1
vote
1answer
97 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
24 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
75 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
30 views

Ideals in specific tensor product of algebras

Let $A$ be a simple central algebra and $B$ a commutative algebra -- what can be said about the 2-sided ideals of $A\otimes_k B$? (I am searching for a situation where the ideals of $A\otimes_k B$ are ...
1
vote
0answers
37 views

Faithfully flat checkable on finitely generated modules

A left $R$-module $_RM$ is said to be faithfully flat if it is flat and, for any $N_R$, $N \otimes_R M = 0$ implies $N = 0$. I would like to show that $M$ is faithfully flat if it is flat and, for any ...
1
vote
0answers
36 views

dimension of tensor products over a submodule

Let $k$ be a field, $A$ be a finite dimensional $k$-algebra (say of dimension $n_A$) and let $B\subset A$ be a sub-algebra. What can be said about the $k$-dimension of $A \otimes_B A$ ? The easiest ...
0
votes
1answer
40 views

A problem in division rings and Brauer group

Suppose that $D, E$ are two division rings in the Brauer group of $F$ ($Br(F)$), where $F$ is local field. Show that $D\otimes_FE$ is a division ring iff $([D:F],[E:F])=1$.
0
votes
1answer
60 views

Tensor product of modules preserve injectiveness and surjectiveness or not?

Let $R$ be a commutative ring with identity and $M$ an $R$-module. If $N_1\longrightarrow N_2$ is injective (resp. surjective), is the induced map $M\otimes_R N_1\longrightarrow M\otimes N_2$ ...
3
votes
1answer
68 views

Prove that $M_{nm}(A \otimes B) \cong M_{n}(A) \otimes M_m(B)$

Prove that $M_{nm}(A \otimes B) \cong M_{n}(A) \otimes M_m(B)$ There's a similar question floating around but I was merely wondering if the result holds in the same way when we let A and B be fields, ...
1
vote
1answer
39 views

When is a simple tensor equivalent to natural multiplication?

Let $R$ be a commutative integral domain; let $M, N$ be $R$-modules. Let $x_1 \in M$ and let $x_2 \in N$. When does the following equivalence hold? $$x_1 \otimes x_2 = x_1x_2$$ The textbook I'm ...
0
votes
1answer
40 views

Linearly independent subsets over extension of scalars

Let $k$ be a field, $D_{1},D_{2}$ division rings containing $k$ and $V$ a $k$-vector space of finite dimension and assume also that $V$ is a $D_{1}-D_{2}$ bimodule. Suppose $W$ is a $k$ basis of ...
0
votes
0answers
26 views

Minimal spectrum of graded rings

Let $R$ be a left Noetherian ($\mathbb N$-)graded ring and let $R_0$ be its $0$-th component. When $R$ is commutative it is well-known, and easy to prove, that the minimal prime ideals of $R$ are ...
1
vote
1answer
63 views

Defining an ideal in the tensor algebra

In the wikipedia article about exterior algebra: The exterior algebra $Λ(V)$ over a vector space $V$ over a field $K$ is defined as the Quotient algebra of the tensor algebra by the two-sided ...
0
votes
0answers
42 views

Extension of multiplication to the tensor algebra.

In this wikipedia article http://en.wikipedia.org/wiki/Tensor_algebra#Construction We construct $T(V)$ as the direct sum of vector spaces $T^kV$ for $k=0,1,2,…$ $$ T(V)= ...
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
1answer
174 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 ...
2
votes
0answers
88 views

When is the canonical extension of scalars map $M\to S\otimes_RM$ injective?

Let $\alpha:R\to S$ be a map of unital rings, and let $M$ be an $R$-module. We have a canonical map of $R$-modules: ...
2
votes
1answer
64 views

Show that $\mathbb{Z}[G\times H]\cong \mathbb{Z}[G]\otimes_\mathbb Z \mathbb{Z}[H]$

This is exercise 5.5 of Rotman's 'Introduction to Homological Algebra'. Let $G$ and $H$ be groups. We must show that $$\mathbb{Z}[G\times H]\cong \mathbb{Z}[G]\otimes_\mathbb Z \mathbb{Z}[H]$$ as ...
0
votes
1answer
64 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) + ...
8
votes
1answer
95 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 ...
12
votes
2answers
285 views

Tensor products of p-adic integers

These are relatively simple questions, but I can't seem to find anything on tensor products and p-adic integers/numbers anywhere, so I thought I'd ask. My first question is: given some ...
5
votes
1answer
73 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
158 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 ...
2
votes
1answer
188 views

Extension of scalars

Let $A\rightarrow B$ be a ring homomorphism, $M$ and $N$ - modules over $A$. How to prove, that $$ (M \otimes_A N) \otimes _A B = (M \otimes_A B) \otimes_B (N\otimes_A B) $$ as $B$-modules in the ...
8
votes
1answer
237 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 ...
3
votes
2answers
117 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 $B[X]$ ...
0
votes
1answer
98 views

A quick question on tensor products of algebras

It's known that for a field $k$, the tensor product of $k$-vector spaces commutes with direct sums. Is it also true that the tensor product of $k$-algebras commutes with finite products ('finite ...
2
votes
1answer
239 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 ...
7
votes
1answer
791 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 ...