Formal affine space - Knowlpedia

Definition

Let $k$ be a field. The formal affine $n$ -space, or formal $n$ -disc, over $k$ is

\widehat{\mathbb A}^{\,n}_k := \operatorname{Spf}k[[X_1,\ldots,X_n]],

where the power-series ring has its $(X_1,\ldots,X_n)$ -adic topology. Its distinguished origin is induced by the augmentation $k[[X_1,\ldots,X_n]]\to k$ , $X_i\mapsto0$ .

Coordinates and tangent space

The elements $X_1,\ldots,X_n$ are formal coordinates. If $\mathfrak m=(X_1,\ldots,X_n)$ , the cotangent space at the origin is $\mathfrak m/\mathfrak m^2$ , and the tangent space is

T_0\widehat{\mathbb A}^{\,n}_k =\operatorname{Hom}_k(\mathfrak m/\mathfrak m^2,k) \cong k^n.

Thus the dimension $n$ is the dimension of the tangent space, even though the underlying topological space of the formal spectrum has only one point.

Pointed maps

A pointed morphism

\widehat{\mathbb A}^{\,n}_k\longrightarrow \widehat{\mathbb A}^{\,m}_k

is represented contravariantly by an $m$ -tuple of series in $(X_1,\ldots,X_n)k[[X_1,\ldots,X_n]]$ . Composition of maps is formal substitution. The map is an isomorphism exactly when its linear term is invertible.

Products

The product of formal discs is again a formal disc:

\widehat{\mathbb A}^{\,n}_k\times_k \widehat{\mathbb A}^{\,m}_k \cong \widehat{\mathbb A}^{\,n+m}_k.

On coordinate rings this uses the completed tensor product $k[[X]]\widehat\otimes_k k[[Y]]\cong k[[X,Y]]$ . Consequently a multiplication map on a formal $n$ -disc is represented by an $n$ -tuple of series in the $2n$ variables $X,Y$ .

References

The Stacks Project Authors, “Formal schemes à la EGA.” Section 87.2, Tag 0AHY. Relevant: affine formal schemes from adic rings.
Michiel Hazewinkel, Formal Groups and Applications, AMS Chelsea Publishing, 2012. AMS book record. Relevant: Appendix A and Chapter 2.