Tensor products allow us to build "linear" objects from "multilinear" ones. It can refer to: basic ones from linear algebra/module theory, or more sophisticated versions from differential/algebraic geometry (bundles/sheaves), functional analysis (Hilbert/Banach/locally convex spaces), or in their ...

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Tensor product with an L on top

Looking at the definition of the Kunneth morphism in SGA4, XVII, 5.4.1.4, there is the notation $$Rf_*K \overset{\mathbb{L}}{\boxtimes}_{\mathcal{A}_0} Rg_* L \rightarrow Rh_*(K ...
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51 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 ...
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24 views

Ring structure on tensor product of two $A$-modules

Let $A, B, C$ be a commutative rings. Suppose I have two ring homomorphisms, $\alpha : A \rightarrow B$ and $\beta : A \rightarrow C$. I am trying to show that $B \otimes_A C$ has a ring structure ...
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53 views

Tensor product of duals

Can we show $V^* \otimes W^* \simeq (V \otimes W)^*$, when $V$ or $W$ is finite-dimensional without referring to the basis? I can inject $V^* \otimes W^*$ into $(V \otimes W)^*$ using the obvious ...
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46 views

Why are these dimensions equal?

For a finite $K$-algebra $A$ and $L\supset K$ fields, why do we have$$\dim_K A=\dim_L(A\otimes_KL)?$$ I ran across this a couple of times and it's always assumed to be quite obvious, which it isn't to ...
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On a basic tensor product question

I am trying to show that $$ \mathbb{Z} / (10) \otimes \mathbb{Z} / (12) \cong \mathbb{Z}/(2) $$ by defining a map $$ h([a]_{10} \otimes [b]_{12}) = [ab]_2 $$ and extend it linearly. I am having ...
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55 views

Why is tensor product of linear maps defined as $(S\otimes T)(v\otimes w)=S(v)\otimes T(w)$?

In my understanding, the definition of tensor product of linear maps cannot be directly derived from the definition of tensor product of vector spaces (or modules), since it's not clear what is the ...
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39 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 ...
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Tensor product of $A_n$ modules/ localisation at ring of differentials

I'm working through Coutinho's "A Primer of Algebraic D-Modules" and I've gotten stuck on the following question: Let $p \in K[x_1, \ldots ,x_n]$ be non-zero, and let $A_n$ be the Weyl Algebra. Show ...
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32 views

can quaternions be expressed in terms of tensor products?

QUESTIONS does this arithmetic check out? if so, is there a geometric interpretation? note: my aim was to try to find a very simple but non-trivial example which might help me begin to understand ...
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21 views

natural isomorphism of polynomial functions on $V$ and $S(V^*)$

In Humphreys, reflection groups and coxeter group book, Humphreys denotes $S(V^*)$ as the ring of polynomial function on the finite dim vector space $V$. Why we are considering $S(V^*)$ rather than ...
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71 views

Is the tensor product of two representations a representation?

I am a little bit uncertain about an argumentation showing that a given map of a topological group is somehow obviously continuous. In the following I will rely on the book of Anthony W. Knapp „Lie ...
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1answer
43 views

Monoidal product is coproduct in category of commutative monoids

If $V$ is a symmetric monoidal category, the category $\text{CMon}(V)$ of commutative monoids in $V$ has binary coproducts given by $\otimes$, the monoidal product of $V$. See for example Johnstone’s ...
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32 views

how do I view the tensor product $X^*\otimes Y$ as a subspace of $\mathcal{L}(X,Y)$?

Background. According to Raymond A. Ryan, in his book Introduction to Tensor Products of Banach Spaces, a Banach ideal $\mathcal{J}$ is an assignment to each pair of Banach spaces $X$ and $Y$ a ...
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Lie algebra: symmetric and exterior power of representation

If $\mathfrak{g}$ is a Lie algebra, $V$ and $W$ are representation of $\mathfrak{g}$ we define the action of $\mathfrak{g}$ on $V \otimes W$ in the following way: $X \cdot (v \otimes w)=(X \cdot v) ...
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119 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 ...
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23 views

Tensor product and free $R$-module construction

I am trying to understand an exposition on Tensor products by Keith Conrad. In the proof of Theorem 3.2 on page 7 it considers the free $R$-module on the set $M \times N$: $$F_R(M \times N) = ...
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53 views

What is the wedge product of multilinear forms?

The construction of $V^* \otimes V^*$ involves creating formal symbols and then adding in relations such as bilinearity by quotienting out. A bilinear form $V\times V\to F$ can be thought of as a ...
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41 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 ...
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57 views

Completion of a torsion-free module

Let $R$ be a Dedekind domain, $K$ its field of fractions, $P$ a non-zero prime ideal of $R$. Let $\hat R_P$ be the completion of $R$ wrt to the valuation $v_P$ induced by $P$ and let $L$ be a ...
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35 views

product between the kronecker product of three matrices and a vector

I need to perform the product between a matrix defined as the kronecker between three matrices and a vector : $$v = (A \otimes B \otimes C)u.$$ I know there is an identity for such a thing with two ...
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1answer
29 views

Operator Tensor Product

Let $S$ and $T$ be bounded operators over a Hilbert space $\mathcal{H}$. Define their tensor product $S\otimes T$ as acting on $\mathcal{H}\otimes\mathcal{H}$ by $S\otimes T(x\otimes y):=Sx\otimes Ty$ ...
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42 views

Localization of a torsion-free module

I have a Dedekind domain $R$ with field of fractions $K$ and a non-zero prime ideal of $P$ of $R$. Let $L$ be a torsion-free $R$-module. How can I show that $R_P\otimes_R L$ is torsion-free as an ...
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39 views

When is $M \times N$ contained in$ M \otimes_{R} N$.

Let $R$ be a ring with 1. $M$ is a right R-module and $N$ is a left R-module. A tensor product of M and N comes with a map from $M \times N \to M\otimes N$ which is actually a composition of maps ...
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21 views

Projections in Tensor Product

Let $X$ and $Y$ be Banach spaces (algebras). If $X\otimes^\pi Y$ denotes the projective tensor product of $X$and $Y$, define $P:X\otimes^\pi Y\rightarrow X$ as follows : for $x\otimes y\in X\otimes ...
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Intuition for a certain tensor product.

Tensor products occur in lots of places and until recently I thought I understood them at least reasonably well. During the past few weeks, however, I've attended several talks where the tensor ...
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1answer
33 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? If this is not true, what can i add to ...
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19 views

Exterior algebra of a direct sum [duplicate]

Why the exterior algebra of a direct sum of subspaces is isomorphic to a Tensorial product of it's exterior algebras, I mean, Why ⋀(V⊕W)≃⋀V⊗⋀W ?
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Let $K$ a field, $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor. Prove that $f \wedge f =0$

Let $K$ a field, with $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor ($f \in {\mathcal T}_n(V):=\Lambda^{n}(V)$), i.e., $f$ is an multilinear ...
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show that $L^+$ is non empty [duplicate]

I want to show that L+ is non-empty where L is a full-rank integer lattice and L+ denotes the set of elements of L having positive coordinates I have an indication that I did not understand L⊗Q⊆Zn⊗Q ...
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30 views

Tensor Product of Hilbert Spaces: incomplete?

Let $\mathcal{H}$ be an infinite dimensional Hilbert space and $\mathcal{H}\otimes_0\mathcal{H}$ its algebraic twofold tensor product. Define a scalar product on it as ...
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$A$ similar to $B$ implies $A\otimes 1 - 1 \otimes A$ is similar to $A\otimes 1 - 1\otimes B$

Let $A$ and $B$ be similar linear transformations on a finite dimensional vector space $V$ over an algebraically closed field. I.e. there exists an invertible transformation $C$ such that ...
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Notation for a vector space: $(\mathbb{C}^\infty)^{\otimes L}$

In a paper, the authors use the notation $(\mathbb{C}^\infty)^{\otimes L}$, where $L$ is a constant, for a vector space, but they do not give a definition. They also implicitly introduce an inner ...
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about the structure of components of tensor product if more than one bipartite graph is taken

I was reading about tensor product of graphs. We know that if we take tensor product of n graphs and want this product to be a connected graph then at most one graph should be bipartite. In the book ...
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distributivity of tensor product and direct sum for Hilbert spaces

Before I ask my actual question about direct sums and tensor products of Hilbert spaces, let's first talk about direct sums and tensor products of vector spaces. We might define direct sums of ...
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28 views

Simple tensors in the dual space

Let $X$ and $Y$ be two Banach spaces and assume, if necessary, that $X^*, Y^*$ have the approximation property (but not necessarily the Radon–Nikodym property). Consider the injective tensor product ...
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1answer
71 views

Introductory questions about tensors

I am trying to understand the concept of tensors. I seem to understand that they are generalization of vectors: They are subject to similar basis transformations with vectors but I am somewhat ...
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Time Evolution of Deformation Gradient Tensor in Lagrangian Frame

I found the following proof in a paper: $\frac{D\mathbf{F}}{Dt} = \frac{D\frac{\delta\mathbf{x}}{\delta\mathbf{X}}}{Dt} = \frac{\delta\frac{D\mathbf{x}}{Dt}}{\delta\mathbf{X}}=\frac{\delta ...
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$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 ...
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70 views

$A\otimes B\cong A$?

I am struggling for quite a while now with tensor products, since I have limeted time to learn to understand them. Now on several occasions I have read that we have $A\otimes_KK\cong A$ for $A$ an ...
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restricting invertible maps to get new maps

For V and W as vector spaces, let we define V ⊗ W and suppose T be a invertible linear map from V ⊗ W to itself with special condition, I want to know whether there exist something like restricted ...
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40 views

Decomposition of tensor product into direct sum of fields

If I have tensor product of two fields $V_1\otimes V_2$, what is the general approach to decompose this product into a direct sum of fields? In particular, I have $\bullet\;\Bbb Q(\sqrt 2) ...
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Proof of injectivity of $\operatorname{Hom}(U, V) \otimes \operatorname{Hom}(U', V') \to \operatorname{Hom}(U \otimes U', V \otimes V')$

Below I have attached a proof of the injectivity of the canonical map $$\theta: \operatorname{Hom}(U, V) \otimes \operatorname{Hom}(U', V') \to \operatorname{Hom}(U \otimes U', V \otimes V').$$ I'm ...
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Prove that when $n + 1$ is a power of $2$, there exists an $(n+1) \times n$ orthogonal array.

I want to prove that when $n + 1$ is a power of $2$, there exists an $(n+1) \times n$ orthogonal array. $\textit{Outline:}$ I want to recursively form $F_{k+1} = F \otimes F_k$, where ...
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47 views

Basis of tensor product

Let $V \neq \{0\}$, $W \neq \{0\}$ be vector spaces over a field $K$, $F=(x_i)_{i \in I} \subset V$, $G=(y_j)_{j \in J} \subset W$ systems of vectors. Let $H$ be defined as $H:=(x_i \otimes ...
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1answer
48 views

Tensor product and direct sums

I have an integral domain $R$, and $R$-modules $M$, $N_1$, $N_2$. I know that there is an $R$- module isomorphism $$M\otimes_R (N_1\oplus N_2)\cong (M\otimes_R N_1)\oplus(M\otimes_R N_2).$$ where ...
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120 views

Natural Isomorphism: how can $A \otimes B \simeq B \otimes A $ and yet $A \otimes B \neq B \otimes A $

I am reading Braided Monoidal Categories by Joyal and Street. They say cateogories with tensor product arise naturally such as the category of Abelian Groups and that of Banach Spaces. Is there any ...
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Tiling a rectangle and tensor products

Consider the following theorem: Whenever a rectangle is tiled by rectangles each of which has at least one integer side, then the tiled rectangle has at least one integer side. There is a paper ...
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34 views

A simple description of $ {C^{*}}(\Gamma) \otimes_{\sigma} {C^{*}}(\Gamma) $ when $ \Gamma $ is finite.

Problem. Let $ \Gamma $ be a discrete group. Denote its full group $ C^{*} $-algebra by $ {C^{*}}(\Gamma) $. If $ \Gamma $ is a finite group, then is it true that $ {C^{*}}(\Gamma) \odot ...
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$2 \otimes_{R} 2 + x \otimes_{R} x$ is not a simple tensor in $I \otimes_R I$

This question is related to Properties of the element $2 \otimes_{R} x - x \otimes_{R} 2$. Another exercise of Dummit-Foote is to show that Let $I = (2, x)$ be the ideal generated by $2$ and $x$ ...