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# 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?