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Deformation theory of objects in homotopy and derived categories. III: Abelian categories. (English) Zbl 1225.18009

[For part I, see ibid. 222, No. 2, 359–401 (2009; Zbl 1180.18006); for part II see ibid. 224, No. 1, 45–102 (2010; Zbl 1197.18003).]
The deformation theory for differential graded modules over a differential graded category developed in the first part of this work is now adapted to the setting of complexes over an abelian category. The pro-representability theorems of the second part are then applied to the study of deformations of quasi-coherent sheaves on a locally Noetherian scheme. Finally, half of the article is devoted to the study of non-commutative Grassmanians. Briefly, a finite-dimensional vector space \(V\) of dimension \(n\) and an integer \(m < n\) define a quadratic \(\mathbb Z\)-algebra, i.e., a linear category with objects \(\mathbb Z\). The category of modules over this algebra modulo torsion “should be thought of as the category of quasi-coherent sheaves on a non-commutative stack”: This is \(NGr(m, V)\). The authors explain then which are the expected properties of such an object and why they are satisfied.

MSC:

18E30 Derived categories, triangulated categories (MSC2010)
18G10 Resolutions; derived functors (category-theoretic aspects)
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