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Matching formatting of algebraic closure in title and body.
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LSpice
LSpice
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Splitting the Witt vectors of $\bar$\overline{\mathbb{F}_p}_p$$

Let $\overline{\mathbb{F}_p}$ be the algebraic closure of $\mathbb{F}_p$. Let $W(-)$$W({-})$ denote the functor of taking $p$-typical Witt vectors. Then the extension $\mathbb{F}_p\rightarrow \overline{\mathbb{F}_p}$ induces a map $f: \mathbb{Z}_p=W(\mathbb{F}_p)\rightarrow W(\overline{\mathbb{F}_p})$.

Is $f$ split as a $\mathbb{Z}_p$-module map?

As far as I understand, $W(\overline{\mathbb{F}_p})$ is built out of $\mathbb{Z}_p$ by adjoining prime-to-$p$-th-power roots of unity, so it seems like you could try sending all those adjoined roots to zero. But some roots live in $\mathbb{Z}_p$, so you better not send those to zero.

Splitting the Witt vectors of $\bar{\mathbb{F}}_p$

Let $\overline{\mathbb{F}_p}$ be the algebraic closure of $\mathbb{F}_p$. Let $W(-)$ denote the functor of taking $p$-typical Witt vectors. Then the extension $\mathbb{F}_p\rightarrow \overline{\mathbb{F}_p}$ induces a map $f: \mathbb{Z}_p=W(\mathbb{F}_p)\rightarrow W(\overline{\mathbb{F}_p})$.

Is $f$ split as a $\mathbb{Z}_p$-module map?

As far as I understand, $W(\overline{\mathbb{F}_p})$ is built out of $\mathbb{Z}_p$ by adjoining prime-to-$p$-th-power roots of unity, so it seems like you could try sending all those adjoined roots to zero. But some roots live in $\mathbb{Z}_p$, so you better not send those to zero.

Splitting the Witt vectors of $\overline{\mathbb{F}_p}$

Let $\overline{\mathbb{F}_p}$ be the algebraic closure of $\mathbb{F}_p$. Let $W({-})$ denote the functor of taking $p$-typical Witt vectors. Then the extension $\mathbb{F}_p\rightarrow \overline{\mathbb{F}_p}$ induces a map $f: \mathbb{Z}_p=W(\mathbb{F}_p)\rightarrow W(\overline{\mathbb{F}_p})$.

Is $f$ split as a $\mathbb{Z}_p$-module map?

As far as I understand, $W(\overline{\mathbb{F}_p})$ is built out of $\mathbb{Z}_p$ by adjoining prime-to-$p$-th-power roots of unity, so it seems like you could try sending all those adjoined roots to zero. But some roots live in $\mathbb{Z}_p$, so you better not send those to zero.

hopefully correctly improving latex
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darij grinberg
darij grinberg
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Let $\bar{\mathbb{F}_p}$$\overline{\mathbb{F}_p}$ be the algebraic closure of $\mathbb{F}_p$. Let $W(-)$ denote the functor of taking $p$-typical Witt vectors. Then the extension $\mathbb{F}_p\rightarrow \bar{\mathbb{F}_p}$$\mathbb{F}_p\rightarrow \overline{\mathbb{F}_p}$ induces a map $f: \mathbb{Z}_p=W(\mathbb{F}_p)\rightarrow W(\bar{\mathbb{F}_p})$$f: \mathbb{Z}_p=W(\mathbb{F}_p)\rightarrow W(\overline{\mathbb{F}_p})$.

Is $f$ split as a $\mathbb{Z}_p$-module map?

As far as I understand, W(\bar{\mathbb{F}_p})$W(\overline{\mathbb{F}_p})$ is buildbuilt out of $\mathbb{Z}_p$ by adjoining prime-to-$p$-th-power roots of unity, so it seems like you could try sending all those adjoined roots to zero. But some roots live in $\mathbb{Z}_p$, so you better not send those to zero.

Let $\bar{\mathbb{F}_p}$ be the algebraic closure of $\mathbb{F}_p$. Let $W(-)$ denote the functor of taking $p$-typical Witt vectors. Then the extension $\mathbb{F}_p\rightarrow \bar{\mathbb{F}_p}$ induces a map $f: \mathbb{Z}_p=W(\mathbb{F}_p)\rightarrow W(\bar{\mathbb{F}_p})$.

Is $f$ split as a $\mathbb{Z}_p$-module map?

As far as I understand, W(\bar{\mathbb{F}_p}) is build out of $\mathbb{Z}_p$ by adjoining prime-to-$p$-th-power roots of unity, so it seems like you could try sending all those adjoined roots to zero. But some roots live in $\mathbb{Z}_p$, so you better not send those to zero.

Let $\overline{\mathbb{F}_p}$ be the algebraic closure of $\mathbb{F}_p$. Let $W(-)$ denote the functor of taking $p$-typical Witt vectors. Then the extension $\mathbb{F}_p\rightarrow \overline{\mathbb{F}_p}$ induces a map $f: \mathbb{Z}_p=W(\mathbb{F}_p)\rightarrow W(\overline{\mathbb{F}_p})$.

Is $f$ split as a $\mathbb{Z}_p$-module map?

As far as I understand, $W(\overline{\mathbb{F}_p})$ is built out of $\mathbb{Z}_p$ by adjoining prime-to-$p$-th-power roots of unity, so it seems like you could try sending all those adjoined roots to zero. But some roots live in $\mathbb{Z}_p$, so you better not send those to zero.

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kiran
kiran
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Splitting the Witt vectors of $\bar{\mathbb{F}}_p$

Let $\bar{\mathbb{F}_p}$ be the algebraic closure of $\mathbb{F}_p$. Let $W(-)$ denote the functor of taking $p$-typical Witt vectors. Then the extension $\mathbb{F}_p\rightarrow \bar{\mathbb{F}_p}$ induces a map $f: \mathbb{Z}_p=W(\mathbb{F}_p)\rightarrow W(\bar{\mathbb{F}_p})$.

Is $f$ split as a $\mathbb{Z}_p$-module map?

As far as I understand, W(\bar{\mathbb{F}_p}) is build out of $\mathbb{Z}_p$ by adjoining prime-to-$p$-th-power roots of unity, so it seems like you could try sending all those adjoined roots to zero. But some roots live in $\mathbb{Z}_p$, so you better not send those to zero.