scholarly journals Grothendieck groups of the Khovanov–Kuperberg algebras

2015 ◽  
Vol 24 (14) ◽  
pp. 1550070
Author(s):  
Louis-Hadrien Robert
Keyword(s):  
Solid Torus ◽  
Projective Modules ◽  
Geometrical Properties ◽  
Finite Dimensional ◽  
Grothendieck Groups

The aim of this paper is to take benefit of the foam nature of the Khovanov–Kuperberg algebras to compute the Grothendieck groups of their categories of finite-dimensional projective modules. The computation relies on the Hattori–Stallings trace and some geometrical properties of foams in the solid torus.

scholarly journals A note on deformations of Gorenstein-projective modules over finite dimensional algebras

2019 ◽  
Vol 53 (supl) ◽  
pp. 245-256 ◽  
Author(s):  
José A. Vélez-Marulanda
Keyword(s):  
Projective Modules ◽  
Finitely Generated ◽  
Universal Deformation Rings ◽  
Finite Dimensional ◽  
Gorenstein Projective Modules ◽  
Deformation Rings ◽  
Finite Dimensional Algebras

In this note, we present a survey of results concerning universal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional algebras.

THE FINITISTIC DIMENSION CONJECTURE AND RELATIVELY PROJECTIVE MODULES

2013 ◽  
Vol 15 (02) ◽  
pp. 1350004 ◽  
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.

Grothendieck groups for categories of complexes

2008 ◽  
Vol 3 (1) ◽  
pp. 165-203 ◽  
Author(s):  
Hans-Bjørn Foxby ◽  
Esben Bistrup Halvorsen
Keyword(s):  
Local Ring ◽  
Finite Length ◽  
Full Subcategory ◽  
Grothendieck Group ◽  
Projective Dimension ◽  
Intersection Theorem ◽  
Projective Modules ◽  
Finitely Generated ◽  
Grothendieck Groups ◽  
Finitely Generated Modules

AbstractThe new intersection theorem states that, over a Noetherian local ring R, for any non-exact complex concentrated in degrees n,…,0 in the category P(length) of bounded complexes of finitely generated projective modules with finite-length homology, we must have n ≥ d = dim R.One of the results in this paper is that the Grothendieck group of P(length) in fact is generated by complexes concentrated in the minimal number of degrees: if Pd(length) denotes the full subcategory of P(length) consisting of complexes concentrated in degrees d,…0, the inclusion Pd(length) → P(length) induces an isomorphism of Grothendieck groups. When R is Cohen–Macaulay, the Grothendieck groups of Pd(length) and P(length) are naturally isomorphic to the Grothendieck group of the category M(length) of finitely generated modules of finite length and finite projective dimension. This and a family of similar results are established in this paper.

ITERATED POWER INTERSECTIONS OF IDEALS IN RINGS OF ITERATED DIFFERENTIAL POLYNOMIALS

2013 ◽  
Vol 12 (07) ◽  
pp. 1350020
Author(s):  
PAVEL PŘÍHODA ◽  
GENA PUNINSKI
Keyword(s):  
Characteristic Zero ◽  
Proper Ideal ◽  
Projective Modules ◽  
Differential Polynomials ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Noetherian Domain ◽  
Solvable Lie Algebra ◽  
Standard Application ◽  
Algebra Over A Field

Let R be an n-iterated ring of differential polynomials over a commutative noetherian domain which is a ℚ-algebra. We will prove that for every proper ideal I of R, the (n + 1)-iterated intersection I(n + 1) of powers of I equals zero. A standard application includes the freeness of non-finitely generated projective modules over such rings. If I is a proper ideal of the universal enveloping algebra of a finite-dimensional solvable Lie algebra over a field of characteristic zero, then we will improve the above estimate by showing that I(2) = 0.

Endomorphism Rings and Gabriel Topologies

1984 ◽  
Vol 36 (2) ◽  
pp. 193-205 ◽  
Author(s):  
Soumaya Makdissi Khuri
Keyword(s):  
Endomorphism Ring ◽  
Lattice Isomorphism ◽  
Close Connection ◽  
Projective Modules ◽  
Finitely Generated ◽  
Basic Tool ◽  
Endomorphism Rings ◽  
Finite Dimensional ◽  
Faithful Module ◽  
Correspondence Theorem

A basic tool in the usual presentation of the Morita theorems is the correspondence theorem for projective modules. Let RM be a left R-module and B = HomR(M, M). When M is a progenerator, there is a close connection (in fact a lattice isomorphism) between left R-submodules of M and left ideals of B, which can be applied to the solution of problems such as characterizing when the endomorphism ring of a finitely generated projective faithful module is simple or right Noetherian. More generally, Faith proved that this connection can be retained in suitably modified form when M is just a generator in R-mod ([4], [2], [3]). In this form the correspondence theorem can be applied to show, e.g., that, when RM is a generator, then (a): RM is finite-dimensional if and only if B is a left finite-dimensional ring and in this case d(RM) = d(BB), and (b): If RM is nonsingular then B is a left nonsingular ring ([6]).

scholarly journals H-joint numerical ranges

2002 ◽  
Vol 66 (1) ◽  
pp. 105-117
Author(s):  
Chi-Kwong Li ◽  
Leiba Rodman
Keyword(s):  
Numerical Range ◽  
Linear Operators ◽  
Dimensional Case ◽  
Geometrical Properties ◽  
Finite Dimensional ◽  
Sesquilinear Form ◽  
Algebraic Properties ◽  
Finite Dimensional Case ◽  
Joint Numerical Range

The notion of the joint numerical range of several linear operators with respect to a sesquilinear form is introduced. Geometrical properties of the joint numerical range are studied, in particular, convexity and angle points, in connection with the algebraic properties of the operators. The main focus is on the finite dimensional case.

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