Questions tagged [tensor-products]

For questions about tensor products, which allow us to build "linear" objects from "multilinear" ones. Add other specific tags to indicate the subject you're referring to.

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Confusion Over Distributive Property in Tensor and External Tensor Products

I've been delving into the properties of tensor ($\otimes$) and external tensor products ($\boxtimes$) within the context of coalgebra, particularly examining how the coproduct $\Delta$ applies to ...
1 vote
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Let $R$ be a commutative ring and $A,B,C$ be $R$-modules, $A$ finitely presented, $C$ flat. Then Hom$(A,B\otimes C)\cong\text{Hom}(A,B)\otimes C$

Let $R$ be a commutative ring with $A$, $B$, and $C$ all $R$--modules. Suppose that $A$ is finitely presented and $C$ is flat (that is, the functor ${-} \otimes C$ preserves short exact sequences). ...
1 vote
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Comultiplication on the tensor algebra

Let $k$ be a commutative base ring. We have a category $\operatorname{Mod}_k$ of $k$-modules and a category $\operatorname{grMod}_k$ of $\mathbb{Z}$-graded $k$-modules. Both of these have monoidal ...
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Module of type $FP_n$

I'm trying to understand the converse of theorem 1.3 in Robert's Bieri Homological dimension of discrete groups which says that a $\Lambda$-module $A$ is of type $FP_n$ if and only if for every direct ...
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What are the names of the conventions for defining the double dot product?

The double dot product of two matrices $A : B$ can be defined as either: $A : B = Tr(AB^T) = A_{ij}B_{ij}$ $A : B = Tr(AB) = A_{ij}B_{ji}$. I've seen the first convention called Frobenius or ...
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Map is almost isomorphism iff tensoring with $\widetilde{\mathfrak{m}} = \mathfrak{m}\otimes_R \mathfrak{m}$ yields isomorphism

Let $R$ be a commutative ring with ideal $\mathfrak m$ satisfying $\mathfrak m^2 = \mathfrak m$, and let $f:M\to N$ be a map of $R$-modules. I want to show that the kernel and cokernel of $f$ are ...
60 views

Induced change of basis on a (p,q) tensor

I'm struggling to simplify the last step of a $(p,q)$ tensor and how its components change with a linear change of basis on the associated vector space. So far I have: Given a vector space $V$ over ...
I am following Qing Liu's textbook "Algebraic Geometry and Arithmetic Curves," and have come upon the following statement (the truth of which is well-known): Theorem: Let $M$ be an $A$-...