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I am reading Serre's Local Fields. In Section V.4, Serre considers a finite totally ramified extension of local fields $L/K$ with the residue field $\bar{L}=\bar{K}$ a perfect field. For $\bar{K}'$ a finite extension of $\bar{K}$, let $K'$ be the corresponding unramified extension of $K$, and let $L'=LK'$. The norm maps (for $n\geq 1$) $$ N_n: U^{\psi(n)}_L/U^{\psi(n)+1}_L \to U^n_K/U^{n+1}_K $$ $$ N'_n: U^{\psi(n)}_{L'}/U^{\psi(n)+1}_{L'} \to U^n_{K'}/U^{n+1}_{K'} $$ can be identified with $N_n:\bar{K}\to \bar{K}$ and $N'_n:\bar{K}'\to \bar{K}'$, given by polynomials that are determined in Section V.3. He remarks that the coefficients of the polynomials are the same for $N_n$ and $N'_n$ (which I believe), and then he writes:

In the language of algebraic geometry, that means we have for each $n\geq 1$ a homomorphism $v_n:G_a\to G_a$ rational over $\bar{K}$, and that $N_n'$ is the restriction of $v_n$ to the points of $G_a$ rational over $\bar{K}'$.

Can some one elaborate on it? I have some exposure to algebraic geometry (from Hartshorne's textbook) but I fail to recognize anything familiar here. I do not even know what the word rational means here.

Thanks!

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1 Answer 1

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$G_a$ is the additive group variety. There are various more and less complicated versions of how to describe it but let me give a very simple one you may believe.

For $F$ a field, the affine line $\mathbb{A}^1_F$ is also equipped with point-wise addition. If we denote this pair of $(\mathbb{A}^1_F, +)$ as $G_a/F$ then because this addition is polynomial one can formulate that $G_a/F$ not just equipped point-wise with an operation, but that this operation can be phrased in terms of the category of $F$-schemes where $G_a/F$ will be a group object.

Here Serre is saying that the Norm maps for $n \geq 1$ are group homomorphisms $(\overline{K},+) \rightarrow (\overline{K}, +)$ which are given by polynomials. Now $(\overline{K},+)$ is the $\overline{K}$ points of $G_a/\overline{K}$ and so one can expect that these polynomials can be lifted to $G_a/\overline{K}$ and made into a morphism of group objects.

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