"NO CO?" NOncommutative meets COmmutative

proposition of a seminar at MIMUW

Idea of seminar: make a meeting place for PhD students (and all others) from algebra, in particular a place where people doing noncommutative stuff can learn/teach people doing commutative algebra and algebraic geometry. The topics should include (non)commutative stuff, but no things requiring advanced machinery of geometry or noncommutative ring theory.

Some inspiration

There was a summer graduate school Noncommutative Algebraic Geometry and Representation Theory, and an introductory workshop Noncommutative Algebraic Geometry and Representation Theory. The school consisted of four series of lectures:

  1. Noncommutative resolutions (notes)
  2. Noncommutative geometry (by Dan Rogalski, notes)
  3. Deformation theory in noncommutative geometry (notes)
  4. Symplectic reflection algebras (notes)
You can watch the lectures on the website of the school.

Some topics suggested:

  1. (done) rings of differential operators (e.g. classical Weyl algebras $A_r$) and their associated graded rings. For example this book by Björk.
  2. Addendum: there are two lectures about $D$-modules in the "introductory" workshop mentioned. The first one is nice, the second is hard; it requires derived cats. There are also notes about Luybeznik's local cohomology finiteness theorems by Hochster, here.
  3. "noncommutative projective geometry" as a study of $\mathbb{N}$-graded algebras, as done by Rogalski-Sierra-Stafford; in particular point modules, projective schemes (or ind-schemes) associated with point modules, Artin-Schelter regular algebras (analogues of projective spaces). They also have striking applications to "classical" noncommutative ring theory, such as Sierra and Walton results on universal enveloping algebras. There will be a workshop on 6-10 July at Toronto, related to these topics. There are great lecture notes from the workshop (2., by Dan Rogalski here). We should focus on these.
  4. "Noncommutative resolutions" (thanks for preparing it, Maciek!).

    (partially done) Noncommutative resolutions are a technique to construct and study crepant resolutions of Gorenstein normal singularities. A Noncommutative Crepant Resolution of $X = \mbox{Spec } R$ is a ring $A$ (usually not commutative) that satisfies some properties (see Def. 2.5 from the notes). The point of this definition is that if $f :Y \to X$ is any projective birational map between Gorenstein normal varieties, $A$ is any ring that is derived equivalent to $Y$, then $f$ is a crepant resolution (in particular, Y is smooth!) if and only if $A$ is an NCCR (see Corollary 4.17). But this theory works best when dimension $\leq 3$; when this holds, then from an NCCR $A$ we can construct a crepant resolution $f: Y \to X$ such that $Y$ is derived equivalent to $A$ (see Theorem 5.17). We do this using quiver GIT quotients. When dimension $\geq 4$, then many theorems still are true, but there are singularities that have an NCCR, but do not have a crepant resolution and vice-versa.

    Proposed schedule:

    1. First meeting. Introduction to quivers, quiver representations, quiver algebra. Motivating examples of NCCR's? Definition of an NCCR. Motivation: Why $\mbox{gldim End}_R(M) = \dim R$ corresponds to smoothness. NCCR's are Morita equivalent in dimension 2 and derived equivalent in dimension 3.
    2. Second meeting. Quiver GIT. Definition of a moduli space of $\theta$-semistable representations of a given dimension vector. Examples.
    3. Third meeting. Tilting. Relative Serre functors. Calabi-Yau categories. Singular derived categories. Auslander-Reiter duality. Rings derived equivalent to a crepant resolution give an NCCR.
    4. Fourth meeting. McKay correspondence (classical version, Auslander version, derived version). NCCR's give crepant resolutions in dimension $\leq 3$.
    This is most probably too much, I maybe we'd better have three meetings. But to shorten the schedule, I need everyone's help (if you are keen on doing this): what do you want to focus on? On examples, or on technical stuff like quivers (which is interesting in its own right)?

Past meetings