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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Geometric generic fibre

I have a question concerning the following exercise in Hartshorne: The inclusion of $k[s]$ into $k[s,t]/(s-t^2)$ induces a morphism of the corresponding affine schemes $X\to Y$. The exercise itself ...
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Does every tensor norm satisfy one half of the crossnorm property

Suppose $(H,\|\cdot\|_{H})$ and $(G,\|\cdot\|_{G})$ are normed vector spaces. Suppose $\|\cdot\|$ is a norm on their algebraic tensor space $H\otimes G$. Do we have $$ \|h\otimes g\|\leq ...
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If $E$ is a finitely presented left $A$-module, then tensor product by E commutes with direct product.

Please can anyone help me to prove this statement: If $E$ is a finitely presented left $A$-module, then for every family $(F_i)$ of right $A$-modules the canonical homomorphism $\phi: ...
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23 views

Associating a $m-1$-tensor on $\mathbb R^m$ to an element of $\mathbb R^m$.

Show that for every alternating $m-1$-tensor on $\mathbb R^m$ there exists a unique $v \in \mathbb R^m$ such that for every linear function from $\mathbb R^m$ into $\mathbb R$ and for every ...
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2answers
29 views

Verify a Tensor Differential Identity

I would like to show that the ID below holds: $$(n \cdot \nabla n)=1/2 \nabla(n \cdot n)-(n \times(\nabla \times n))$$ Using the Einstein notation I've come up with: $$(n \times(\nabla \times ...
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14 views

Matrices over tensor product of Banach algebras

Suppose that $A$ is a Banach algebra, or even a $C^*$-algebra, and $B$ is a closed subalgebra of $B(L^p)$ for some $L^p$ space. In particular, $M_n(B)$ has a canonical matrix norm. Is there some ...
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57 views

Tensor product group representations and spaces of intertwiners.

Let $V_{1}$, $V_{2}$, $W_{1}$, and $W_{2}$ be the carrier spaces of representations of some finite group $G$. Suppose also that $G$ acts trivially on $V_{1}$ and $V_{2}$. I would like to prove the ...
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2answers
36 views

Atiyah and Macdonald's proof of the existence of the tensor product

I have a question regarding the proof of the proposition 2.12 in Atiyah and Macdonald's book. They say: " Let C denote the free A-module $A^{(M \times N)}$. The elements of C are formal linear ...
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89 views

In which sense is composition a tensor product

Let $\Phi\colon U\to V$ and $\Psi\colon V \to W$ be linear operators, and consider their composition $$ \Psi\circ \Phi $$ The operation, $$\circ:\mathcal{L}(U,V)\times\mathcal{L}(V,W)\to ...
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49 views

Is $M_R\otimes _R {_R}N\cong M_{\mathbb Z}\otimes_{\mathbb Z} {_{\mathbb Z}}N$?

Suppose $M$ is a right $R-$module and $N$ is a left $R-$ module. Also $M$ and $N$ are naturally $Z-$ module, both in left and right side. So we will denote $M_R$, $M_{\mathbb Z}$, and $_RN, _{\mathbb ...
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54 views

Tensor product of purely inseparable extension

I would like to understand the tensor product of $A=\Bbb F_2(\sqrt{t})\otimes_{\Bbb F_2(t)}\Bbb F_2(\sqrt{t})$. The extension $L/k:=\Bbb F_2(\sqrt{t})/\Bbb F_2(t)$ is a finite extension of degree ...
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47 views

Extension of scalars and completions

Suppose that $A$ is a Noetherian regular (added later) local domain. Moreover $\widehat A$ is $\mathfrak m$-adic completion $\widehat A$ w.r.t the maximal ideal and $K$ is the fraction field of $A$. ...
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$\text{Hom}(M \otimes_A N, L) \approx \mathscr{L}(M,N; L)$ The $A$-linear homs from the tensor product into $L$ are isomorphic with bilinear maps.

Let $M,N, L$ be two $A$-modules over a commutative ring $A$. Let $\mathscr{L}(M,N;L)$ be the $A$-module of bilinear maps $M \times N \to L$. Then $\text{Hom}_A(M \otimes_A N, L) \approx ...
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44 views

Is the sum of the tensor product of a linear operator, the tensor of the sum?

According to the notes I'm working from $$ \sum_{y\in \mathbf{B}_{M},z \in \mathbf{B}_{M}}\big(\left|z\right>\!\!\left<z\right|\otimes\left|y\oplus f(z)\right>\!\!\left<y\right|\big) ...
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What triple “tensor product” is this? Is it just isomorphic to a double tensor product?

Consider the abelian groups $A = \Bbb{Q}^{\times}, B = \Bbb{Q}^{\times}, C = \Bbb{Z}^+$. What if we formed a product like: $A \star B \star C = \text{Free}_{\Bbb{Z}}(A \times B \times C)$ ...
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$\{a x^{z}: a\in \Bbb{Q}, z \in \Bbb{Z}\} \approx \Bbb{Q}^{\times} \otimes_{\Bbb{Z}} \Bbb{Z}^+ \implies$? what about $\Bbb{Q}$-linear sums?

Consider all functions $f: \Bbb{Q} \to \Bbb{Q}$ of the form $f(x) = a x^z$ where $a \in \Bbb{Q}, z \in \Bbb{Z}$, call it $G$. It forms an abelian group under usual multiplication. I think it's ...
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34 views

Tensor products, direct sums and products of modules

We get the following problem in our algebra class. Let $ (F_i)_{i \in I} $ be a family of left $R$-modules. a) Define an isomorphism $ \sigma: \oplus_{i \in I} (E \otimes_R F_i) \to E \otimes_R ...
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38 views

How to prove A is a flat R-module [duplicate]

Let A be a right R-module. Suppose for every left ideal J of R, the homomorphism $f:A\otimes J\to A$ defined by $f(x\otimes y)=xy$ is injective, then A is a flat R-module.(the identity 1 is in R) I ...
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81 views

When can we say elements of tensor product are equal to $0$?

I am learning about tensor products of modules, but there is a question which makes me very confused about it! If $E$ is a right $R$-module and $F$ is a left $R$-module, then suppose we have a ...
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81 views

Show that a Z-module A is flat if and only if it is torsion free?

I found a question in my textbook which really confuses me! Show that a $\mathbb Z$-module $A$ is flat if and only if it is torsion free? Over here, torsion free means if abelian group A is ...
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how to prove the following sequence is exact

Suppose E be a right R-module, and E is flat, then how can I prove that for any exact sequence of left R-modules $A\to B\to C$, the sequence $E\otimes A\to E\otimes B\to E\otimes C$ is exact? I am ...
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71 views

In Kunneth Formula for Cohomology, the finitely generated condition is necessary.

Kunneth formula for cohomology: The cross product $H^*(X;\mathbb Z)\otimes H^*(Y;\mathbb Z)\to H^*(X\times Y;\mathbb Z)$ is an isomoprhism of rings if $X$ and $Y$ are CW complexes and $H^k(Y,R)$ is ...
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19 views

Toric varieties

I've started to read about toric varieties and I have a couple of questions about the definition. There is an example that says the following: "Given a lattice $N$, an isomorphism $N \simeq ...
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1answer
28 views

Show that if E is a free right R-module,then $\pi$ is injective

Let $(F_{i})_{i\in I}$ be a family of left R-modules and E is a right R-module. (a) Define a homomorphism $\pi :E\otimes_{R}(\prod_{i\in I} F_{i})\to \prod_{i\in I}(E\otimes_{R} F_{i})$ question: ...
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54 views

Exercising elementary toolkit for quantum computing

One of the major challenges for me (and I expect for many as a beginning students with only general maths skills) in studying quantum computation is that while the background and calculations required ...
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64 views

Text recommendations for linear algebra (tensors, jordan forms)

I'm having extreme difficulty trying to understand to topic of tensor products, freespaces, and jordan forms. Are there any text books that take an elementary approach to these topics that you may ...
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Proof of Tensor Products from Atiyah and Macdonald's Commutative Algebra

I apologize in advance if this question is stupid, but I am self studying and I am having trouble deciphering the following notation in Atiyah and Macdonald's "Introduction to Commutative Algebra." ...
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37 views

Question about injectivity of tensor products

Let $A$ be a commutative ring, $M$ a module over it and $k_1,k_2$ fields such that we have the following maps $A \to k_1\to k_2$. Construct the natural map: $$f: M\otimes_A k_1 \to M\otimes_A k_2.$$ ...
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Compute the trace of $\text{Sym}^2 \left(f \right)$ and that of $\text{Sym}^3 \left(f \right)$.

Consider the linear map $f: \mathbb{C}^3 \to \mathbb{C}^3$ defined by the matrix $$\begin{pmatrix} 1 & 0 & 3 \\ 2 & 1 & -1 \\ 0 & 1 & 2 \end{pmatrix}.$$ Compute the trace of ...
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1answer
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Schur Index Divisibility Question: Ind A^E divides Ind A

Background notation: If $A\in\mathcal{F}$ is a central simple algebra, $A\cong M_n(D)$, where $D$ is a division algebra. The Schur index of $A$ is defined as $Ind(A)=Deg(D)$. How do we prove $A^E$ ...
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Second derivative of a multi-variate composition?

This is an extension of a former question (titled "Second derivative of a composition?"). Consider the functions $f:\mathbb{R}^p\to\mathbb{R}^n$ and $g:\mathbb{R}^n\to\mathbb{R}^p$ and define the ...
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1answer
140 views

Second derivative of a composition?

Let $g:\mathbb{R}^n\to\mathbb{R}^p$, $f:\mathbb{R}^p\to\mathbb{R}$ and define the composition $h(x) = f(g(x))$. The gradient of $h$ with respect to $x$, $\nabla h\in\mathbb{R}^{1\times n}$, is given ...
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Tensor product with endomorphism ring

Let $k$ be a commutative ring, $M$ a $k$-module and $k\longrightarrow A$ a $k$-algebra. Is it true that $$A\otimes_k\operatorname{End}_k(M)\cong\operatorname{End}_A(A\otimes_kM)?$$ If not, under what ...
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1answer
22 views

Adding and Multiplication of Pure Tensors

Consider the tensor product $V\bigotimes W$, and $v_1\otimes w_1$, $v_2\otimes w_2$ in $V\bigotimes W$, where $V$ and $W$ are $F$-algebras. Can we add or multiply the pure tensors as follows? ...
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1answer
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symmetric tensors

I am reading a paper ,,The Gelfand map and symmetric product" by V.M. Buchstaber and E.G. Rees http://arxiv.org/abs/math/0109122 On the page 6 in the proof of Theorem 2.8 there is considered a ...
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Does $f\otimes \operatorname{Id} = \operatorname{Id}$ imply $f= \operatorname{Id}$?

Let $R$ be a commutative ring, and $X$ an $R$-module. If an $R$-endomorphism of $X$ satisfies $f\otimes \operatorname{Id}_X = \operatorname{Id}_{X\otimes X}$, is it true that ...
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1answer
47 views

On characterizing modules that don't annihilate any module under tensor product.

Let $R$ be a commutative ring and $M$ an $R$-module. Then under what condition can we deduce that for any nonzero $R$-module $N$, $M\otimes_RN\neq0$?
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Computing an explicit tensor product

So I think this question is trivial but I can't seem to be able to do it so here we go : what is the tensor product $$k[x,y]/(y^2-x^3) \otimes_{k[y]} k[x,y]/(y^2-x^3)\ ?$$ My guess is that it is ...
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Is a tensor product of maps an isomorphism iff maps themselves isomorphisms?

Working over vector spaces, let $f:U \longrightarrow U'$, $g: V \longrightarrow V'$. Define $f \otimes g : U \otimes V \longrightarrow U' \otimes V'$ by $(f \otimes g)(u\otimes v) = f(u)\otimes g(v)$. ...
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24 views

Divergence of outer product in polar coordinates

Right now I am trying to solve Euler's conservation equations for circular domain. Due to several factors, I am restricted to polar coordinates. I can't manage to correctly calculate divergence of ...
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62 views

Differential forms on a scheme: unclear equation

Disclaimer: In this question I assume that the reader is familiar with the construction of the module of differentials $\Omega^1_{B|A}$ where $B$ is an $A$-algebra. (If you need more details about ...
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Calculating Recommendations based on HOSVD

Two-dimensional case Given a user-movie matrix $\mathbf{M}$ that contains ratings, it can be decomposed using SVD to the product of $\mathbf{U}$, $\Sigma$ and $\mathbf{V^T}$. Now, given a new users ...
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1answer
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Doing Symbolic Computations With Tensors And Differential Operators

Motivation Consider the following expression $${\varepsilon}= \frac{1}{2} \left( \nabla \otimes u + \nabla \otimes u^\text{T} \right) \tag{1}$$ where $u:\mathbb{R^3} \to \mathbb{R^3}$ is a vector ...
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36 views

Showing that the tensor product of vector spaces is closed under addition

Let $V$ and $W$ be vector spaces over the field $F$. We know that $V\otimes W$ is a vector space over $F$. But how do we show closure under addition? For instance let $(a\otimes b), (c\otimes d)\in ...
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A conjecture on Schatten 1-norm

I have a conjecture on Schatten 1-norm. Before presenting the conjecture, let us first specify the notions used here. A matrix $A$ is said to be a density operator if $A$ is positive semidefinite ...
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Tensoring a connective chain complex with a simplicial set

Let $\mathrm{Ch}_{\geq 0}(R)$ be the category of chain complexes of $R$-modules concentrated in nonnegative degrees, equipped with the projective model structure. By a general theorem about model ...
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Matrix of a linear transformation of tensor products

Suppose I have a linear transformation $T: V \to V'$. How can I easily obtain the matrix of the linear transformation from $V^{\otimes n}$ to $V'^{\otimes n}$ induced by $T$? What if I have ...
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48 views

Matrix acting on a tensor product

What does it mean for a matrix to act on a tensor product? I think there is a disconnect between vocabulary I am using and vocabulary the professor is using. Specifically, I have a $2 \times 2$ matrix ...
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1answer
35 views

Tensoring Groupring with field of fractions

Is it true that $\mathbb Z[G]\otimes_\mathbb Z \mathbb Q$ is isomorphic to $\mathbb Q[G]$ as a $\mathbb Q$-algebra? Context: If I have the Jacobian of a smooth curve $X$, call it $J_X$, then I've ...
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28 views

Is A⊗B invertible if A and B are invertible?

If two matrices A and B are invertible. Is the tensor product A⊗B also invertible? I believe the answer to be yes but I am struggle to give a reasons for why it is.