## A proof of the absolute purity conjecture (after Gabber).(English)Zbl 1059.14026

Usui, Sampei (ed.) et al., Algebraic geometry 2000, Azumino. Proceedings of the symposium, Nagano, Japan, July 20–30, 2000. Tokyo: Mathematical Society of Japan (ISBN 4-931469-20-5/hbk). Adv. Stud. Pure Math. 36, 153-183 (2002).
From the introduction: This article is an edited version of O. Gabber’s talk on his proof of the absolute purity conjecture of A. Grothendieck given at the $$p$$-adic conference held in Toulouse in 1994.
Conjecture. A. Grothendieck [Lect. Notes Math. 589, 1–72 (1977; Zbl 0356.14004)]. Let $$Y\overset {i} {\hookrightarrow} X$$ be a closed immersion of noetherian regular schemes of pure codimension $$c$$. Let $$n$$ be an integer which is invertible on $$X$$, and let $$\Lambda=\mathbb{Z}/n$$. Then ${\mathcal H}_Y^q(\Lambda) \simeq\begin{cases} 0& \text{for }q\neq 2c,\\ \Lambda_Y(-c)& \text{for }q=2c. \end{cases}$ The proof of Gabber consists of three key parts:
The first is the construction of the global cycle class $$\text{cl}(Y)\in H_Y^{2c} \bigl(X,\Lambda (c)\bigr)$$ for a noetherian scheme $$X$$, $$\Lambda=\mathbb{Z}/n$$ with $$n$$ invertible on $$X$$, and $$Y \subset X$$ a closed local complete intersection subscheme. Now, absolute purity is reduced to punctual purity: For a regular strict local ring $${\mathcal O}$$ of dimension $$d$$ with the closed point $$i_x:x\to\text{Spec}\,{\mathcal O}$$, the cycle class gives an isomorphism $$\text{cl}(x): \Lambda_x \simeq i^!_x \Lambda(d)[2d].$$ By induction this is reduced to showing the vanishing $H^p\bigl(\text{Spec} \,{\mathcal O}[f^{-1}], \Lambda\bigr)= 0\quad\text{for }p\neq 0,1,$ where $$f\in m\setminus m^2$$ with $$m\subset{\mathcal O}$$, the maximal ideal. The second key step is the affine Lefschetz theorem on vanishing of cohomology of affine schemes of arithmetic type. It is used to show that $H^p\bigl(\text{Spec}\,{\mathcal O}[f^{-1}], \Lambda\bigr)= 0\quad\text{for }p\neq 0,1,d-1,d,$ where $${\mathcal O}$$ and $$f\in m$$ are the same as before. The final key step uses the Atiyah-Hirzebruch type spectral sequence for the étale $$K$$-theory. Thanks to A. A. Suslin’s computation of the $$K$$-theory of a separably closed field [J. Pure Appl. Algebra, 34, 301–318 (1984; Zbl 0548.12009)] and O. Gabber’s rigidity theorem for algebraic $$K$$-theory [Contemp. Math., 126, 59–70 (1992; Zbl 0791.19002)], $$(K/\ell^\nu)_*({\mathcal O} [f^{-1}])[\beta^{-1}]$$ is computed. This implies the degeneracy of the spectral sequence from which the remaining vanishing of étale cohomology follows.
For the entire collection see [Zbl 1007.00031].

### MSC:

 14F20 Étale and other Grothendieck topologies and (co)homologies 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry

### Citations:

Zbl 0356.14004; Zbl 0548.12009; Zbl 0791.19002