Mohamed Barakat - Academia.edu (original) (raw)
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Papers by Mohamed Barakat
Journal of Algebra, 2014
To develop a constructive description of Ext in categories of coherent sheaves over certain schem... more To develop a constructive description of Ext in categories of coherent sheaves over certain schemes, we establish a binatural isomorphism between the Ext-groups in Serre quotient categories A/C and a direct limit of Ext-groups in the ambient Abelian category A. For Ext 1 the isomorphism follows if the thick subcategory C ⊂ A is localizing. For the higher extension groups we need further assumptions on C. With these categories in mind we cannot assume A/C to have enough projectives or injectives and therefore use Yoneda's description of Ext.
In this paper we follow the converse question: When are c composable 1-cocycles induced by a modu... more In this paper we follow the converse question: When are c composable 1-cocycles induced by a module E together with a chain of submodules as above? We call such modules c-extension modules. The case c = 1 is the classical correspondence between 1-extensions and 1cocycles. For c = 2 we prove an existence theorem stating that a 2-extension module exists for two composable 1-cocycles η M L ∈ Ext 1 (M, L) and η L N ∈ Ext 1 (L, N), if and only if their Yoneda product η M L • η L N ∈ Ext 2 (M, N) vanishes. We further prove a modelling theorem for c = 2: In case the set of all such 2-extension modules is non-empty it is an affine space modelled over the abelian group that we call the first extension group of 1-cocycles, Ext 1 (η M L , η L N
Journal of Algebra, 2014
To develop a constructive description of Ext in categories of coherent sheaves over certain schem... more To develop a constructive description of Ext in categories of coherent sheaves over certain schemes, we establish a binatural isomorphism between the Ext-groups in Serre quotient categories A/C and a direct limit of Ext-groups in the ambient Abelian category A. For Ext 1 the isomorphism follows if the thick subcategory C ⊂ A is localizing. For the higher extension groups we need further assumptions on C. With these categories in mind we cannot assume A/C to have enough projectives or injectives and therefore use Yoneda's description of Ext.
In this paper we follow the converse question: When are c composable 1-cocycles induced by a modu... more In this paper we follow the converse question: When are c composable 1-cocycles induced by a module E together with a chain of submodules as above? We call such modules c-extension modules. The case c = 1 is the classical correspondence between 1-extensions and 1cocycles. For c = 2 we prove an existence theorem stating that a 2-extension module exists for two composable 1-cocycles η M L ∈ Ext 1 (M, L) and η L N ∈ Ext 1 (L, N), if and only if their Yoneda product η M L • η L N ∈ Ext 2 (M, N) vanishes. We further prove a modelling theorem for c = 2: In case the set of all such 2-extension modules is non-empty it is an affine space modelled over the abelian group that we call the first extension group of 1-cocycles, Ext 1 (η M L , η L N