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Question. Let $B \twoheadrightarrow C$ be a fully faithful morphismhomomorphism of finite connected commutative group schemes over a perfect field $k$. Let $T = k[x^{1/p^\infty}]/(x) = \varinjlim k[t]/(t^p)$. Is the map $B(T) \rightarrow C(T)$ surjective?

For $m < n$, the underlying space of natural quotients $\mu_{p^n} \twoheadrightarrow \mu_{p^m}$ and $\alpha_{p^n} \twoheadrightarrow \alpha_{p^m}$ are induced by the natural ring monomorphism $k[t]/(t^{p^m}) \hookrightarrow k[t]/(t^{p^n})$. Since every element in $T$ has a unique $p$-th root of unity, any ring homomorphism $k[t]/(t^{p^m}) \rightarrow T$ naturally lifts to $k[t]/(t^{p^n}) \rightarrow T$.

I know that that the underlying space of a connected finite group scheme is a product of $\mathrm{spec}\,k[t]/(t^{p^n})$ for various $n$. However, I do not know whether the morphisms between the underlying spaces of $B \rightarrow C$ are given by a product of $k[t]/(t^{p^m}) \hookrightarrow k[t]/(t^{p^n})$$\mathrm{spec}\,k[t]/(t^{p^n}) \twoheadrightarrow \mathrm{spec}\,k[t]/(t^{p^m})$. If this is true, the earlier argument implies that $B(T) \rightarrow C(T)$ is surjective.

Question. Let $B \twoheadrightarrow C$ be a fully faithful morphism of finite connected commutative group schemes over a perfect field $k$. Let $T = k[x^{1/p^\infty}]/(x) = \varinjlim k[t]/(t^p)$. Is the map $B(T) \rightarrow C(T)$ surjective?

For $m < n$, the underlying space of natural quotients $\mu_{p^n} \twoheadrightarrow \mu_{p^m}$ and $\alpha_{p^n} \twoheadrightarrow \alpha_{p^m}$ are induced by the natural ring monomorphism $k[t]/(t^{p^m}) \hookrightarrow k[t]/(t^{p^n})$. Since every element in $T$ has a unique $p$-th root of unity, any ring homomorphism $k[t]/(t^{p^m}) \rightarrow T$ naturally lifts to $k[t]/(t^{p^n}) \rightarrow T$.

I know that that the underlying space of a connected finite group scheme is a product of $\mathrm{spec}\,k[t]/(t^{p^n})$ for various $n$. However, I do not know whether the morphisms between the underlying spaces of $B \rightarrow C$ are given by a product of $k[t]/(t^{p^m}) \hookrightarrow k[t]/(t^{p^n})$. If this is true, the earlier argument implies that $B(T) \rightarrow C(T)$ is surjective.

Question. Let $B \twoheadrightarrow C$ be a fully faithful homomorphism of finite connected commutative group schemes over a perfect field $k$. Let $T = k[x^{1/p^\infty}]/(x) = \varinjlim k[t]/(t^p)$. Is the map $B(T) \rightarrow C(T)$ surjective?

For $m < n$, the underlying space of natural quotients $\mu_{p^n} \twoheadrightarrow \mu_{p^m}$ and $\alpha_{p^n} \twoheadrightarrow \alpha_{p^m}$ are induced by the natural ring monomorphism $k[t]/(t^{p^m}) \hookrightarrow k[t]/(t^{p^n})$. Since every element in $T$ has a unique $p$-th root of unity, any ring homomorphism $k[t]/(t^{p^m}) \rightarrow T$ naturally lifts to $k[t]/(t^{p^n}) \rightarrow T$.

I know that that the underlying space of a connected finite group scheme is a product of $\mathrm{spec}\,k[t]/(t^{p^n})$ for various $n$. However, I do not know whether the morphisms between the underlying spaces of $B \rightarrow C$ are given by a product of $\mathrm{spec}\,k[t]/(t^{p^n}) \twoheadrightarrow \mathrm{spec}\,k[t]/(t^{p^m})$. If this is true, the earlier argument implies that $B(T) \rightarrow C(T)$ is surjective.

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HJK
HJK
  • 199
  • 5

Is the functor $\mathrm{Hom}(\mathrm{spec}\,k[x^{1/{p^\infty}}]/(x), -)$ from the category of finite commutative group schemes exact?

Question. Let $B \twoheadrightarrow C$ be a fully faithful morphism of finite connected commutative group schemes over a perfect field $k$. Let $T = k[x^{1/p^\infty}]/(x) = \varinjlim k[t]/(t^p)$. Is the map $B(T) \rightarrow C(T)$ surjective?

For $m < n$, the underlying space of natural quotients $\mu_{p^n} \twoheadrightarrow \mu_{p^m}$ and $\alpha_{p^n} \twoheadrightarrow \alpha_{p^m}$ are induced by the natural ring monomorphism $k[t]/(t^{p^m}) \hookrightarrow k[t]/(t^{p^n})$. Since every element in $T$ has a unique $p$-th root of unity, any ring homomorphism $k[t]/(t^{p^m}) \rightarrow T$ naturally lifts to $k[t]/(t^{p^n}) \rightarrow T$.

I know that that the underlying space of a connected finite group scheme is a product of $\mathrm{spec}\,k[t]/(t^{p^n})$ for various $n$. However, I do not know whether the morphisms between the underlying spaces of $B \rightarrow C$ are given by a product of $k[t]/(t^{p^m}) \hookrightarrow k[t]/(t^{p^n})$. If this is true, the earlier argument implies that $B(T) \rightarrow C(T)$ is surjective.