4.2 Tensor products (over commutative rings)

KNOW: the defining universal property of tensor products, the fact that tensor products uniquely exist; easy rules for tensor products (commutativity, associativity, $R \otimes_{R} M=M$, distributivity for direct sums); tensor products of $R$-linear maps; the fact that injectivity is in general not preserved by tensoring with $M$; the definition of flat modules ( $M \otimes_{R}$ preserves injectivity nevertheless); the fact that a module is flat if all its finitely generated submodules are flat; the implications "free $\Rightarrow$ projective $\Rightarrow$ flat" for all (commutative) rings; "flat $\Rightarrow$ torsion free" for integral domains; the equivalences "free $\Leftrightarrow$ projective" and "flat $\Leftrightarrow$ torsion free" for PID’s; scalar extensions and the general definition of "rank" for an integral domain $R: \operatorname{rk}_{R}(M):=\operatorname{dim}_{F}\left(F \otimes_{R} M\right)$ ( $M$ an $R$-module, $F:=$ field of fractions of $R$ ).

Know the basic Lemma for tensor products of free modules:
If $\left\{e_{i} \mid i \in I\right\}$ is an $R$-basis of $M$ and $\left\{f_{j} \mid j \in J\right\}$ is an $R$-basis of $N$, then $\left\{e_{i} \otimes f_{j} \mid i \in I\right.$ and
$j \in J\}$ is an $R$ - basis of $M \otimes_{R} N$.
Consequence: $\operatorname{dim}\left(V \otimes_{F} W\right)=\operatorname{dim} V \cdot \operatorname{dim} W$ for $F$-vector spaces $V$ and $W$.
Know the basic relation between tensor products and modules of fractions:
If $S$ is a multiplicatively closed subset of $R$ and $M$ is an $R$-module, then $S^{-1} R \otimes_{R} M$ and $S^{-1} M$ are isomorphic, as $R$-modules and as $S^{-1} R$-modules.

Related Problems

1. (Aug 03 #4) (a) Determine the following tensor products and explain your answers.
   (i) $\mathbb{R} \bigotimes_{\mathbb{Z}} \mathbb{Z}_{2003}$;
   (ii) $\mathbb{Q}[x] /\left(x^{2}+1\right) \bigotimes_{\mathbb{Q}[x]} \mathbb{Q}[x] /\left(x^{2}+2\right)$.
   (b) Let $V$ and $W$ be finite-dimensional vector spaces over a field $F$, and $\left\{v_{1}, \ldots, v_{n}\right\}$ be a basis of $V$. Prove that if

$$v_{1} \otimes w_{1}+\ldots+v_{n} \otimes w_{n}=0$$

in $V \bigotimes_{F} W$ for $w_{1}, \ldots w_{n} \in W$, then $w_{1}=\ldots=w_{n}=0$.
2. (Jan 04 #6) Let $V$ and $W$ be finite-dimensional vector spaces over a field $k$, and let $f: V \rightarrow V$ and $g: W \rightarrow W$ be linear operators. One can define a linear operator $f \otimes g: V \bigotimes_{k} W \rightarrow V \otimes_{k} W$. Show that $\operatorname{Tr}(f \otimes g)=\operatorname{Tr}(f) \cdot \operatorname{Tr}(g)$.
3. (Aug 04 #8) Let $K$ be a field, $V$ a finite-dimensional vector space over $K$, and $v, w$ two non-zero vectors in $V$. Show that $v \otimes w=w \otimes v$ in $V \otimes_{K} V$ if and only if there exists a $c \in K^{*}$ such that $w=c v$.
4. (Aug 06 #9) Let $M / K$ be a field extension and let $\zeta \in M \backslash K$ be algebraic over $K$. Let $L:=K(\zeta)$ denote the intermediate field generated by $\zeta$. Show that $L \bigotimes_{K[x]} K[[x]]=\{0\}$. Here, we consider $L$ as a $K[x]$-module where we let act $x$ as multiplication by $\zeta$.
5. (Aug 08 #2) Prove the following statements about tensor products.
(a) $\mathbb{Z}_{5}[x] \bigotimes_{\mathbb{Z}} \mathbb{Q}[x]=0$.
(b) $\mathbb{Q}(i) \bigotimes_{\mathbb{Q}} \mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(i+\sqrt{2})$ as $\mathbb{Q}$-vector spaces.
6. (Jan 09 #8) Let $R$ be a commutative integral domain with 1 , and let $I$ be a principal ideal of $R$. Prove that the $R$-module $I \bigotimes_{R} I$ is torsionfree.
In parts (b) - (d) of this problem let $R=\mathbb{Z}[x]$ and $I=(2, x)$.
(b) Let $m=2 \otimes x-x \otimes 2 \in I \bigotimes_{R} I$. Find a nonzero element $r \in R$ such that $r m=0$.
(c) Consider the mapping $\phi: I \times I \rightarrow \mathbb{Z} / 2 \mathbb{Z}$ given by

$$\phi(p(x), q(x))=\frac{p(0)}{2} q^{\prime}(0) \quad \bmod 2$$

where $q^{\prime}$ is the formal derivative of $q$. Also consider $\mathbb{Z} / 2 \mathbb{Z}$ as an $R$-module via the canonical projection $R \rightarrow \mathbb{Z} / 2 \mathbb{Z}$, i.e. $f . a:=f(0) a \bmod 2$ for $f \in R$ and $a \in \mathbb{Z} / 2 \mathbb{Z}$. Prove that $\phi$ is $R$-bilinear.
(d) Use (c) to prove that $2 \otimes x \neq x \otimes 2$ in $I \bigotimes_{R} I$ (and thus, by (b) $I \bigotimes_{R} I$ is not trosionfree).
7. (Aug $09 \# 7$ ) Let $F$ be a field and $K$ a finite-dimensional vector space over $F$. Let $n=\operatorname{dim}_{F} K$, and assume that $n>1$.
(a) Is it always true that $K \bigotimes_{F} K \cong M_{n}(F)$ as $F$-modules?
(b) Now assume that $K$ also has the structure of a commutative ring with 1 , so being an $F$-vector space, $K$ becomes an $F$-algebra. Recall that in this case $K \bigotimes_{F} K$ possesses a unique $F$-algebra structure such that $(a \otimes b) \cdot(c \otimes d)=a c \otimes b d$ for $a, b, c, d \in K$. Prove that $K \otimes_{F} K$ cannot be a field.
Hint: Construct a non-trivial $F$-algebra homomorphism $K \bigotimes_{F} K \rightarrow K$.
8. (Aug $11 \# 8$ ) Let $K$ be a field and let $A, B$ be commutative $K$-algebras. We do not assume $A$ or $B$ is finite dimensional over $K$.
(a) If the $K$-algebra $A \otimes_{K} B$ is a field, show that $A$ and $B$ must be fields, too. (Partial credit is given if you have to assume that $A$ or $B$ is finite dimensional over $K$.)
(b) Provide an example of two field extensions $A$ and $B$ of degree 2 over $K$ such that $A \otimes_{K} B$ is a field of degree 4 over $K$.
(c) Compute $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C}$ explicitly.
(d) Suppose $K$ has characteristic $p>0$ and that $A \mid K$ is a field extension such that there exists $\xi \in A$ such that $\xi \notin K$, but $\xi^{p} \in K$. Prove $A \otimes_{K} A$ is not a field.
9. (Aug $12 \# 5$ ) Recall that if $R$ is a commutative ring with 1 and $A$ and $B$ are $R$-algebras, then $A \otimes_{R} B$ also has the natural structure of an $R$-algebra.
(a) Let $K$ and $L$ be fields of different characteristics. Prove that $K \otimes_{\mathbb{Z}} L=\{0\}$.
(b) Let $K$ and $L$ be fields of the same positive characteristic $p$. Prove that $K \otimes_{\mathbb{Z}} L$ can be provided in a natural way with the structure of an $\mathbb{F}_{p}$-algebra, and that this $\mathbb{F}_{p}$-algebra is isomorphic to $K \otimes_{\mathbb{F}_{p}} L$. Deduce that $K \otimes_{\mathbb{Z}} L$ is nonzero.
(c) Find an example of commutative rings $A$ and $B$ which are NOT fields such that $A \otimes_{\mathbb{Z}} B$ is a field. Hint: Use a suitable property of tensor products involving direct sums.
10. (Aug $13 \# 2$ ) Let $K$ and $L$ be fields of characteristic 0 . Prove that $K \bigotimes_{\mathbb{Z}} L$ is nonzero.
11. (Jan 14 #5)
(a) Consider the abelian group

$$A=\prod_{n \geq 2} \mathbb{Z} / n \mathbb{Z}$$

Show that this is not a torsion group by exhibiting an element of infinite order.
(b) Show that $\mathbb{Q} \otimes_{\mathbb{Z}} A \neq 0$. (Hint: this is $S^{-1} A$ for $S=\mathbb{Z}-\{0\}$.) Bonus: Determine $\operatorname{dim}_{\mathbb{Q}}\left(\mathbb{Q} \otimes_{\mathbb{Z}} A\right)$.
12. (Aug 14 #2) Compute
(a) $\mathbb{Q} / \mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Q} / \mathbb{Z}$;
(b) $\mathbb{Z}_{2014} \otimes_{\mathbb{Z}} \mathbb{Z}_{2013}$;
(c) $\operatorname{Hom}_{\mathbb{Z}}\left(\mathbb{Z}_{2014}, \mathbb{Z}_{10}\right)$ (Here $\mathbb{Z}_{n}$ is regarded as a $\mathbb{Z}$-module).
13. (Aug $14 \# 5$ ) Let $K$ be a field and $a \in K$. Consider $K$ as a $K[x]$-module (denoted by $K_{a}$ ) via the homomorphism $\mathrm{ev}_{a}: K[x] \rightarrow K$ which is the identity on $K$ and sends $x$ to $a$. Let $K[[x]]$ be the power series ring, which is regarded as a $K[x]$-algebra in a natural way. Determine with proof the tensor product $K_{a} \otimes_{K[x]} K[[x]]$. (Hint: keep in mind if one needs to divide into cases depending on $a$.)
14. (Aug $15 \# 7$ ) Let $V$ be a finite dimensional vector space over a field $F$ and let $V^{*}$ be its dual. For $v \in V$ and $f \in V^{*}$, denote by $\phi_{v, f}$ the endomorphism of $V$ defined by $\phi_{v, f}(w)=f(w) v$ for $w \in V$. Prove that there exists a well-defined $F$-linear map $\Phi: V \otimes_{F} V^{*} \rightarrow \operatorname{End}_{F}(V)$ satisfying $\Phi(v \otimes f)=\phi_{v, f}$ for all $v \in V$ and $f \in V^{*}$. Prove that $\Phi$ is an isomorphism.
15. (Jan $16 \# 8$ ) Let $K$ and $L$ be finite extension fields of a field $F$ of characteristic 0 . Prove that $K \otimes_{F} L$ has no nonzero nilpotent elements.
16. (Aug $16 \# 5$ ) Let $V$ and $W$ be vector spaces over a field $F$, and let $\left\{v_{1}, \ldots, v_{\ell}\right\}$ and $\left\{w_{1}, \ldots, w_{\ell}\right\}$ be elements of these vector spaces. Assume that the vectors $\left\{w_{1}, \ldots, w_{\ell}\right\}$ are linearly independent. Show that $\sum_{i=1}^{\ell} v_{i} \otimes w_{i}=0$ implies that $v_{1}=\cdots=v_{\ell}=0$.
17. (Jan $17 \# 2$ ) Consider a field $K$, and two finite extensions $L, M$ of $K$. Consider the $K$-algebra $L \otimes_{K} M$ (with the usual multiplication $(a \otimes b)(c \otimes d)=a c \otimes b d$ ). Prove that $L \otimes_{K} M$ is a field if and only if any time an extension $E / K$ contains subfields $L^{\prime}$ and $M^{\prime}$ which are $K$-isomorphic to $L$ and $M$, the composite $L^{\prime} M^{\prime}$ has degree $\left[L^{\prime} M^{\prime}: K\right]=\left[L^{\prime}: K\right]\left[M^{\prime}: K\right]$.
18. (Jan $17 \# 3$ ) Let $R$ be a commutative ring, and $M, N$ be $R$-modules. Show that for any submodules $M^{\prime} \subset M$ and $N^{\prime} \subset N$, the induced map $M \otimes_{R} N \rightarrow\left(M / M^{\prime}\right) \otimes_{R}\left(N / N^{\prime}\right)$ has kernel given by $M^{\prime} \otimes N+M \otimes N^{\prime}$. Here $M^{\prime} \otimes N$ and $M \otimes N^{\prime}$ denote, respectively, the images of $M^{\prime} \otimes_{R} N$ and $M \otimes_{R} N^{\prime}$ in $M \otimes_{R} N$.
19. (Aug $17 \# 7$ ) Consider the polynomial ring $R=F[x, y]$ in two variables over the field $F$ and the ideal $I=(x, y)$ of $R$. Let $\phi: R \longrightarrow F$ be the $F$-algebra homomorphism with $\phi(x)=\phi(y)=0$, which turns $F$ into an $R$-module.
(a) Show that the $R$-modules $F \otimes_{R} F$ and $F$ are isomorphic.
(b) Define maps $s, t: I \longrightarrow F$ by $s(f)=c_{1,0}$, respectively, $t(f)=c_{0,1}$ if $f=\sum_{i, j} c_{i, j} x^{i} y^{j} \in I$ (with $c_{i, j} \in F$ for all $i, j \in \mathbb{N}_{0}$ ). Verify that $s$ and $t$ are $R$-module homomorphisms.
(c) Prove that $x \otimes y-y \otimes x$ is not 0 in $I \otimes_{R} I$.
(d) Prove that $I$ is not a flat $R$-module.
20. (Jan 18 #4) Let $A$ be a finite abelian group of order $n, p$ a prime divisor of $n$ and $n=p^{k} m$ with $k, m \in \mathbb{N}$ such that $(p, m)=1$. Denote by $A_{p}$ the Sylow $p$-subgroup of A .
(a) Show that the abelian groups $A_{p}$ and $\mathbb{Z} / p^{k} \mathbb{Z} \otimes_{\mathbb{Z}} A$ are isomorphic.
(b) Describe $\mathbb{Z} / p \mathbb{Z} \otimes_{\mathbb{Z}} A$ as an abelian group without using tensor products but (certain) invariants of $A$.
21. (Aug $19 \# 4$ ) Let $R$ be a commutative ring with 1 , let $S$ be a subring of $R$ with 1 , and let $M$ and $N$ be $R$-modules.
(a) Prove that there exists a surjective $S$-module homomorphism $\phi: M \otimes_{S} N \rightarrow M \otimes_{R} N$ such that $\phi\left(m \otimes_{S} n\right)=m \otimes_{R} n$ for all $m \in M$ and $n \in N$. Also prove that such $\phi$ is unique.
(b) Assume that $R$ and $S$ are both fields, $R \neq S$ and $M$ and $N$ are both nonzero. Prove that $\phi$ in part (a) is not injective. Warning: $M$ and $N$ are not assumed to be finitely generated. (c) Suppose that in (b) we remove the assumption ’ $S$ is a field’ (but keep all other assumptions). Does the conclusion of (b) remain true?
22. (Jan $20 \# 7$ ) Let $F, K_{1}, K_{2}$ and $L$ be fields with $F \subseteq K_{1} \subseteq L$ and $F \subseteq K_{2} \subseteq L$. Recall that the tensor product $K_{1} \otimes_{F} K_{2}$ has a natural structure of a ring where multiplication of simple tensors is given by $(a \otimes b) \cdot(c \otimes d)=a c \otimes b d$ for all $a, c \in K_{1}$ and $b, d \in K_{2}$.
(a) Prove that there exists a unique ring homomorphism $\pi: K_{1} \otimes_{F} K_{2} \rightarrow L$ such that $\pi(a \otimes b)=a b$ for all $a \in K_{1}, b \in K_{2}$.
(b) Assume that $K_{1} \otimes_{F} K_{2}$ is a field. Prove that $\pi$ is injective.
(c) Assume that $K_{1} \cap K_{2} \neq F$. Prove that $\pi$ is not injective.
23. (Aug $20 \# 6$ ) Let $K$ be a field of characteristic 0 , and fix some algebraic closure $\bar{K}$ of $K$. Let $L / K$ be a finite extension, with $L \subseteq \bar{K}$, and let $\alpha$ be an element of $\bar{K}$.
(a) Prove that there is a surjective $K$-algebra homomorphism $\rho: L \otimes_{K} K(\alpha) \rightarrow L(\alpha)$.
(b) Prove that there exists an injective $K$-algebra homomorphism (not necessarily sending 1 to 1) $\iota: L(\alpha) \rightarrow L \otimes_{K} K(\alpha)$. Hint: $K(\alpha) \cong K[x] /\left(\mu_{\alpha, K}(x)\right)$.
24. (Jan 21 #6) In both parts of this problem, $R$ is a commutative ring with 1 , assume that $|R|>1$, and $M$ is a flat $R$-module, that is, assume that
Whenever we have an injective homomorphism $N \xrightarrow{f} N^{\prime}$ of $R$-modules, the induced map $f \otimes 1_{M}: N \otimes_{R} M \rightarrow N^{\prime} \otimes_{R} M$ is also injective.
(a) Assume that $R$ is a domain. Prove that $M$ must be torsion-free.
(b) Now let $R$ be arbitrary. Prove that the following are equivalent:
i. for every nonzero $R$-module $N \neq 0$ we have $N \otimes_{R} M \neq 0$
ii. for every maximal ideal $\mathfrak{m}$ of $R$ we have $\mathfrak{m} M \neq M$.

Note: You may use without proof standard isomorphisms of the form $(R / I) \otimes_{R} M \cong \ldots$ where $I$ is an ideal of $R$.
25. (Aug $21 \# 5$ ) Let $R$ be a commutative ring with 1 and let $M$ and $N$ be finitely generated $R$-modules.
(a) Prove that $M \otimes_{R} N$ is finitely generated.
(b) Assume in addition that $M$ is Noetherian. Prove that $M \otimes_{R} N$ is Noetherian. Note: You can use standard properties of Noetherian modules (unless they are equivalent or almost equivalent to the statement of (b)), but state clearly what you are using.

Two more problems about tensor products:
26. Prove or disprove:
(a) $\mathbb{Q}(\sqrt{10}) \otimes_{\mathbb{Q}} \mathbb{Q}(\sqrt{10})$ is isomorphic to $\mathbb{Q}(\sqrt{10})$ as a $\mathbb{Q}$-vector space.
(b) $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q}$ is isomorphic to $\mathbb{Q}$ as a $\mathbb{Z}$-module.
27. Let $I$ and $J$ be ideals of a commutative ring $R$ with 1 . Show that $R / I \otimes_{R} R / J$ is isomorphic (as $R$-algebra) to $R /(I+J)$.