Math 520B
Written Assignment No. 4

due Friday, November 18, 2005

Directions. Although you may refer to books for definitions and standard theorems, searching for solutions to these written exercises is not permitted. You may not seek help from others.

Bear in mind that rings are always assumed to have a multiplicative identity, and a homomorphism of rings is always assumed to carry the multiplicative identity of its domain to that of its target. Recall that if $T$ is a ring, the term $T$-algebra indicates, by definition, a pair where $R$ is a ring and $\rho :T\to R$ is a ring homomorphism.

1. Let $F$ be a field, and let $V$ and $W$ be finite-dimensional vector spaces over $F$. Recall that the ring of endomorphisms of an $F$-vector space is an $F$-algebra.

   1. Explain why for $f\in {\mathrm{End}}_F\left(V\right)$ and $g\in {\mathrm{End}}_F\left(W\right)$ there is a unique $h=f\otimes g\in {\mathrm{End}}_F\left(V{\otimes }_{F}W\right)$ for which $$h\left(x\otimes y\right)=f\left(x\right)\otimes g\left(y\right)\text{.}$$

   2. Show that the map $${\mathrm{End}}_F\left(V\right)\times {\mathrm{End}}_F\left(W\right)\to {\mathrm{End}}_F\left(V{\otimes }_{F}W\right)$$ given by $\left(f,g\right)\mapsto f\otimes g$ is $F$-bilinear.

   3. Prove that the bilinear map in the previous part provides an isomorphism $${\mathrm{End}}_F\left(V\right){\otimes }_F{\mathrm{End}}_F\left(W\right)\to {\mathrm{End}}_F\left(V{\otimes }_{F}W\right)\text{.}$$

2. Let $F$ be a field, $V$ an $n$-dimensional vector space over $F$, ${\otimes }^{p}V$ the vector space $${\otimes }^{p}V=\underset{p\text{times}}{V{\otimes }_{F}V{\otimes }_F\dots {\otimes }_{F}V}$$ with the convention ${\otimes }^{0}V=F$, and $T\left(V\right)$ the direct sum $$T\left(V\right)=\underset{p\ge 0}{\oplus }{\otimes }^{p}V\text{.}$$ Endow $T\left(V\right)$ with the structure of a non-commutative $F$-algebra as follows:

   1. Define canonical bilinear maps $${\otimes }^{p}V\times {\otimes }^{q}V\to {\otimes }^{p+q}V$$

   2. Use the bilinear maps of the previous item with standard facts about direct sums to define multiplication $$T\left(V\right)\times T\left(V\right)\to T\left(V\right)\text{.}$$

3. With $F$, $V$, and $T\left(V\right)$ as in the previous exercise, do the following:

   1. Prove that if $V$ is $1$-dimensional over $F$, then $T\left(V\right)$ is isomorphic to the polynomial ring $F\left[t\right]$.

   2. State and prove a universal (initial) mapping property for the tensor algebra $T\left(V\right)$.

4. Let $A$ be a commutative ring and $I,J$ ideals in $R$. Prove that $$A/I{\otimes }_{A}A/J\cong A/\left(I+J\right)\text{.}$$

5. Let $F$ be a field, $i:F\left[t\right]\to F\left[x,y\right]$ the unique $F$-algebra homomorphism for which $i\left(t\right)=x$ and $j:F\left[t\right]\to F\left[x,y\right]$ the unique $F$-algebra homomorphism for which $j\left(t\right)=y$.

   Let $V$ be the $F$-vector space having basis $\left\{X,Y\right\}$ which may be canonically identified with the subspace ${\otimes }^{1}V\subset T\left(V\right)$. Let $i\prime :F\left[t\right]\to T\left(V\right)$ be the unique $F$-algebra homomorphism for which $i\prime \left(t\right)=X$ and $j\prime :F\left[t\right]\to T\left(V\right)$ be the unique $F$-algebra homomorphism for which $j\prime \left(t\right)=Y$.

   1. Show that $\left(F\left[x,y\right],i,j\right)$ is universal-initial among triples $\left(X,f,g\right)$ where $X$ is a centered $F$-algebra, and $f,g$ are $F$-algebra homomorphisms satisfying $f\left(p\right)g\left(q\right)=g\left(q\right)f\left(p\right)$ for all $p,q\in F\left[t\right]$.

   2. Show that $\left(T\left(V\right),i\prime ,j\prime \right)$ is universal-initial among triples $\left(X,f,g\right)$ where $X$ is a centered $F$-algebra, and $f,g$ are any $F$-algebra homomorphisms.