MATH 236B, S 2015. Further remarks on direct limits (generally useful)
Let Ai , fij
i,jI, ij
be a directed system of left R-modules, with lim Ai =
iI
Ai /U ,
where U = ij (inj fij ini )(Ai ) iI Ai (here inj : Aj iI Ai is the canonical
injection). Moreover
MATH 236 B, S 2015, Second Homework, due Monday, June 1, in class
Do 3 out of the 4 problems.
Throughout, R denotes a ring.
1. Let 0 X1 X2 X3 0 be an exact sequence in R-Mod. Show that, if two of the
Xi have nite projective dimension, then so does the thi
MATH 236 A, W 2015, Projective Modules over Local Rings
Second installment: Showing freeness of projectives over local rings. Ill start by reminding
you of the background weve assembled earlier.
Kaplanskys Theorem (Theorem 11). Let R be any ring and M an
MATH 236B, S 2015. A brief introduction to limits of functors
Let I be a small category, I its set of objects. Moreover, let F : I C be a contravariant
functor. A limit of F is an object C of C, together with a family i HomC (C, F (i) for
i I, with F (f )
MATH 236 B, S 2015, First Homework, due Friday, May 1, in class
Throughout R denotes a ring.
1. Let I be a directed partially ordered set. Then the functor lim from the category of
I-indexed directed systems in R-Mod to the category R-Mod is exact.
Prove
MATH 236 B, S 2015
An example of a ring with diering right and left global dimensions
Let R :=
ZQ
0 Q
, short for the subring
a b
0 c
a Z, b, c Q . We claim
M2 (Q)
that r.gl.dim R = 1 while l.gl.dim R 2. Obviously R is not semisimple (it has a nonzero
ni
MATH 236B, S 2015. Supplement re colimits: Pushouts and direct limits
These notes pick up where our nal 236A class, afternoon of March 13, left o.
A. Pushouts
I was already rushed when I sketched the fact that pushouts are colimits. So Ill back
up a littl
Application to Topology:
Theorem 36: [Mayer-Vietoris] Suppose X is a toplological space and X1 , X2
are subspaces with X1 X2 , where Xi is the interior of Xi . Then the singular
homology groups of X2 X2 , of X and of the Xi are related by a long exact
seq
Math 236B, Spring 2015, MWF 9-9:50, HSSB 1223
Homological Algebra
Instructor: Birge Huisgen-Zimmermann, SH 6518, Oce hours M, F 11 - 12, W 12:30 1:30.
Accompanying texts, as for the winter quarter:
The manuscript I will put on the board will again serve