Nisnevich topology on non-(locally) Noetherian schemes

2011-10-18T07:35:18Z

Background

Lurie has in DAG XI a definition (given below) of a Nisnevich cover for arbitrary commutative rings, which reduces to the usual one for Noetherian rings. It boils down to being a etale covering family admitting local sections over a different, finite, cover of a particular sort. In particular, we can assume wlog it is a finite etale family. The usual definition of a Nisnevich cover of schemes uses the (locally) Noetherian assumption and I would like to translate this into the Lurie style definition, but my algebraic geometry-foo is not up to scratch. This possibly is a simple question, but bear with me.

Details

We will work initially in $Ring$, so that what we are describing is really a cocovering family, but the translation to $Aff = Ring^{op}$ is of course trivial. Consider the following definition:

Definition (Lurie) Let $R$ be a commutative ring. A (finite) etale covering family $\{\phi_\alpha : R \to R_\alpha\}$ is a Nisnevich covering if there is a finite sequence $a_1,\ldots,a_n\in R$ such that

$(a_1,\ldots,a_n)$ is the unit ideal in $R$

For each $1 \leq i \leq n$ there is an index $\alpha$ and a ring homomorphism $$\psi_i : R_\alpha \to R[a_i^{-1}]/(a_1,\ldots,a_{i-1})$$ such that $\psi_i\circ \phi_\alpha:R \to R[a_i^{-1}]/(a_1,\ldots,a_{i-1})$ is the canonical map to the quotient of the localisation.

When interpreted in $Aff$ the maps $\psi_i$ are just local sections of the etale maps $Spec R_\alpha \to Spec R$, for a given notion of 'local'.

What should the extension to non-affine schemes be?

My guess is that we just ask for an affine cover an then a Nisnevich cover of each affine, but I'm not sure of the subtleties.

I suspect we should be able to combine the maps $\psi_i$ into a single map

$$\coprod_i Spec R[a_i^{-1}]/(a_1,\ldots,a_{i-1}) \to \coprod_\alpha Spec R_\alpha,$$

which is a 'local' section of $\coprod_\alpha Spec R_\alpha \to Spec R$.

Can we describe the Nisnevich topology on schemes as being just etale covers which admit local sections over another sort of cover?

Answer by philip314 for Nisnevich topology on non-(locally) Noetherian schemes

2011-10-18T10:51:42Z

A key feature of the Nisnevich topology is that as a cd-structure (cf. [Voevodsky, Homotopy theory of simplicial sheaves in completely decomposable topologies]) it is complete and regular. This implies what Lurie calls Nisnevich excision in DAG XI. The proof of this "excision" relies on the existence of a "splitting sequence" (cf. [Morel-Voevodsky, A^1-Homotopy Theory of Schemes, Lemma 3.1.5]) for any given Nisnevich covering.

Def.:(MV) A splitting sequence for a covering family $\{p_{\alpha}:Spec(R_\alpha)\to Spec(R)\}$ is a sequence of closed subsets of $Spec(R)$ of the form $$ \emptyset = Z_{n+1}\subset Z_n\subset \ldots \subset Z_0=Spec(R) $$ such that for $i=0,\ldots,n$ the morphism $\coprod_\alpha (p_\alpha)^{-1}(Z_i\setminus Z_{i+1})\to Z_i\setminus Z_{i+1}$ splits.

This existence statement needs the space which is covered to be noetherian. Lurie drops the noetherian requirement and pays for the splitting sequence, which doesn't come for free any longer. You find the non-affine situation in section 3.1 of the Morel-Voevodsky paper.

However, here we have $Z_i:=V(a_1,\ldots,a_{i-1})$ and the first condition above says that $Z_{n+1}= \emptyset$ and the second condition gives a splitting on

$$ Z_i\setminus Z_{i+1}= V(a_1,\ldots,a_{i-1})\cap (V(a_1,\ldots,a_i))^c$$ $$= V(a_1,\ldots,a_{i-1})\cap (V(a_1,\ldots,a_{i-1})\cap V(a_i))^c$$ $$= V(a_1,\ldots,a_{i-1})\cap (V(a_1,\ldots,a_{i-1})^c\cup D(a_i))$$ $$= V(a_1,\ldots,a_{i-1})\cap D(a_i)$$ $$= Spec(R[a_i^{-1}]/(a_1,\ldots,a_{i-1}))$$

Nisnevich topology on non-(locally) Noetherian schemes - MathOverflow

Nisnevich topology on non-(locally) Noetherian schemes

Answer by philip314 for Nisnevich topology on non-(locally) Noetherian schemes