Homological theory of idempotent ideals

Auslander, Maurice; Platzeck, María Inés; Todorov, Gordana

doi:10.1090/s0002-9947-1992-1052903-5

Cited by 82 publications

(131 citation statements)

References 2 publications

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“…First we recall the definitions of quasi-hereditary algebras ( [CPS1]) and of left properly stratified algebras (compare with [CPS2,APT,ADL,KlM]). We fix an arbitrary ground field k. In this section, by an algebra we mean a finite-dimensional associative k-algebra.…”

Section: Quasi-hereditary Algebras and Left Properly Stratified Algebrasmentioning

confidence: 99%

Enright’s completions and injectively copresented modules

König

Mazorchuk

2002

Trans. Amer. Math. Soc.

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Abstract. Let A be a finite-dimensional simple Lie algebra over the complex numbers. It is shown that a module is complete (or relatively complete) in the sense of Enright if and only if it is injectively copresented by certain injective modules in the BGG-category O. Let A be the finite-dimensional algebra associated to a block of O. Then the corresponding block of the category of complete modules is equivalent to the category of eAe-modules for a suitable choice of the idempotent e. Using this equivalence, a very easy proof is given for Deodhar's theorem (also proved by Bouaziz) that completion functors satisfy braid relations. The algebra eAe is left properly and standardly stratified. It satisfies a double centralizer property similar to Soergel's "combinatorial description" of O. Its simple objects, their characters and their multiplicities in projective or standard objects are determined.

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Section: Quasi-hereditary Algebras and Left Properly Stratified Algebrasmentioning

confidence: 99%

Enright’s completions and injectively copresented modules

König

Mazorchuk

2002

Trans. Amer. Math. Soc.

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“…This result is known for modules over Artin algebras, where one may use minimal projective resolutions (see [1]). For general rings, projective covers of modules may not exist.…”

Section: And (2) M Has a Finite-type Resolution That Is There Is Anmentioning

confidence: 92%

Higher Algebraic K-theory of Ring Epimorphisms

Chen

2016

Algebr Represent Theor

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We establish formulas for computation of the higher algebraic K-groups of the endomorphism rings of objects linked by a morphism in an additive category. Let C be an additive category, and let Y → X be a covariant morphism of objects inEnd C ,Y (X) is the quotient ring of the endomorphism ring End C (X) of X modulo the ideal generated by all those endomorphisms of X which factorize through Y . Moreover, let R be a ring with identity, and let e be an idempotent element in R. If J := ReR is homological and R J has a finite projective resolution by finitely generated projective R-modules, then K n (R) ≃ K n (R/J) ⊕ K n (eRe) for all n ∈ N. This reduces calculations of the higher algebraic K-groups of R to those of the quotient ring R/J and the corner ring eRe, and can be applied to a large variety of rings: Standardly stratified rings, hereditary orders, affine cellular algebras and extended affine Hecke algebras of typeÃ.

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“…Applying pT, q to the above sequence we get an exact sequence (4) ¤ ¤ ¤ Ñ pT, C 1 q Ñ pT, C 0 q Ñ pT, DΛq Ñ 0. since T C. Consider the short exact sequence 0 Ñ pT, Y j 1 q Ñ pT, C j q Ñ pT, Y j q Ñ 0. Applying ppT, I C pFqq, q we get the following commutative diagram by Lemma 4.4:…”

Section: 1mentioning

confidence: 99%

“…Note that the factor category mod Λ{I, in Proposition 3.5, is not necessarily closed under extensions in mod Λ [4]. However, if C is closed under extensions, then mod Λ{I is also closed under extensions in mod Λ (by using the functor G ϕ above).…”

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confidence: 99%

Relative theory in subcategories

Mohamed¹

2009

Colloq. Math.

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Abstract. We generalize the relative (co)tilting theory of Auslander-Solberg in the category mod Λ of finitely generated left modules over an artin algebra Λ to certain subcategories of mod Λ. We then use the theory (relative (co)tilting theory in subcategories) to generalize one of the main result of Marcos et al. [Comm. Algebra 33 (2005)].Introduction. Let Λ be an artin algebra, and let mod Λ denote the category of finitely generated left Λ-modules. Auslander and Solberg [9, 10] developed a relative (co)tilting theory in mod Λ which is a generalization of standard (co)tilting theory [3], [12], [14], [23]. In this paper we develop a relative (co)tilting theory in extension-closed functorially finite subcategories of mod Λ.Let T be an ordinary tilting module over Λ. Then the module DT , where D is the usual duality between left and right modules, is a cotilting module

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Homological theory of idempotent ideals

Cited by 82 publications

References 2 publications

Enright’s completions and injectively copresented modules

Enright’s completions and injectively copresented modules

Higher Algebraic K-theory of Ring Epimorphisms

Relative theory in subcategories

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