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In this note we give a careful exposition of the basic properties of derived categories of quasi-coherent sheaves on a scheme. This includes Neeman's version of Grothendieck duality [Nee96] and the proof that every complex with quasi-coherent cohomology is isomorphic to a complex of quasi-coherent sheaves in the derived category.
We give an introduction to computation and logic tailored for algebraists, and use this as a springboard to discuss geometric models of computation and the role of cut-elimination in these models, following Girard's geometry of interaction program. We discuss how to represent programs in the λ-calculus and proofs in linear logic as linear maps between… (More)
We define a notion of total acyclicity for complexes of flat quasi-coherent sheaves over a semi-separated noetherian scheme, generalising complete flat resolutions over a ring. By studying these complexes as objects of the pure derived category of flat sheaves we extend several results about totally acyclic complexes of projective modules to schemes; for… (More)
Abelian categories are the most general category in which one can develop homological algebra. The idea and the name " abelian category " were first introduced by MacLane [Mac50], but the modern axiomitisation and first substantial applications were given by Grothendieck in his famous Tohoku paper [Gro57]. This paper was motivated by the needs of algebraic… (More)
Triangulated categories are important structures lying at the confluence of several exciting areas of mathematics (and even physics). Our notes on the subject are divided into three parts which, if named by the major construction occurring within them, would be titled " Verdier quotients " , " Thomason localisaton " and " Brown representability ". There are… (More)
Disclaimer: I am by no means a set theorist or any kind of expert. This is the record of my search for peace of mind with respect to foundations in category theory. Perhaps my experience will save another graduate student some frustration. There is a large literature on this subject, of which I am largely ignorant. The reader might find [Isb66], [Kru65],… (More)
We study cocoverings of triangulated categories, in the sense of Rouquier, and prove that for any regular cardinal α the condition of α-compactness, in the sense of Neeman, is local with respect to such cocoverings. This was established for ordinary compactness by Rouquier. Our result yields a new technique for proving that a given triangulated category is… (More)
Let k be a field. We have seen earlier that if A = a b c d ∈ M 2 (k) is an invertible 2 × 2 matrix over k then ϕ : k[x, y] −→ k[x, y] defined by ϕ(x) = ax + by, ϕ(y) = cx + dy is an automorphism of k-algebras, and this extends to polynomial rings over any number of variables. We wish to establish an analgous result for power series rings.
These notes closely follow Matsumura's book [Mat80] on commutative algebra. Proofs are the ones given there, sometimes with slightly more detail. Our focus is on the results needed in algebraic geometry, so some topics in the book do not occur here or are not treated in their full depth. In particular material the reader can find in the more elementary… (More)
Stable equivalences of Morita type preserve many interesting properties and is proved to be the appropriate concept to study for equivalences between stable categories. Recently the singularity category attained much attraction and Xiao-Wu Chen and Long-Gang Sun gave an appropriate definition of singular equivalence of Morita type. We shall show that under… (More)