"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)

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.
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.
2. "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.

3. "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

• First meeting: 12.03.2015 at 14:15.
  Joachim Jelisiejew Derivations of rings, inner derivations, skew polynomial rings and their universal property, consequences of surjectivity of derivations, construction of Weyl algebra $A_1(k)$ and $A_r(k) = \bigotimes^r_k A_1(k)$. Proof that $A_r(k)$ is a simple ring for a field $k$ of characteristic zero. The degree filtration of $A_r$ and the associated graded algebra.
• Second meeting: 24.04.2015 at 11:00
  Joachim Jelisiejew Filtered algebras and modules. Associated graded objects. Behaviour of annihilators on the family associated to a filtration. Associated graded is left Noetherian implies that the algebra itself is left Noetherian (so $A_r$ is left (and right) Noetherian, enveloping algebras of finite dimensional Lie algebras are left (and right) Noetherian). Filtrations and Hilbert polynomials. The degree of Hilbert polynomial of every nontrivial $A_r(k)$-module is at least $n$. Associated graded is flat implies that the algebra itself is flat (if time permits).
• Third meeting (end of $A_r$ session): 30.04.2015 at 14:15
  Joachim Jelisiejew Weak (or flat) dimension of a module, weak global dimension of a $k$-algebra. Filtered complexes, graded complexes and their exactness. The inequality $\mbox{w.dim}(gr(M))\leq \mbox{w.dim}(M)$ for filtered modules. Corollary: $\mbox{w.gl.dim}(A_r(k)) \leq 2r$ for any field $k$. Proof that $\mbox{w.gl.dim}(A_r(k)) = r$ for a field $k$ of characteristic zero. Koszul resolutions.
• Fourth meeting (beginning of noncommutative resolutions): 19.05.2015 at 14:15
  Maciej Gałązka Algebra over a complete local Gorenstein ring of dimension $2$, such as $R = \mathbb{C}[[x, y]]^G$ for a finite subgroup of $SL_n$ acting on $\langle x, y\rangle$. Maximal Cohen-Macaulay modules. Representations of $G$ vs maximal CM-modules. Endomorphism ring of the sum of all indecomposable CM modules has global dimension at most $\dim R$. The quiver = (indecomposable CM-modules, $R$-morphisms).

Information:

• Proposed meeting time: Thursdays 14:15.
• Tentative set of participants: Maciej Gałązka, Łukasz Gołębiowski, Maksymilian Grab, Joachim Jelisiejew, Arkadiusz Męcel, Łukasz Sienkiewicz, Maciej Zdanowicz.