Tagged Questions

3
votes
1answer
76 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 ...
4
votes
2answers
56 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 ...
1
vote
1answer
111 views

Tensor product is zero

If $R$ is a local ring and $M$ and $N$ are finitely generated $R$-modules such that $M\otimes N=0$ then how does it follow from Nakayama's lemma that either $M=0$ or $N=0$? This is an exercise ...
4
votes
1answer
112 views

Defining a surjective $\mathbb{Q}$-algebra homomorphism

Let $p,q$ be prime numbers. I want to define a surjective $\mathbb{Q}$-algebra homomorphism $\phi:\mathbb{Q}(\sqrt{p})[X]\rightarrow\mathbb{Q}(\sqrt{p})\otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt{q})$, where ...
4
votes
1answer
94 views

Inverse image of a sheaf

In Hartshorne, Algebraic geometry it's written, that for every scheme morphism $f: Spec B \to Spec A$ and $A$-module $M$ $f^*(\tilde M) = \tilde {(M \otimes_A B)}$. And that it immediately follows ...
2
votes
1answer
79 views

Tensors in math and physics

I know how tensor product f two modules is defined in communtative algebra. But there is also a concept of tensors used in physics. Are these two concepts related? If yes, can someone explain me ...
1
vote
2answers
80 views

Basic Questions on Tensor Product of Modules

Let $M$, $N$ be $A$-modules where $A$ is a commutative ring with $1$. I am studying $M \otimes_{A} N \simeq A^{(M \times N)}/(4 \text{ generators that makes $\otimes$ bilinear})$, as module ...
5
votes
2answers
172 views

Tensor product of domains is a domain

I'm reading Milne's Algebraic Geometry course notes, version 5.22, as a companion to an algebraic geometry course I'm taking now. Proposition 4.15 states: Let $A$ and $B$ be $k$-algebras, which are ...
0
votes
2answers
192 views

Tensor products of fields

Let $K/F$ be a field extension. I am interested in the situation where there exists a field extension $L/F$ such that the ring $L \otimes_FK$ is not a field. If there exists $z\in K \setminus F$ ...
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 ...
1
vote
2answers
72 views

A criterion of flat modules

Let $R$ be a commutative ring and $M$ a $R$-module such that for every ideal $I \subset R$, the natural map $I \otimes_R M \rightarrow I.M$ is an isomorphism. Why is $M$ flat ? This result is ...
4
votes
1answer
67 views

The image of a map of separable algebras

I'm reading Lenstra's notes on the étale fundamental group, and I've got stuck on his exercise 3.9(c). He says that if $A$ and $B$ are separable algebras over a field $K$, and $f:A \to B$ a ...
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 ...
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 ...
5
votes
1answer
133 views

Tensor product of Noetherian modules

Let $L$ and $N$ be two Noetherian $R$-modules ($R$ is a commutative ring with 1). Is it right that $L \otimes_R N$ is Noetherian? If not, what additional conditions on $L$ and $N$ are required in ...
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 ...
3
votes
1answer
112 views

Ideals of the tensor product $R\otimes_{k} S$?

Let $R$ and $S$ be commutative rings over a field $k$. Let $I$ be an ideal of the tensor ring $R\otimes_{k} S$. It is true that there exist ideals $I_{1}$ and $I_{2}$ of $R$ and $S$ respectively such ...
2
votes
1answer
100 views

Endomorphism algebra of the tensor product of modules

Let $k$ be a commutative ring, and let $M$ and $N$ be $k$-modules. Let $\mathrm{End}(M) = \mathrm{Hom}_k (M,M)$ be the endomorphism algebra. Is it true that $\mathrm{End}(M) \otimes \mathrm{End}(N) ...
3
votes
1answer
120 views

Proving something is the basis of a quotient space

Let $k$ be a field which does not have characteristic 2. Let $M$ be the free $k$-vector space generated by two elements $\{ c, x \}$. Let $T(M)$ be the tensor algebra of $M$ and let $I$ be the ideal ...
0
votes
1answer
34 views

Binomial/Tensor Identity

Let $k$ be a a field and consider the space $k[x] \otimes_k k[x]$. I would like to verify the equation $$ \sum_{k=0}^{m+n} {m+n \choose k} x^k \otimes x^{(n+m)-k}= \sum_{i=0}^n \sum_{j=0}^m{n \choose ...
4
votes
2answers
102 views

Tensor product of faithful modules

In commutative algebra, is it true that the tensor product of two faithful modules is a faithful module? I have written for myself a proof for the case of finitely generated modules over reduced ...
3
votes
1answer
117 views

An $(R,S)$-bimodule is a left $R \otimes_k S^{\text{op}}$-module

Let $k$ be a commutative ring, and let $R,S$ be $k$-algebras. To me "$R$ is a $k$-algebra" means that $R$ is a $k$-module such that $a(rs)=(ar)s=r(as)$ for all $a\in k$ and $r,s \in R$. Let $M$ be a ...
6
votes
3answers
194 views

Are bimodules over a commutative ring always modules?

Let $R$ be a commutative ring. It is true that every module over $R$ is an $(R,R)$-bimodule. Is the converse true? In other words is it possible that there is an $R$-module where left multiplication ...
8
votes
4answers
246 views

Equality of two notions of tensor products over a commutative ring

Let $R$ be a ring (not necessarily commutative), let $M$ be a right $R$-module and let $N$ be a left $R$-module. Then the tensor product $M \otimes_R N$ is an abelian group satisfying the universal ...
7
votes
5answers
272 views

Question about proof of $A[X] \otimes_A A[Y] \cong A[X, Y] $

As far as I understand universal properties, one can prove $A[X] \otimes_A A[Y] \cong A[X, Y] $ where $A$ is a commutative unital ring in two ways: (i) by showing that $A[X,Y]$ satisfies the ...
3
votes
2answers
109 views

$B \otimes_A A[X] \cong B[X]$

Let $A$ be a subring of a commutative unital ring $B$. Can you tell me if my proof of the following claim is correct? Claim: $B \otimes_A A[X] \cong B[X]$ Proof: It's enough to show that $B[X]$ ...
2
votes
0answers
105 views

Completion and Tensor Product of Algebras

Let $A$ be a commutative ring with 1, $I$ an ideal in $A$, $B$ an $A$-algebra. I am trying to prove the following isomorphism of $A$-algebras: $$ \big( A^* \otimes _A B \big) ^* \cong B^* $$ "$^*$" ...
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, ...
2
votes
1answer
163 views

Proving $(U \otimes V) \otimes W \cong U \otimes (V \otimes W)$ without the universal property

Let $F$ be a commutative field, and let $U$, $V$, and $W$ be finite dimensional vector spaces over $F$. How can one prove $(U \otimes V) \otimes W \cong U \otimes (V \otimes W)$ without using the ...
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: $$ S^{-1} R\otimes_R S^{-1}M \cong S^{-1} M$$ ...
3
votes
2answers
167 views

Tensor product of $R$-algebras

Let $f: R \to S$ and $g: R \to T$ be two $R$-algebras. To show that $S \otimes_R T$ is an $R$-algebra I need to define a ring structure (multiplication) on it and a ring homomorphism $h : R \to S ...
3
votes
3answers
224 views

$R \otimes_R M \cong M$

Let $R$ be a commutative unital ring and $M$ an $R$-module. I'm trying to prove $R \otimes_R M \cong M$ but I'm stuck. If $(R \otimes M, b)$ is the tensor product then I thought I could construct an ...
3
votes
1answer
119 views

Example computation of $\operatorname{Tor_i}{(M,N)}$

Let $M = \mathbb Z / 284 \mathbb Z$ and $N = \mathbb Z / 2 \mathbb Z$. Can you tell me if my computation of $\operatorname{Tor_i}{(M,N)}$ is correct: (i) First we want a projective resolution of ...
2
votes
1answer
150 views

Question about a proof of $f$ injective $\implies$ $f \otimes \operatorname{id}$ injective

I'd like to prove (i) implies (ii) where: (i) Whenever $f: A \to B$ is injective and $A,B$ are finitely generated then $f \otimes \operatorname{id}: A \otimes P \to B \otimes P$ is injective. (ii) ...
1
vote
1answer
122 views

Question about flat modules and exact sequences

I have a basic question about exact sequences. I want to show that if I have that whenever $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ is exact then $0 \to A \otimes N \to B \otimes N \to C ...
2
votes
2answers
105 views

Confusion about unique isomorphism $M \otimes N \to N \otimes M$

This is a follow up question to my previous question here. I'm confused about the following: in Atiyah-Macdonald they state that there exists a unique isomorphism $M \otimes N \to N \otimes M$, $m ...
4
votes
1answer
235 views

There exists a unique isomorphism $M \otimes N \to N \otimes M$

I want to show that there is a unique isomorphism $M \otimes N \to N \otimes M$ such that $x\otimes y\mapsto y\otimes x$. (Prop. 2.14, i), Atiyah-Macdonald) My proof idea is to take a bilinear $f: M ...
3
votes
2answers
133 views

Is this computation of the tensor product correct?

I'm reading the proof of the existence of the tensor product. If $M,N$ are two $R$-modules then we can construct the tensor product $T$ as the quotient $C/D$ where $C$ is the free module over $M ...
6
votes
4answers
264 views

Tensor product of $\mathbb R$ and $\mathbb C$ over $\mathbb R$.

$$ \mathbb{C} \otimes_{\mathbb{R}} \mathbb{R} = \;? $$ I guess this guy is just $\mathbb{C}$, is this correct?
7
votes
1answer
195 views

Is the image of a tensor product equal to the tensor product of the images?

Let $S$ be a commutative ring with unity, and let $A,B,A',B'$ be $S$-modules. If $\phi:A\rightarrow A'$ and $\psi:B\rightarrow B'$ are $S$-module homomorphisms, is it true that ...
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?
0
votes
0answers
124 views

Turning the tensor product of algebras into an algebra

Let $B, C$ be $A$-algebras, where $A$ is a commutative ring, i.e. $B, C$ are rings and we have ring homomorphisms $f:A\rightarrow B, g:A \rightarrow C$. Since both $B, C$ are $A$-modules, we define ...
1
vote
2answers
177 views

$(B \otimes C) \otimes (D \otimes E)$ is isomorphic to $B \otimes C \otimes D \otimes E$

Let $B, C, D, E$ be $A$-modules. Is there a way to show that $(B \otimes C) \otimes (D \otimes E)$ is isomorphic to $B \otimes C \otimes D \otimes E$ using the result that $(M \otimes N) \otimes P$ ...
2
votes
1answer
79 views

How to show that $M_B = B \otimes_{A} M$ is a $B$-module?

Let $A,B$ be commutative rings with identity. Let $f:A \rightarrow B$ be a ring homomorphism and let $M$ be an $A$-module. Since $B$ can be viewed as an $A$-module with the operation $A \times B ...
6
votes
3answers
155 views

How to prove that $(M \oplus N) \otimes P$ is isomorphic to $(M \otimes P) \oplus (N \otimes P)$?

Let $M, N, P$ be $A$-modules, where $A$ is a commutative ring with identity. I want to prove that $(M \oplus N) \otimes P$ is isomorphic to $(M \otimes P) \oplus (N \otimes P)$. I start by defining ...
4
votes
2answers
260 views

Why this element in this tensor product is not zero?

$R=k[[x,y]]/(xy)$, $k$ a field. This ring is local with maximal ideal $m=(x,y)R$. Then the book proves that $x\otimes y\in m\otimes m$ is not zero, but I don't understand what's going on, if the ...
12
votes
2answers
739 views

Atiyah-Macdonald Exercise 2.20 (tensor products commute with direct limits)

How to prove that $\varinjlim (M_i \bigotimes N) \cong (\varinjlim M_i) \bigotimes N$ ? Atiyah give a suggestion, he says that one should obtain a map $g \colon (\varinjlim M_i )\times N ...
2
votes
2answers
76 views

How can one see that $\operatorname{tr}(f\otimes g)=\operatorname{tr}f\operatorname{ tr }g$?

Suppose you have two free modules $M$ and $N$ of finite rank over a commutative ring $R$. Let's also take some $f\in\operatorname{End}_R(M)$ and $g\in\operatorname{End}_R(N)$, which gives a ...
5
votes
4answers
189 views

Why does $(A/I)\otimes_R (B/J)\cong(A\otimes_R B)/(I\otimes_R 1+1\otimes_R J)$?

While reading, there is an isomorphism that I'm having trouble fulling seeing. If you have two algebras $A$ and $B$ over a commutative ring $R$, with $I$ and $J$ two sided ideals in $A$ and $B$, ...
4
votes
1answer
131 views

Corollary 2.13 of Atiyah - Macdonald

I just started learning about tensor products and I have some trouble understanding this corollary in Atiyah - Macdonald. All modules are assumed to be $A$ - modules for $A$ a commutative ring. ...

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