Multivariable formal power series ring

Let $R$ be a commutative ring with $1$ and let $X=(X_1,\ldots,X_n)$ . The multivariable formal power series ring

R[[X_1,\ldots,X_n]]

consists of all formal sums $f=\sum_{\alpha\in\mathbb N^n}a_\alpha X^\alpha$ , where $X^\alpha=X_1^{\alpha_1}\cdots X_n^{\alpha_n}$ . Equality and addition are coefficientwise, and multiplication is determined by

[X^\gamma](fg)=\sum_{\alpha+\beta=\gamma}a_\alpha b_\beta.

Formal multiplication

The sum is finite for each $\gamma$ , so no analytic convergence is involved.

Augmentation-adic description

The constant-term map

\epsilon:R[[X_1,\ldots,X_n]]\longrightarrow R,\qquad f\longmapsto a_0

has augmentation ideal $\mathfrak m=(X_1,\ldots,X_n)$ . Truncating by total degree gives a natural isomorphism of topological rings

R[[X_1,\ldots,X_n]] \cong \varprojlim_N R[X_1,\ldots,X_n]/\mathfrak m^N.

Consequently the ring is complete and separated for its $\mathfrak m$ -adic topology. The one-variable case is $R[[X]]$ .

Universal property for pointed substitutions

For $Y=(Y_1,\ldots,Y_m)$ , a tuple

g=(g_1,\ldots,g_n)\in(Y_1,\ldots,Y_m)R[[Y]]^n

determines a continuous $R$ -algebra homomorphism

R[[X]]\longrightarrow R[[Y]],\qquad X_i\longmapsto g_i.

This is the algebraic form of substituting formal power series with zero constant term. The direction reverses when the rings are viewed as coordinate rings of formal affine spaces.

References

Michiel Hazewinkel, Formal Groups and Applications, AMS Chelsea Publishing, 2012. AMS book record. Relevant: Appendix A, power-series rings.
Nicolas Bourbaki, Algebra II: Chapters 4–7, Springer, 1990. Relevant: Chapter 4, formal series.