Purity in the tame cohomology

Technical Summary: Purity in the Tame Cohomology

Problem Statement
The paper addresses the failure of the absolute cohomological purity theorem for $p$-torsion sheaves in the étale topology over schemes of characteristic $p$. While purity holds for coefficients prime to $p$ (Gabber-Fujiwara theorem), it fails for logarithmic de Rham-Witt sheaves $\nu_{m}(n)$ in the étale topology. Specifically, for a regular pair $(X, Z)$ of codimension $r$, the higher direct image $R^{r+1}i_!\nu_{X,m}(n)$ does not necessarily vanish, obstructing the existence of a purity isomorphism $Ri_!\nu_{X,m}(n) \cong \nu_{Z,m}(n-r)[-r]$. The obstruction arises because the inverse Cartier operator (or Artin-Schreier map) is not surjective in the étale topology due to the presence of wildly ramified covers.

Methodology
The author, Amine Koubaa, resolves this obstruction by shifting the framework from the étale topology to the tame topology, introduced by Hübner and Stalder. The methodology relies on the following structural components:

1. Adic Spaces and Discretely Ringed Spaces: The paper utilizes discretely ringed adic spaces, denoted $X = \text{Spa}(X, S)$, to provide a geometric setting where tame ramification is well-behaved. The proof leverages the equivalence between the tame cohomology of a scheme and the cohomology of its associated adic space.
2. Logarithmic Geometry and Differentials: The core technical tool is the sheaf of logarithmic differentials. The author introduces and analyzes the sheaf $\Omega^+_{X/k}$, defined via Kähler seminorms and "rangers" (a concept from Hübner and Temkin), which captures differentials with bounded valuation.
3. Resolution of Singularities: The proofs assume the existence of resolution of singularities and embedded resolution of singularities (specifically for log smooth schemes). This allows the reduction of general cases to log smooth pairs $(X, \bar{X})$ where the boundary is a simple normal crossing divisor.
4. The Cartier Operator: A crucial step involves establishing the exactness of the sequence involving the Cartier operator $C$ on the sheaves of closed differentials $Z\Omega^+$. In the tame topology, the inverse Cartier operator becomes surjective, unlike in the étale topology where it fails due to wild ramification.
5. Popescu's Theorem: To extend results from smooth schemes over perfect fields to regular pairs over arbitrary bases of characteristic $p$, the paper employs Popescu's theorem, expressing regular algebras as filtered colimits of smooth algebras.

Key Contributions and Results

• Purity Theorem for Tame Cohomology (Theorem 1.3 & 5.9): The main result establishes that for a smooth pair $(X, Z)$ of pure codimension $r$ over a perfect field $k$ of characteristic $p$, there is a natural isomorphism in the derived category of tame sheaves:
  $$Ri_!\nu_{X,m}(n) \cong \nu_{Z,m}(n-r)[-r]$$
  This holds under the assumption of resolution of singularities. The isomorphism is constructed via a Gysin morphism.

• Exactness of the Cartier Sequence (Theorem 4.27): The paper proves that on discretely ringed adic spaces, the sequence
  $$0 \to \nu(n) \to Z\Omega^+_{X/k} \xrightarrow{C^{-1}} \Omega^+_{X/k} \to 0$$
  is exact in the tame topology. This exactness is the mechanism that restores purity, as it ensures the vanishing of the obstruction term $R^{r+1}i_!$.

• Gysin Morphism Construction (Section 5.1): The author constructs a Gysin morphism for the sheaves $\Omega^+_{n}$ and extends it to the logarithmic de Rham-Witt sheaves $\nu_{m}(n)$. This involves proving that the higher direct images $R^ji_!\Omega^+_{n}$ vanish for $j \neq r$ and that $R^ri_!\Omega^+_{n} \cong \Omega^+_{n-r}[-r]$.

• Generalization to Regular Pairs (Theorem 5.15): Using Popescu's theorem, the purity result is generalized from smooth pairs over perfect fields to regular pairs $(X, Z)$ over a base scheme $S$ of characteristic $p$. The result states:
  $$Ri_!\nu_{X,m}(n) \cong \nu_{Z,m}(n-r)[-r]$$
  in the derived category $D((Z/S)_t)$.

• Comparison with Étale Cohomology (Proposition 5.10): The paper clarifies the relationship between the tame and étale results. It shows that the tame purity isomorphism is the "correct" version, and the failure of étale purity is due to the non-surjectivity of the Cartier operator in the presence of wild ramification. The tame sheafification of the étale direct image recovers the tame result.

Significance and Claims
The paper claims to provide a complete solution to the purity problem for logarithmic de Rham-Witt sheaves in positive characteristic by utilizing the tame topology.

• Restoration of Purity: It demonstrates that the obstruction to purity in the étale topology (the non-vanishing of $R^{r+1}i_!$) is an artifact of wild ramification, which is excluded in the tame topology.
• Methodological Advancement: By integrating the theory of adic spaces (specifically discretely ringed ones) with logarithmic geometry and the Cartier operator, the paper provides a robust framework for handling $p$-torsion coefficients where classical étale methods fail.
• Consistency with Known Results: The results generalize the semi-purity theorems of Milne and Gros and align with the absolute purity theorem for coefficients prime to $p$.
• Modesty on Assumptions: The author explicitly notes that the results rely on the assumption of resolution of singularities. The paper mentions that future work aims to remove this assumption, particularly regarding the existence of the Cartier operator on $\Omega^+$ without relying on resolution.

In summary, the paper establishes that "purity holds in the tame topology," offering a natural isomorphism for the direct image with proper support of logarithmic de Rham-Witt sheaves, thereby resolving a long-standing issue in the cohomology of schemes in characteristic $p$.