1
vote
1answer
99 views

$k[x]/(x^n) \otimes_{k[x]} k[y]/(y^m)$

This is part of an exercise from Eisenbud: $k$ is a field, describe as explicitly as possible a) $k[x]/(x^n) \otimes_{k[x]} k[y]/(y^m)$ b) $k[x] \otimes_{k} k[y]$ Any hint ?
0
votes
1answer
27 views

Are the generators of the subgroup defining tensor products linearly independent over $\mathbb Z$?

Let $S$ be a (commutative) ring with identity, and let $M$, $N$ be $S$-modules. (I guess if $S$ isn't commutative, I want $M$ to be a right $S$-module an $N$ a left $S$-module.) In the definition of ...
0
votes
2answers
76 views

Tensor product of quotient rings [duplicate]

$A$ is a commutative ring with unit and $\mathfrak a$, $\mathfrak b$ ideals. I have to show that $$A/\mathfrak a \otimes_{A} A/\mathfrak b \cong A/(\mathfrak{a+b}).$$ Any hint ?
2
votes
1answer
35 views

Example of $\sum_i a_i\otimes b_i\in M\otimes_AN$ which cannot be written as $a\otimes b$

In the appendix of my commutative algebra text: Note that in general the element of $M\otimes_AN$ is a sum of the form $\sum_i a_i\otimes b_i$ and cannot be necessarily written as $a\otimes b$. ...
0
votes
0answers
23 views

Examples where $\hat{A}$ is flat as an $A$-algebra (and $A$ is not noetherian)?

Lately I've looked a bit at $f$-adic completions of commutative rings (see for example my last 2 questions), so here's another question concerning the topic: Let $A$ a commutative ring, $f \in A$ not ...
1
vote
0answers
40 views

How can we show that $\operatorname{Tor}_1^A(\hat{A},G_f/G)=0$?

This question is quite closely related to my last question: Is it always true that $\hat{A} \otimes_A A_f/A \cong \hat{A}_f/ \hat{A}$? Let $A$ a commutative ring, $f \in A$ a non zerodivisor. Let ...
1
vote
1answer
35 views

Homomorphisms from the base change of a module

Let $A, B$ be commutative rings with one and let $M$ be an $A$-module, $f: A \rightarrow B$ a ring homomorphism. Consider the (right) $B$-module $M \otimes_A B$. What can we say about ...
2
votes
3answers
105 views

What is $\mathbb{Z}/n\mathbb{Z}\otimes_\mathbb{Z} m\mathbb{Z}$?

I would like to know what $\mathbb{Z}/n\mathbb{Z}\otimes_\mathbb{Z} m\mathbb{Z}$ is isomorphic to, where $n,m\in\mathbb{N}$. Of course there will likely be cases depending on coprimeness and whatnot; ...
0
votes
0answers
43 views

$\frac{\mathbb{Z}}{m\mathbb{Z}}\otimes_{Z}\frac{\mathbb{Z}}{n\mathbb{Z}} \cong \frac{\mathbb{Z}}{d\mathbb{Z}}$ [duplicate]

I want to prove that $$\frac{\mathbb{Z}}{m\mathbb{Z}}\otimes_{\mathbb{Z}}\frac{\mathbb{Z}}{n\mathbb{Z}} \cong \frac{\mathbb{Z}}{d\mathbb{Z}}$$ where $m ,n \in \mathbb{N}$ and $d = \gcd(m,n)$. Any ...
3
votes
2answers
47 views

$\mathbb{Z}/n\mathbb{Z}$ is not flat

On the flat module Wikipedia page, it's stated that $\mathbb{Z}/n\mathbb{Z}$ is not flat over $\mathbb{Z}$. But I don't understand their explanation of why. It is said that ...
1
vote
1answer
191 views

Example of non-noetherian algebras which are tensor products of noetherian algebras

We suppose all rings are commutative with unity. I am looking for examples of a tensor product $B\otimes_A C$ which is not noetherian, where $A$ is a noetherian ring and $B, C$ are noetherian ...
1
vote
1answer
24 views

Given ring $A$, ideal $I$, and $A$-module $M$, show that $A/I \otimes_A M$ is isomorphic to $M/IM$.

The question is stated as in the title; the hint I am given is to "tensor the exact sequence" $0 \rightarrow I \rightarrow A \rightarrow A/I \rightarrow 0$, which I take to mean using that sequence ...
3
votes
1answer
73 views

Prove $A_\mathfrak{p} \otimes_A B_\mathfrak{q} = B_\mathfrak{q}$, where $\mathfrak{q}$ prime in $B$

$\require{AMScd}$ Hi, I think I have the answer for this question, but I'm not sure if it's correct. So I would be very glad if someone could have a quick look through it. Let $A$, $B$ be ...
3
votes
1answer
38 views

Unramification stable under change base

I want to show that if $f:X\to Y$ is an unramified scheme morphism (ie $m_y\mathcal{O}_{X,x}=m_x\mathcal{O}_{X,x}$ and $k(x)\leftarrow k(y)$ finite and separable) then any base change $X\times_Y Z\to ...
0
votes
1answer
59 views

Examples proving that the tensor product does not commute with direct products

Examples proving why the tensor product does not distribute over direct products? In fact the canonical map is not surjective; can you give me a simple example?
1
vote
3answers
84 views

Tensor products and monomorphisms of $A$-modules.

This problem addresses the same question that has been asked in $A^m\hookrightarrow A^n$ implies $m\leq n$ for a ring $A\neq 0$, which is exercise 2.11 of Atiyah and Macdonald's Introduction to ...
1
vote
2answers
96 views

If for $A$ a commutative nonzero ring $A^m ≅ A^n$ as $A$-modules, then $m = n$

This is the problem I need to solve: Let $A$ be a nonzero ring. Show that if $A^m ≅ A^n$, then $m = n$. The book I got this problem from suggests using the following method to solve it: Let ...
1
vote
0answers
39 views

Question about tensor product of homomorphisms

I've come to think about this problem when reading a proof in Commutative Algebra by N. Bourbaki. Say, let $R$ be a commutative ring, given 3 $R-$modules $A$, $B$, $C$, and the $R$-homomorphism $f:B ...
7
votes
2answers
216 views

Origin of the modern definition of the tensor product

Due to whom is the modern (i.e. via its universal property) definition of the tensor product, and in which article was it communicated?
0
votes
1answer
43 views

On scalar extension of module and annihilator

Let $A, B$ be commutative rings with identity, $f: A \longrightarrow B$ a ring morphism, $M$ an $A$-module. Given $b\in B, x\in M$, does the following statement hold? $b\otimes x=0$ in $B ...
0
votes
2answers
58 views

Prime ideals in tensor products of algebras and their pullbacks

Suppose $\mathfrak{p}$ is a prime ideal in $B\otimes_CA$, and $\mathfrak{p}_A,\mathfrak{p}_B,\mathfrak{p}_C$ are its pullbacks in $A,B,C$. Does it hold: $(B\otimes_CA)_{\mathfrak{p}}\cong ...
2
votes
1answer
38 views

Homomorphism of modules and Tensor Product.

Let $\phi: A \rightarrow B$ be a ring homomorphism. Let $M$ be an $A$-module. We can think $B$ as $A$-module via the map $\phi$ defined by $\phi:A\times B \rightarrow B$, $(a,b)\mapsto\phi(a)\cdot ...
3
votes
1answer
70 views

A criterion for an extension to be Galois

This is an exercise given during my Commutative Algebra course. I reached to solve just the "if" arrow, but not the "only if". The question is: Let $F\subseteq L$ be a finite degree extension of ...
0
votes
1answer
88 views

Tensor product of a module and a localized ring

Let $A$ be a commutative ring with unity. Let $S$ be a multiplicative subset of $A$. Let $M$ be an $A$-module. Let $x \in M$. Suppose $x\otimes 1 = 0$ in $M\otimes_A S^{-1}A$. Then there exists $s \in ...
4
votes
0answers
66 views

Tensor product, Artin-Rees lemma and Krull intersection theorem

I asked another question about tensor product, but can't conclude from the answer, so here is another more concrete question. Let $(A,m)$ be a local ring then by Artin-Rees Lemma $m^k \bigcap I ...
0
votes
1answer
52 views

Question about tensor product of modules and ideals

Trying to prove some properties of tensor product with a given module, I came up with questions some of them I can't prove. Maybe it is also because Im not very used to work with tensor products and I ...
2
votes
2answers
91 views

Eisenbud's proof of right-exactness of the exterior algebra

I'm trying to understand the proof in Eisenbud's Commutative Algebra that, given a right exact sequence $$K \to N \to M \to 0$$ of $R$-modules, we have an exact sequence $$K \otimes \wedge N \to ...
3
votes
1answer
54 views

$M\otimes_A N = M\otimes_{A/a} N $?

Let $A$ be a commutative ring and $a$ an ideal. Let $M$ and $N$ be $A$-modules. Now suppose $a\subset Ann(M)\cap Ann(N)$. I'm hoping that $M\otimes_A N = M\otimes_{A/a} N$. Is it true?
2
votes
0answers
39 views

Mistake in the proof that a domain is flat as a module over any subring

Where is the mistake in the following argument? I feel that there has to be one, for example by the very existence of this article. Let $R$ be an integral domain and $S \subseteq R$ be a subring ...
5
votes
1answer
62 views

Determining whether a certain element in a tensor product is zero

Let $I = (x,y) \subset k[x,y]$, where $k$ is a field. Prove that a) $x \otimes y - y \otimes x =0$ inside $k[x,y] \otimes_{k[x,y]} k[x,y]$. b) $x \otimes y - y \otimes x \not= 0$ inside ...
4
votes
1answer
142 views

What is the kernel of the tensor product of two maps?

Assume that $f_1\colon V_1\to W_1, f_2\colon V_2\to W_2$ are $k$-linear maps between $k$-vector spaces (over the same field $k$, but the dimension may be infinity), then the tensor product $f_1\otimes ...
6
votes
2answers
191 views

Idempotents in $\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{Z}[i]$

Letting $\mathbb{Z}[i]=\left\{a+bi:a,b \in \mathbb{Z} \right\}$ be the ring of Gaussian integers, how many idempotents are there in $\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{Z}[i]$? I came ...
2
votes
1answer
44 views

Tensor product of modules

Let $R$ be a polynomial ring over $\mathbb{C}$. Let $R_1=R/I$ for some ideal $I \subset R$. Let $M_1, M_2$ be $R_1$-modules. So, they are $R$-modules as well. Is it true that $M_1 \otimes_{R_1} M_2 ...
0
votes
1answer
66 views

Decomposable Tensors over Rings

Suppose $R$ is a commutative ring and $M$ is a $R$-module. Then we can define the tensor product $M\otimes_R M$ and more generally the $k$-fold tensor powers $\otimes_R^kM$ for any $k\in\mathbb{N}$, ...
5
votes
3answers
121 views

When does there exist a commutative ring $C$ that contains rings $A$ and $B$ as a subring?

The statement I'm trying to prove is the following: Let $A$ and $B$ be commutative rings, both of characteristic $0$. Then there exists a commutative ring $C$ that contains both $A$ and $B$ as ...
4
votes
2answers
141 views

About the injection $M \hookrightarrow \mathbb Q \otimes_{\mathbb Z} M$.

I want to prove that every abelian group can be embedded in a divisible abelian group. So I tried $M \rightarrow \mathbb Q \otimes_{\mathbb Z} M, m \mapsto 1 \otimes m$. It is obvious that $\mathbb Q ...
4
votes
1answer
85 views

Is injectivity of algebras preserved by tensor products?

Suppose $R' \subset R$, $S'\subset S$ are inclusion of $k$-algebras. Does it hold that $R'\otimes_kS' \rightarrow R \otimes_k S$ is injective ? I know there're counterexamples for modules, but ...
4
votes
1answer
175 views

Example of rings of the same positive characteristic that do not embed into their tensor product?

I'm overcoming my fear of tensor products, and the following exercise got me wondering: Give an example of commutative rings $A$ and $B$ with $\operatorname{char}A=\operatorname{char}B$ such that ...
6
votes
2answers
114 views

Embedding a module into its quotient module

I've got a very basic question on tensor products. Let $R$ be a commutative integral domain, $K$ its quotient field and let $M$ be a $R$-module. Is the map $M \rightarrow K\otimes_R M$ given by ...
7
votes
1answer
136 views

Understanding the right-exactness of the tensor product using *only* its universal property and the Yoneda lemma

I would like to get an intuition for why $(-)\otimes N$ is right-exact using its universal property involving bilinear maps, not by appealing to higher-level observations such as "left-adjoints ...
4
votes
1answer
154 views

Could I see that the tensor product is right-exact using its universal property and the Yoneda lemma?

I have been doing some review with the goal of trying to understand as much as I can via universal properties and category theory (already feeling comfortable with the mundane way of doing things). ...
2
votes
2answers
131 views

$A/ I \otimes_A A/J \cong A/(I+J)$

For commutative ring with unit $A$, ideals $I, J$ it holds $$A/ I \otimes_A A/J \cong A/(I+J).$$ A proof can be found here (Problem 10.4.16) for example. However, I'd be interested in a less ...
4
votes
5answers
105 views

Direction of map between tensor products

Suppose $A$ and $B$ are commutative rings, $A\to B$ is a ring map, and $M, N$ are $B$-modules. Is there a map $M\otimes_A N \to M\otimes_B N$, or in the other direction? This should be very ...
3
votes
1answer
67 views

Atiyah-Macdonald, Proposition 2.12, uniqueness of the tensor product.

The following is a result from Atiyah-Macdonald, defining and showing existence and uniqueness of tensor product of modules over a commutative ring. Proposition 2.12. Let $M, N$ be $A$-modules. ...
21
votes
3answers
384 views

My proof of $I \otimes N \cong IN$ is clearly wrong, but where have I gone wrong?

Ok, I'm reading some thesis of some former students, and come up with this proof, but it doesn't really look good to me. So I guess it should be wrong somewhere. So, here it goes: Let $R$ be a ...
5
votes
1answer
101 views

Length of the tensor product of two modules

I'm trying to prove the following statement: Let $R$ be a commutative ring. If $M$ and $N$ are finite length $R$-modules, then $M \otimes_{R} N$ has finite length, and ...
2
votes
0answers
150 views

Quick question on localization of tensor products

All rings are commutative with unit. Let $\rho:A\rightarrow B$ be a ring homomorphism. Suppose $\mathfrak q$ is a prime ideal of $B$, and let $\mathfrak p=\rho^{-1}(\mathfrak q)$. My question: Is ...
5
votes
2answers
135 views

a flatness criterion

I'm having trouble with part (b) of Exercise 10.5.25 from Dummit & Foote (the goal of the problem is to prove that $A$ is a flat $R$-module iff $A\otimes_R I\to A\otimes_R R$ is one-to-one for all ...
4
votes
1answer
126 views

The natural map $M \to M \otimes_R K$ is injective iff $M$ is torsion free

I'm reading some lecture notes of Pete L. Clark, and there's one problem that I cannot solve. It's on page 45 of this book: Commutative Algebra. The problem reads as follow: Exercise 3.42 Let ...
5
votes
2answers
92 views

Does $A\!\leq\!M$ and $B\!\leq\!N$ imply $A\!\otimes_R\!B\hookrightarrow M\!\otimes_R\!N$? (tensor product of modules)

Let $R$ be a commutative unital ring. What would be an example of a $R$-modules $M,N$ with submodules $A,B$, such that there does not exist an embedding of $R$-modules $$A\!\otimes_R\!B\hookrightarrow ...