Commutative algebra homework

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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, $$0\subseteq \cdots \subseteq G_{n+1}\subseteq G_n\subseteq \cdots \subseteq G_2\subseteq G_1.$$ Show that the completion of $G$ $$\hat{G}\simeq \underleftarrow{\lim }\frac{G}{G_n}$$ by showing that a sequence in $G$ is coherent if and only if it is Cauchy.

HW: Let $S$ be a set and let $n\in {\mathbb{Z}}_{>0}.$

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 $X\times Y$.

		Hint:
	Consider the graph of $f$ and write down precise conditions for a subset $S$ of $X\times Y$ 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 $\underset{x\to a}{\lim }f\left(x\right)$ and give some illustrative examples.
3. Prove the following fundamental theorem: $$f \; is\; continuous\; at\; x=a\iff \underset{x\to a}{\lim }f\left(x\right)=f\left(a\right).$$

Week 1 Problems

Lecture 2 - 29/02/2012

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

HW: Expand $\tan t$ 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 $$+:A\left[S^{-1}\right]\times A\left[S^{-1}\right]\to A\left[S^{-1}\right]$$ is well defined, that $$\cdot :A\left[S^{-1}\right]\times A\left[S^{-1}\right]\to A\left[S^{-1}\right]$$ 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 $$\begin{array}{rrcl}{S}^{-1}u:& S^{-1}M:& \to & S^{-1}N:\\ & \frac{m}{s}& \mapsto & \frac{u\left(m\right)}{s}\end{array}$$ is an $A\left[S^{-1}\right]-$module homomorphism.

HW: Show that $S^{-1}$ is a functor (i.e. $S^{-1}(u_1\circ u_2)=S^{-1}\left(u_1\right)\circ S^{-1}\left(u_2\right)$).

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 $G\simeq \mathbb{Z}$ or $G\simeq \mathbb{Z}/r\mathbb{Z}$ for some $r\in {\mathbb{Z}}_{\ge 0}.$

HW: Show that the group $G_{r,r,2}$ can be presented by generators $s_1,s_2$ with relations $$s_1^2=1,s_2^2=1,{\displaystyle \underset{r \; factors}{\underbrace{s_1{s}_2{s}_1\cdots }}}={\displaystyle \underset{r \; factors}{\underbrace{s_2{s}_1{s}_2\cdots }}}$$

Notes and References

References

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