Points in algebraic geometry

Abstract

We give scheme-theoretic descriptions of the category of fibre functors on the categories of sheaves associated to the Zariski, Nisnevich, étale, rh, cdh, ldh, eh, qfh, and h topologies on the category of separated schemes of finite type over a separated noetherian base. Combined with a theorem of Deligne on the existence of enough points, this provides an algebro-geometric description of a conservative family of fibre functors on these categories of sheaves. As an example of an application we show that direct image along a closed immersion is exact for all these topologies except qfh. The methods are transportable to other categories of sheaves as well.1 The second is valid in many other situations which the reader can work out according to their needs. For example, the étale, qfh, Nisnevich, Zariski, etc topologies admit various "small" sites, and when C is taken to be such a small site instead of Sch/S, we obtain the full subcategory of the category of (C, τ )-local schemes consisting of those which are obtainable as inverse limits of pro-objects of C. 2 By abuse of notation, by f (P ) we mean lim − − → (P \Sch/S) f (X), where (P \Sch/S) is the category of factorizations P → X → S with X ∈ Sch/S.

Motivic and real étale stable homotopy theory

“…Let be a point of . Then corresponds to a pro-object in , which is to say that there is a filtered family such that for we have [GK15, Proposition 1.4 and Remark 1.5].…”

K-theory of valuation rings

“…Let be a Noetherian scheme of finite Krull dimension. A conservative family of points for the cdh site is given by the spectra of Henselian valuation rings [GK15], and hence Theorem 3.4(iii) implies that for , where denotes the sheafification of the abelian presheaf on . In fact, this follows from the earlier result of Goodwillie and Lichtenbaum [GL01] that a conservative family of points for the rh site is given by the spectra of valuation rings.…”

Grothendieck topologies on posets

“…One example is the category C = Comm op fp , the opposite category of the category of finitely presented commutative rings. In [4], very explicit descriptions are given for the points of various Grothendieck topologies. 1 As a demonstration of why this is useful, they point out an application to sheaf cohomology [4,Proposition 4.5].…”

“…In [4], very explicit descriptions are given for the points of various Grothendieck topologies. 1 As a demonstration of why this is useful, they point out an application to sheaf cohomology [4,Proposition 4.5]. For this category Comm op fp , it is already an open problem to give a complete…”

“…description for the points of the flat topology (some partial results are in [4,Lemma 3.3] and in [19]).…”