AI 中文总结
该研究针对半直群$C_{p^s}\rtimes C_m$，给出将$k[G]$-模提升为$R[G]$-模的充要准则，修正文献结果，证明曲线上$G$作用的提升障碍，应用于Harbater-Katz-Gabber覆盖的相关情形。
AI 中文摘要
对于素数$p$、满足$m$与$p$互素的正整数对$(s,m)$、同态$\chi:C_m\rightarrow\operatorname{Aut}(C_{p^s})$以及特征为$p$的代数闭域$k$，我们考虑半直积$G=C_{p^s}\rtimes_{\chi} C_m$，将其$p$-西罗子群$C_{p^s}$记为$H$，并考虑$k[G]$-模$V$。设$R$为剩余域为$k$、混合特征为$(0,p)$且包含本原$p^s$次单位根的完备离散赋值环。若$\chi$是单同态，我们给出将$V$提升为自由$R$-模的$R[G]$-模$\widetilde{V}$的两个充要准则：(i) 对$\widetilde{V}$无额外要求；(ii) 要求$\widetilde{V}^{C_{p^s}}=\{0\}$。这些准则修正了文献中的若干结果，我们用其证明：若$\chi$是单同态且$G$忠实作用于$k$上连通光滑射影曲线$X$，在$X\rightarrow X/G$为Harbater-Katz-Gabber覆盖所满足的温和假设下，$k[G]$-模$H^0(X,\Omega_X)$存在提升$\widetilde{V}$至$R$且满足$\widetilde{V}^{C_{p^s}}=\{0\}$。取$B$为$R$的分式域，我们证明：当$p$为奇数、$G/\operatorname{Ker}(\chi)$阶为偶数且$X/C_{p^s}\cong\mathbb{P}^1_k$时，若不存在这样的提升$\widetilde{V}$使得$B[H]$-模$\widetilde{V}\otimes_R B$定义在$\mathbb{Q}$上，则$G$在$X$上的作用无法提升至$R$。
英文摘要
For a prime $p$, a pair $(s,m)\in\mathbb{N}^2$ with $m$ relatively prime to $p$, a homomorphism $χ:C_m\rightarrow\operatorname{Aut}(C_{p^s})$, and an algebraically closed field $k$ of characteristic $p$, we consider the semidirect product $G=C_{p^s}\rtimes_χ C_m$, denote its $p$-Sylow subgroup $C_{p^s}$ by $H$, and consider a $k[G]$-module $V$. Let $R$ be a complete discrete valuation ring of residue field $k$ and mixed characteristic $(0,p)$ that contains a primitive $p^s$-th root of unity. If $χ$ is injective, we present two necessary and sufficient criteria for lifting $V$ to an $R[G]$-module $\widetilde{V}$ which is a free $R$-module: (i) when no extra requirement is made on $\widetilde{V}$ and (ii) when we require $\widetilde{V}^{C_{p^s}}=\{0\}$. The criteria correct several results in the literature and we use them to prove that, if $χ$ is injective and $G$ acts faithfully on a connected smooth projective curve $X$ over $k$, then, under mild hypotheses satisfied if $X\rightarrow X/G$ is a Harbater--Katz--Gabber cover, the $k[G]$-module $H^0(X,Ω_X)$ has a lift $\widetilde{V}$ to $R$ with $\widetilde{V}^{C_{p^s}}=\{0\}$. With $B$ as the field of fractions of $R$, we prove the following obstruction when $p$ is odd, $G/\operatorname{Ker}(χ)$ has even order, and $X/C_{p^s}\cong\mathbb{P}^1_k$: if no such lift $\widetilde{V}$ exists with the $B[H]$-module $\widetilde{V}\otimes_R B$ defined over $\mathbb{Q}$, then the action of $G$ on $X$ does not lift to $R$.
