1
vote
0answers
20 views

Tensor Algebra and Isomorphism

Consider the standard tensor algebra over a vector space: $$T(V)=\bigoplus_{n\in N}V^{\otimes n}$$, I am trying to define a linear map from $T(V)\times T(V)\rightarrow T(V)$. Up to this time, I have a ...
0
votes
1answer
29 views

Constructing nontrivial $\mathbb{Z}$-bilinear map from $\mathbb{R} \times (\mathbb{R} / \mathbb{Z})$

I'm trying to show $\mathbb{R} \otimes_\mathbb{Z} (\mathbb{R} / \mathbb{Z})$ is nontrivial, where my tensor product is defined using the universal property. This problem can be easily reduced ...
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 ...
1
vote
3answers
105 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
102 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 ...
3
votes
1answer
130 views

Prove that $M_{n}(F)\otimes _{F}M_{m}(F)\simeq M_{nm}(F)$ .

Suppose $F$ is a field. Then prove that $$M_{n}(F)\otimes _{F}M_{m}(F)\simeq M_{nm}(F)$$ as $F$-algebras. I know that I should take $$\alpha :M_{n}(F)\otimes _{F}M_{m}(F)\rightarrow M_{nm}(F)$$ ...
0
votes
0answers
22 views

If $A$ and $B$ are $F$-algebras, then $A\otimes _{F}B $ is $F$-algebra. [duplicate]

Suppose $A$ and $B$ are two $F$-algebras ($F$ is a field). Prove that $A\otimes _{F}B $ is an $F$-algebra with the multiplication: $$(a\otimes b )({a}'\otimes {b}')=(a{a}')\otimes (b{b}').$$ With ...
2
votes
2answers
175 views

Show that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is isomorphic to V + iV

Let V be a real n-dimensional vector space. Show that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is isomorphic to V + iV. Note that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is a real vector space and is ...
2
votes
0answers
64 views

Complete reducibility of tensor product

Let $L$ be a Lie algebra (over a algebraically closed field, not sure if it is relevant). If $V$ and $W$ are two completely reducible $L$-modules, can anyone give a hint on how to show that $V\otimes ...
0
votes
2answers
288 views

What is a Tensor Product?

If you were to explain the concept of a tensor product to an undergraduate(post linear algebra), how would you do so? I would like to hear your definition, your take, on the definition of a tensor ...
5
votes
3answers
129 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 ...
3
votes
1answer
61 views

Help with notation in second order tensor.

I have seen recently this notation: $$F=F_{ij}\,e_i\otimes e_j $$ Where $F_{ij}=\frac{\partial x_i}{\partial X_j}$ is the tensor matrix: $$ F=\left[ {\begin{array}{cc} ...
1
vote
1answer
54 views

What does it mean to have an $L$-basis of $L\otimes_K V$?

Exercise: I have got a vector space $V$ over $K$, and $L$ a field extension of $K$. The task is to show that if $(v_1, ..., v_n)$ is a basis of $V$, then $(1\otimes_K v_1, ... ,1\otimes_K v_n)$ is an ...
0
votes
1answer
64 views

Extension of the field of scalars.

I want to make sure that I understand this correctly: $V- \mathbb{Q}$ vector space. Then $V_\mathbb{R} := V \otimes \mathbb{R} $ is naturally an R-vector space (next to being a $\mathbb{Q}$-vector ...
2
votes
1answer
52 views

Clifford Algebra and quadratic forms

I was wondering if someone can help me with the following. The question is, How can I show that $C_0(q) = R[t]/(t^2 - bt + ac)$, given that $q(x, y) = ax^2 + bxy + cy^2$ is a quadratic form in two ...
3
votes
1answer
90 views

Showing that $k(1\otimes 1) \neq 0$ when $k<\gcd(n,m)$ in $\mathbb{Z}/n\mathbb{Z}\otimes_{\mathbb{Z}} \mathbb{Z} / m \mathbb{Z}$.

I only recently learned about the tensor product of modules and came across the following exercise: Show that $$\mathbb{Z}/n\mathbb{Z}\otimes_{\mathbb{Z}} \mathbb{Z} / m \mathbb{Z} \cong ...
3
votes
2answers
85 views

Proving $\otimes_{i=1}^{i=n}\mathcal{B}_{X_{i}}=\mathcal{B}_{\Pi X_{i}}$

I am given the following exercise: Let $X_{\alpha}$ be a measureable space with $\sigma$-algebra $M_{\alpha}$ , mark $X\triangleq{\displaystyle \prod_{\alpha\in A}X_{\alpha}}$ and ...
6
votes
1answer
722 views

Postitive Definiteness of Kronecker Product of Two Positive Definite Matrix

Let $A$ and $B$ both be positive definite matrices. How do I show that their Kronecker product is also positive definite? I know we can use the fact that the eigenvalues of the Kronecker product is ...
6
votes
1answer
73 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 ...
17
votes
2answers
2k views

Tensor products commute with direct limits

This is Exercise 2.20 in Atiyah-Macdonald. How can we prove that $\varinjlim (M_i \bigotimes N) \cong (\varinjlim M_i) \bigotimes N$ ? Atiyah gives a suggestion, he says that one should obtain a map ...
2
votes
1answer
179 views

Linearly disjoint field extensions and the tensor product

The problem: Let $K$ and $L$ be subfields of a field $\Omega$, and let $k\subset K\cap L$ be a common subfield. (a) Show that there exists a unique ring homomorphism $f_{K,L}:K\otimes_k L\to ...
6
votes
1answer
240 views

A characterization of finite purely inseparable extensions of fields

Let $K/k$ be a finite extension of fields. Let $A=K \otimes_k K$. An exercise: Show that $K/k$ is purely inseparable $\Leftrightarrow A/J(A) \cong k$, where $J(A)$ is the Jacobson radical of $A$. It ...
13
votes
2answers
1k 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 cmmutative ring), the tensor product $M\otimes_A N$ is zero iff $\operatorname{Ann}(M)+\operatorname{Ann}(N)=A$. ...
16
votes
2answers
2k 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
2answers
208 views

what is the tensor product $\mathbb{H\otimes_{R}H}$

I'm looking for a simpler way of thinking about the tensor product: $\mathbb{H\otimes_{R}H}$, i.e a more known algbera which is isomorphic to it. I have built the algebra and played with it for a ...