Definition 85.4.7. Let $R$ be a topological ring. Let $M$ and $N$ be linearly topologized $R$-modules. The tensor product of $M$ and $N$ is the (usual) tensor product $M \otimes _ R N$ endowed with the linear topology defined by declaring

$\mathop{\mathrm{Im}}(M_\mu \otimes _ R N + M \otimes _ R N_\nu \longrightarrow M \otimes _ R N)$

to be a fundamental system of open submodules, where $M_\mu \subset M$ and $N_\nu \subset N$ run through fundamental systems of open submodules in $M$ and $N$. The completed tensor product

$M \widehat{\otimes }_ R N = \mathop{\mathrm{lim}}\nolimits M \otimes _ R N/(M_\mu \otimes _ R N + M \otimes _ R N_\nu ) = \mathop{\mathrm{lim}}\nolimits M/M_\mu \otimes _ R N/N_\nu$

is the completion of the tensor product.

Comment #4905 by MABUD ALI SARKAR on

In the definition completed tensor product, what does the limit "lim" mean? Is it the inverse limit? Thanking you,

Comment #4906 by on

Yes, in the stacks project we use $\lim$ for the limit as in Section 4.14 and we use $\colim$ for the colimit.

There are also:

• 2 comment(s) on Section 85.4: Topological rings and modules

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).