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1
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Left sections and the left part of an artin algebra

100%
Assem I.
Colloquium Mathematicum
|
2009
|
tom 116
|
nr 2
273-300
EN
We define a notion of left section in an Auslander-Reiten component, by weakening one of the axioms for sections. We derive a generalisation of the Liu-Skowroński criterion for tilted algebras, then apply our results to describe the Auslander-Reiten components lying in the left part of an artin algebra.
2
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On tame repetitive algebras

64%
Assem I., Skowroński A.
Fundamenta Mathematicae
|
1993
|
tom 142
|
nr 1
59-84
EN
Let A be a finite dimensional algebra over an algebraically closed field, and denote by T(A) (respectively, Â) the trivial extension of A by its minimal injective cogenerator bimodule (respectively, the repetitive algebra of A). We characterise the algebras A such that  is tame and exhaustive, that is, the push-down functor mod  → mod T(A) associated with the covering functor  → T(A)\nsimto Â/(ν_A)$ is dense. We show that, if  is tame and exhaustive, then A is simply connected if and only if A is not an iterated tilted algebra of type $Â_m$. Then we prove that  is tame and exhaustive if and only if A is tilting-cotilting equivalent to an algebra which is either hereditary of Dynkin or Euclidean type or is tubular canonical.
3
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Endomorphism algebras of exceptional sequences over path algebras of type $\widetilde{\sym A}_{n}$

64%
Assem I., Zhang Y.
Colloquium Mathematicum
|
1998
|
tom 77
|
nr 2
271-292
4
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Full embeddings of almost split sequences over split-by-nilpotent extensions

64%
Assem I., Zacharia D.
Colloquium Mathematicum
|
1999
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tom 81
|
nr 1
21-31
EN
Let R be a split extension of an artin algebra A by a nilpotent bimodule $_A Q_A$, and let M be an indecomposable non-projective A-module. We show that the almost split sequences ending with M in mod A and mod R coincide if and only if $Hom_A (Q, τ_A M)$ = 0 and $M ⊗ _A Q = 0$.
5
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On split-by-nilpotent extensions

64%
Assem I., Zacharia D.
Colloquium Mathematicum
|
2003
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tom 98
|
nr 2
259-275
EN
Let A and R be two artin algebras such that R is a split extension of A by a nilpotent ideal. We prove that if R is quasi-tilted, or tame and tilted, then so is A. Moreover, generalizations of these properties, such as laura and shod, are also inherited. We also study the relationship between the tilting R-modules and the tilting A-modules.
6
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On the tameness of trivial extension algebras

64%
Assem I., de la Peña J. A.
Fundamenta Mathematicae
|
1996
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tom 149
|
nr 2
171-181
EN
For a finite dimensional algebra A over an algebraically closed field, let T(A) denote the trivial extension of A by its minimal injective cogenerator bimodule. We prove that, if $T_A$ is a tilting module and $B=End T_A$, then T(A) is tame if and only if T(B) is tame.
7
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Special biserial algebras with no outer derivations

51%
Assem I., Bustamante J., Le Meur P.
Colloquium Mathematicum
|
2011
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tom 125
|
nr 1
83-98
EN
Let A be a special biserial algebra over an algebraically closed field. We show that the first Hohchshild cohomology group of A with coefficients in the bimodule A vanishes if and only if A is representation-finite and simply connected (in the sense of Bongartz and Gabriel), if and only if the Euler characteristic of Q equals the number of indecomposable non-uniserial projective-injective A-modules (up to isomorphism). Moreover, if this is the case, then all the higher Hochschild cohomology groups of A vanish.
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