Completed tensor product

Definition

Let $k$ be a commutative ring, let $M$ have a decreasing filtration $M=M_0\supseteq M_1\supseteq\cdots$ , and let $N$ have a decreasing filtration $N=N_0\supseteq N_1\supseteq\cdots$ . Their completed tensor product for these filtrations is

M\mathbin{\widehat\otimes}_k N := \varprojlim_{r,s} \bigl(M/M_r\bigr)\otimes_k\bigl(N/N_s\bigr).

It receives the ordinary tensor product $M\otimes_k N$ through the compatible quotient maps. The symbol $\widehat\otimes$ is therefore meaningful only after the filtrations or topologies being completed have been specified.

Adic algebras

If $A$ and $B$ are complete separated $k$ -algebras with ideals of definition $I\subseteq A$ and $J\subseteq B$ , the adic completed tensor product is

A\mathbin{\widehat\otimes}_k B = \varprojlim_{r,s} (A/I^r)\otimes_k(B/J^s).

Under standard hypotheses this is the completion of $A\otimes_k B$ for the ideal generated by $I\otimes B$ and $A\otimes J$ . For example,

k[[x_1,\ldots,x_m]] \mathbin{\widehat\otimes}_k k[[y_1,\ldots,y_n]] \cong k[[x_1,\ldots,x_m,y_1,\ldots,y_n]]

with the adic topologies defined by the variables.

Continuous bilinear maps

The ordinary tensor product represents $k$ -bilinear maps before completion. After equipping $M\otimes_kN$ with the linear topology generated by the images of

M_r\otimes_kN+M\otimes_kN_s,

its separated completion is the displayed inverse limit. Consequently a continuous bilinear map $b:M\times N\to P$ into a complete separated linearly topologized $k$ -module factors uniquely through $M\widehat\otimes_kN$ provided that, for every open submodule $P_0\subseteq P$ , there are $r,s$ such that

b(M_r,N)+b(M,N_s)\subseteq P_0.

This is the universal property in the linearly topologized category fixed by the filtrations above.

Convention warning

There is no single completed tensor product for every kind of topological module. Functional analysis distinguishes projective, injective, Hilbert, and other tensor topologies, while adic algebra uses linear topologies arising from ideals or submodules. This knowl uses the inverse-limit, linearly topologized convention needed for formal geometry. It should not be silently identified with a Banach- or $C^*$ -algebra tensor product.

Role in formal geometry

If $X=\operatorname{Spf}A$ and $Y=\operatorname{Spf}B$ are affine adic formal schemes over $k$ , their fiber product has coordinate algebra $A\widehat\otimes_k B$ in the standard affine setup. Consequently, a multiplication $G\times G\to G$ on an affine formal group pulls back to a continuous comultiplication $\mathcal O(G)\to\mathcal O(G)\widehat\otimes_k\mathcal O(G)$ .

References

The Stacks Project Authors, Formal Algebraic Spaces, Definition 87.4.7: Completed tensor product.
Alexandre Grothendieck and Jean Dieudonné, Éléments de géométrie algébrique I: Le langage des schémas, Publications Mathématiques de l’IHÉS 4 (1960). Numdam record. Relevant: Chapter 0, §7.7 on completed tensor products.