AI 中文总结
本文证明正常光滑$p$-进形式概形一般纤维的Hodge多边形位于特殊纤维广义Hodge--Witt多边形之上，结合棱镜上同调与Ekedahl对角线理论，允许负Hodge--Witt数和晶体挠。
AI 中文摘要
我们证明了，在$\mathcal O_C$上的一个正常光滑$p$-进形式概形的特殊纤维的广义Hodge--Witt多边形之上或之上，其一般纤维的Hodge多边形成立。该论证结合了棱镜上同调与Ekedahl的相干Raynaud复形的对角线理论。对于每个整数截断和每个有限迭代长度，一个有限的迭代Nygaard构造产生一个完美的$A_{\mathrm{inf}}$-复形。其沿一般除子分支的逐次长度是加权Hodge和。一个极大-主子式特殊化论证将其和限制在相应的晶体长度上。在特殊纤维上，相干Raynaud复形的精确导出修改具有由加权Hodge--Witt数控制的长度增长，且误差有界。在这些数值长度中取极限得到比较，允许负的Hodge--Witt数和晶体挠。不假设投影性、代数化或下降到离散值域。
英文摘要
We prove that the Hodge polygon of the generic fiber of a proper smooth $p$-adic formal scheme over $\mathcal O_C$ lies on or above the generalized Hodge--Witt polygon of its special fiber. The argument combines prismatic cohomology with Ekedahl's diagonal theory of coherent Raynaud complexes. For each integral cutoff and each finite iteration length, a finite iterated Nygaard construction produces a perfect $A_{\mathrm{inf}}$-complex. Its individual-degree lengths along the generic divisor branches are weighted Hodge sums. A maximal-minor specialization argument bounds their sum by the corresponding crystalline length. On the special fiber, an exact derived modification of coherent Raynaud complexes has length growth governed by weighted Hodge--Witt numbers, with bounded error. Passing to the limit in these numerical lengths gives the comparison, allowing negative Hodge--Witt numbers and crystalline torsion. No projectivity, algebraization, or descent to a discretely valued field is assumed.
Comments14 pages
