Modules Of Finite Type Research Articles

Finiteness of cohomology groups of stacks of shtukas as modules over Hecke algebras, and applications

In this paper we prove that the cohomology groups with compact support of stacks of shtukas are modules of finite type over a Hecke algebra. As an application, we extend the construction of excursion operators, defined by V. Lafforgue on the space of cuspidal automorphic forms, to the space of automorphic forms with compact support. This gives the Langlands parametrization for some quotient spaces of the latter, which is compatible with the constant term morphism.

Spectral theory of von Neumann algebra valued differential operators over non-compact manifolds

We provide criteria for self-adjointness and \tau -Fredholmness of first and second order differential operators acting on sections of infinite dimensional bundles, whose fibers are modules of finite type over a von Neumann algebra A endowed with a trace \tau . We extend the Callias-type index to operators acting on sections of such bundles and show that this index is stable under compact perturbations.

C∗-Algebras Associated with HilbertC∗-Quad Modules of Finite Type

A HilbertC∗-quad module of finite type has a multistructure of HilbertC∗-bimodules with two finite bases. We will construct aC∗-algebra from a HilbertC∗-quad module of finite type and prove its universality subject to certain relations among generators. Some examples of theC∗-algebras from HilbertC∗-quad modules of finite type will be presented.

Supports of $$(\mathfrak{g},\mathfrak{k})$$ -modules of finite type

Let \(\mathfrak{g}\) be a semisimple Lie algebra and \(\mathfrak{k}\) be a reductive subalgebra in \(\mathfrak{g}\). We say that a \(\mathfrak{g}\)-module M is a \((\mathfrak{g},\mathfrak{k})\)-module if M, considered as a \(\mathfrak{k}\)-module, is a direct sum of finite-dimensional \(\mathfrak{k}\)-modules. We say that a \((\mathfrak{g},\mathfrak{k})\)-module M is of finite type if all \(\mathfrak{k}\)-isotopic components of M are finite-dimensional. In this paper we prove that any simple \((\mathfrak{g},\mathfrak{k})\)-module of finite type is holonomic. A simple \(\mathfrak{g}\)-module M is associated with the invariants V(M), V(LocM), and L(M) reflecting the “directions of growth of M.” We also prove that for a given pair \((\mathfrak{g},\mathfrak{k})\) the set of possible invariants is finite.

On the structure of the fundamental series of generalized Harish-Chandra modules

We continue the study of the fundamental series of generalized Harish-Chandra modules initiated in Generalized Harish-Chandra modules with generic minimal $\mathfrak{k}$-type. Generalized Harish-Chandra modules are $(\mathfrak{g}, \mathfrak{k})$-modules of finite type where $\mathfrak{g}$ is a semisimple Lie algebra and $\mathfrak{k} \subset \mathfrak{g}$ is a reductive in $\mathfrak{g}$ subalgebra. A first result of the present paper is that a fundamental series module is a $\mathfrak{g}$-module of finite length. We then define the notions of strongly and weakly reconstructible simple $(\mathfrak{g}, \mathfrak{k})$-modules $M$ which reflect to what extent $M$ can be determined via its appearance in the socle of a fundamental series module. In the second part of the paper we concentrate on the case $\mathfrak{k} \simeq sl(2)$ and prove a sufficient condition for strong reconstructibility. This strengthens our main result from Generalized Harish-Chandra modules with generic minimal $\mathfrak{k}$-type for the case $\mathfrak{k} = sl(2)$. We also compute the $sl(2)$-characters of all simple strongly reconstructible (and some weakly reconstructible) $(\mathfrak{g}, sl(2))$-modules. We conclude the paper by discussing a functor between a generalization of the category $\mathcal{O}$ and a category of $(\mathfrak{g}, sl(2))$-modules, and we conjecture that this functor is an equivalence of categories.

Orsions for Manifolds with Boundary and Glueing Formulas

We extend the definition of analytic and Reidemeister torsion for closed compact iemannian manifolds, to compact Riemannian manifolds with boundary (M, δ M), given a parallel at bundle F of A-Hilbert modules of finite type and a decomposition of the boundary δM = -M ⊔ δ+M into disjoint components. When F is induced from the universal covering of M, and the fundamental group of M is infinite (cf. [BFKM]), these torsions are known as the L2-analytic resp. 2-Reidemeister torsions. If the system (M, δ-M, δ+M, F) is of determinant class we compute he quotient of the analytic and the Reidemeister torsion and prove gluing formulas for both of hem. In particular we answer positively Conjecture 7.6 in [LL]. If F is induced from a Γ-principal overing where Γ is a residually finite group, we derive from work of Luck (cf. [L3]), that the system (M, δ-M, δ+M, F) is of determinant class (cf. Theorem 5.1 in Appendix A).

A Construction of Generalized Harish-Chandra Modules for Locally Reductive Lie Algebras

We study cohomological induction for a pair \( {\left( {\mathfrak{g},\mathfrak{k}} \right)} \), \( \mathfrak{g} \) being an infinitedimensional locally reductive Lie algebra and \( \mathfrak{k} \subset \mathfrak{g} \) being of the form \( \mathfrak{k}_{0} \subset C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} \), where \( \mathfrak{k}_{0} \subset \mathfrak{g} \) is a finite-dimensional reductive in \( \mathfrak{g} \) subalgebra and \( C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} \) is the centralizer of \( \mathfrak{k}_{0} \) in \( \mathfrak{g} \). We prove a general nonvanishing and \( \mathfrak{k} \)-finiteness theorem for the output. This yields, in particular, simple \( {\left( {\mathfrak{g},\mathfrak{k}} \right)} \)-modules of finite type over k which are analogs of the fundamental series of generalized Harish-Chandra modules constructed in [PZ1] and [PZ2]. We study explicit versions of the construction when \( \mathfrak{g} \) is a root-reductive or diagonal locally simple Lie algebra.

DECONSTRUCTING MONOPOLES AND INSTANTONS

We give a unifying description of the Dirac monopole on the 2-sphere S2, of a graded monopole on a (2, 2)-supersphere S2, 2 and of the BPST instanton on the 4-sphere S4, by constructing a suitable global projector p via equivariant maps. This projector determines the projective modules of finite type of sections of the corresponding vector bundle. The canonical connection ∇ = p ◦ d is used to compute the topological charge which is found to be equal to -1 for the three cases. The transposed projector q = pt gives the value +1 for the charges; this showing that transposition of projectors, although an isomorphism in K-theory, is not the identity map. We also study the invariance under the action of suitable Lie groups.

Coherent Functors

We develop Auslander's theory of coherent functors in the case of functors on modules of finite type over a noetherian ringA. In particular, the duality of coherent functors, which interchanges representable functors and tensor products, plays a special role. We apply these coherent functors to study cohomology of a flat family of sheaves on projective space over an affine base schemeT=SpecA. These results form a basic tool which is used in forthcoming work on the variation of the Rao module in a flat family of curves in P3.

ENERGY REPRESENTATIONS OF INFINITE DIMENSIONAL GAUGE GROUPS IN NONCOMMUTATIVE GEOMETRY

Let M be a compact smooth manifold, let [Formula: see text] be a unital involutive subalgebra of the von Neumann algebra £ (H) of bounded linear operators of some Hilbert space H, let [Formula: see text] be the unital involutive algebra [Formula: see text], let [Formula: see text] be an hermitian projective right [Formula: see text]-module of finite type, and let [Formula: see text] be the gauge group of unitary elements of the unital involutive algebra [Formula: see text] of right [Formula: see text]-linear endomorphisms of [Formula: see text]. We first prove that noncommutative geometry provides the suitable setting upon which a consistent theory of energy representations [Formula: see text] can be built. Three series of energy representations are constructed. The first consists of energy representations of the gauge group [Formula: see text], [Formula: see text] being the group of unitary elements of [Formula: see text], associated with integrable Riemannian structures of M, and the second series consists of energy representations associated with (d, ∞)-summable K-cycles over [Formula: see text]. In the case where [Formula: see text] is a von Neumann algebra of type II 1 a third series is given: we introduce the notion of regular quasi K-cycle, we prove that regular quasi K-cycles over [Formula: see text] always exist, and that each of them induces an energy representation.

A counterexample to the rigidity conjecture for rings

An example is constructed of a local ring and a module of finite type and finite projective dimension over that ring such that the module is not rigid. This shows that the rigidity conjecture is false.

Definition of separability in alternative and jordan algebras

One of the classical results of the theory of separable associative algebras over a field is the Wedderburn-Mal'tsev theorem on the decomposition of associative algebras over a field into a sum of a semisimple subalgebra and a radical and on the uniqueness of this decomposition under the condition of separability of the quotient algebra by the radical. In 1951, Azumaya [3] generalized this theorem, proving its analog for associative algebras over a local Hensel ring, which are modules of finite type. Analogs of Azumaya's theorem for alternative algebras and of its first part for Jordan algebras were found by the author in [2, 9, I0].

Differential criteria on the permissibility of a blowing-up

Probably one of the most conceptual descriptions of nonsingularity in algebraic geometry is given by the Jacobian criterion, i.e., by a differential criterion. It seems of interest to investigate the extent to which differential criteria enter into the algebraic machinery around the resolution of singularities. We shall provide some criteria concerning the concept of a permissible center for a blowing-up. The results are as follows: Let k be a fixed ground field of characteristic 0, and let R be a noetherian k-algebra which is a G-ring; i,e., all of its localizations at prime ideals have everywhere smooth formal tibres. If P E Spec R is maximally differential for some set 3 of k-derivations on R, then P is permissible (as a center for a blowing-up), and any reasonably defined differential module of finite type for (R, a) is normally flat along P. A rather simple example shows that permissibility actually is a weaker concept than being maximally differential. On the other hand, we obtain (at least in the local case) a “graded” differential criterion which characterizes in traditional algebro-geometric situations the permissibility of P by the fact that the augmentation ideal in gr,(R) is maximally differential for some set of homogeneous k-derivations (of degree 0) on gr,(R).

Concerning filtrations on commutative rings

We discuss the following three questions concerning multiplicative filtrations on commutative rings: (1) Must every filtration with finite power type be an AP-filtration? (2) Is the intersection of two AP-filtrations always an AP-filtration?, and (3) Does the multiplicity limit for a module of finite type with respect to a O-dimensional filtration on a noetherian ring always exist? We give examples to show that all three questions have in general negative answers for noetherian rings. However, in case of Dedekind domains the questions all have affirmative answers.

An Extension of the Noether-Deuring Theorem

Let $R$ be a commutative semilocal noetherian ring, $\Lambda$ a left noetherian $R$-algebra and $M,N$ finitely generated left $\Lambda$-modules such that ${\operatorname {End} _\Lambda }(M)$ is of finite type over $R$. By $\hat R$ we denote the $(\operatorname {rad} R)$-adic completion of $R$. Theorem. $M$ is $\Lambda$-isomorphic to a direct summand of $N$ iff $\hat R{ \otimes _R}M$ is $\hat R{ \otimes _R}\Lambda$-isomorphic to a direct summand of $\hat R{ \otimes _R}N$. This result is used to prove a generalization of the Noether-Deuring theorem. Let $S$ be a commutative $R$-algebra which is a faithful projective $R$-module of finite type; then $M$ is $\Lambda$-isomorphic to direct summand of $N$ iff $S{ \otimes _R}M$ is $S{ \otimes _R}\Lambda$-isomorphic to a direct summand of $S{ \otimes _R}N$.