A GENERALIZATION OF THE THEORY OF STANDARDLY STRATIFIED ALGEBRAS I: STANDARDLY STRATIFIED RINGOIDS

Glasgow Mathematical Journal ◽

10.1017/s0017089520000476 ◽

2020 ◽

pp. 1-36

Author(s):

O. MENDOZA ◽

M. ORTÍZ ◽

C. SÁENZ ◽

V. SANTIAGO

Keyword(s):

Classical Theory ◽

General Class ◽

General Setting ◽

Finite Dimensional ◽

Classical Notion ◽

Hereditary Algebras

Abstract We extend the classical notion of standardly stratified k-algebra (stated for finite dimensional k-algebras) to the more general class of rings, possibly without 1, with enough idempotents. We show that many of the fundamental results, which are known for classical standardly stratified algebras, can be generalized to this context. Furthermore, new classes of rings appear as: ideally standardly stratified and ideally quasi-hereditary. In the classical theory, it is known that quasi-hereditary and ideally quasi-hereditary algebras are equivalent notions, but in our general setting, this is no longer true. To develop the theory, we use the well-known connection between rings with enough idempotents and skeletally small categories (ringoids or rings with several objects).

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Global Perturbation of Nonlinear Eigenvalues

Advanced Nonlinear Studies ◽

10.1515/ans-2021-2127 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Julián López-Gómez ◽

Juan Carlos Sampedro

Keyword(s):

Perturbation Theory ◽

Spectral Parameter ◽

Classical Theory ◽

General Setting ◽

Perturbation Parameter ◽

Linear Operators ◽

Fredholm Operators ◽

Index Zero ◽

Finite Dimensional ◽

Nonlinear Eigenvalues

Abstract This paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface 𝔏 : [ a , b ] × [ c , d ] → Φ 0 ⁢ ( U , V ) {\mathfrak{L}:[a,b]\times[c,d]\to\Phi_{0}(U,V)} , ( λ , μ ) ↦ 𝔏 ⁢ ( λ , μ ) {(\lambda,\mu)\mapsto\mathfrak{L}(\lambda,\mu)} , depends continuously on the perturbation parameter, μ, and holomorphically, as well as nonlinearly, on the spectral parameter, λ, where Φ 0 ⁢ ( U , V ) {\Phi_{0}(U,V)} stands for the set of Fredholm operators of index zero between U and V. The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see [T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Class. Math., Springer, Berlin, 1995, Chapter 2, Section 5]).

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Free Extensions of Chiral Polytopes

Canadian Journal of Mathematics ◽

10.4153/cjm-1995-033-7 ◽

1995 ◽

Vol 47 (3) ◽

pp. 641-654 ◽

Cited By ~ 12

Author(s):

Egon Schulte ◽

Asia Ivić Weiss

Keyword(s):

Classical Theory ◽

Convex Polytope ◽

Regular Polytopes ◽

Geometric Structures ◽

Maximal Symmetry ◽

Abstract Polytopes ◽

Chiral Polytopes ◽

Classical Notion ◽

Extension Result ◽

Amalgamated Product

AbstractAbstract polytopes are discrete geometric structures which generalize the classical notion of a convex polytope. Chiral polytopes are those abstract polytopes which have maximal symmetry by rotation, in contrast to the abstract regular polytopes which have maximal symmetry by reflection. Chirality is a fascinating phenomenon which does not occur in the classical theory. The paper proves the following general extension result for chiral polytopes. If 𝒦 is a chiral polytope with regular facets 𝓕 then among all chiral polytopes with facets 𝒦 there is a universal such polytope 𝓟, whose group is a certain amalgamated product of the groups of 𝒦 and 𝓕. Finite extensions are also discussed.

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Torsion-Free and Divisible Modules Over Finite-Dimensional Algebras

Canadian Mathematical Bulletin ◽

10.4153/cmb-1996-014-9 ◽

1996 ◽

Vol 39 (1) ◽

pp. 111-114

Author(s):

F. Okoh

Keyword(s):

Rational Functions ◽

Indecomposable Module ◽

Torsion Theory ◽

Infinite Dimensional ◽

Finite Dimensional ◽

Dynkin Diagrams ◽

The Status ◽

Hereditary Algebras ◽

Divisible Modules ◽

Torsion Free

AbstractIf R is a Dedekind domain, then div splits i.e.; the maximal divisible submodule of every R-module M is a direct summand of M. We investigate the status of this result for some finite-dimensional hereditary algebras. We use a torsion theory which permits the existence of torsion-free divisible modules for such algebras. Using this torsion theory we prove that the algebras obtained from extended Coxeter- Dynkin diagrams are the only such hereditary algebras for which div splits. The field of rational functions plays an essential role. The paper concludes with a new type of infinite-dimensional indecomposable module over a finite-dimensional wild hereditary algebra.

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PURE-INJECTIVES RELATIVE TO A COTORSION PAIR: APPLICATIONS

Glasgow Mathematical Journal ◽

10.1017/s001708951200033x ◽

2012 ◽

Vol 55 (1) ◽

pp. 59-68

Author(s):

SERGIO ESTRADA ◽

PEDRO A. GUIL ASENSIO

Keyword(s):

Classical Theory ◽

General Class ◽

Cotorsion Pair ◽

Coherent Sheaves ◽

Accessible Categories

AbstractFinitely accessible categories naturally arise in the context of the classical theory of purity. In this paper we generalise the notion of purity for a more general class and introduce techniques to study such classes in terms of indecomposable pure injectives related to a new notion of purity. We apply our results in the study of the class of flat quasi-coherent sheaves on an arbitrary scheme.

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THE FINITISTIC DIMENSION CONJECTURE AND RELATIVELY PROJECTIVE MODULES

Communications in Contemporary Mathematics ◽

10.1142/s0219199713500041 ◽

2013 ◽

Vol 15 (02) ◽

pp. 1350004 ◽

Cited By ~ 7

Author(s):

CHANGCHANG XI ◽

DENGMING XU

Keyword(s):

Left Ideal ◽

General Setting ◽

Projective Dimension ◽

Homological Dimension ◽

Projective Modules ◽

Finitistic Dimension ◽

Artin Algebra ◽

Finite Dimensional ◽

New Formulation ◽

Dimension Conjecture

The famous finitistic dimension conjecture says that every finite-dimensional 𝕂-algebra over a field 𝕂 should have finite finitistic dimension. This conjecture is equivalent to the following statement: If B is a subalgebra of a finite-dimensional 𝕂-algebra A such that the radical of B is a left ideal in A, and if A has finite finitistic dimension, then B has finite finitistic dimension. In the paper, we shall work with a more general setting of Artin algebras. Let B be a subalgebra of an Artin algebra A such that the radical of B is a left ideal in A. (1) If the category of all finitely generated (A, B)-projective A-modules is closed under taking A-syzygies, then fin.dim (B) ≤ fin.dim (A) + fin.dim (BA) + 3, where fin.dim (A) denotes the finitistic dimension of A, and where fin.dim (BA) stands for the supremum of the projective dimensions of those direct summands of BA that have finite projective dimension. (2) If the extension B ⊆ A is n-hereditary for a non-negative integer n, then gl.dim (A) ≤ gl.dim (B) + n. Moreover, we show that the finitistic dimension of the trivially twisted extension of two algebras of finite finitistic dimension is again finite. Also, a new formulation of the finitistic dimension conjecture in terms of relative homological dimension is given. Our approach in this paper is completely different from the one in our earlier papers.

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Nonnegative Matrices

Hidden Markov Processes ◽

10.23943/princeton/9780691133157.003.0003 ◽

2014 ◽

Author(s):

M. Vidyasagar

Keyword(s):

Markov Processes ◽

General Class ◽

Transition Matrix ◽

Stochastic Matrix ◽

Nonnegative Matrix ◽

General Setting ◽

Special Kind ◽

State Transition Matrix ◽

Nonnegative Matrices ◽

Frobenius Theorem

This chapter deals with nonnegative matrices, which are relevant in the study of Markov processes because the state transition matrix of such a process is a special kind of nonnegative matrix, known as a stochastic matrix. However, it turns out that practically all of the useful properties of a stochastic matrix also hold for the more general class of nonnegative matrices. Hence it is desirable to present the theory in the more general setting, and then specialize to Markov processes. The chapter first considers the canonical form for nonnegative matrices, including irreducible matrices and periodic irreducible matrices, before discussing the Perron–Frobenius theorem for primitive matrices and for irreducible matrices.

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On semi-infinite cohomology of finite-dimensional graded algebras

Compositio Mathematica ◽

10.1112/s0010437x09004382 ◽

2010 ◽

Vol 146 (2) ◽

pp. 480-496 ◽

Cited By ~ 1

Author(s):

Roman Bezrukavnikov ◽

Leonid Positselski

Keyword(s):

Quantum Group ◽

General Setting ◽

Derived Categories ◽

Graded Algebras ◽

Root Of Unity ◽

Finite Dimensional ◽

Small Quantum ◽

Definition Of

AbstractWe describe a general setting for the definition of semi-infinite cohomology of finite-dimensional graded algebras, and provide an interpretation of such cohomology in terms of derived categories. We apply this interpretation to compute semi-infinite cohomology of some modules over the small quantum group at a root of unity, generalizing an earlier result of Arkhipov (posed as a conjecture by B. Feigin).

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