Quotients of Incidence Algebras and the Euler Characteristic

Assem, Ibrahim; Castonguay, Diane; Marcos, Eduardo N.; Trepode, Sonia

doi:10.1080/00927870500243072

Cited by 5 publications

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“…By Theorem A, it is strongly simply connected. Hence there exists an incidence algebra kΣ such that Q A = Q kΣ (see, for instance, [2]). But then A contains no bypass, a contradiction.…”

Section: Corollary Let a Be A Simply Connected Schurian Algebra Contmentioning

confidence: 99%

Strongly simply connected schurian algebras and multiplicative bases

et al. 2005

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In this paper, we define concepts of crowns and quasi-crowns, valid in an arbitrary schurian algebra, and which generalise the corresponding concepts in an incidence algebra. We show first that a triangular schurian algebra is strongly simply connected if and only if it is simply connected and contains no quasi-crown. We then prove that the absence of quasi-crowns in a triangular schurian algebra implies the existence of a multiplicative basis.

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“…By Theorem A, it is strongly simply connected. Hence there exists an incidence algebra kΣ such that Q A = Q kΣ (see, for instance, [2]). But then A contains no bypass, a contradiction.…”

Section: Corollary Let a Be A Simply Connected Schurian Algebra Contmentioning

confidence: 99%

Strongly simply connected schurian algebras and multiplicative bases

et al. 2005

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“…Recall that min I = {1, 2}. Write G = F 1 ∩ F 2 = U (1,2). Denote by H the set of elements of G which are G-connected with 1.…”

Section: Proposition 3 If a Poset I Satisfies Conditions (1)-(4) Andmentioning

confidence: 99%

“…Let I be a set with a partial order ¢, considered as a small category, in particular, we interpret i ¢ j as a (unique) arrow from i to j. Denote by Set the category of sets. In analogy with the case of directed posets (see [12,Chapter VIII]), by an I-spectrum 1 Alternatively, an I-spectrum M I consists of (possibly empty) sets M i (i ∈ I) and maps ϕ ij : M i → M j (i ¢ j) such that ∀i ∈ I ϕ ii = id (the identity map),…”

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confidence: 99%

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On Colimits Over Arbitrary Posets

Dokuchaev

Novikov

2015

Glasgow Math. J.

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We characterize those partially ordered sets I for which the canonical maps M i → colim M j into colimits of abstract sets are always injective, provided that the transition maps are injective. We also obtain some consequences for colimits of vector spaces.2010 Mathematics Subject Classification. 18A30, 06A06. Introduction.Crowns arise in various problems related to partially ordered sets (posets). Thus, for example, they appear in the study of retracts and fixed points (see [7]), in calculation of the cohomological dimension (see [5]), in applications to homotopy theory (see [11]) and in the investigation of incidence algebras and their quotients (see [1] and [6]). At the same time, quite often they play a "negative" role: the absence of crowns of some kind ensures the existence of certain good properties of posets or constructions related to them. For instance, an incidence algebra κ[S] of a finite poset S is completely separating if and only if S contains no crowns [6]. It is not surprising that such a situation arises in a problem of colimits which is discussed in this note: roughly speaking, the crowns are antagonists of directed sets for which colimits are usually considered and well understood (note that colimits over directed posets are called directed colimits, or direct limits, or inductive limits).More precisely, if one takes a directed colimit colim M i (also denoted by lim − → M i ), where i runs over a directed poset I, such that the transition maps ϕ ij : M i → M j are injective, then the canonical maps M i → colim M j are also injective, which is a crucial property for applications. Thus, one may wonder which are the posets I for which this always happens. We completely characterize such I in the case when the M i 's are abstract sets and obtain consequences for the colimits of vector spaces.This problem is related to a similar question about the ring-theoretic version of cross-sectional algebras of Fell bundles over inverse semigroups studied in [8], since such algebras are epimorphic images of colimits of vector spaces over non-necessarily directed posets. In the C * -algebraic context, it is proved that the fibres are canonically embedded into the cross-sectional algebra; however, the abstract ring theoretic version

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“…In particular, A is schurian. (4) In [4] are given criteria for the strong simple connectedness of quotients of incidence algebras.…”

Section: Theoremmentioning

confidence: 99%

Special biserial algebras with no outer derivations

Assem¹,

Bustamante²,

Meur³

2011

Colloq. Math.

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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.Comment: 13 pages, submitte

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Quotients of Incidence Algebras and the Euler Characteristic

Cited by 5 publications

References 14 publications

Strongly simply connected schurian algebras and multiplicative bases

Strongly simply connected schurian algebras and multiplicative bases

On Colimits Over Arbitrary Posets

Special biserial algebras with no outer derivations

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