2021/10/15 by Leonid Positselski, Positselski, Leonid
Mathematics · Medicine · #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Intracranial Aneurysms: Treatment and Complications #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2110.08237
openalex publication_date 2021/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct an "almost involution" assigning a new DG-category to a given one, and use this construction to recover, say, the abelian category of graded modules over the graded ring R^* from the DG-category of DG-modules over a DG-ring (R^*,d). This provides an appropriate technical background for the definition and discussion of abelian and exact DG-categories. In the setting of exact DG-categories, derived categories of the second kind are defined in the maximal natural generality. We develop the related abstract category-theoretic language and use it to formulate and prove several full-and-faithfulness theorems for triangulated functors induced by the inclusions of fully exact DG-subcategories. Such functors are fully faithful for derived categories of the second kind more often than for the conventional derived categories. Examples and applications range from the categories of complexes in abelian/exact categories to matrix factorization categories, and from curved DG-modules over curved DG-rings to quasi-coherent CDG-modules over quasi-coherent CDG-quasi-algebras over schemes.