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Throughout I am looking at one parameter group laws.

Let $R$ be a commutative ring with identity. Let $\mathbb{G}_a(X,Y) \in R[[X,Y]]$ be the formal additive group law, i.e., $\mathbb{G}_a(X,Y)= X+Y$.

I proved that $\operatorname{End}(\mathbb{G}_a) \cong R$, when $R$ has characteristic $0$.

I am trying to figure out what $\operatorname{End}(\mathbb{G}_a)$ will be when $R$ is a field of positive characteristic $p$.

I think I managed to show that any element of $\operatorname{End}(\mathbb{G}_a)$ in this case is of the form $a_oT + a_1T^{p^1} + a_2T^{p^2}+...$ But, I am not quite sure what the ring $\operatorname{End}(\mathbb{G}_a)$ should be.

Any help would be appreciated.

It is possible that the elements of $\operatorname{End}(\mathbb{G}_a)$ do not have the form I think they do. In that case it would be helpful to know what form the elements of $\operatorname{End}(\mathbb{G}_a)$ have.

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1 Answer 1

up vote 3 down vote accepted

When $k$ is a field of positive characteristic $p$, then $\text{End}(\mathbb{G}_{a,k})$ is isomorphic to the non-commutative polynomial ring $k \langle \tau \rangle$, where the multiplication is induced by $\tau b = b^p \tau$ for $b \in k$. In fact, $\tau$ is the frobenius $T^p$. This result follows easily from your observation.

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It's not the non-commutative power series ring in $\tau?$ –  jspecter Apr 30 '12 at 14:03
Oh, you consider the formal scheme. Ok then it's the power series ring. –  Martin Brandenburg Apr 30 '12 at 18:30

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