Tagged Questions

0
votes
1answer
26 views

Tensor product of two cyclic modules

Given a commutative ring $A$ (with identity) and two cyclic $A$-modules, $M$ and $N$ with generators $x$ and $y$, respectively. How do you show that $\mathrm{Ann}(x\otimes_A y) = \mathrm{Ann}(x) + ...
2
votes
1answer
37 views

$G$-invariants of tensor products of $G$-modules

Given two $G$-modules $A$ and $B$, are there general criteria which induce $$ (A \otimes B)^G = A^G \otimes B^G $$ where $\sigma(a \otimes b) = \sigma a \otimes \sigma b$? I saw a note stating, that ...
7
votes
1answer
60 views

If a tensor product is free, what can we say about the tensor factors?

Here is what I'd like to prove: Let $R$ be a commutative, noetherian ring, and let $M$ and $N$ be finitely generated $R$-modules. Suppose $M\otimes_RN\cong R$. Does it follow that $M\cong N\cong ...
3
votes
1answer
39 views

Question on Criterion for Flat Modules

I'm reading Categories and Modules with K-Theory in View by A. J. Berrick, M. E. Keating, on Flat Modules and there's one part that I'm not quite sure that I fully grasp it. It's on page 155. ...
3
votes
1answer
68 views

If $x\otimes y=0$, then $x\otimes y=0$ in the tensor product of finitely generated submodules

Let $A$ be a commutative ring, let $E,F$ be $A$-modules. Let $x\in E$ and $y\in F$. If $x\otimes y=0$ in $E\otimes F$, then there exist finitely generated submodules $M\subset E$ containing $x$ and ...
4
votes
2answers
96 views

Tensor product over different rings are not equal

I'm looking for an example of modules $A,B$ such that $A\otimes_{R}B\neq A\otimes_{\mathbb{Z}}B$. I've thought about some tensor products I know but I haven't been able to produce an example. What ...
2
votes
1answer
53 views

Tensor product of module homomorphisms

This is an exercise problem from Hungerford's Algebra but first I'll state a result that forms the background to the problem. Let $R$ be a ring. Let $A,A'$ be right $R$-modules. Let $B,B'$ be left ...
6
votes
0answers
116 views

Alternative construction of the tensor product (or: pass this secret)

The paper Tensor products and bimorphisms by B. Banachewski and E. Nelson studies tensor products (defined by classifying bimorphisms) in concrete categories. It is quite interesting that their main ...
6
votes
2answers
125 views

Restriction of scalars and tensor product

All rings I'll consider will be commutative with identity. Given a homomorphism $f:R \to S$ we can give an $S$-module an $R$-module structure via restriction of scalars. In particular, $S$ can be ...
3
votes
1answer
51 views

If Hom(M,-) is stable under base change, is then M f.g. projective?

Let $R$ be a commutative ring. Let $M$ be an $R$-module with the following property: For every commutative $R$-algebra $A$ and every $R$-module $N$ the canonical map $\mathrm{Hom}_R(M,N) \otimes_R A ...
0
votes
0answers
81 views

Checking the universal property of tensor products of modules

I'm wondering about a general procedure for checking that an object satisfies the universal property of tensor products. (Similar questions already asked seem to consider only special cases and use, ...
3
votes
1answer
96 views

How does one show that this tensor product is not torsion-free?

I am having trouble showing that a particular tensor product is not torsion-free. Let $ R = k[[x,y]] $, where $ k $ is a field (this is the ring of formal power series in $ x $ and $ y $ with ...
0
votes
1answer
76 views

tensor product over local ring

If $R$ is a local ring and $M$ and $N$ are finitely generated R-modules such that $M\otimes N = 0$ then it follows from Nakayama's lemma that either $M=0$ or $N=0$. This I know. But now I am looking ...
0
votes
2answers
98 views

How to compute $\mathbb{Z}/m\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}$? [duplicate]

Possible Duplicate: Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$ How to compute $\mathbb{Z}/m\mathbb{Z} ...
1
vote
1answer
127 views

A question regarding tensor product and isogenies of elliptic curves

Let $E_1$ and $E_2$ be elliptic curves and $T_l(E_i)\cong \mathbb{Z}_l \oplus \mathbb{Z}_l$ the $l$-adic Tate module. Given $ \varphi \in Hom(E_1,E_2)$ this induces $\varphi_l \in ...
1
vote
1answer
70 views

$\mathbb{Q}$-dimension of f. g. $\mathbb{Q}[\mathbb{Z}/p^l]$-modules.

My question arises from the previous question Let $M$ be a finitely generated $\mathbb{Q}[\mathbb{Z}/p^l]$-module, where $p$ is a prime number. Is it true that \begin{equation}\dim_\mathbb{Q} ...
1
vote
1answer
53 views

Is this valid reasoning? Exactness question

I am wondering if this is valid reasoning: Let $N$ and $N'$ be $R$-modules. I know that $0\rightarrow N\rightarrow N'$ is exact and want to show for a given multiplicative subset $S$ and that ...
2
votes
1answer
63 views

Basis for a tensor product of group algebras

Let $G$ and $H$ be groups, and $R$ a commutative ring. Then elements of $RG$ look like finite sums $\sum\limits_{g\in G}r_g\,g$, and similarly for $RH$. So $RG$ and $RH$ are $R$-modules with bases $G$ ...
5
votes
2answers
263 views

Universal property of tensor product [duplicate]

Possible Duplicate: Equality of two notions of tensor products over a commutative ring Let $A$ be a commutative ring. There are two common definitions of tensor product of two $A$-modules ...
8
votes
1answer
149 views

Tensor product of simple modules

Let $M$ a right simple module and $N$ be a left simple module over a ring $R$. My questions are: How can we describe $M \otimes_R N$ explicitly? Well, I guess that it is a quotient of $R$ by a sum of ...
18
votes
3answers
306 views

$\operatorname{Ann}(M\otimes_A N)=\operatorname{Ann}M+\operatorname{Ann}N$?

In the course of working on an exercise in Atiyah-MacDonald (exercise 3 on p. 31), I've come to the belief that, for $A$ an arbitrary commutative ring and $M,N$ arbitrary $A$-modules, ...
4
votes
1answer
165 views

$C \otimes A \cong C \otimes B$ does not imply $A \cong B$

Let $R$ be a commutative unital ring and let $M$ be an $R$-module and let $S$ be a multiplicative subset of $R$. Today I proved both of the following: $$ S^{-1} R\otimes_R S^{-1}M \cong S^{-1} M$$ ...
0
votes
1answer
114 views

Tensor product of a finitely generated modul and a finite length module is finite length

Let $R$ be a commutative ring and $M,N$ $R$-modules finitely generated with $M$ of finite length. How can I prove that $M\otimes_R N$ is of finite length?
6
votes
1answer
66 views

r-th transvectants and $\mathbb{C}G$-module maps

Suppose $V=\mathbb{C}^2$ and $G=SL(V)=SL_2(\mathbb{C})$. We define $C_n = H_{\mathbb{C},n}(V,\mathbb{C}) \cong S^n(V^*)$, the n-th symmetric power of the dual of $V$, i.e. the homogeneous polynomials ...
1
vote
0answers
49 views

Prove that $Fun(X)\otimes_\Bbb{C}Fun(Y)\cong Fun(X\oplus Y)$

Let $X,Y$ be finite sets, $Fun(X),Fun(X),Fun(X\oplus Y)$ be $\Bbb{C}$-modules of functions mapping the corresponding sets to $\Bbb{C}$. It's obvious that $dim(Fun(X))=|X|$ and ...
2
votes
1answer
87 views

Why is the image of $M\times N\rightarrow M\otimes N, (m,n)\mapsto m\otimes n$ not a submodule?

$M,N$ are $A$-modules. I don't see why the statement is true. Can you explain please?
2
votes
1answer
85 views

Why does $M$ free imply $M\otimes_K N$ is free also?

I don't understand the following implication in some notes I'm reading, either I'm overlooking something simple, or some details are being swept under the rug. Let $K$ be a field, and ...
1
vote
2answers
121 views

Prove that $A^{\oplus n} \otimes_A N \cong N^{\oplus n}$

Let $A,N$ be $A$-modules. I'm trying to prove $A^{\oplus n} \otimes_A N \cong N^{\oplus n}$. First we define $f: F(A,N) \rightarrow N^{\oplus n}$ where $F(A,N)$ is a free module with a basis ...
5
votes
2answers
279 views

Examples proving why the tensor product does not distribute over direct products?

I recently read about the result that the tensor product distributes over direct sums. I was curious if it also distributes over direct products, but google tells me it doesn't. What are some simple ...
15
votes
1answer
373 views

Modules with $m \otimes n = n \otimes m$

Let $R$ be a commutative ring. Which $R$-modules $M$ have the property that the symmetry map $$M \otimes_R M \to M \otimes_R M, ~m \otimes n \mapsto n \otimes m$$ equals the identity? In other ...
2
votes
1answer
56 views

Finding a $\mathbb{C}$-balanced map from $\mathbb{C}^2\to\mathbb{R}^2$?

I was trying to show $\mathbb{C}\otimes_\mathbb{C}\mathbb{C}\cong\mathbb{R}^2$ as $\mathbb{R}$-modules. At one point I would like to prove the existence of an $\mathbb{R}$-module homomorphism from ...
2
votes
0answers
71 views

Isomorphism modulo torsion is stable under tensor products

Let $\phi : A \to B$ be a homomorphism of grades $k$-algebras and $N$ be a graded $A$-right module. Assume that $\mathrm{ker}(\phi)$ and $\mathrm{coker}(\phi)$ are torsion $A$-right modules, i.e. ...
2
votes
1answer
135 views

Extending an ideal of a polynomial ring to a polynomial ring with more indeterminates. Is it a tensor product?

Let $\mathbb{k}$ be a field, let $S'=\mathbb{k}[x_1,x_2,\dots,x_m]$, and let $I'\subseteq S'$ be an ideal. For some $n>m$, let $$S=\mathbb{k}[x_1,x_2,\dots,x_n]\ \ \ ...
9
votes
1answer
475 views

What does a zero tensor product imply?

I'm trying to prove that for two finitely generated $A$-modules $M,N$ ($A$ being any ring), the tensor product $M\otimes_A N$ is zero iff $Ann(M)+Ann(N)=A$. The if direction is of course easy- just ...
10
votes
2answers
672 views

Proving that the tensor product is right exact

Let $A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0$ a exact sequence of left $R$-modules and $M$ a left $R$-module ($R$ any ring). I am trying to prove that ...
2
votes
1answer
59 views

Equivalent conditions for existence of an invertible element in an exterior algebra

I am working on old qualifying problems involving tensor products. I am stuck on a statement about invertible elements in an exterior algebra and was wondering if this was a well known fact in a book ...
5
votes
1answer
232 views

A question about the tensor product of $\mathbb{Q}$

I'm reading this blog post about $\mathbb{Q} \otimes_\mathbb{Z} \mathbb{Q}$ and I have two questions about it: Is a simple tensor a tensor that cannot be written as a sum of tensors? On the first ...
2
votes
1answer
136 views

Isomorphism involving tensor products of homomorphism groups

This post is related to the following question: A counterexample to the isomorphism $M^{*} \otimes M \rightarrow Hom_R( M,M)$. I have been trying to isolate what hypothesis can be eliminated in these ...
2
votes
1answer
215 views

construction of the tensor product, generating submodule which is factored out

The tensor product of modules $M_0, M_1$ is a quotient of a free module $F$, …, by a submodule $F'$. I found 2 definitions of this $F'$, and the difference is in this generating rules: ...
0
votes
1answer
80 views

Superscript wedge on an $R$-module

I'm reading these lecture notes here about the tensor product and the author uses $M^\vee$ where $M$ is an $R$-module. Look at page 2 for an example (example 2.1). What does it mean? I looked through ...
2
votes
1answer
216 views

Tensoring a monomorphism of free modules with an identity map

Suppose $R$ is a commutative ring, $f\colon F_1\to F_2$ is a homomorphism of free modules, and $M$ is an $R$-module. If $f$ is a surjective homomorphism, then $f\otimes_R \mathrm{id}_M$ is ...
5
votes
1answer
613 views

Given a commutative ring $R$ and an epimorphism $R^m \to R^n$ is then $m \geq n$?

If $\varphi:R^{m}\to R^{n}$ is an epimorphism of free modules over a commutative ring, does it follow that $m \geq n$? This is obviously true for vector spaces over a field, but how would one show ...