**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 H_{i}(f) is not surjective.
Letting C' be the kernel of f, compute the connecting
homomorphism from H_{i}(C'') to H_{i-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 H_{i}(f) is not injective.
Letting C'' be the cokernel of f, compute the connecting
homomorphism from H_{i+1}(C'') to H_{i}(C') in this case.

**2**.
(Marni Mishna) Let (C,d) be a bounded positive
complex of finite-dimensional F-vector spaces (i.e., C_{i}=0
for i<0 and
for all i sufficiently large).
Let r_{i} and s_{i} denote the dimensions of
C_{i} and H_{i}(C) respectively.
Show that the alternating sum

r_{0} - r_{1}+r_{2} -r_{3} + ...

is equal to

s_{0} - s_{1}+s_{2} -s_{3} + ...

**3**. (Dan Segal)
Let X be a smooth region in R^{n}.
Define a co-chain complex by letting C^{k} be the
R vector space of k-differentials on X, i.e., linear combinations
of expressions of the form

f(x_{1},...,x_{n})
dx_{i1}
dx_{i2} ...
dx_{ik}

where the f are infiinitely differentiable functions on
R^{n},
and letting
d^{k} :C^{k} --> C^{k+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 R^{2}.