Commutative algebra homework

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 02 March 2012

Homework problems

HW: Let G be a (commutative?) topological group with a fundamental system of neighborhoods of 0 which are subgroups, 0 Gn+1 Gn G2 G1. Show that the completion of G G^ lim G Gn by showing that a sequence in G is coherent if and only if it is Cauchy.

HW: Let S be a set and let n>0.

  1. Define relation on S, equivalence relation on S, and equivalence class and give some illustrative examples.
  2. Define partition of S and partition of n and give some illustrative examples.
  3. State and prove a theorem which makes precise the concept that equivalence relations and partitions are interchangable.

HW: Define the notion of a function f:XY as a subset of X×Y.

Consider the graph of f and write down precise conditions for a subset S of X×Y to be the graph of a function.


  1. Define isomorphism of sets, and f is bijective and give some illustrative examples.
  2. Prove that a function f:XY is an isomorphism of sets if and only if f is bijective.

HW: Let X and Y be topological spaces. Let aX and let f:XY be a function.

  1. Define f is continuous at x=a and give some illustrative examples.
  2. Define lim xa f(x) and give some illustrative examples.
  3. Prove the following fundamental theorem: f   is continuous at   x=a lim xa f(x) = f(a) .

Week 1 Problems

Lecture 2 - 29/02/2012

HW: Show that (0) = .

HW: Expand tant as a power series beginning with t.

Lecture 3 - 02/03/2012

HW: Do the exercises on the p-adic numbers page.

HW: Using the definition of the ring of fractions with denominators in S from the lectures, prove that = is an equivalence relation, that +:A[S-1]×A[S-1] A[S-1] is well defined, that :A[S-1]×A[S-1] A[S-1] is well defined, and A[S-1] with + and is a ring.

Week 2 Problems

Lecture 2 - 07/03/2012

HW: Show that S-1u: S-1M: S-1N: m s u(m) s is an A[S-1]-module homomorphism.

HW: Show that S-1 is a functor (i.e. S-1(u1u2) = S-1(u1) S-1(u2) ).

Lecture 3 - 09/03/2012

HW: Let f:VE be a linear transformation from a vector space V into an algebra E. Suppose that if vV then f(v)2=0. Show that if v1,v2V then f(v1)f(v2) = -f(v2)f(v1).

HW: Compute the dimension of Sk(V).

HW: Determine (make precise) the definition of sgn(σ), as used in the construction of Λ(V).

Week 3 Problems

Lecture 1 - 12/03/2012

HW: Show that if G is a cyclic group then G or G/r for some r0.

HW: Show that the group Gr,r,2 can be presented by generators s1,s2 with relations s12=1, s22=1, s1s2s1 r  factors = s2s1s2 r  factors

Notes and References


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