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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Is it possible that $1\otimes 1 = 0$?

Let $R$ be a commutative ring. Let $A,B$ be $R$-algebras and consider their product $A\otimes_R B$. Is it possible that $1\otimes 1=0$? What is an example? If $R$ is a field, $1\otimes 1$ is never ...
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32 views

If M and N are nonzero, finitely generated R-modules with M projective, then $M\otimes N$ is nonzero

I am trying to work through the following problem: If M and N are nonzero, finitely generated R-modules with M projective, then $M\otimes N$ is nonzero. My thought on how to approach this problem is ...
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29 views

How does a 4-tensor act as a linear trasformation of 2-tensors?

I'm trying to understand tensors by looking at this table and thinking about the various types of transformations the tensors represent. From the linked table, I tried looking up some of the less ...
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29 views

Tensor product in dual-space

I am a bid confused regarding the notation for tensor products when going into dual-space If $\left| \Psi \rangle \right. = \left| A \rangle \right. \left| B \rangle \right.$ is $ \left. \langle \Psi ...
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20 views

Going into dual space for a vector product [duplicate]

I am a bid confused regarding the notation for tensor products when going into dual-space If $\left| \Psi \rangle \right. = \left| A \rangle \right. \left| B \rangle \right.$ is $ \left. \langle \Psi ...
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50 views

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$ This is not homework, it is part of an answer of Show that $\mathbb{A}_\mathbb{C}^2 \ncong ...
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46 views

The adjoint of left exterior multiplication by $\xi$ for hodge star operator

As we know, for $V$ vectoral space and a orientation $\mathcal{O}$ on $V$ and $e_{1},...,e_{n}$, the hodge star operator $\ast:\wedge V^*\rightarrow\wedge V^*$ is defined for ...
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37 views

Is there a way to factor out the middle tensor product?

Given a state $(|1\rangle \otimes |0\rangle \otimes |1\rangle) + (|0\rangle \otimes |0\rangle \otimes |1\rangle)$, is it possible to factorise out the $|0\rangle$ in the middle of both of them? ...
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70 views

Looking For a Coordinate Free Way to Prove This Linear Algebra 'Fact'

$$\newcommand{\mc}{\mathcal}$$ Let $V$ be an $n$-dimensional vector space over a field $F$. (We use $\mc L(V)$ to denote $End(V)$). For each $v\in V$, define $\Theta_v:\mc L(V)\to V$ as ...
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27 views

Does the dual behave well with base change?

Let $R$ be a ring, $M$ an $R$-module and $R \to S$ a ring homomorphism. I wondered under which conditions we have an isomorphism $$ Hom_R(M,R) \otimes_R S \xrightarrow{\sim} Hom_S(M \otimes_R S, S). ...
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12 views

Why is $\mathbb{P}(Sym_2(\mathbb{C}^2))$ isomorphic to $\mathbb{P}^2(\mathbb{C})$?

Let $Sym_2(\mathbb{C}^2)$ denote the space of symmetric 2-tensors on $\mathbb{C}^2.$ I want to understand why is $\mathbb{P}(Sym_2(\mathbb{C}^2)) \cong \mathbb{P}^2(\mathbb{C})$. Any help please?
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39 views

Computing tensor products of $\mathbb{Z}$-modules.

I'd like to compute $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Z}^{n}$, for some natural number $n$, and $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}$.
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35 views

unitary transformation of $\mathbb{C^2} \otimes \mathbb{C^2}$ that preserves the decomposability

I have some doubts about the relationship between tensor product and unitary transformations... Take $\mathbb{C^2} \otimes \mathbb{C^2}$ and think about it as a inner product space with the canonical ...
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39 views

Tensor product of the Heaviside distribution

I would like to prove that: \begin{equation} H_{(a,b)}=H_a \otimes H_b \end{equation} So far I have: \begin{equation} \langle H_a(x) \otimes H_b(y), \phi\rangle=\langle H_a(x),\langle ...
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34 views

Direct Limit and Tensor Product

Let $I$ be a directed order set. And let $\{ (M_i,\mu_{ij}) | i\leq j\} $ be a directed system of modules with $M = \lim M_i$ the direct limit. With maps $\mu_i :M_i\to M$ satisfying the required ...
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54 views

Conjugation in the complexification of a vector space switches its type

Let $V$ be a real vector space with an almost complex structure $J$ and consider its complexification $V^\mathbb{C}$ where we extend $\mathbb{C}$-linearly the linear maos of $V$, in particular $J$. In ...
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Does the Boardman-Vogt tensor product of operads commute with their W-construction [migrated]

I have absolutely no idea whether this is true or not but it could well be useful for me in the future if it is. If we have topological operads $\mathcal{P}$ and $\mathcal{Q}$ and we let $W$ denote ...
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10 views

Normalizers of subgroups of simple tensors

Inside $GL(2^n,\mathbb{C})$, we have subgroups that are formed by simple tensors. For example, in $GL(16,\mathbb{C})$, we've got subgroups like $H \leq G \leq GL(16,\mathbb{C})$ where $H = \{A_1 ...
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32 views

Product of characters in representation theory

Is it true that if $\phi_1$ and $\phi_2$ are characters then $\phi_1\phi_2$ is a character of a representation? I think this is true, say for instance if $R_1$ is a matrix representation with ...
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23 views

Understanding linearly disjoint fields and how their rings of integers interact in a proof

Let $L,L'$ be linearly disjoint number fields (i.e. finite-degree extension of $\mathbb{Q}$). Their rings of integers are denoted $O_L,O_{L'}$. I am trying to understand a proof of how if $p$ is ...
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47 views

A question about the definition of tensor product

Let $M$ and $N$ be modules over a ring $R$. Generally, the tensor product $M\otimes N$ is defined to be an abelian group with a balanced map $j:M\times N\to M\otimes N$ such that for any abelian group ...
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Prove that $I_k \otimes_k \Omega \rightarrow I$ is injective

Let $\Omega$ be an algebraically closed field, $k$ a subfield of $\Omega$, $I$ an ideal of $\Omega[X_1, ... , X_n]$, and $I_k = I \cap k[X_1, ... , X_n]$. Then $I_k$ is an ideal of $k[X_1, ... , ...
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22 views

Dimension of subspace of $(\mathbb{C}^2)^{\otimes n}$

Consider the space $V = (\mathbb{C}^2)^{\otimes n}$ with $n$ even. Let $(v_+, v_-) = ((1,0), (0,1))$ be a basis of $\mathbb{C}^2$. Then the pure tensors $v_{\pm} \otimes \cdots \otimes v_{\pm}$ form a ...
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18 views

Stabilizers of Segre varieties

What, if anything, is known about maps in PGL(V) that preserve Segre varieties? I am specifically interested in linear maps preserving the Segre embeddings of $\mathbb{P}^{15} \times ...
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1answer
22 views

zero element in tensor product of a localization ring and a module

Let $R$ be a commutative ring with $1$. Let $f$ be a non-nilpotent element of $R$ and let $R_f$ be a localization of $R$ by the multiplicative set $\{ f^i \mid i=0,1,2,\dots\}$. Let $M$ be an ...
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33 views

Can the following simple tensor preserving map exist?

In this question here, I asked if there could exist a $U \in U(4)$ such that $U$ itself was not the tensor product of two matrices, but such that $U(A \otimes B)U^{-1} = A' \otimes B'$ for all $A,B ...
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56 views

Can all “standard” properties of the tensor product be proven from the universal property?

The tensor product is typically constructed in an existence proof by referring to a rather esoteric quotient space which "feels" hard to work with in general. The universal property of bilinear ...
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66 views

Commuting of Hom and Tensor Product functors?

Let $V_i,W_i$ be finite dimensional vector spaces, for $i=1,2$. Assume we have homomorphisms $\phi_i:V_i\rightarrow W_i$. Then, there is an induced map $\widehat{\phi_1 \times \phi_2} \in Hom(V_1 ...
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28 views

Tensor algebra becomes a graded $R$-algebra short proof verification

I had a post proving that the tensor algebra becomes a graded ring, i have come up with a simple approach that goes as follows: Proposition: The tensor algebra $T(M)$ with multiplication defined ...
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45 views

Tensor algebra becomes an R-algebra. Theorem, Proof, Dummit and Foote

I have the definition of tensor algebra as follows: $T(M) = \bigoplus_{i =1}^{\infty} T^i(M)$, where $M$ is an $R$ module, where $R$ is commutative and contains the element $1$. Finally $T^k(M) = ...
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Symmetric kernel of tensor product

Let $V,W$ be two real vector spaces, and let $L_i:V\rightarrow W$, $i=1,\ldots,n$ be $n$ linear maps with distinct kernels $K_i$ of dimension $1$. Consider the tensor product of these maps ...
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Maps that preserve tensor rank

Suppose we have some tensor product of vector spaces. By tensor rank, I mean the minimal number of simple tensors required to write down an element of this tensor product of spaces. Is there much ...
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10 views

Tensor products of Lipschitz functions

I have encountered a problem on which I am sure there is some background, which unfortunately I don't know anything about (so that I don't even know where to start). Let $(M, d_M)$, $(N, d_N)$ be ...
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Injective homomorphism on tensor product

I am currently attempting the following: Find (cyclic) $\mathbb{Z}$-modules $M, N, P$ and an injective homomorphism $f: M \rightarrow N$ s.t. $g: M \otimes_{\mathbb{Z}} P \rightarrow N ...
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14 views

Explicit example of tensor norms

I can't find any example anywhere on the web where someone actually evaluates a non-trivial tensor norm. So I'm wondering about the simplest non-trivial case. Let $X$ be $\mathbb R^2$ with the ...
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1answer
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Is there an “internal” definition of the tensor product?

We have the following "internal" definition of the direct sum: A vector space $V$ with subspaces $S,T$ is said to be the direct sum of $S$ and $T$ if $S + T = V$ and $S \cap T = \{0\}$. (Of course ...
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1answer
44 views

Prove that the following is a non zero tensor.

I'm asked to prove that the ideal $I=(x,y)$ in $R=k[x,y]$ is not a flat R-module. My approach was to use the exact sequence $$0\rightarrow I \to R \to R/I \to 0$$ to induce a non injective map ...
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When do injective and projective tensor norms agree?

For $C^*$-algebra tensor products, one talks about the min and max tensor norms, and they agree when one of the $C^*$-algebras is nuclear. For general Banach algebras, what is the analog of ...
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1answer
22 views

Module homomorphism on the algebraic closure of $\mathbb{Z}_p$

Let $k$ be the algebraic closure of $\mathbb{Z}_p$ and $k^*=k-\{0\}$ be a multiplicative group. For a finite abelian group (i.e. a finite $\mathbb{Z}$-module) $G$, compute the following: ...
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24 views

derivative of a tensor function

If e is a second order tensor and it's symmetrical, assuming abs() is a tensor function such that returns all the components of e be positive, I am interested the derivatives of the function abs(e) ...
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Pullbacks of symmetric tensors commute with products

The problem: Show that $$F*(AB)=(F*A)(F*B)$$ where F is a smooth map from a smooth manifold M to another smooth manifold N, A and B are symmetric tensor fields on N, and $F*$ denotes the pullback ...
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How are these definitions of the inertia tensor the same?

I'm looking for some help in understanding the inertia tensor (not the physics, just the math). I'm trying to figure out how to convert between the wedge product and tensor product definitions. ...
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Proving existence of Hermitian Adjoint in unusual way

For a map $T:V\rightarrow V$, we define the Hermitian adjoint to be the unique $T^*:V\rightarrow V$ such that $\langle Tu,v\rangle = \langle u, T^*v\rangle$. There are two things I'm required to ...
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1answer
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How do I show this relation between exterior product and the projection of a tensor product

I have troubles understanding this whole problem starting at the definition. We have defined the exterior product as follows: If $\alpha = \pi (a) \in \bigwedge^pV$ and $\beta = \pi(b) \in ...
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1answer
19 views

injectivity. identity map of a ring to its tensor product

Let $B$ be an $A$-algebra, $f: B \rightarrow B \otimes_{A} B$ is defined by $f(b)=b\otimes 1$. Is $f$ injective? I know the definition of tensor product and started from representing as $(b,1)=\sum ...
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1answer
53 views

Length of tensor product of finite length modules is finite

Let $R$ be a commutative ring. If $M$ and $N$ are finite length $R$-modules, then $M\otimes_R N$ has finite length, and $l(M\otimes_R N) \le l(M)l(N)$. I know the question has been posted ...
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1answer
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Isomorphism of tensor product involving a principal ideal

This question arose when dealing with a long exact sequence of Tor. Let $R$ be a (not necessarily commutative) ring, $g$ a central element of $R$ and $M$ a right $R$-module. We have an exact sequence ...
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54 views

Module structure of base extension via tensor product

Let $A,B$ be commutative rings. Defining a product of $B\otimes_{A}B$ as $(b_1 \otimes b_2)\cdot (b_3 \otimes b_4)=(b_1b_3)\otimes(b_2b_4)$, this becomes a commutative ring. Defining $b\cdot(b_1 ...
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1answer
117 views

What does it mean for a category to be “tensored over” another category?

What does it mean for a category to be "tensored over" another category? I was reading "Stable model categories are categories of modules" by Schwede and Shipley ...
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20 views

rank 3 tensor product

A can be any elements you choose. I suppose the identity matrix would look like this and the augmented 3-hypermatrix would look like A below.