Topics in Noncommutative Algebraic Geometry, Homological Algebra and K-theory

This text is based on my lectures delivered at the School on Algebraic K-Theory and Applications which took place at the International Center for Theoretical Physics (ICTP) in Trieste during the last two weeks of May of 2007. It might be regarded as an introduction to some basic facts of noncommutative algebraic geometry and the related chapters of homological algebra and (as a part of it) a non-conventional version of higher K-theory of noncommutative ’spaces’. Arguments are mostly replaced by sketches of the main steps, or references to complete proofs, which makes the text an easier reading than the ample accounts on different topics discussed here indicated in the bibliography. Lecture 1 is dedicated to the first notions of noncommutative algebraic geometry – preliminaries on ’spaces’ represented by categories and morphisms of ’spaces’ represented by (isomorphism classes of) functors. We introduce continuous, affine, and locally affine morphisms which lead to definitions of noncommutative schemes and more general locally affine ’spaces’. The notion of a noncommutative scheme is illustrated with important examples related to quantized enveloping algebras: the quantum base affine spaces and flag varieties and the associated quantum D-schemes represented by the categories of (twisted) quantum D-modules introduced in [LR] (see also [T]). Noncommutative projective ’spaces’ introduced in [KR1] and more general Grassmannians and flag varieties studied in [KR3] are examples of smooth locally affine noncommutative ’spaces’ which are not schemes. In Lecture 2, we recover some fragments of geometry behind the pseudo-geometric picture outlined in the first lecture. We start with introducing underlying topological spaces (spectra) of ’spaces’ represented by abelian categories and describing their main properties. One of the consequences of these properties is the reconstruction theorem for commutative schemes [R4] which can be regarded as one of the major tests for the noncommutative theory. It says, in particular, that any quasi-separated commutative scheme can be canonically reconstructed uniquely up to isomorphism from its category of quasi-coherent sheaves. The noncommutative fact behind the reconstruction theorem is the geometric realization of a noncommutative scheme as a locally affine stack of local categories on its underlying topological space. The latter is a noncommutative analog of a locally affine locally ringed topological space, that is a geometric scheme. Lecture 3 complements this short introduction to the geometry of noncommutative ’spaces’ and schemes with a sketch of the first notions and facts of pseudo-geometry (in particular, descent) and spectral theory of ’spaces’ represented by triangulated categories. This is a simple, but quite revelative piece of derived noncommutative algebraic geometry. Lectures 4, 5, 6 are based on some parts of the manuscript [R8] created out of attempts to find natural frameworks for homological theories which appear in noncommutative algebraic geometry. We start, in Lecture 4, with a version of non-abelian (and