Structure theorem for log de Rham-Witt sheaves with vanishing
Fei Ren
TL;DR
This paper provides a precise structural description of log de Rham-Witt sheaves with vanishing along an effective divisor $D$ on a smooth $k$-scheme $X$. By exploiting a $p$-divisibility decomposition of $D$, it proves an explicit global formula: $W_n\Omega^q_{(X,-D),\log} = \sum_{i=0}^{n-1} p^i \operatorname{dlog} \operatorname{Ker}(\mathcal{O}_X^\times \to \mathcal{O}_{\lceil D/p^i \rceil}^\times) \wedge (\operatorname{dlog} j_{i*}\mathcal{O}_{U_i}^\times)^{\wedge (q-1)}$, as a Nisnevich subsheaf of $W_n\Omega^q_X$, with the restriction map $R$ surjective. The result clarifies the relation between vanishing and pole theories, compares GK/JSZ-type subsheaves, and yields a Bloch-Gabber-Kato-type isomorphism for Milnor $K$-theory with vanishing. The proofs reduce to a formal local analysis over $R=k[[T_1,...,T_d]]$ with explicit basis calculations for log differentials, complemented by Hartogs-type and CM-structure arguments, and are extended by an appendix containing a Hartog lemma for CM sheaves, a RamFil1 generalization, and a Frobenius-invariance statement for log forms modulo a filtration. Collectively, the work lays a solid foundation for studying Milnor $K$-theory with vanishing along $D$ and deepens the structural understanding of log de Rham-Witt sheaves in the modulus setting.
Abstract
We prove an elegant structure theorem for log de Rham-Witt sheaves with vanishing along an effective Cartier divisor $D$ defined in arXiv:2403.18763, answering a question of Shuji Saito during the Mainz conference and a question of Yigeng Zhao during a short visit of the author last summer. Our structural result for the log forms also lays the foundation for the study of Milnor $K$-theory with vanishing along $D$ in the paper to come.