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].


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