2019/07/11 by Jin, Haibo
#16E35 #16E45 #18E30 #Category Theory (math.CT) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1907.05114
Silting and Calabi-Yau reductions are important process in representation theory to construct new triangulated categories from given one, which are similar to Verdier quotient. In this paper, first we introduce a new reduction process of triangulated category, which is analogous to the silting (Calabi-Yau) reduction. For a triangulated category \cal T with a pre-simple-minded collection (=pre-SMC) \cal R, we construct a new triangulated category \cal U such that the SMCs in \cal U bijectively correspond to those in \cal T containing \cal R. Secondly, we give an analogue of Buchweitz's theorem for the singularity category \cal T\rm sg of a SMC quadruple (\cal T,\cal T\rm p,\mathbb S, \cal S): the category \cal T\rm sg can be realized as the stable category of an extriangulated subcategory \cal F of \cal T. Finally, we show the SMS (simple-minded system) reduction due to Coelho Simões and Pauksztello is the shadow of our SMC reduction. This is parallel to the result that Calabi-Yau reduction is the shadow of silting reduction due to Iyama and Yang.