Toric Geometry and Combinatorial Commutative Algebra
MATH 7290

Fall 2008
Lecture 2. Algebraic constructions and categories
We dene some categories and display some constructions that will play a role later.
M sets
Denition. Let M be a monoid. A left M set is a set X equipped with a function
M X X ; (m, x) mx such that ex = x
Math 7290
Quiver Varieties
Spring 2015
P. Achar
Lecture Notes
Lecture 1 (Jan. 14) Quiver: Q = (Q0 , Q1 ). Repk (Q): category of finitedimensional representations of
Q over a field k. For s Q0 , let Vs be the representation with k at vertex s and 0s elsew
Math 7290
Quiver Varieties
Spring 2015
P. Achar
InClass Worksheet
March 6, 2015
Q = (type A1 ), = + = cfw_.
Q=
dim. vector d
d()
n
dim. of
Repd (Q)
0
orbit/module n
n()
n
rep.
of Q
kn
dim On
0
rep.
of Q
k0
0k
dim On
0
0
0
2
0
0
0
0
2
k2 0
1
1
1
1
0
1
kk
Math 7290
Quiver Varieties
Spring 2015
P. Achar
Problem Set 1
Due: January 26, 2015
1. (Not to hand in) Let Q be a quiver, and let k be a field. Show the category Repk (Q) is equivalent to
the category of finitedimensional kQmodules.
2. Let Q be a quive
Math 7290
Quiver Varieties
Spring 2015
P. Achar
Problem Set 3
Due: March 13, 2015
F
1. (Optional) Let X = sJ Xs be a stratified complex variety, with dense stratum Xs0 . Let f : Y X
be a surjective proper map, with Y smooth. Assume that f is a small map,
Math 7290
Quiver Varieties
Spring 2015
P. Achar
Problem Set 4
Due: May 8, 2015
For this problem set, let n be a positive integer, and let Q be the following quiver with n 1 vertices:
x1 / x2 / x3 / xn2 /
Let
v = (1, 2, . . . , n 1),
= + = (1, 1, . . .
Math 7290
Quiver Varieties
Spring 2015
P. Achar
Problem Set 2
Due: February 13, 2015
1. Let k = Fq be the finite field with q elements. Let n m 0. Prove that the number of mdimensional
[n]q !
n
. (Recall that this is defined to be [m]q ![nm]
subspaces of