# Tagged Questions

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 ...

32 views

### Showing that $A \otimes_B A \to A$ is a surjective homomorphism.

Let us define a homomorphism $\phi: A \otimes_B A \to A$ by $a \otimes a' \to aa'$ where $A$ is a $B$-algebra, and both $A$ and $B$ are commutative rings. I want to show that this is a surjective ...
35 views

64 views

### If $R\otimes_\mathbb R\mathbb C$ is finitely generated $\mathbb C$ - algebra then $R$ is a finitely generated $\mathbb R$ - algebra?

Let $R$ be an $\mathbb R$ - algebra. Suppose $A=R\otimes_\mathbb R\mathbb C$ is a finitely generated $\mathbb C$ - algebra then is $R$ a finitely generated $\mathbb R$ - algebra? I thought along the ...
49 views

### Symmetric decompositions of $SU(2)$ representations.

Let us consider the representation theory of $SU(2)$. There is a unique irreducible representation of dimension $n$ for each $n \ge 1$, which we will denote $\mathbf{n}$, with the defining $2$-...
56 views

### Dimension of polynomial rings and tensor products of residue fields

In Matsumura textbook to show that $\dim A[x] = \dim A + 1$, first it states that $A[x] \otimes k(\mathfrak{p}) = k(\mathfrak{p})[x]$ which is one dimensional. Then it uses the theorem 15.1.(ii) since ...
170 views

### Corollaries of the Yoneda Lemma in Analysis?

I am looking for some simple examples of how the Yoneda Lemma can be applied in analysis and probability theory and related fields. A simple candidate example that I can think of and somewhat ...
80 views

### Tensor products and Residue fields

Given a ring homomorphism between two Noetherian rings, $f:A \to B$. Let $P$ be a prime ideal in $B$ and let $\mathfrak{p}$ be an ideal in $A$ such that $f^{-1}(P) = \mathfrak{p}$. How can we prove ...
37 views

51 views

36 views

### Tensor Product and Flat Module

While doing some exercises about flat modules I got to this particular one: Let $I$ be an ideal of $A$ (commutative ring with 1), and $M$ an $A$-module. Show that $I \otimes_{A} M \simeq IM$ if $M$...
19 views

29 views

### Linear algebra textbook for quantum computing?

I'm looking for an recommendation for a linear algebra textbook specifically to give me the background for learning about quantum computing, and quantum mechanics more generally. In particular, none ...
25 views

### General procedure to prove something is a tensor product of modules

I'm trying to understand some proofs of statements of the form: Show that some module is the tensor product of two other modules. When I'm looking at these proofs I always see that they start ...
So I have this expression $\sum_{l,m} c_{lmn} a_l a_m$. Where c is rank 3 tensor, and a is a rank 1 tensor. I would like to express this without specific reference to the tensor indices. Is there a ...