## Commutative algebra homework

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 $n\in {ℤ}_{>0}.$

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 $f:X\to Y$ as a subset of $X×Y$.

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

HW:

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

HW: Let $X$ and $Y$ be topological spaces. Let $a\in X$ and let $f:X\to Y$ be a function.

1. Define $f$ is continuous at $x=a$ and give some illustrative examples.
2. Define $\underset{x\to a}{\mathrm{lim}}f\left(x\right)$ and give some illustrative examples.
3. Prove the following fundamental theorem:

## Week 1 Problems

### Lecture 2 - 29/02/2012

HW: Show that ${ℤ}_{\left(0\right)}=ℚ.$

HW: Expand $\mathrm{tan}t$ 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 $A\left[{S}^{-1}\right]$ with $+$ and $\cdot$ 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 $A\left[{S}^{-1}\right]-$module homomorphism.

HW: Show that ${S}^{-1}$ is a functor (i.e. ${S}^{-1}\left({u}_{1}\circ {u}_{2}\right)={S}^{-1}\left({u}_{1}\right)\circ {S}^{-1}\left({u}_{2}\right)$).

### Lecture 3 - 09/03/2012

HW: Let $f:V\to E$ be a linear transformation from a vector space $V$ into an algebra $E.$ Suppose that if $v\in V$ then $f{\left(v\right)}^{2}=0.$ Show that if ${v}_{1},{v}_{2}\in V$ then $f\left({v}_{1}\right)f\left({v}_{2}\right)=-f\left({v}_{2}\right)f\left({v}_{1}\right).$

HW: Compute the dimension of ${S}^{k}\left(V\right).$

HW: Determine (make precise) the definition of $\mathrm{sgn}\left(\sigma \right),$ as used in the construction of $\Lambda \left(V\right).$

## Week 3 Problems

### Lecture 1 - 12/03/2012

HW: Show that if $G$ is a cyclic group then $G\simeq ℤ$ or $G\simeq ℤ/rℤ$ for some $r\in {ℤ}_{\ge 0}.$

HW: Show that the group ${G}_{r,r,2}$ can be presented by generators ${s}_{1},{s}_{2}$ with relations