Holmstrom Nisnevich topology

May useful things in Morel-Voevodsky: A1-homotopy theory of schemes, see for example start of chapter 3 for a list of useful properties comparing it to Zariski and etale. Section 3.2.3 contains results on “gluing, homotopy purity, and the blowup square” which require the topology to be (at least as strong as?) the Nisnevich top.

Voevodsky’s Nordfjordeid lecture - supernice (Voevodsky folder). Has a serious appendix on the Nisnevich topology, Nisnevich descent, and model structures.

http://mathoverflow.net/questions/78431/nisnevich-topology-on-non-locally-noetherian-schemes

---

Memo notes from Jardine: Generalized etale… pp 282

Let $X$ be a locally Noetherian scheme. The Nisnevich site (or cd-site, completely decomposed site) of $X$ are defined as follows. Objects are separated etale morphisms $U \to X$ of finite type (i.e. as in the separated etale site). Morphisms are $X$-scheme morphisms. A collection $\{ U_i \to U \}$ of morphisms is a covering family if for every point $y \in U$, the map $Spec(k(y)) \to U$ lifts to some $U_i$.

If $X = Spec(k)$ for a field $k$, then the above objects are finite disjoint unions of Spec of finite separable extensions of $k$. A family $\{ U_i \to Spec(L) \}$ is a covering family iff at least one of the $U_i$ has a component equal to $Spec(L)$. Hence there is only one covering sieve for each finite separable extension $L/k$, and the sheaf condition is simply that of “additivity” for disjoint unions of fields. More details on the case of a field.

For $x$ a point of $X$, have the morphism $i_x: Spec(k(x)) \to X$. Composition with the induced functor $et|_X \to et|_{k(x)}$ defines a direct image functor $(i_x)_*: PreShv(Nis|_{k(x)}) \to PreShv(Nis|_{X})$ which preserves sheaves. It has a left adjoint $i_x^*$ defined by left Kan extension. Some consequences of this.

If a presheaf $F$ on $Nis|_X$ is the restriction of a contravariant functor defined on a sufficiently large category of schemes, and if $F$ is continuous, then there is an isomorphism $i_x^*(k(y)) = F(\mathcal{O}_y^h)$.

Lemma: Suppose that $F$ is an additive presheaf on $Nis|_X$. Then $i_x^* F$ is an additive presheaf on $Nis|_{k(x)}$. (Additive means: taking disjoint unions to products). Also, if $F$ is a sheaf, so is $i_x^* F$.

The above gives us two different points of view on stalks in the Nisnevich topology. Spelling out of these things, and conditions for maps of sheaves to be IMs (skipped here). In many cases, maps of presheaves arise from natural transformations which are continuous and “globally defined”. Then the induced map of associated sheaves is an isomorphism “iff it is so on henselizations of local rings”. Example: Presheaves of stable homotopy groups arising from K-theory presheaves of spectra.

Def: Nisnevich excision property, for a presheaf of spectra on the Nisnevich site of a scheme. This condition says that (1) $F( \emptyset)$ is contractible and (2) For each map $\phi: U \to V$ in $Nis|_X$ and a closed subscheme $Z \subset V$ such that $\phi$ induces an isomorphism $Z \times_V U \to Z$, the obvious commutative diagram of spectra involving $F(V)$, $F(V-Z)$, $F(U)$ and $F(\phi^{-1}(V-Z))$ is homotopy cartesian.

Examples: If a Noetherian scheme is a also separated and regular, then the K-theory presheaf of spectra on the Nisnevich site has this property (need hyps to invoke Quillen’s localization thm). Smashing with a constant presheaf of spectra preserves the Nisnevich excision property. The Nisnevich excision property is stable under pointwise stable equivalence.

Lemma: Roughly: Given a presheaf $F$ of spectra satisfying Nis excision, there is a pointwise weak equivalence from this presheaf to a presheaf $G_{Zar} F$ for which “open subset inclusions induces stable fibrations”.

Let $F$ be as in the lemma. Then $F$ satisfies Zariski excision, so $F$ is additive up to stable equivalence, meaning that the corresponding presheaves of stable homotopy groups are additive.

For a closed subset $Z$ of $U$, where $U \to X$ is in $Nis|_X$, we define a presheaf $\Gamma_Z F$ as the fibre of $F(U) \to F(U-Z)$. Various properties and applications of this (several pages).

Thm (Kato-Saito): Let $X$ be a Noetherian scheme of Krull dimension $d$, and let $F$ be a sheaf of abelian groups on the Nisnevich site of $X$. Then $H^i_{Nis}(X,F) = 0$ for $i > d$.

Thms of this kind “forces descent spectral sequences to converge”.

An important idea in the above seems to be to prove that under suitable hyps, any choice of globally fibrant model is a pointwise stable equivalence. Actually, this seems to be true in general for $X$ of finite Krull dimension and $F$ satisfying Nisnevich excision. This is the Nisnevich descent theorem, essentially, and can also be formulated in terms of a “Godement resolution” for $F$. Details on this, including a construction $F \to Tot \Pi^* G^* F$. A theorem saying that this is a pointwise stable equivalence, under some hyps.

Corollary: Roughly: Smash the K-theory presheaf with a constant presheaf. A globally fibrant model for this for the Nisnevich topology is a pointwise stable equivalence.

nLab page on Nisnevich topology