On semigroups of endomorphisms of biregular algebras

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.

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

Discrete Mathematics and Applications ◽

10.1515/dma-2018-0030 ◽

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.

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Relative Quotient Triangulated Categories

Algebra Colloquium ◽

10.1142/s1005386714000157 ◽

2014 ◽

Vol 21 (02) ◽

pp. 195-206 ◽

Cited By ~ 1

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.

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THE HOCHSCHILD COHOMOLOGY RING MODULO NILPOTENCE OF A MONOMIAL ALGEBRA

Journal of Algebra and Its Applications ◽

10.1142/s0219498806001648 ◽

2006 ◽

Vol 05 (02) ◽

pp. 153-192 ◽

Cited By ~ 13

Author(s):

EDWARD L. GREEN ◽

NICOLE SNASHALL ◽

ØYVIND SOLBERG

Keyword(s):

Hochschild Cohomology ◽

Cohomology Ring ◽

Finite Dimensional Algebra ◽

Finitely Generated ◽

Dimensional Algebra ◽

Monomial Algebra ◽

Hochschild Cohomology Ring ◽

Finite Dimensional ◽

Algebra Over A Field ◽

The Ideal

For a finite dimensional monomial algebra Λ over a field K we show that the Hochschild cohomology ring of Λ modulo the ideal generated by homogeneous nilpotent elements is a commutative finitely generated K-algebra of Krull dimension at most one. This was conjectured to be true for any finite dimensional algebra over a field in [13].

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Semigroup ideals in rings

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700016633 ◽

1979 ◽

Vol 27 (1) ◽

pp. 51-58

Author(s):

David A. Hill

Keyword(s):

Principal Ideal ◽

Finite Dimensional Algebra ◽

Dimensional Algebra ◽

Finite Dimensional ◽

Structure Theorems ◽

Semigroup Ideal ◽

Algebra Over A Field

AbstractA ringRis called anl-ring (r-ring) in caseRcontains an indentity and every left (right) semigroup ideal is a left (right) ring ideal. A number of structure theorems are obtained forl-rings whenRis left noetherian and left artinian. It is shown that left noetherianl-rings are local left principal ideal rings. WhenRis a finite dimensional algebra over a field, the property of being anl-ring is equivalent to being anr-ring. However, examples are given to show that these two concepts are in general not equivalent even in the artinian case.

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Interlacing methods and large indecomposables

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700046037 ◽

1970 ◽

Vol 3 (3) ◽

pp. 337-348 ◽

Cited By ~ 4

Author(s):

S. E. Dickson ◽

G. M. Kelly

Keyword(s):

Artinian Ring ◽

Finite Dimensional Algebra ◽

Finitely Generated ◽

Dimensional Algebra ◽

Infinite Dimensional ◽

Direct Sum ◽

Number Of Generators ◽

Finite Dimensional ◽

Factor Module ◽

Algebra Over A Field

The method of interlacing of modules, like amalgamation of groups, is a way of getting new objects from old. Briefly, the interlacing module we consider is a certain factor module of a direct sum of copies (finite or infinite) of an original module M. The conditions given in a previous paper by the first author in order that the interlacing module (using finitely many copies) be indecomposable are here greatly weakened, and we further allow the number of copies of the original to be infinite. R. Colby has shown that if R is a left artinian ring, the existence of a bound on the number of generators required for any indecomposable finitely-generated left R-module implies that R has a distributive lattice of two-sided ideals. This result is extended to rings whose identity is a sum of orthogonal local idempotents.For these rings the same distributivity is proved in case every indecomposable interlacing module of the above type which begins with an indecomposable projective M is finitely-generated. A consequence is that any finite-dimensional algebra over a field having infinitely many two-sided ideals has infinite-dimensional indecomposables.

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Residually finite dimensional algebras and polynomial almost identities

Journal of Algebra and Its Applications ◽

10.1142/s0219498822500384 ◽

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,

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Rings whose Indecomposable Injective Modules are Uniserial

Canadian Journal of Mathematics ◽

10.4153/cjm-1982-055-3 ◽

1982 ◽

Vol 34 (4) ◽

pp. 797-805 ◽

Cited By ~ 7

Author(s):

David A. Hill

Keyword(s):

Artinian Ring ◽

Dimensional Algebra ◽

Artinian Rings ◽

Direct Sum ◽

Injective Modules ◽

Finite Dimensional ◽

Uniserial Modules ◽

Left And Right ◽

Finitely Generated Modules ◽

Algebra Over A Field

A module is uniserial in case its submodules are linearly ordered by inclusion. A ring R is left (right) serial if it is a direct sum of uniserial left (right) R-modules. A ring R is serial if it is both left and right serial. It is well known that for artinian rings the property of being serial is equivalent to the finitely generated modules being a direct sum of uniserial modules [8]. Results along this line have been generalized to more arbitrary rings [6], [13].This article is concerned with investigating rings whose indecomposable injective modules are uniserial. The following question is considered which was first posed in [4]. If an artinian ring R has all indecomposable injective modules uniserial, does this imply that R is serial? The answer is yes if R is a finite dimensional algebra over a field. In this paper it is shown, provided R modulo its radical is commutative, that R has every left indecomposable injective uniserial implies that R is right serial.

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

Mathematische Zeitschrift ◽

10.1007/s00209-020-02475-y ◽

2020 ◽

Vol 296 (3-4) ◽

pp. 1157-1183 ◽

Cited By ~ 1

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.

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Involutions on finite-dimensional algebras over real closed fields

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700010193 ◽

2004 ◽

Vol 77 (1) ◽

pp. 123-128 ◽

Cited By ~ 1

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.

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