3
votes
0answers
79 views

Tensor products over monoids : Element structure

Let $A$ be a (commutative) monoid. Let $M$ be a right $A$-set and let $N$ be a left $A$-set. Then we can construct the tensor product $M \otimes_A N$, which is a set (of even $A$-set when $A$ is ...
1
vote
0answers
56 views

Tensor product of free modules over free algebra

Suppose $M$ and $N$ are modules over a (commutative, unital) ring $S$. Let $R$ be a subring of $S$ such that $S,M$ and $N$ are all free, finitely generated modules over $R$. Question: Under what ...
1
vote
0answers
60 views

Surjectivity implies injectivity of finitely generated modules, localization?

The following problem is canonical: Suppose $A$ is a commutative unitary ring, and $M$ is a finitely generated module over $A$. If an endomorphism $f\colon M\to M$ is surjective, then it's also ...
0
votes
0answers
41 views

Exercise 7.10 Atiyah, $M[x] $ is a noetherian $A[x] $-module [duplicate]

The exercise is: Let $M$ be a noetherian $A$-module. Then $M[x] $ is a noetherian $A[x] $ module. The action of $A[x] $ on $M[x] $ is the obvious one. In a previous exercise it was shown that ...
4
votes
2answers
125 views

Proving that tensoring a projective module with a flat module gives a projective module?

If $P$ is a projective module and $M$ is a flat module, both over some commutative ring $R$, then how do you prove that $M\otimes_R P$ is flat? I tried using the fact that $P$ is the direct ...
2
votes
2answers
48 views

Relation between $\operatorname{Coker}(f)$ and $\operatorname{Coker}(f \otimes 1_P)$

Let $M,N,P$ be $R$-modules ($R$ commutative ring with $1$) and let $f:M\to N$ be a $R$-module homormorphism. Let tensor the homomorphism to get $ f \otimes 1_P : M \otimes P \to N \otimes P $. I ...
0
votes
1answer
45 views

tensor product of R-algebra and f.g module [closed]

$R$ is a commutative noetherian ring. If $S$ is an $R$-algebra, and $M$ a finitely generated $R$-module, is $M\otimes_RS$ finitely generated $S$-module? I only need a hint. Thanks!
2
votes
0answers
52 views

$M_{\mathfrak{p}} \otimes_{R_{\mathfrak{p}}} N_{\mathfrak{p}} = 0$ implies $M_{\mathfrak{p}} = 0$ or $N_{\mathfrak{p}} = 0$ [duplicate]

Studying commutative algebra I've found this statement: If $M$ and $N$ are finitely generated $R$-modules, with $R$ a commutative ring, and $\mathfrak{p} \subset R $ is a prime ideal, then ...
0
votes
1answer
53 views

$\mathcal{O}_{X_y,x}=\mathcal{O}_{X,x}/\mathfrak{m}_y\mathcal{O}_{X,x}$

In a proof (proof of theorem 4.3.36 in Liu's book) I need the equality $\mathcal{O}_{X_y,x}=\mathcal{O}_{X,x}/\mathfrak{m}_y\mathcal{O}_{X,x}$. The hypothesis of the theorem are the following: $Y$ ...
7
votes
1answer
104 views

When does the tensor product consist of elementary tensors only?

The question is: Assume that $R$ is a (commutative) ring. Under what conditions on $R$-modules $M,N$ does the tensor product $M\otimes_RN$ consist of elementary tensors only? That is, every ...
0
votes
1answer
41 views

Number of generators in a tensor product

'Commutative Algebra' by Atiyah and Mcdonald, mentions if $ \lbrace x_{i} \rbrace_{i ∈ I}$ and $\lbrace y_{j} \rbrace_{j ∈ J}$ generate $M$ and $N$ as $A-$modules, respectively, then $x_{i} \otimes ...
3
votes
0answers
54 views

When is $N\otimes_A B \to N$ an isomorphism?

Let $A, B$ be commutative (unital) rings and $f\colon A \to B$ an $A$-algebra. There then exists a canonical functor $f_*\colon \mathbf{Mod}_B \to \mathbf{Mod}_A$ such that, for every morphism of ...
0
votes
1answer
78 views

$\biggl( \prod_{i \in \mathbb{N}}\mathbb{Z}/p^i\mathbb{Z}\biggr)\otimes_{\mathbb{Z}} \mathbb{Q}\ne 0$

I've found this claim $$\biggl( \prod_{i \in \mathbb{N}}\mathbb{Z}/p^i\mathbb{Z}\biggr)\otimes_{\mathbb{Z}} \mathbb{Q} \not\cong \prod_{i \in \mathbb{N}}\biggl( ...
1
vote
1answer
112 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
35 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
169 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 ?
1
vote
1answer
49 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
31 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
43 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
42 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
109 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
44 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
52 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 ...
3
votes
1answer
223 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
26 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
86 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
46 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 ...
4
votes
1answer
124 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
106 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
108 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
46 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 ...
5
votes
2answers
258 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
51 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
90 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
51 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
73 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
99 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
73 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
55 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
126 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
55 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
42 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 ...
4
votes
1answer
71 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
228 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
205 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
47 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
73 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
131 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
171 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
87 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 ...