scholarly journals τ-Complexity and Tilting Modules

2016 ◽  
Vol 2016 ◽  
pp. 1-4 ◽  
Author(s):  
Lijing Zheng ◽  
Chonghui Huang ◽  
Qianhong Wan
Keyword(s):  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Tilting Modules ◽  
Dimensional Algebra ◽  
Finite Dimensional

Let A be a finite dimensional algebra over an algebraic closed field k. In this note, we will show that if T is a separating and splitting tilting A-module, then τ-complexities of A and B are equal, where B=EndA(T).

Separating Splitting Tilting Modules and Hereditary Algebras

1987 ◽  
Vol 30 (2) ◽  
pp. 177-181 ◽  
Author(s):  
Ibrahim Assem
Keyword(s):  
Exact Sequence ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Finitely Generated ◽  
Tilting Module ◽  
Tilting Modules ◽  
Dimensional Algebra ◽  
Direct Sums ◽  
Finite Dimensional ◽  
Hereditary Algebras

AbstractLet A be a finite-dimensional algebra over an algebraically closed field. By module is meant a finitely generated right module. A module T^ is called a tilting module if and there exists an exact sequence 0 → A^ → T' → T" → 0 with T'. T" direct sums of summands of T. Let B = End T^·T^ is called separating (respectively, splitting) if every indecomposable A-module M (respectively, B-module N) is such that either Hom^(T,M) = 0 or (respectively, N ⊗ T = 0 or . We prove that A is hereditary provided the quiver of A has no oriented cycles and every separating tilting module is splitting.

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.

scholarly journals -tilting theory

2013 ◽  
Vol 150 (3) ◽  
pp. 415-452 ◽  
Author(s):  
Takahide Adachi ◽  
Osamu Iyama ◽  
Idun Reiten
Keyword(s):  
Direct Summand ◽  
Triangulated Category ◽  
Finite Dimensional Algebra ◽  
Tilting Module ◽  
Tilting Modules ◽  
Dimensional Algebra ◽  
Tilting Theory ◽  
Finite Dimensional ◽  
Cluster Tilting ◽  
Tilting Objects

AbstractThe aim of this paper is to introduce $\tau $-tilting theory, which ‘completes’ (classical) tilting theory from the viewpoint of mutation. It is well known in tilting theory that an almost complete tilting module for any finite-dimensional algebra over a field $k$ is a direct summand of exactly one or two tilting modules. An important property in cluster-tilting theory is that an almost complete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly two cluster-tilting objects. Reformulated for path algebras $kQ$, this says that an almost complete support tilting module has exactly two complements. We generalize (support) tilting modules to what we call (support) $\tau $-tilting modules, and show that an almost complete support $\tau $-tilting module has exactly two complements for any finite-dimensional algebra. For a finite-dimensional $k$-algebra $\Lambda $, we establish bijections between functorially finite torsion classes in $ \mathsf{mod} \hspace{0.167em} \Lambda $, support $\tau $-tilting modules and two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$. Moreover, these objects correspond bijectively to cluster-tilting objects in $ \mathcal{C} $ if $\Lambda $ is a 2-CY tilted algebra associated with a 2-CY triangulated category $ \mathcal{C} $. As an application, we show that the property of having two complements holds also for two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$.

scholarly journals THE UNIVERSAL COVER OF A MONOMIAL TRIANGULAR ALGEBRA WITHOUT MULTIPLE ARROWS

2008 ◽  
Vol 07 (04) ◽  
pp. 443-469 ◽  
Author(s):  
PATRICK LE MEUR
Keyword(s):  
Fundamental Group ◽  
Universal Cover ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
Triangular Algebra ◽  
Underlying Space ◽  
Finite Dimensional ◽  
Ordinary Quiver

Let A be a basic connected finite dimensional algebra over an algebraically closed field k. Assuming that A is monomial and that the ordinary quiver Q of A has no oriented cycle and no multiple arrows, we prove that A admits a universal cover with group the fundamental group of the underlying space of Q.

scholarly journals Torsion Pairs and Ringel Duality for Schur Algebras

2021 ◽  
Author(s):  
Karin Erdmann ◽  
Stacey Law
Keyword(s):  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Schur Algebras ◽  
Morita Equivalent ◽  
Finite Dimensional ◽  
Endomorphism Algebras ◽  
Hereditary Algebras ◽  
Torsion Pairs ◽  
Functorial Approach

AbstractLet A be a finite-dimensional algebra over an algebraically closed field. We use a functorial approach involving torsion pairs to construct embeddings of endomorphism algebras of basic projective A–modules P into those of the torsion submodules of P. As an application, we show that blocks of both the classical and quantum Schur algebras S(2,r) and Sq(2,r) in characteristic p > 0 are Morita equivalent as quasi-hereditary algebras to their Ringel duals if they contain 2pk simple modules for some k.

Cocovers and tilting modules

1989 ◽  
Vol 106 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Jeremy Rickard ◽  
Aidan Schofield
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Finite Dimensional Algebra ◽  
Tilting Module ◽  
Tilting Modules ◽  
Dimensional Algebra ◽  
Direct Sums ◽  
Finite Dimensional ◽  
Dimension One

We recall [5] that a module T for a finite-dimensional algebra Λ is called a tilting module if(i) T has protective dimension one;(ii) (iii) there is a short exact sequence 0 → Λ → T0 → T1 → 0 with T0 and T1 in add (T), the category of direct summands of direct sums of copies of T.

Degrees in Auslander–Reiten components with almost split sequences of at most two middle termss

2015 ◽  
Vol 14 (07) ◽  
pp. 1550106 ◽  
Author(s):  
Claudia Chaio
Keyword(s):  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Finite Degree ◽  
Artin Algebra ◽  
Dimensional Algebra ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Finite Dimensional ◽  
Almost Split Sequences ◽  
Irreducible Morphisms

We consider A to be an artin algebra. We study the degrees of irreducible morphisms between modules in Auslander–Reiten components Γ having only almost split sequences with at most two indecomposable middle terms, that is, α(Γ) ≤ 2. We prove that if f : X → Y is an irreducible epimorphism of finite left degree with X or Y indecomposable, then there exists a module Z ∈ Γ and a morphism φ ∈ ℜn(Z, X)\ℜn+1(Z, X) for some positive integer n such that fφ = 0. In particular, for such components if A is a finite dimensional algebra over an algebraically closed field and f = (f1, f2)t : X → Y1 ⊕ Y2 is an irreducible epimorphism of finite left degree then we show that dl(f) = dl(f1) + dl(f2). We also characterize the artin algebras of finite representation type with α(ΓA) ≤ 2 in terms of a finite number of irreducible morphisms with finite degree.

Invariants of finite-dimensional algebras recognized by the semigroup of conjugacy classes of left ideals

2017 ◽  
Vol 16 (10) ◽  
pp. 1750182
Author(s):  
Arkadiusz Mȩcel ◽  
Jan Okniński
Keyword(s):  
Jacobson Radical ◽  
Semigroup Algebra ◽  
Conjugacy Classes ◽  
Maximal Subgroups ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Ordinary Quiver

We study the semigroup structure on the set [Formula: see text] of conjugacy classes of left ideals of a finite-dimensional algebra [Formula: see text] over an algebraically closed field [Formula: see text], equipped with the natural multiplication inherited from [Formula: see text], and the structure of the contracted semigroup algebra [Formula: see text]. It is shown that [Formula: see text] has a finite chain of ideals with either nilpotent or completely [Formula: see text]-simple factors with trivial maximal subgroups, so in particular it is locally finite. The ordinary quiver [Formula: see text] of [Formula: see text] is proved to be a subquiver of [Formula: see text], if [Formula: see text] is finite. Moreover, in this case, the structure of [Formula: see text] determines, up to isomorphism, the structure of the algebra [Formula: see text] modulo its Jacobson radical. Combining these results we show that if the semigroup [Formula: see text] is finite, then it determines the structure of any (not necessarily basic) triangular algebra [Formula: see text] which admits a normed presentation.

DEGENERATION-LIKE ORDERS IN TRIANGULATED CATEGORIES

2005 ◽  
Vol 04 (05) ◽  
pp. 587-597 ◽  
Author(s):  
BERNT TORE JENSEN ◽  
XIUPING SU ◽  
ALEXANDER ZIMMERMANN
Keyword(s):  
Partial Order ◽  
Natural Generalization ◽  
Derived Categories ◽  
Derived Category ◽  
Closed Field ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional

In an earlier paper we defined a relation ≤Δ between objects of the derived category of bounded complexes of modules over a finite dimensional algebra over an algebraically closed field. This relation was shown to be equivalent to the topologically defined degeneration order in a certain space [Formula: see text] for derived categories. This space was defined as a natural generalization of varieties for modules. We remark that this relation ≤Δ is defined for any triangulated category and show that under some finiteness assumptions on the triangulated category ≤Δ is always a partial order.

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.

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