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694
Abelian Categories
, 2006
"... modern axiomitisation and first substantial applications were given by Grothendieck in his famous Tohoku paper [Gro57]. This paper was motivated by the needs of algebraic geometry, where the ..."
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modern axiomitisation and first substantial applications were given by Grothendieck in his famous Tohoku paper [Gro57]. This paper was motivated by the needs of algebraic geometry, where the
SemiAbelian Categories
, 2000
"... The notion of semiabelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abeliangroup and module theory. In modern terms, semiabelian categories ar ..."
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Cited by 80 (9 self)
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The notion of semiabelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abeliangroup and module theory. In modern terms, semiabelian categories
Semiabelian Categories and Exactness
, 2009
"... We show that every semiabelian category, as defined by Palamodov, possesses a maximal exact structure in the sense of Quillen and that the exact structure of a quasiabelian category is a special case thereof. ..."
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We show that every semiabelian category, as defined by Palamodov, possesses a maximal exact structure in the sense of Quillen and that the exact structure of a quasiabelian category is a special case thereof.
Comparison of Abelian Categories Recollements
 DOCUMENTA MATH.
, 2004
"... We give a necessary and sufficient condition for a morphism between recollements of abelian categories to be an equivalence. ..."
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Cited by 9 (1 self)
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We give a necessary and sufficient condition for a morphism between recollements of abelian categories to be an equivalence.
On Moduli Spaces for Abelian Categories
, 711
"... We show that if A is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. This idea is applied to several important abelian categories in representation theory, ..."
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We show that if A is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. This idea is applied to several important abelian categories in representation theory
Deformation theory of abelian categories
 in Math. 198 (2005
"... Abstract. In this paper we develop the basic infinitesimal deformation theory of abelian categories. This theory yields a natural generalization of the wellknown deformation theory of algebras developed by Gerstenhaber. As part of our deformation theory we define a notion of flatness for abelian cat ..."
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Cited by 8 (1 self)
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Abstract. In this paper we develop the basic infinitesimal deformation theory of abelian categories. This theory yields a natural generalization of the wellknown deformation theory of algebras developed by Gerstenhaber. As part of our deformation theory we define a notion of flatness for abelian
INJECTIVE OBJECTS IN ABELIAN CATEGORIES
"... It is known that the category of modules over a ring has enough injectives, i.e. any object can be imbedded as a subobject of an injective object. This is one of the first things one learns about injective modules, though it is a nontrivial fact and requires some work. Similarly, when introducing sh ..."
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sheaf cohomology, one has to show that the category of sheaves on a given topological space has enough injectives, which is a nottoodifficult corollary of the first fact. I recently learned, however, of a much more general result that guarantees that an abelian category nonsense posts). I learned
GORENSTEIN COHOMOLOGY IN ABELIAN CATEGORIES
, 2007
"... We investigate relative cohomology functors on subcategories of abelian categories via AuslanderBuchweitz approximations and the resulting strict resolutions. We verify that certain comparison maps between these functors are isomorphisms and introduce a notion of perfection for this context. Our ..."
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Cited by 3 (2 self)
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We investigate relative cohomology functors on subcategories of abelian categories via AuslanderBuchweitz approximations and the resulting strict resolutions. We verify that certain comparison maps between these functors are isomorphisms and introduce a notion of perfection for this context. Our
The Künneth Formula in Abelian Categories
, 2004
"... In algebraic homology the Künneth Formula and its implications are well known. Given an additive functor t of two variables defined for modules and modules as values, there is a canonical way to extend this functor to chain complexes of modules. The Künneth Formula then states under certain conditio ..."
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is introduced in the literature using modules, it is not a specific property of the category of left (or right) Rmodules due to which the formula is valid. The formula proves to be valid in a more general environment: in abelian categories. Just a few additional assumptions (such as the existence of coproducts
Simplicial homotopy in semiabelian categories
 J. Ktheory
, 2008
"... The theory of abelian categories proved very useful, providing an axiomatic framework for homology and cohomology of modules over a ring (in particular, abelian groups) [5]. A similar framework has been lacking for nonabelian (co)homology, the subject of which includes the categories of groups and ..."
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Cited by 4 (3 self)
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The theory of abelian categories proved very useful, providing an axiomatic framework for homology and cohomology of modules over a ring (in particular, abelian groups) [5]. A similar framework has been lacking for nonabelian (co)homology, the subject of which includes the categories of groups
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