Geodesic
On the homotopy finiteness of DG categories
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 74 (2019) no. 3, pp. 431-460

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This paper gives a short overview of results related to homotopy finiteness of DG categories. A general plan is explained for proving homotopy finiteness of derived categories of coherent sheaves and coherent matrix factorizations on separated schemes of finite type over a field of characteristic zero. Bibliography: 39 titles.
Keywords: derived categories, differential graded categories, homotopy finiteness, Verdier localization, resolution of singularities. 
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A. I. Efimov. On the homotopy finiteness of DG categories. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 74 (2019) no. 3, pp. 431-460. http://geodesic.mathdoc.fr/item/RM_2019_74_3_a2/