Cdh descent, cdarc descent, and Milnor excision
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Mathematische Annalen
Cdh descent, cdarc descent, and Milnor excision Elden Elmanto1 · Marc Hoyois2 · Ryomei Iwasa3 · Shane Kelly4 Received: 12 March 2020 / Revised: 9 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We give necessary and sufficient conditions for a cdh sheaf to satisfy Milnor excision, following ideas of Bhatt and Mathew. Along the way, we show that the cdh ∞topos of a quasi-compact quasi-separated scheme of finite valuative dimension is hypercomplete, extending a theorem of Voevodsky to nonnoetherian schemes. As an application, we show that if E is a motivic spectrum over a field k which is n-torsion for some n invertible in k, then the cohomology theory on k-schemes defined by E satisfies Milnor excision.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Complements on the cdh topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The cdp, rh, and cdh topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Riemann–Zariski spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Valuative dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The homotopy dimension of the cdh ∞-topos . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Milnor excision and cdarc descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The cdarc topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Milnor squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Criteria for Milnor excision and cdarc descent . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Application to motivic spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Communicated by Vasudevan Srinivas.
B
Elden Elmanto [email protected] https://www.eldenelmanto.com/
Extended author information available on the last page of the article
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E. Elmanto et al.
1 Introduction A Milnor square is a cartesian square of rings of the form A
B
A/I
B/J.
This class of squares was introduced by Milnor in [28, Sect. 2]. In modern language, he proved that such a square induces a cartesian square of categories Proj(A)
Proj(B)
Proj(A/I)
Proj(B/J),
where Proj(A) denotes the category of finitely generated projective left A-modules. We will say that a functor of rings satisfies Milnor excision if it sends Milnor squares to cartesian squares. Thus, Proj(−) satisfies Milnor excision. A closely related invariant that satisfies Milnor excision is Weibel’s homotopy K-theory spectrum KH(−) [42, Theorem 2.1]. Algebraic K-theory itself does not satisfy Milnor excisi
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