Math 582F/583F
Non-commutative Rings
Paul Smith
Winter 2007/Spring 2007, Monday/Wednesday/Friday, 1:30
Roughly speaking, a graded ring is one in wich we cn speak of the degree of
an element. The standard example is the polynomial ring S =
C[x0,...,xn] where deg
xi=1 for all i. If I is an ideal of
S generate by homogeneous elements, then A = S/I is also
a graded ring and there is an associated projective algebraic variety
Proj A. For example, if L is a very ample line bundle on
a projective variety X there is a graded ring structure on A
:= ⊕n≥0H0(X,L⊗n)
and X ≅ Proj A.
Serre's Thoerem: If X is a closed subvariety of
Pn with homogeneous coordinate ring
A, then there is an equivalence
(*) Qcoh X ≡ GrMod A/T
where
QchoX is the category of quasi-coherent
OX-modules and the right-hand side is the quotient of
the category of graded A-modules by the full subcategory of
modules supported at the origin.
This theorem lies at the heart of projective algebraic geometry. It is a
bridge between algebra and geometry.
We will prove Serre's theorem, but this course will also treat rings graded
by groups other than Z and rings that are not necessarily
commutative. In these more exotic cases, the cateogry on the right-hand
side of (*) can still be formed and an analogue of Serre's thoerem still
holds although the X will not usually be Proj
A. Nevertheless, there is still a geometric object X whose
category of quasi-coherent sheaves is equivalent to the category on the
right hand side of (*). However, the perspective in this course will be to
define X implicitely as the geometric object whose category of
quasi-coherent sheaves is the category on the right-hand side of (*). Such
geometric objects include:
- weighted projective space-A is the polynomial ring with a
Z-grading in which some of the xi's have degree
>1;
- toric varieties---the polynomial ring now has a grading by--Zn;
- orbifold Riemann spheres--the polynomial ring in two variables graded
--by a rank one abelian group possibly with
some torsion;
- Deligne-Mumford stacks related to the previous three examples;
- non-commutative projective varieties--A is no longer commutative.
Although we will never use sheaves, everything we proved about the category
on the right-hand side of (*) will, via the equivalence of categories, be a
result about quasi-coherent sheaves. Two nice examples illustrating the
utility of the graded algebra perspective will be (i) a proof that every
vector bundle on P1 is a direct sum of line bundles and a
classification of those line bundles, and (ii) a proof of Bézout's
theorm for curves in P2 using homological algebra over
the polynomial ring in three variables. Those algebraic proofs will allow
us to extend those two results to slightly more exotic settings---some
orbifolds on the Riemann sphere and some non-commutative analgoues of
P2,
The precise content of the course, particularly the ratio of algebra to
geometry, will depend on the intersets and background of the students. The
algebraic parts of the course will be suitable for students who have
completed the standard graduate algebra sequence (504/5/6). I will treat
the geometric parts of th ecourse from an algebraic perspective by using
the homogeneous coordinate ring to study the associated geometric
space. The course will also be aimed at those who have taken the 2005-2006
algebraic geometry courses.
Likely Topics:
- Graded rings and graded modules, Hilbert series, minimal projective
resolutions, Ext and Tor.
- Graded rings and actions of abelian groups on guasi-affine varieties.
- Homogeneous coordinate rings of projective varieties. Cox's homogeneous
coordinate ring of a toric variety and coordinate rings of some
orbifolds. Non-commutative homogeneous coordinate rings.
- A little about adjoint pairs of functors with an eye towards
localization functors and direct and inverse image functors.
- Quotient categories of the category of graded modules.
- Serre's theorem as proved by Artin-Zhang so it also applies to
non-commutative projective varieties.
- The relation between the geometry of X and graded A-modules.
- Classification of quasi-coherent sheaves on P1.
- Bézout's theorem.
- Sheaf cohomology as the derivied functor of the global sections functor
devined using the right-hand side of the equivalence (*) and its
computation via local cohomology.
- Homomorphism between grded rings and morphisms between the associated varieties.
- Hirzebruch surfaces as toric varieties.
- Relation between the grading group and the Picard group.
- Weighted projective varieties and weighted projective stacks.