theorem to the situation of rings Integral over their centres. Each chapter begins with a summary of the results contained in that chapter.

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Let ( R , ) be a commutative Noetherian local ring. We investigate conditions for a non-finitely generated R -module M to have a system of parameters. We prove that if then any system of parameters for R /An R ( M ) is a system of parameters for M . As an application we characterize by means of systems of parameters those balanced big Cohen–Macaulay R -modules M for which Supp R ( M ) = supp R ( M ).

A number of results are proved concerning the Quillen ^-theory K+(S*G) of the skew group ring S*G, where S is a Noetherian ring and G is a finite group of automorphisms of 5. Applications are given to the computation of AT-groups of group algebras and of equivariant /^-theory for affine varieties.