Definition:Tensor Product of Abelian Groups/Family

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Definition

Let $I$ be an indexing set.

Let $\left\langle{G_i}\right\rangle_{i \mathop \in I}$ be a family of abelian groups.

Let $G = \displaystyle \prod_{i \mathop \in I} G_i$ be their direct product.


Definition 1: by universal property

Their tensor product is a pair $\left( \displaystyle \bigotimes_{i \in I} G_i, \theta \right)$ where:

such that, for every pair $(C, \omega)$ where:

there exists a unique group homomorphism $g : \displaystyle \bigotimes_{i \in I} G_i \to C$ with $\omega = g \circ \theta$.

$\xymatrix{ G \ar[d]_\theta \ar[r]^\omega & C\\ \displaystyle \bigotimes_{i \in I} G_i \[email protected]{.>}[ru]_g }$


Definition 2: construction

Their tensor product is the pair $\left( \displaystyle \bigotimes_{i \in I} G_i, \theta \right)$ where:

  • $\displaystyle \bigotimes_{i \in I} G_i$ is the quotient of the free abelian group $\Z\left[ G \right]$ on $G$, by the subgroup generated by the elements of the form $\left( x + y, (z_i)_{i \neq j} \right) - \left( x , (z_i)_{i \neq j} \right) - \left( y, (z_i)_{i \neq j} \right)$
    for $j\in I$, $x, y \in G_j$, $(z_i)_{i \neq j} \in \displaystyle \prod_{i \neq j} G_i$, where we denote $(x, (z_i)_{i \neq j})$ for:
  • $\theta : G \to \bigotimes_{i \in I} G_i$ is the composition of the canonical mapping $G \to \Z\left[ G \right]$ with the quotient group epimorphism $\Z\left[ G \right] \to \displaystyle\bigotimes_{i \in I} G_i$:
    $G \hookrightarrow \Z\left[ G \right] \twoheadrightarrow \displaystyle\bigotimes_{i \in I} G_i$


Also see


Special case


Generalization


Sources

  • 1998: N. Bourbaki: Algebra I Chapter II. Linear Algebra $\S3$. Tensor Products. 9. Tensor product of families of multimodules