# Nisnevich topology on non-(locally) Noetherian schemes

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?

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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

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

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I'm not interested in the non-Noetherian situation per se, just Lurie's formulation of the definition. – David Roberts Oct 18 '11 at 19:41
I'm not sure what you intend to trigger with that comment, so just made some things more explicit. – user2146 Oct 19 '11 at 9:39
In that I don't really care about Nisnevich excision, or problems of what the definition does or does not imply for non-Noetherian schemes. I asked how to extend the definition given by Lurie to non-affine schemes, and if this definition can be condensed. – David Roberts Oct 19 '11 at 21:50
David, the purpose of archiving system of question/answers in MO is not only to satisfy the guy who asked but to have canonical balanced answers to most readers. Nisnevich excision is quite central in this business (hence important for many future users of MO answers) and I find your comments in this line at least superfluous. If you want some other aspect still to be added then say precisely what is still missing from the answers, rather than attacking part of the genuine content which you "do not really care". – Zoran Skoda Oct 21 '11 at 11:17
"the guy who asked", or "the girl who asked". After all, MO is for female mathematicians also. – Matthieu Romagny Jan 3 at 11:11

In his recently posted draft of Spectral Algebraic Geometry, Lurie defines Nisnevich covers for quasi-compact, quasi-separated spectral algebraic spaces as follows.

Consider a family of étale morphisms $\{X_\alpha \to X\}$. They generate a Nisnevich covering if there is a sequence of open immersions $$\emptyset = U_0 \hookrightarrow \ldots \hookrightarrow U_n \simeq X$$ and for $i=1,\ldots,n$, the composite $K_i \hookrightarrow U_i \hookrightarrow X$ factors through some $X_\alpha \to X$, where $K_i \subset U_i$ is the reduced closed substack [DR: why substack and not subspace? I'm not sure] of $U_i$ complementary to $U_{i-1}$.

One can see fairly immediately that the definition reduces to that given for affine ordinary schemes in my question. Again, one can interpret this as saying that the surjective étale map $\coprod_\alpha X_\alpha \to X$ is Nisnevich is there are local sections over some locally closed subspaces $K_i$ arising from a filtration $\emptyset = U_0 \hookrightarrow \ldots \hookrightarrow U_n \simeq X$.

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Not incredibly deep, but for completeness' sake so that a) rather than making people chase down Morel-Voevodsky, they can see it here and b) it does work in vastly more general detail. – David Roberts Jan 3 at 10:27