Lifting idempotents and Clifford theory

Extended plus closure in complete local rings

Let Q be a complete local ring which has the same characteristic as its residue field P, and, for the present, let us denote by A the image of a subset A of Q under the natural homomorphism of Q onto P. Then a subfield F of Q is called a coefficient field if = P. It has been shown in [2] and in [3] that a complete equicharacteristic local ring, such as the above, always possesses at least one coefficient field; this is the embedding theorem for the equicharacteristic case.

<p>The goal of this thesis is to study non-commutative deformation rings of representations of algebras. The main motivation is to provide a generalization of the deformation theory over commutative local rings studied by B. Mazur, M. Schlessinger and others. The latter deformation theory has played an important role in number theory, and in particular in the proof of Fermat's Last Theorem.</p> <p>The thesis is divided into two parts.</p> <p>In the first part, <em>A</em> is an arbitrary λ-algebra for a complete local commutative Noetherian ring λ with residue field <em>k</em>. A category Ĉ is defined whose objects are complete local λ-algebras <em>R</em> with residue field <em>k</em> such that <em>R</em> is a quotient ring of a power series algebra over λ in finitely many non-commuting variables. If <em>V</em> is a finite dimensional <em>k</em>-vector space that is also a left <em>A</em>-module and that satisfies a natural finiteness condition, it is proved that <em>V</em> has a so-called versal deformation ring <em>R(A,V)</em>. More precisely, <em>R(A,V)</em> is an object in Ĉ such that the isomorphism class of every lift of <em>V</em> over an object <em>R</em> in Ĉ arises from a morphism α : <em>R(A,V)</em>→ <em>R</em> in Ĉ and α is unique if <em>R</em> is the ring of dual numbers <em>k</em>[ϵ].</p> <p>In the second part, two particular examples of λ, <em>A</em> and <em>V</em> are studied and the versal deformation ring <em>R(A,V)</em> is determined in each of these cases. In the first example, λ=<em>k</em>, <em>A</em> is a series of non-commutative <em>k</em>-algebras depending on a parameter <em>r≥2</em>, and <em>V</em> is a particular quotient module of <em>A</em>; it is shown that <em>R(A,V)</em> is isomorphic to <em>A</em>. The second example builds on the first example when <em>r=2</em> and uses that, if additionally the characteristic of <em>k</em> is 2, then <em>A</em> is isomorphic to the group ring <em>k[D<sub>8</sub>]</em> of a dihedral group <em>D<sub>8</sub></em> of order 8.</p> <p>It is shown that if <em>k</em> is perfect and <em>W</em> is the ring of infinite Witt vectors over <em>k</em>, then <em>R(W[D<sub>8</sub>],V)</em> is isomorphic to <em>W[D<sub>8</sub>]</em>.</p>

Contents: Introduction. §1: Computable rings and modules. §2: Ideal membership relation. §3: Effective structured domains. §4: Completion of a local ring as a domain. §5: The recursive completion. Epilogue. References.Introduction. Completion is an important general mathematical device. Often, but not always, a completion takes the following form. Let A be a topological algebraic structure whose topology is derived from a metric. For A, a topological algebra  and an embedding i: A →  are constructed such that  is a complete metric space in which A is densely embedded by i. The long list of structures for which such completions exist begins with Cantor's construction of the real number field and includes objects like the p-adic integers, Baire space, and Boolean algebras. In Bourbaki [6] a careful and thorough account of completions for arbitrary topological groups and fields is given, for which it is important to note that the topological structures need not be metrizable, but must possess a uniformity.The effectiveness of the completion process of a computable structure A cannot be readily studied using the tools of computable algebra, simply because the resulting structure  is almost invariably uncountable. However, in particular cases, it has been possible to define and study the substructure Ak of computable elements of Â; this has been done for the structures mentioned above, starting with the field of recursive real numbers.In this paper we analyse the effectivity of the completion of a local ring R. We do this using structured Scott-Ershov domains. Our study may be considered as a prototype containing methods applicable to a broad class of completions, including all the examples mentioned above, except for the real number field, which needs a generalisation of the domain concept.A Scott-Ershov domain D formalises how a set Dt of possibly “infinite” elements, called total elements, is constructed from a set Dc of “finite” elements, called compact elements. This is achieved by means of an approximation ordering which determines a topology on D and, in particular, on Dt. Our methodology is to associate to a given topological algebra A a structured domain D(A) such that the total elements D(A)t form a topological algebra topologically isomorphic to A. In such circumstances A is said to be domain definable by D(A). The theory of computability for domains is now applied to study the effectivity of the topological algebra A.

Small Cofinite Irreducibles

Let $(R,m)$ be a complete Noetherian local ring, $I$ a proper ideal of R and $M$, $N$ two finitely generated R-modules such that Supp $(N)\subseteq V(I)$. Let $t\geq 0$ be an integer such that for each $0\leq i\leq t$, the $R$-module $H^i_I(M)$ is in dimension $<n$. Then we show that each element $L$ of the set $\mathfrak{J}$, which is defined as: $$ \{{Ext}^j_R(N,H^i_I(M)):j\ge0 \ and \ 0\le i \le t\} \\ \cup\{{Hom}_R(N,H^{t+1}_I(M)),{Ext}^1_R(N,H^{t+1}_I(M))\} $$ is in dimension $<n-2$ and so as a consequence, it follows that the set $$ {Ass}_R(L)\cap\{{\frak{p}} \in Spec(R): :\,\dim (R/{\frak{p}})\ge n − 2\} $$ is finite. In particular, the set $$ {Ass}_R(\oplus_{i=0}^{t+1}H^i_I(R))\cap \{{\frak{p}}\in \,{Spec}(R)\,:\,\dim(R/{\frak{p}})\geq n-2\} $$ is finite. Also, as an immediately consequence of this result it follows that the $R$-modules ${Ext}^j_R(N,H^i_I(M))$ are in dimension $<n-1$, for all integers $i,j\geq 0$, whenever $\dim(M/IM)=n$. These results generalizes the main results of Huneke-Koh [17], Delfino [10], Chiriacescu [9], Asadollahi-Naghipour [1], Quy [18], Brodmann-Lashgari [7], Bahmanpour-Naghipour [5] and Bahmanpour et al. [6] in thecase of complete local rings.

The versal deformation ring R ( G , V ) R(G,V) of a mod p p representation V V of a profinite group G G encodes all isomorphism classes of lifts of V V to representations of G G over complete local commutative Noetherian rings. We introduce a new technique for determining R ( G , V ) R(G,V) when G G is finite which involves Brauer’s generalized decomposition numbers.

Let M be a finitely generated module of rank r and finite projective dimension over a commutative Noetherian local ring S. We show that M has a free direct summand of rank r if and only if its Buchsbaum–Eisenbud invariant J(M) is a principal ideal. As an application, we consider the problem of finding rigid triples. Recall that the module M is said to be rigid if for each i ≥ 0 and each finitely generated S-module N the condition implies for every j ≥ i. A triple of positive integers b = (b 0, b 1, b 2) is said to be rigid if for every commutative Noetherian local ring S one has that every S-module with minimal free resolution of the form 0 → S b 2 → S b 1 → S b 0 → 0 is rigid. We use our criterion for free direct summands of maximal rank to derive a necessary and sufficient condition for a triple b to be rigid in terms of the rigidity of the universal modules of type b over complete regular local rings. In particular, we exhibit the rigidity of any triple of the form (k, m + 1,m), thus providing a new class of rigid modules.

We give a constructive sufficient condition for a matrix over a commutative ring to be von Neumann regular, and we show that it is also necessary over local rings. Specifically, we prove that a matrix A over a local commutative ring is von Neumann regular if and only if A has an invertible -submatrix if and only if the determinantal rank and the McCoy rank of A coincide. We deduce consequences to (products of local) commutative rings, and we determine the number of von Neumann regular matrices over some finite rings of residue classes and group algebras.

Two Clifford's theorems for strongly group-graded rings

Introduction. In the investigation of functors on the category of preschemes, one is led, by Grothendieck [3], to consider the following situation.Let A be a complete noetherian local ring, u, its maximal ideal, and k = A/u.the residue field.(In most applications A is k itself, or a ring of Witt vectors.)Let C be the category of Artin local A-algebras with residue field k.A covariant functor F from C to Sets is called pro-representable if it has the formwhere R is a complete local A-algebra such that R/mn is in C, all n.(m is the maximal ideal in R.)In many cases of interest, F is not pro-representable, but at least one may find an R and a morphism Hom(7?, )-> F of functors such that Hom(.R, A) -> F(A) is surjective for all A in C. If R is chosen suitably "minimal" then R is called a "hull" of F; R is then unique up to noncanonical isomorphism.Theorem 2.11, 2, gives a criterion for F to have a hull, and also a simple criterion for pro-representability which avoids the use of Grothendieck's techniques of nonflat descent [3], in some cases.Grothendieck's program is carried out by Levelt in [4].3 contains a few geometric applications of these results.To avoid awkward terminology, I have used the word " pro-representable " in a more restrictive sense than Grothendieck [3] has.He considers the category of A-algebras of finite length and allows R to be a projective limit of such rings.The methods of this paper are a simple extension of those used by David Mumford in a proof (unpublished) of the existence of formal moduli for polarized Abelian varieties.I am indebted to Mumford and to John Tte for many valuable suggestions.1.The category CA.Let A be a complete noetherian local ring, with maximal ideal u. and residue field k = A/u.. We define C= CA to be the category of Artinian local A-algebras having residue field k. (That is, the "structure morphism" A -> ,4 of such a ring A induces a trivial extension of residue fields.)Morphisms in C are local homomorphisms of A-algebras.

Let A be a noetherian commutative ring which is a homomorphic image of a Gorenstein ring.The first result in this note is that A is a Cohen-Macaulay ring if and only if every finitely generated /I-module can be embedded in a finitely generated Amodule T which is pointwise of finite injective dimension (i.e.all localizations of 7"are of finite injective dimension).The second result is that if A is Cohen-Macaulay, then every finitely generated Amodule can be embedded in a finitely generated /(-module of finite projective dimension if and only if A is Gorenstein.Let A denote a commutative noetherian ring.As a running hypothesis we will assume that A is a homomorphic image of a Gorenstein ring (a commutative noetherian ring R is Gorenstein if Rv has finite injective dimension as an /?p-module, for every prime ideal p of R; see [2] for other definitions equivalent to this one).However, this class of noetherian rings is rather extensive, since it contains most of the rings occurring in algebraic geometry, and, in particular, it contains the complete local rings, by Cohen's structure theorem [5].All modules will be finitely generated (f.g.), and for a f.g.A-module M and p g Spec A, depth Mp denotes (as usual) the maximal length of a Mpregular sequence in pAp, while grade M is the maximal length of an /(-regular sequence in Ann M. Furthermore, recall that a f.g.nonzero Amodule Q is said to be a Gorenstein module if Í2 pointwise is of finite injective dimension (i.e.inj dim Op< co for all p e Supp O) and depth Í2p= depth Ap for all p e Supp O. Since A is a homomorphic image of a Gorenstein ring, by assumption, there exists a Gorenstein /1-module which is of rank one (i.e.Hom(i2, D.) = A), if A is a Cohen-Macaulay ring (cf.Sharp [II], and see [10], [6] and [7] for further properties of Gorenstein modules).Theorem I.A is a Cohen-Macaulay ring if and only if every f.g.Amodule can be embedded in a f.g.A-module which pointwise is of finite injective dimension.

A certain set of associated primes of the Matlis duality of any top local cohomology module of a complete filter local ring is characterized. Also, it is proved that the set of associated primes of a finitely generated module over a four-dimensional local ring is finite.

4 - Indecomposable Modules

Let O be a complete local noetherian ring, whose field of fractions has characteristic zero and residue field has non-zero characteristic. A block algebra over O is an indecomposable summand of the algebra of a finite group over O.