Substitution of formal power series

Core idea

Let $R$ be a commutative ring, $f\in R[[X_1,\ldots,X_n]]$ , and

g_i\in(Y_1,\ldots,Y_m)R[[Y_1,\ldots,Y_m]] \qquad(1\leq i\leq n).

The substitution of formal power series is

f(g_1,\ldots,g_n) := \sum_{\alpha\in\mathbb N^n}a_\alpha g_1^{\alpha_1}\cdots g_n^{\alpha_n}, \qquad f=\sum_\alpha a_\alpha X^\alpha.

It is a well-defined element of $R[[Y_1,\ldots,Y_m]]$ : modulo $(Y_1,\ldots,Y_m)^N$ , only the terms with $|\alpha|<N$ can contribute.

Topological meaning

The vanishing constant terms make every $g_i$ topologically nilpotent for the augmentation-adic topology. Substitution is the unique continuous $R$ -algebra homomorphism

R[[X_1,\ldots,X_n]] \longrightarrow R[[Y_1,\ldots,Y_m]], \qquad X_i\longmapsto g_i.

It can equivalently be constructed on every finite quotient and then passed to the inverse limit.

Composition and pointed maps

Formal substitution is associative:

f(g_1(h),\ldots,g_n(h)) = \bigl(f(g_1,\ldots,g_n)\bigr)(h).

The coordinate tuple $g$ has zero constant term precisely when it sends the formal origin to the formal origin. Consequently pointed maps between formal affine spaces are represented by such tuples, with composition given by substitution.

Scope warning

Over a ring containing nilpotents, a series with nonzero nilpotent constant term may also admit some substitutions. The zero-constant-term convention is the clean, base-preserving pointed construction used for formal group laws; it should not be misread as a classification of every possible continuous map between arbitrary adic rings.

References

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