scholarly journals On the finiteness of the derived equivalence classes of some stable endomorphism rings

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1157-1183 ◽  
Author(s):  
Jenny August
Keyword(s):  
Equivalence Class ◽  
Minimal Model ◽  
Equivalence Classes ◽  
Finite Dimensional Algebra ◽  
Derived Equivalence ◽  
Endomorphism Rings ◽  
Dimensional Algebra ◽  
Rigid Objects ◽  
Finite Dimensional ◽  
Frobenius Category

Abstract We prove that the stable endomorphism rings of rigid objects in a suitable Frobenius category have only finitely many basic algebras in their derived equivalence class and that these are precisely the stable endomorphism rings of objects obtained by iterated mutation. The main application is to the Homological Minimal Model Programme. For a 3-fold flopping contraction $$f :X \rightarrow {\mathrm{Spec}\;}\,R$$ f : X → Spec R , where X has only Gorenstein terminal singularities, there is an associated finite dimensional algebra $$A_{{\text {con}}}$$ A con known as the contraction algebra. As a corollary of our main result, there are only finitely many basic algebras in the derived equivalence class of $$A_{\text {con}}$$ A con and these are precisely the contraction algebras of maps obtained by a sequence of iterated flops from f. This provides evidence towards a key conjecture in the area.

Relative Quotient Triangulated Categories

2014 ◽  
Vol 21 (02) ◽  
pp. 195-206 ◽  
Author(s):  
Shengyong Pan
Keyword(s):  
Triangulated Category ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Stable Category ◽  
Gorenstein Algebra ◽  
Finite Dimensional ◽  
Frobenius Category ◽  
Relative Version ◽  
Relative Derived Category ◽  
Algebra Over A Field

Let A be a finite dimensional algebra over a field k. We consider a subfunctor F of [Formula: see text], which has enough projectives and injectives such that [Formula: see text] is of finite type, where [Formula: see text] denotes the set of F-projectives. One can get the relative derived category [Formula: see text] of A-mod. For an F-self-orthogonal module TF, we discuss the relation between the relative quotient triangulated category [Formula: see text] and the relative stable category of the Frobenius category of TF-Cohen-Macaulay modules. In particular, for an F-Gorenstein algebra A and an F-tilting A-module TF, we get a triangle equivalence between [Formula: see text] and the relative stable category of TF-Cohen-Macaulay modules. This gives the relative version of a result of Chen and Zhang.

scholarly journals On Quiver Grassmannians and Orbit Closures for Gen-Finite Modules

2021 ◽  
Author(s):  
Matthew Pressland ◽  
Julia Sauter
Keyword(s):  
Orbit Closure ◽  
Finite Dimensional Algebra ◽  
Tilting Module ◽  
Endomorphism Rings ◽  
Dimensional Algebra ◽  
Indecomposable Modules ◽  
Quiver Grassmannians ◽  
Finite Dimensional ◽  
Orbit Closures ◽  
Endomorphism Algebras

AbstractWe show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module, whose tilted algebra B is related to A by a recollement. Let M be a gen-finite A-module, meaning there are only finitely many indecomposable modules generated by M. Using the canonical tilts of endomorphism algebras of suitable cogenerators associated to M, and the resulting recollements, we construct desingularisations of the orbit closure and quiver Grassmannians of M, thus generalising all results from previous work of Crawley-Boevey and the second author in 2017. We provide dual versions of the key results, in order to also treat cogen-finite modules.

scholarly journals Duality and socle generators for residual intersections

2019 ◽  
Vol 2019 (756) ◽  
pp. 183-226 ◽  
Author(s):  
David Eisenbud ◽  
Bernd Ulrich
Keyword(s):  
Gorenstein Ring ◽  
Jacobian Determinant ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Duality Results ◽  
Finite Dimensional ◽  
Algebra Over A Field ◽  
Residue Theory

AbstractWe prove duality results for residual intersections that unify and complete results of van Straten, Huneke–Ulrich and Ulrich, and settle conjectures of van Straten and Warmt.Suppose that I is an ideal of codimension g in a Gorenstein ring, and {J\subset I} is an ideal with {s=g+t} generators such that {K:=J:I} has codimension s. Let {{\overline{I}}} be the image of I in {{\overline{R}}:=R/K}.In the first part of the paper we prove, among other things, that under suitable hypotheses on I, the truncated Rees ring {{\overline{R}}\oplus{\overline{I}}\oplus\cdots\oplus{\overline{I}}{}^{t+1}} is a Gorenstein ring, and that the modules {{\overline{I}}{}^{u}} and {{\overline{I}}{}^{t+1-u}} are dual to one another via the multiplication pairing into {{{\overline{I}}{}^{t+1}}\cong{\omega_{\overline{R}}}}.In the second part of the paper we study the analogue of residue theory, and prove that, when {R/K} is a finite-dimensional algebra over a field of characteristic 0 and certain other hypotheses are satisfied, the socle of {I^{t+1}/JI^{t}\cong{\omega_{R/K}}} is generated by a Jacobian determinant.

Necessary conditions for power commuting in finite-dimensional algebras over a field

2018 ◽  
Vol 28 (5) ◽  
pp. 339-344
Author(s):  
Andrey V. Zyazin ◽  
Sergey Yu. Katyshev
Keyword(s):  
Necessary Conditions ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional ◽  
Algebra Over A Field ◽  
Finite Dimensional Algebras

Abstract Necessary conditions for power commuting in a finite-dimensional algebra over a field are presented.

scholarly journals Residually finite dimensional algebras and polynomial almost identities

2020 ◽  
pp. 2250038
Author(s):  
Michael Larsen ◽  
Aner Shalev
Keyword(s):  
Lie Algebra ◽  
Finite Index ◽  
Open Group ◽  
Finite Codimension ◽  
Finite Dimensional Algebra ◽  
Nilpotent Ideal ◽  
Dimensional Algebra ◽  
Residually Finite ◽  
Finite Dimensional ◽  
Theoretic Problem

Let [Formula: see text] be a residually finite dimensional algebra (not necessarily associative) over a field [Formula: see text]. Suppose first that [Formula: see text] is algebraically closed. We show that if [Formula: see text] satisfies a homogeneous almost identity [Formula: see text], then [Formula: see text] has an ideal of finite codimension satisfying the identity [Formula: see text]. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra [Formula: see text] over [Formula: see text] is almost [Formula: see text]-Engel, then [Formula: see text] has a nilpotent (respectively, locally nilpotent) ideal of finite codimension if char [Formula: see text] (respectively, char [Formula: see text]). Next, suppose that [Formula: see text] is finite (so [Formula: see text] is residually finite). We prove that, if [Formula: see text] satisfies a homogeneous probabilistic identity [Formula: see text], then [Formula: see text] is a coset identity of [Formula: see text]. Moreover, if [Formula: see text] is multilinear, then [Formula: see text] is an identity of some finite index ideal of [Formula: see text]. Along the way we show that if [Formula: see text] has degree [Formula: see text], and [Formula: see text] is a finite [Formula: see text]-algebra such that the probability that [Formula: see text] (where [Formula: see text] are randomly chosen) is at least [Formula: see text], then [Formula: see text] is an identity of [Formula: see text]. This solves a ring-theoretic analogue of a (still open) group-theoretic problem posed by Dixon,

scholarly journals Involutions on finite-dimensional algebras over real closed fields

2004 ◽  
Vol 77 (1) ◽  
pp. 123-128 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Characteristic Zero ◽  
Real Closed Field ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
Real Closed Fields ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Finite Dimensional Algebras

AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

Isotopes of simple algebras of arbitrary dimension

2019 ◽  
Vol 13 (06) ◽  
pp. 2050108 ◽  
Author(s):  
Vita Glizburg ◽  
Sergey Pchelintsev
Keyword(s):  
Jordan Algebra ◽  
Matrix Algebra ◽  
Dimensional Space ◽  
Arbitrary Dimension ◽  
Simple Algebra ◽  
Symmetric Bilinear Form ◽  
Finite Dimensional Algebra ◽  
Ground Field ◽  
Dimensional Algebra ◽  
Finite Dimensional

It is proved that every finite-dimensional algebra is embeddable in a simple finite-dimensional algebra (a suitable isotope of a matrix algebra). An isotope of the 2nd order matrix algebra over an infinite extension of the ground field may contain a trivial ideal. Every one-sided isotope of a simple unital alternative or Jordan algebra is a simple algebra. Besides, any isotope of a central simple non-Lie Maltsev algebra of characteristic other than 2 and 3 is a simple algebra. But an isotope of a simple Jordan algebra of the symmetric bilinear form on the infinite dimensional space may contain a trivial ideal.

Hopf algebra actions and transfer of Frobenius and symmetric properties

2020 ◽  
Vol 126 (1) ◽  
pp. 32-40
Author(s):  
S. Dăscălescu ◽  
C. Năstăsescu ◽  
L. Năstăsescu
Keyword(s):  
Hopf Algebra ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional ◽  
Finite Dimensional Hopf Algebra ◽  
Symmetric Properties

If $H$ is a finite-dimensional Hopf algebra acting on a finite-dimensional algebra $A$, we investigate the transfer of the Frobenius and symmetric properties through the algebra extensions $A^H\subset A\subset A\mathbin{\#} H$.

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