Thick subcategories of perfect complexes over a commutative ring

0 → Ps → · · · → Pi → 0 where each Pi is a finitely generated projective R-module. Let P the full subcategory of D consisting of complexes isomorphic to perfect complexes. These are precisely the compact objects, also called small objects, in D. These notes are an abstract of two lectures I gave at the workshop. The main goal of the lectures was to present various proofs of a theorem of Hopkins [7] and Neeman [8], Theorem 1 below, that classifies the thick subcategories of P, and to discuss results from [5], which is inspired by this circle of ideas. As usual Spec R denotes the set of prime ideals in R with the Zariski topology; thus, the closed subsets are precisely the subsets Var(I) = {p ⊇ I |p ∈ Spec R}, where I is an ideal in R. A subset V of Spec R is specialization closed if it is a (possibly infinite) union of closed subsets; in other words, if p and q are prime ideals such that p is in V and q ⊇ p, then q is in V . For a prime ideal p, we write k(p) for Rp/pRp, the residue field of R at p. The support of a complex of R-modules M is the set of prime ideals