Finite Number Of Isomorphism Classes Research Articles

On the category of Harish-Chandra block modules

If Γ is a subalgebra of A, then an A-module is called a Harish-Chandra module if it is the direct sum of its generalized weight spaces with respect to Γ. Drozd, Futorny, and Ovsienko [4] defined a generalization of a central subalgebra called a Harish-Chandra subalgebra and showed that when Γ is a Harish-Chandra subalgebra of A the structure of Harish-Chandra A-modules can be described using information about the relationship between A and the cofinite maximal ideals of Γ.We extend the results of [4] by dropping the assumption that Γ is quasicommutative. We facilitate this by introducing an equivalence relation ∼ on the set cfs(Γ) of cofinite maximal ideals of Γ. We define Harish-Chandra block modules with respect to ∼ to be A-modules that are the direct sum of so called block spaces corresponding to the equivalence classes cfs(Γ)/∼. If Γ is a Harish-Chandra block subalgebra of A with respect to ∼, then the structure of Harish-Chandra block modules can be described based on the relationship between A and cfs(Γ)/∼. In particular, we give a decomposition of the category of Harish-Chandra block modules and the collection of isomorphism classes of irreducible Harish-Chandra block modules. Furthermore, we define a category A on cfs(Γ)/∼, and show the category of profinite A-modules is equivalent to the category of Harish-Chandra block modules. Taking Γ to be noetherian and quasicommutative, and ∼ to be the equality relation, we recover (in fact, a slight refinement of) results from [4]. Lastly, we provide a sufficient condition for when there are a finite number of isomorphism classes of simple Harish-Chandra block modules with a given support.

Tame and wild theorem for the category of modules filtered by standard modules

We introduce the notion of interlaced weak ditalgebras and apply reduction procedures to their module categories to prove a tame-wild dichotomy for the category F(Δ) of Δ-filtered modules for an arbitrary finite homological system (P,≤,{Δi}i∈P). This includes the case of standardly stratified algebras. Moreover, in the tame case, we show that given a fixed dimension d, for every d-dimensional indecomposable module M∈F(Δ), with the only possible exception of those lying in a finite number of isomorphism classes, the module M coincides with its Auslander-Reiten translate in F(Δ). Our proofs rely on the equivalence of F(Δ) with the module category of some special type of ditalgebra.

Swampland conjectures and infinite flop chains

We investigate swampland conjectures for quantum gravity in the context of M-theory compactified on Calabi-Yau threefolds which admit infinite sequences of flops. Naively, the moduli space of such compactifications contains paths of arbitrary geodesic length traversing an arbitrarily large number of K\"ahler cones, along which the low-energy spectrum remains virtually unchanged. In cases where the infinite chain of Calabi-Yau manifolds involves only a finite number of isomorphism classes, the moduli space has an infinite discrete symmetry which relates the isomorphic manifolds connected by flops. This is a remnant of the 11D Poincare symmetry and consequently gauged, as it has to be by the no-global symmetry conjecture. The apparent contradiction with the swampland distance conjecture is hence resolved after dividing by this discrete symmetry. If the flop sequence involves infinitely many non-isomorphic manifolds, this resolution is no longer available. However, such a situation cannot occur if the Kawamata-Morrison conjecture for Calabi-Yau threefolds is true. Conversely, the swampland distance conjecture, when applied to infinite flop chains, implies the Kawamata-Morrison conjecture under a plausible assumption on the diameter of the K\"ahler cones.

Geometry of varieties for graded maximal Cohen–Macaulay modules

We study a variety for graded maximal Cohen–Macaulay modules, which was introduced by Dao and Shipman. The main result of this paper asserts that there are only a finite number of isomorphism classes of graded maximal Cohen–Macaulay modules with fixed Hilbert series over Cohen–Macaulay algebras of graded countable representation type.

One combinatorial construction in representation theory

Let K be an extension of the field Qp of p-adic rationals, let OK be its ring of integers, and let G be a finite group. According to the classical Jordan–Zassenhaus Theorem, if K∕Qp is finite, every isomorphism class of KG-representation modules splits in a finite number of isomorphism classes of OKG-representation modules. We consider a p-group G of a given nilpotency class k>1 and the extension K∕Qp where K=Qp(ζp∞) obtained by adjoining all roots ζpi,i=1,2,3,... of unity, and we use an explicit combinatorial construction of a faithful absolutely irreducible KG-module M to show that the number of OKG-isomorphism classes of OKG-representation modules, which are KG-equivalent to M, is infinite, and there are extra congruence properties for these OKG-modules.

A Tool for Deadlock Analysis of Parameterized-chain Networks

This paper studies algorithmic aspects of deadlock analysis for parameterized networks of discrete-event systems. A parameterized network consists of interacting finite-state subsystems, including finite but arbitrarily large numbers of subsystems within each of a finite number of isomorphism classes. While deadlock analysis of such systems is generally undecidable, decidable subproblems have recently been identified. The decision procedure of Zibaeenejad and Thistle (2017) rests on the construction of a finite dependency graph for the network, and the computation of its full, consistent subgraphs. We present a software tool that takes the template of a Parameterized Chain Network (PCN) and outputs the set of all full, consistent subgraphs of the dependency graph. These subgraphs represent infinite set of deadlocked states of the PCN for all parameter values. As a case study, we investigate deadlock in a complex train network that extends beyond the current theoretical framework. The results suggest ways in which the framework could be extended.

Hilbert–Samuel sequences of homogeneous finite type

This paper deals with the problem of the classification of the local graded Artinian quotients K[x,y]/I where K is an algebraically closed field of characteristic 0. They have a natural invariant called Hilbert–Samuel sequence. We say that a Hilbert–Samuel sequence is of homogeneous finite type, if it is the Hilbert–Samuel sequence of a finite number of isomorphism classes of graded local algebras. We give the list of all the Hilbert–Samuel sequences of homogeneous finite type in the case of algebras generated by 2 elements of degree 1.

A bound for the splitting of smooth Fano polytopes with many vertices

The classification of toric Fano manifolds with large Picard number corresponds to the classification of smooth Fano polytopes with large number of vertices. A smooth Fano polytope is a polytope that contains the origin in its interior and is such that the vertex set of each facet forms a lattice basis. Casagrande showed that any smooth d-dimensional Fano polytope has at most 3d vertices. Smooth Fano polytopes in dimension d with at least $$3d-2$$3d-2 vertices are completely known. The main result of this paper deals with the case of $$3d-k$$3d-k vertices for k fixed and d large. It implies that there is only a finite number of isomorphism classes of toric Fano d-folds X (for arbitrary d) with Picard number $$2d-k$$2d-k such that X is not a product of a lower-dimensional toric Fano manifold and the projective plane blown up in three torus-invariant points. This verifies the qualitative part of a conjecture in a recent paper by the first author, Joswig, and Paffenholz.

On the maximal superalgebras of supersymmetric backgrounds

In this paper we give a precise definition of the notion of a maximal superalgebra of certain types of supersymmetric supergravity backgrounds, including the Freund–Rubin backgrounds, and propose a geometric construction extending the well-known construction of its Killing superalgebra. We determine the structure of maximal Lie superalgebras and show that there is a finite number of isomorphism classes, all related via contractions from an orthosymplectic Lie superalgebra. We use the structure theory to show that maximally supersymmetric waves do not possess such a maximal superalgebra, but that the maximally supersymmetric Freund–Rubin backgrounds do. We perform the explicit geometric construction of the maximal superalgebra of and find that it is isomorphic to . We propose an algebraic construction of the maximal superalgebra of any background asymptotic to and we test this proposal by computing the maximal superalgebra of the M2-brane in its two maximally supersymmetric limits, finding agreement.

On Modules and Complexes Without Self-Extensions

Let Λ be an Artin algebra over a commutative Artinian ring, k. If M is a finitely generated left Λ -module, we denote by Ω (M) the kernel of η M : P M → M a minimal projective cover. We prove that if M and N are finitely generated left Λ -modules and Ext Λ 1 (M, M) = 0, Ext Λ 1 (N, N) = 0, then M≅ N if and only if M/rad M≅ N/rad N and Ω (M)≅ Ω (N). Now if k is an algebraically closed field and (d i ) i∈ℤ is a sequence of nonnegative integers almost all of them zero, then we prove that the family of objects X ∈ 𝒟 b (Λ), the bounded derived category of Λ, with Hom𝒟 b (Λ)(X,X[1]) = 0 and dim k H i (X) = d i for all i ∈ ℤ, has only a finite number of isomorphism classes (see Huisgen-Zimmermann and Saorín, 2001).

Derived endo-discrete artin algebras

Let ${\mit\Lambda} $ be an artin algebra. We prove that for each sequence $(h_{i})_{i\in \mathbb{Z}}$ of non-negative integers there are only a finite number of isomorphism classes of indecomposables $X\in \mathcal{D}^{\rm b}({\mit\Lambda} ),$ the boun

One-Parameter Families of Modules for Tame Algebras and Bocses

Let Λ be a finite-dimensional algebra over an algebraically closed field k. We denote by mod Λ the category of finitely generated left Λ-modules. Consider the family ℱ(u) of the indecomposables M∈mod Λ such that \(\mathrm{dim}_{k}\,\mathrm{Hom}_{\Lambda}(M,\mathrm{Dtr}\,M)/\mathcal{S}(M,\mathrm{Dtr}\,M)=u\) , where \(\mathcal{S}(M,\mathrm{Dtr}\,M)\) is the subspace of morphisms which factorize through semisimple modules. If P,Q are projectives in mod Λ, ℱ(u)(P,Q) is the family of those modules M∈ℱ(u) such that a minimal projective presentation is of the formfM: P→Q. We prove that if Λ is of tame representation type then each ℱ(P,Q) has only a finite number of isomorphism classes or is parametrized by μ(u,P,Q) one-parameter families. We give an upper bound for this number in terms of u,P and Q. Then we give some sufficient conditions for tame of polynomial growth type. For the proof we consider similar results for bocses.

Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles

Let X be a Fano manifold of Picard number 1 admitting a rational curve with trivial normal bundle and f : X′ → X be a generically finite surjective holomorphic map from a projective manifold X′ onto X. When the domain manifold X′ is fixed and the target manifold X is a priori allowed to deform we prove that the holomorphic map f : X′ → X is locally rigid up to biholomorphisms of target manifolds. This result complements, with a completely different method of proof, an earlier local rigidity theorem of ours (see J. Math. Pures Appl. 80 (2001), 563– 575) for the analogous situation where the target manifold X is a Fano manifold of Picard number 1 on which there is no rational curve with trivial normal bundle. In another direction, given a Fano manifold X′ of Picard number 1, we prove a finiteness result for generically finite surjective holomorphic maps of X′ onto Fano manifolds (necessarily of Picard number 1) admitting rational curves with trivial normal bundles. As a consequence, any 3-dimensional Fano manifold of Picard number 1 can only dominate a finite number of isomorphism classes of projective

Global Matric Massey Products and the Compactified Jacobian of the E6-Singularity

In this paper we compute the compactified Jacobian of the singularity E6. By G. M. Greuel and H. Knörrer (1985, Math. Ann.270, 417–425) this singularity has only a finite number of isomorphism classes of rank 1 torsionfree modules. Using the theory of Matric Massey products, in an earlier work we computed the local formal moduli with its local versal family for each local module, and we studied the degeneracy of each local module. Here we prove a result showing how the local theory connects to the global theory; i.e., we prove that the morphism from the local formal moduli of a local module to the local ring at the point corresponding to the module on the compactified Jacobian is a smooth morphism. In the case where M=E6, i.e., the normalization, this morphism is an isomorphism. Thus the degeneracy (stratification) diagram for the compactified Jacobian can be found from the degeneracy of the normalization in the local case.

The Bruhat–Chevalley Order of Parabolic Group Actions in General Linear Groups and Degeneration for Δ-Filtered Modules

Let k be an algebraically closed field and V a finite dimensional k-space. Let GL(V) be the general linear group of V and P a parabolic subgroup of GL(V). Now P acts on its unipotent radical Pu and on pu=LiePu, the Lie algebra of Pu, via the adjoint action. More generally, we consider the action of P on the lth member of the descending central series of pu denoted by p(l)u. All instances when P acts on p(l)u for l⩾0 with a finite number of orbits are known. In this note, we give a complete combinatorial description of the closure relation on the set of P-orbits on p(l)u, i.e., the Bruhat–Chevalley order, for every finite case. There is a canonical bijection between the set of P-orbits on p(l)u and the set of isomorphism classes of Δ-filtered modules of a particular dimension vector e of a certain quasi-hereditary algebra A(t, l). These isomorphism classes in turn are given by the orbits of a reductive group G(e) on the variety R(Δ)(e) of all A(t, l)-modules with Δ-filtration and dimension vector e. The subcategory of A(t, l)-mod of all Δ-filtered A(t, l)-modules of dimension vector e is denoted by F(Δ)(e). In our chief result, Theorem 1.1, we show that provided there is only a finite number of isomorphism classes of indecomposable modules in F(Δ) the following three posets coincide: (1)the Bruhat–Chevalley order on the set of P-orbits on p(l)u; (2)the Bruhat–Chevalley order on the set of G(e)-orbits on R(Δ)(e); (3)the poset opposite to the so called hom-order on the set of isomorphism classes of F(Δ)(e). The advantage of this hom-order is that it is given purely by discrete invariants and that it can be computed explicitly for any given finite case. Theorem 1.1 then in turn allows us to explicitly determine the closure relations for the P-orbits on p(l)u with the aid of this hom-order. We present some examples of Hasse diagrams in an Appendix.

The Module Structure of a Group Action on a Polynomial Ring

For any representation of a p-group G on a vector space of dimension 3 over a finite field k of characteristic p, we show how the symmetric algebra, regarded as a kG-module, can be expressed as a direct sum of kG-modules, each one of which is isomorphic to a summand in low degree. It follows that, for any group G, only a finite number of isomorphism classes of summands can occur.

Invariants of three-dimensional manifolds

O. Introduction There are several approaches to the construction of invariants of a threedimensional manifold using quantum groups, and, more generally, monoidal categories. In this paper we consider the combinatorial approach which was suggested by Turaev and Viro (see [TV] and later generalizations in [T2], [KS], [Po], [Ro] among others) and construct an invariant starting from a monoidal category C without any braiding conditions. The starting point of a combinatorial approach is a triangulation of a given three-dimensionai M, and the invariant is constructed using the combinatorics of this triangulation. More precisely, fix a monoidal category C (a category with the multiplication fimctor X ® Y on objects) over a field k, which is semisimple, has a finite number of isomorphism classes of simple objects, and possesses a duality X ~-~ X* such that X** is isomorphic to X. To define Ic(M), we must fix a balancing, i.e. functorial isomorphisms X ~ X**, X E Ob C, satisfying certain natural conditions. On the other hand, we do not assume that C satisfies any kind of braiding conditions, i.e. commutativity conditions of the form X®Y ~ Y®X. A similar construction, which also does not use braiding, was recently suggested by Kuperberg [Ku], who used the la~lguage of Hopf algebras. For any three-dimensional manifold M with boundary S and a triangulation D of M, we construct a finite-dimensional linear space W(S, D ~) depending on the restriction D ~ of D to S (a triangulation of S) and a vector It(M, S, D) C W(S, D'). The main result of the paper (Theorem 1) is that Ic(M,S,D) depends only on D ~ and not on D itself. Taking the inductive limit of spaces W(S, D ~) over all triangulations D ~ of S, we construct a space K(S) and a vector Ic(M, S) C K(S), which is our invariant of a three-dimensionM manifold M with boundary S. (See §6 of the paper, where this is explained in slightly different terms.) In particular, when M is a closed manifold, S = O, we have K(O) = k, and our construction gives an invariant Ic(M) e k.

Composition factors for indecomposable modules

Let k k be a field and A A be a finite-dimensional algebra over k k having only a finite number of isomorphism classes of indecomposable A A -modules. Let M M , N N be two indecomposable A A -modules. Then a homomorphism f : M → N f:M \to N is said to be irreducible if for every factorization f = g h f = gh , g g is split mono or h h is split epi [2]. The aim of this note is to give an elementary proof of the fact that the indecomposable A A -modules are completely determined, up to isomorphism, by their composition factors if there is no chain of irreducible maps from an indecomposable module to itself. This theorem was first proved in [5] involving the theory of tilted algebras.

Partial Coxeter functors and right pure semisimple hereditary rings

An important problem in the representation theory of artinian rings is to obtain a charcterization of rings of finite representation type (see [2, 7, 16, 201). We recall that a ring R is of finite representation type if R is both left and right artinian and there is only a finite number of isomorphism classes of indecomposable finitely generated R-modules. The main tools for studying the representation theory of hereditary artinian rings are partial Coxeter functors and Coxeter functors (see [4, 8, 10, 141). They are successfuly applied in the investigation of hereditary rings of finite representation type [8, 9-12, 141. In the present paper we study partial Coxeter functors for hereditary artinian rings and we use them as a tool for studying right pure semisimple hereditary rings (which are close to rings of finite representation type). We recall that for any ring R the right pure global dimension r.P.gl.dim R and the left pure global dimension l.P.gl.dim R are defined [19-21, 261. There is an interesting problem how to characterize rings R with r.P.gl.dim R = 0 which are exactly those right artinian rings R all of whose right R-modules are algebraically compact [ 19, 20, 261 or, equivalently, all of whose right R-modules are direct sums of modules of finite length. A ring R with r.P.gl.dim R = 0 is called right pure semisimple [25-271. It is well known that any ring of finite representation type is both left and right pure semisimple 12, 20, 23, 251 and the converse implication also holds true [2, 15, 201. However, it is still an open question if a right pure semisimple ring R is of finite representation type or, equivalently, if r.P.gl.dim R = 0 implies l.P.gl.dim R = 0 (see [29]). A positive solution of this problem is given by Auslander [3] for artin algebras. He also discusses the problem for arbitrary left artinian rings (see also 1181). We recall that a 195 0021.8693/81/070195~24$02.00/0