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189-571B: Higher Algebra II
Assignment 9
To be presented: Wednesday, March 21.
The problem marked with a (*) is for everyone to do and to hand in on
the day of the problem session.
I will return this problem to you on Friday of the same week,
so late problems will not be accepted!
If you are stuck on your problem, come see me during my Monday
office hours (or make an appointment)
to get a hint...
1. (*)
Reading assignment: Read carefully chapter
6.3 of the handout (The Long Exact Homology
Sequence) and master all the steps in the
proof of theorem 6.1. You will find it helpful, while reading the
proof, to draw the relevant commutative diagram and translate the
somewhat cumbersome algebraic formulae into a
"diagram chase".
Written assignment:
a) Give an example of a ring R, a
surjective morphism
f: C --> C''
in the category of complexes of R-modules, and an integer i for which
the morphism Hi(f) is not surjective.
Letting C' be the kernel of f, compute the connecting
homomorphism from Hi(C'') to Hi-1(C') in this case.
b) Give an example of a ring R, an
injective morphism
f: C' --> C
in the category of complexes of R-modules, and an integer i for which
the morphism Hi(f) is not injective.
Letting C'' be the cokernel of f, compute the connecting
homomorphism from Hi+1(C'') to Hi(C') in this case.
2.
(Marni Mishna) Let (C,d) be a bounded positive
complex of finite-dimensional F-vector spaces (i.e., Ci=0
for i<0 and
for all i sufficiently large).
Let ri and si denote the dimensions of
Ci and Hi(C) respectively.
Show that the alternating sum
r0 - r1+r2 -r3 + ...
is equal to
s0 - s1+s2 -s3 + ...
3. (Dan Segal)
Let X be a smooth region in Rn.
Define a co-chain complex by letting Ck be the
R vector space of k-differentials on X, i.e., linear combinations
of expressions of the form
f(x1,...,xn)
dxi1
dxi2 ...
dxik
where the f are infiinitely differentiable functions on
Rn,
and letting
dk :Ck --> Ck+1 be the differential
map
that arose in advanced calculus during your discussion
of Stoke's theorem. (If it did not, or your memory is hazy,
come see me...)
a) Show that C is indeed a cochain complex.
b) Compute the cohomology of this complex in the case where
X is the punctured unit disk in R2.