AI 中文总结
针对混合特征非分歧正则局部环，建立了深度与局部上同调维数的关系，给出局部上同调消失的判定条件。
AI 中文摘要
设$(R,\boldsymbol{\frak m})$为混合特征$(0,p)$、维数$d$的非分歧正则局部环，$I\boldsymbol{\trianglelefteq}R$为理想。我们证明：若$depth(R/I)\boldsymbol{\trianglelefteq}3$，则$cd(I)\boldsymbol{\trianglelefteq}d-3$；若$R$在离散赋值环（DVR）上本质有限型，则$depth(R/I)\boldsymbol{\trianglelefteq}4$时$cd(I)\boldsymbol{\trianglelefteq}d-4$。更一般地，当$j>d-depth(R/I)$时，$H_I^j(R)$为$\boldsymbol{\frak Q}$-向量空间，此范围内局部上同调的消失完全由特征零纤维决定。
英文摘要
Let $(R,\mathfrak m)$ be an unramified regular local ring of mixed characteristic $(0,p)$ and dimension $d$ and let $I\subseteq R$ be an ideal. We prove that $depth(R/I)\geq 3$ implies $cd(I)\leq d-3$, and if $R$ is essentially of finite type over a DVR, then $depth(R/I)\geq 4$ implies $cd(I)\leq d-4$. More generally, $H_I^j(R)$ is a $\mathbb{Q}$-vector space whenever $j>d-depth(R/I)$, thus vanishing of local cohomology in this range is determined completely by the characteristic zero fiber.
