Preprojective modules over artin algebras

Auslander, Maurice; Smalø, Sverre O.

doi:10.1016/0021-8693(80)90113-1

Cited by 418 publications

(324 citation statements)

References 6 publications

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“…Moreover, one says that C is contravariantly finite in A if every object in A has a right C-approximation, covariantly finite in A if every object in A has a left C-approximation, and functorially finite if it is both contravariantly finite and covariantly finite in A; see [7].…”

Section: Preliminariesmentioning

confidence: 99%

Almost Split Sequences and Approximations

Liu

Paquette

2012

Algebr Represent Theor

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Abstract. Let A be an exact category, that is, an extension-closed full subcategory of an abelian category. First, we give new characterizations of an almost split sequence in A, which yields some necessary and sufficient conditions for A to have an almost split sequence with prescribed end terms. Then, we study when an almost split sequence in A induces an almost split sequence in an exact subcategory C of A. In case A has almost split sequences and C is Ext-finite and Krull-Schmidt, we obtain a necessary and sufficient condition for C to have almost split sequences. Finally, we show some applications of these results.

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Section: Preliminariesmentioning

confidence: 99%

Almost Split Sequences and Approximations

Liu

Paquette

2012

Algebr Represent Theor

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“…In particular, we will see that, in this situation, the map φ constructed in Proposition 15 is a group isomorphism Out X (A) → OutX(B), provided that B X A is a twosided tilting complex inverse to X. Moreover, we will obtain U = V, which will provide us with an alternate description of φ as p 2 …”

Section: Proof (1) =⇒ (2) One Readily Verifies That (1) Is Equivalementioning

confidence: 90%

“…But A and B fail to be derived equivalent, since they do not have the same number of simple modules. More specific examples of such modules M are the preprojective modules (in the sense of [2]) which are devoid of projective direct summands. In case Out(A) is finite, all modules M with vanishing A-dual Hom A (M, A) qualify, of course.…”

Section: -Mod) and Analogously That In D(a-mod) As Well As Its Out-mentioning

confidence: 99%

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Geometry of chain complexes and outer automorphisms under derived equivalence

Huisgen-Zimmermann

Saorı́n

2001

Trans. Amer. Math. Soc.

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Abstract. The two main theorems proved here are as follows: If A is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of A is invariant under derived equivalence. This invariance is obtained as a consequence of the following generalization of a result of Voigt. Namely, given an appropriate geometrization Comp A d of the family of finite A-module complexes with fixed sequence d of dimensions and an "almost projective" complex X ∈ Comp A d , there exists a canonical vector space embedding, where G is the pertinent product of general linear groups acting on Comp A d , tangent spaces at X are denoted by T X (−), and X is identified with its image in the derived category D b (A-Mod).

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confidence: 99%

Preprojective Modules and Auslander-Reiten Components

Liu

2003

Communications in Algebra

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In [2], Auslander and Smalø introduced and studied extensively preprojective modules and preinjective modules over an artin algebra. We now call a module hereditarily preprojective or hereditarily preinjective if its submodules are all preprojective or its quotient modules are all preinjective, respectively. In [4], Coelho studied Auslander-Reiten components containing only hereditarily preprojective modules and gave a number of characterizations of such components. We shall study further these modules by using the description of shapes of semi-stable Auslander-Reiten components; see [6,7]. Our results will imply the result of Coelho [4, (1.2)] and that of Auslander-Smalø [2, (9.16)]. As an application, moreover, we shall show that a stable Auslander-Reiten component with "few" stable maps in TrD-direction is of shape Z ZA ∞ . Preliminaries on Auslander-Reiten componentsThroughout this note, A denotes an artin algebra, mod A the category of finitely generated right A-modules, and rad ∞ (mod A) the infinite radical of mod A. Let Γ A be the Auslander-Reiten quiver of A which is defined in such a way that its vertices form a complete set of the representatives of isoclasses of the indecomposables of mod A. We denote by τ the Auslander-Reiten translation DTr. The reader is referred to [7] for notions not defined here. We first reformulate a result stated in [7, (2.3)] for later use. Its proof can be found in the proofs of [7, (2.2), (2.3)]. (1) τ n X lies on an oriented cycle in Γ A for infinitely many n > 0. Proposition. Let Γ be a left stable component of Γ A with no τ -periodic module. If Γ contains an oriented cycle, then every module in Γ admits at most two immediate successors in Γ and there exists an infinite sectional path(2) A module is a predecessor of τ n X, for infinitely many n > 0, in Γ A . Proof. Assume that either (1) or (2) occurs. In particular, X is left stable. It suffices to consider the case where X is not τ -periodic. Then there exists s ≥ 0 such that the τ n X with n ≥ s lie in an infinite non τ -periodic left stable component Γ of Γ A . Suppose that Γ contains no oriented cycle. Then Γ admits a section ∆, and hence Γ is embedded in Z Z∆; see [7, (3.1), (3.4)]. For modules M, N in Γ , there exist at most finitely many n ≥ 0 such that N is predecessor in Γ of τ n M . Moreover, applying some power of τ , we may assume 1

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Preprojective modules over artin algebras

Cited by 418 publications

References 6 publications

Almost Split Sequences and Approximations

Almost Split Sequences and Approximations

Geometry of chain complexes and outer automorphisms under derived equivalence

Preprojective Modules and Auslander-Reiten Components

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