While studying cohomology theories on the stable homotopy setting, I have come up with the following basic question:

Consider the additive formal group law, $F_a$, and the multiplicative formal group law, $F_m$, both defined over a ring $R$.

If $R$ is a $\mathbb{Q}$-algebra, the exponential series defines a morphism of formal group laws $$F_a \to F_m \quad , \quad x \mapsto e^x := \sum_{k=0}^\infty x^k / k! $$ and it is not difficult to check that any other morphism $\,f\colon F_a \to F_m\,$ is of the form $f (x) = e^{ax}$, for some $a\in R$.

I can extend the previous reasoning in case $R$ is an integral domain–of any characteristic–, and I am wondering how many morphisms of formal group laws $F_a \to F_m$ are there over a general ring. That is to say:

The question is to describe all the series $\,f \in R[[x]]\,$ such that $$ f (s+t) = f(s)\cdot f(t) \qquad \mbox{ and } \qquad f(0) = 1 \ . $$

On the other hand, I am also interested in endomorphisms of the multiplicative group law $F_m$. The obvious ones are rising to the $k^{th}$ power, for $k \in \mathbb{Z}$:

$$ F_m \to F_m \quad , \quad x \mapsto (1+x)^k \ , $$ and the analogous question would be:

Describe all the series $\,f \in R[[x]]\,$ such that $$ f( x + y + xy) = f(x) \cdot f(y) \quad \mbox{ and } \quad f(0) = 1 \ . $$

I think that there only exist these "rising to the $k^{th}$-power" maps, no matter the ring of coefficients.Impr

While studying cohomology theories on the stable homotopy setting, I have come up with the following basic question:

Consider the additive formal group law, $F_a$, and the multiplicative formal group law, $F_m$, both defined over a ring $R$.

If $R$ is a $\mathbb{Q}$-algebra, the exponential series defines a morphism of formal group laws $$F_a \to F_m \quad , \quad x \mapsto e^x := \sum_{k=0}^\infty x^k / k! $$ and it is not difficult to check that any other morphism $\,f\colon F_a \to F_m\,$ is of the form $f (x) = e^{ax}$, for some $a\in R$.

I can extend the previous reasoning in case $R$ is an integral domain–of any characteristic–, and I am wondering how many morphisms of formal group laws $F_a \to F_m$ are there over a general ring. That is to say:

The question is to describe all the series $\,f \in R[[x]]\,$ such that $$ f (s+t) = f(s)\cdot f(t) \qquad \mbox{ and } \qquad f(0) = 1 \ . $$

On the other hand, I am also interested in endomorphisms of the multiplicative group law $F_m$. The obvious ones are rising to the $k^{th}$ power, for $k \in \mathbb{Z}$:

$$ F_m \to F_m \quad , \quad x \mapsto (1+x)^k \ , $$ and the analogous question would be:

Describe all the series $\,f \in R[[x]]\,$ such that $$ f( x + y + xy) = f(x) \cdot f(y) \quad \mbox{ and } \quad f(0) = 1 \ . $$

I think that there only exist these "rising to the $k^{th}$-power" maps, no matter the ring of coefficients.

While studying cohomology theories on the stable homotopy setting, I have come up with the following basic question:

Consider the additive formal group law, $F_a$, and the multiplicative formal group law, $F_m$, both defined over a ring $A$.

If $A$ is a $\mathbb{Q}$-algebra, the exponential series defines a morphism of formal group laws $$F_a \to F_m \quad , \quad x \mapsto e^x -1 $$ and I can check that any other morphism $\,f\colon F_a \to F_m\,$ is of the form $f (x) = e^{ax}-1$, for some $a\in A$.

The previous reasoning extends to the case when $A$ is an integral domain–of any characteristic–, and I am wondering what happens if there are zero divisors. That is to say:

Over an arbitrary ring, how many morphisms of formal group laws $F_a \to F_m$ are there?

I am also interested in endomorphisms of the multiplicative group law $F_m$. The obvious ones are rising to the $k^{th}$ power, for $k \in \mathbb{Z}$:

$$ F_m \to F_m \quad , \quad x \mapsto (1+x)^k - 1 \ , $$ and the analogous question would be:

Over an arbitrary ring, how many morphisms of formal group laws $F_m \to F_m$ are there?

Thanks in advance for any help.

While studying cohomology theories on the stable homotopy setting, I have come up with the following basic question:

Consider the additive formal group law, $F_a$, and the multiplicative formal group law, $F_m$, both defined over a ring $R$.

If $R$ is a $\mathbb{Q}$-algebra, the exponential series defines a morphism of formal group laws $$F_a \to F_m \quad , \quad x \mapsto e^x := \sum_{k=0}^\infty x^k / k! $$ and it is not difficult to check that any other morphism $\,f\colon F_a \to F_m\,$ is of the form $f (x) = e^{ax}$, for some $a\in R$.

I can extend the previous reasoning in case $R$ is an integral domain–of any characteristic–, and I am wondering how many morphisms of formal group laws $F_a \to F_m$ are there over a general ring. That is to say:

The question is to describe all the series $\,f \in R[[x]]\,$ such that $$ f (s+t) = f(s)\cdot f(t) \qquad \mbox{ and } \qquad f(0) = 1 \ . $$

On the other hand, I am also interested in endomorphisms of the multiplicative group law $F_m$. The obvious ones are rising to the $k^{th}$ power, for $k \in \mathbb{Z}$:

$$ F_m \to F_m \quad , \quad x \mapsto (1+x)^k \ , $$ and the analogous question would be:

Describe all the series $\,f \in R[[x]]\,$ such that $$ f( x + y + xy) = f(x) \cdot f(y) \quad \mbox{ and } \quad f(0) = 1 \ . $$

I think that there only exist these "rising to the $k^{th}$-power" maps, no matter the ring of coefficients.Impr