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The Thin Whistler
The Thin Whistler
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I am still studying Deligne and Illusie's paper (https://eudml.org/doc/143480), and I am again stuck, this time on pages 262/263.

Assume $Y\longrightarrow S$$X\longrightarrow S$ is a smooth morphism of $\mathbb{F}_{p}$-schemes, then $\operatorname{Lif}(Y,\tilde{S})$$\operatorname{Lif}(X,\tilde{S})$ is the gerbe of liftings to $\tilde{S}=S(\mathbb{Z}/p^{2})$, a morphism between two liftings $\tilde{U}'\longrightarrow\tilde{U}''$ (defined over an open $U\subseteq X$) being a morphism that is compatible with reduction mod $p$.

(1.) The authors make the claim that the sheaf of automorphisms of any given lifting of $U\subseteq X$ is the $\operatorname{Hom}$-sheaf $\operatorname{Hom}(\Omega^{1}_{U/S},\mathcal{O}_{U})$. Why is that the case?

(2.) While proving Theorem 3.5, the authors further claim that, if $a$ is any such automorphism, from the fact that $dF=0$ (where $F$ is the Frobenius on $X$) it follows that $\tilde{F}\circ a=\tilde{F}$ for any Lifting $\tilde{F}$ of $F$. Why?

I might not be seeing the wood for all the trees here, or maybe there is something that I just don't see. Can somebody explain this to me in detail?

I am still studying Deligne and Illusie's paper (https://eudml.org/doc/143480), and I am again stuck, this time on pages 262/263.

Assume $Y\longrightarrow S$ is a smooth morphism of $\mathbb{F}_{p}$-schemes, then $\operatorname{Lif}(Y,\tilde{S})$ is the gerbe of liftings to $\tilde{S}=S(\mathbb{Z}/p^{2})$, a morphism between two liftings $\tilde{U}'\longrightarrow\tilde{U}''$ (defined over an open $U\subseteq X$) being a morphism that is compatible with reduction mod $p$.

(1.) The authors make the claim that the sheaf of automorphisms of any given lifting of $U\subseteq X$ is the $\operatorname{Hom}$-sheaf $\operatorname{Hom}(\Omega^{1}_{U/S},\mathcal{O}_{U})$. Why is that the case?

(2.) While proving Theorem 3.5, the authors further claim that, if $a$ is any such automorphism, from the fact that $dF=0$ (where $F$ is the Frobenius on $X$) it follows that $\tilde{F}\circ a=\tilde{F}$ for any Lifting $\tilde{F}$ of $F$. Why?

I might not be seeing the wood for all the trees here, or maybe there is something that I just don't see. Can somebody explain this to me in detail?

I am still studying Deligne and Illusie's paper (https://eudml.org/doc/143480), and I am again stuck, this time on pages 262/263.

Assume $X\longrightarrow S$ is a smooth morphism of $\mathbb{F}_{p}$-schemes, then $\operatorname{Lif}(X,\tilde{S})$ is the gerbe of liftings to $\tilde{S}=S(\mathbb{Z}/p^{2})$, a morphism between two liftings $\tilde{U}'\longrightarrow\tilde{U}''$ (defined over an open $U\subseteq X$) being a morphism that is compatible with reduction mod $p$.

(1.) The authors make the claim that the sheaf of automorphisms of any given lifting of $U\subseteq X$ is the $\operatorname{Hom}$-sheaf $\operatorname{Hom}(\Omega^{1}_{U/S},\mathcal{O}_{U})$. Why is that the case?

(2.) While proving Theorem 3.5, the authors further claim that, if $a$ is any such automorphism, from the fact that $dF=0$ (where $F$ is the Frobenius on $X$) it follows that $\tilde{F}\circ a=\tilde{F}$ for any Lifting $\tilde{F}$ of $F$. Why?

I might not be seeing the wood for all the trees here, or maybe there is something that I just don't see. Can somebody explain this to me in detail?

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The Thin Whistler
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Automorphisms of Frobenius liftings and degeneration of the Hodge-de Rham spectral sequence

I am still studying Deligne and Illusie's paper (https://eudml.org/doc/143480), and I am again stuck, this time on pages 262/263.

Assume $Y\longrightarrow S$ is a smooth morphism of $\mathbb{F}_{p}$-schemes, then $\operatorname{Lif}(Y,\tilde{S})$ is the gerbe of liftings to $\tilde{S}=S(\mathbb{Z}/p^{2})$, a morphism between two liftings $\tilde{U}'\longrightarrow\tilde{U}''$ (defined over an open $U\subseteq X$) being a morphism that is compatible with reduction mod $p$.

(1.) The authors make the claim that the sheaf of automorphisms of any given lifting of $U\subseteq X$ is the $\operatorname{Hom}$-sheaf $\operatorname{Hom}(\Omega^{1}_{U/S},\mathcal{O}_{U})$. Why is that the case?

(2.) While proving Theorem 3.5, the authors further claim that, if $a$ is any such automorphism, from the fact that $dF=0$ (where $F$ is the Frobenius on $X$) it follows that $\tilde{F}\circ a=\tilde{F}$ for any Lifting $\tilde{F}$ of $F$. Why?

I might not be seeing the wood for all the trees here, or maybe there is something that I just don't see. Can somebody explain this to me in detail?