The Cohomological Support And Classification Of Thick Subcategories Over Complete Intersections

This is a doctoral dissertation on the structure of thick subcategories in the bounded derived category of a complete intersection ring.The article mainly contains the following three parts.1.For compactly generated triangulated category:the derived category of rings,the homotopy category of complexes of injective modules,the homotopy category of acyclic complexes of injective modules,in Chapter 3,we realize these three examples as derived categories of differential graded categories.By comparing the differential graded BGG correspondence with the classical BGG correspondence,we explicitly establish their connection in Chapter 3.2.We study homological properties of a complete intersection ring by importing facts from exterior algebra.We prove in Chapter 4 that the inclusion among thick subcategories in the bounded derived category of a complete intersection ring can be determined by the inclusion among cohomological supports.This positively answers a conjecture raised by Iyengar[51].As an application,we get:(1)Thick subcategories in the bounded derived category over a complete intersection ring are closed under Grothendieck duality.This strengthens a theorem of Stevenson[89].In particular,we recover a theorem of Avramov-Buchwetiz[5]:(2)For finitely generated modules M,N over a complete intersection ring R,then ExtR>>0(M,N)=0 if and only if Tor>>0R(M,N)=0.3.Let R be a regular local ring modulo a regular sequence.Inspired by results of Burke-Stevenson[28]and Takahashi[90],we calculate the union of all the cohomological support.Combining this with the main result in second part,we prove in Chapter 5 that there is a one-to-one correspondence between thick subcategories in the bounded derived category of R and specialization closed subsets in the singular locus of the generic hypersurface corresponding to R.This classification is implicitly contained in two papers just mentioned.We emphasize that we construct a precise bijection of the corresponding classification and the proof is direct.As an application,we recover a theorem of Stevenson[88]:There is a one-toone correspondence between thick subcategories in the singularity category of R and specialization closed subsets in the singular locus of a projective scheme of the generic hypersurface corresponding to R.