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Quasi-coherent sheaves and contraherent cosheaves
According to Enochs-Estrada, quasi-coherent sheaves on a scheme are representations of a certain quiver with relations (otherwise known as additive functors from a fixed preadditive category into abelian groups, or modules over a big ring) satisfying the additional "quasi-coherence" condition.
What is this quasi-coherence condition? What is its place among the general concepts of homological algebra known to the contemporary algebraists? It is the condition of left Ext^{0,1}-perpendicularity to a certain set of objects of injective dimension 1, as in the paper of Geigle and Lenzing. The quasi-coherent sheaves are the left perpendicular class to a set of objects of injective dimension 1 in the category of modules over a big ring.
According to yours truly, contraherent cosheaves on a scheme are representations of a certain quiver with relations ( = modules over a certain big ring) satisfying the additional "contraherence" and "contraadjustness" conditions. What are these contraherence and contraadjustness conditions?
These are the conditions of right perpendicularity to a certain set of objects of projective dimension 2. The contraherent cosheaves are the right Ext^{0,1,2}-perpendicular class to a set of objects of projective dimension 2 in the category of modules over a big ring.
That is why the quasi-coherent sheaves are an abelian category, while the contraherent cosheaves are an exact category. The left/right perpendicular class to a set/class of objects of injective/projective dimension 1 in an abelian category is an abelian category (as Geigle and Lenzing already observed), while the Ext^*-perpendicular class to a set/class of objects of homological dimension more than 1 in an abelian category is an exact category (since it is a full subcategory closed under extensions).
What is this quasi-coherence condition? What is its place among the general concepts of homological algebra known to the contemporary algebraists? It is the condition of left Ext^{0,1}-perpendicularity to a certain set of objects of injective dimension 1, as in the paper of Geigle and Lenzing. The quasi-coherent sheaves are the left perpendicular class to a set of objects of injective dimension 1 in the category of modules over a big ring.
According to yours truly, contraherent cosheaves on a scheme are representations of a certain quiver with relations ( = modules over a certain big ring) satisfying the additional "contraherence" and "contraadjustness" conditions. What are these contraherence and contraadjustness conditions?
These are the conditions of right perpendicularity to a certain set of objects of projective dimension 2. The contraherent cosheaves are the right Ext^{0,1,2}-perpendicular class to a set of objects of projective dimension 2 in the category of modules over a big ring.
That is why the quasi-coherent sheaves are an abelian category, while the contraherent cosheaves are an exact category. The left/right perpendicular class to a set/class of objects of injective/projective dimension 1 in an abelian category is an abelian category (as Geigle and Lenzing already observed), while the Ext^*-perpendicular class to a set/class of objects of homological dimension more than 1 in an abelian category is an exact category (since it is a full subcategory closed under extensions).