Meta
- CiteKey: “CWZ19”
- Type: report
- Author: “Campbell, Jonathan; Wolfson, Jesse; Zakharevich, Inna;”\
- Publisher: “arXiv,”
- Year: 2019
- Zotero Tags: “14F43, 11S40”; “Mathematics - Algebraic Geometry”; “Mathematics - Algebraic Topology”; “Mathematics - K-Theory and Homology”; “Mathematics - Number Theory”
- Original URL: http://arxiv.org/abs/1703.09855
- Open in Zotero: Zotero
Abstract
We lift the classical Hasse–Weil zeta function of varieties over a finite field to a map of spectra with domain the Grothendieck spectrum of varieties constructed by Campbell and Zakharevich. We use this map to prove that the Grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map \(\mathbb{S} \to K(Var_k)\) induced by the inclusion of \(0\)-dimensional varieties is not surjective on \(\pi_1\) for a wide range of fields \(k\). The methods used in this paper should generalize to lifting other motivic measures to maps of \(K\)-theory spectra.
Extracted Annotations
Annotations(6/9/2022, 2:42:42 PM)
File:CWZ19_76MDLUY4.png (Campbell et al., 2019, p. 1)
File:CWZ19_748ZMDKN.png (Campbell et al., 2019, p. 1)
File:CWZ19_PJRIRRKW.png (Campbell et al., 2019, p. 2)
File:CWZ19_RW23D4W3.png (Campbell et al., 2019, p. 2)
File:CWZ19_WVB7BGVK.png (Campbell et al., 2019, p. 2)
File:CWZ19_ZCKKHRWR.png (Campbell et al., 2019, p. 3)
File:CWZ19_XZYNFK2M.png (Campbell et al., 2019, p. 4)
File:CWZ19_ST8NWVUG.png (Campbell et al., 2019, p. 4)
File:CWZ19_ZL9SH858.png (Campbell et al., 2019, p. 4)
File:CWZ19_DNGJZQS8.png (Campbell et al., 2019, p. 5)
File:CWZ19_JULHLVJS.png (Campbell et al., 2019, p. 5)
File:CWZ19_8ILR9YRS.png (Campbell et al., 2019, p. 6)
File:CWZ19_9UZKZVTI.png (Campbell et al., 2019, p. 6)
File:CWZ19_MI7Y4W9P.png (Campbell et al., 2019, p. 7)
File:CWZ19_ZF4V9VV9.png (Campbell et al., 2019, p. 7)
File:CWZ19_LRPHJ2D2.png (Campbell et al., 2019, p. 7)
File:CWZ19_GJLU5D59.png (Campbell et al., 2019, p. 8)
File:CWZ19_AIVL5UIA.png (Campbell et al., 2019, p. 8)
File:CWZ19_XKET4VVS.png (Campbell et al., 2019, p. 8)
File:CWZ19_ELMTGC3Y.png (Campbell et al., 2019, p. 8)
- Most important example of a Waldhausen category for this paper.
File:CWZ19_TN424ICE.png (Campbell et al., 2019, p. 9)
File:CWZ19_JJAYWW5H.png (Campbell et al., 2019, p. 9)
File:CWZ19_CEZPQDGT.png (Campbell et al., 2019, p. 9)
File:CWZ19_9UZZ2EN3.png (Campbell et al., 2019, p. 10)
File:CWZ19_A22SEYAS.png (Campbell et al., 2019, p. 10)
File:CWZ19_CBPY9SQW.png (Campbell et al., 2019, p. 10)
File:CWZ19_WMD7YQ47.png (Campbell et al., 2019, p. 10)
File:CWZ19_Y5DAXYUX.png (Campbell et al., 2019, p. 11)
File:CWZ19_KYERY8FY.png (Campbell et al., 2019, p. 11)
File:CWZ19_ZVN6U2AJ.png (Campbell et al., 2019, p. 12)
File:CWZ19_L8UUTCET.png (Campbell et al., 2019, p. 14)
File:CWZ19_R8MTLLL9.png (Campbell et al., 2019, p. 16)
File:CWZ19_ITK93V3G.png (Campbell et al., 2019, p. 17)
File:CWZ19_B4XBL9L7.png (Campbell et al., 2019, p. 17)
File:CWZ19_5YAPLHMY.png (Campbell et al., 2019, p. 17)
File:CWZ19_5EL3ENCD.png (Campbell et al., 2019, p. 17)
File:CWZ19_R2EQTYSZ.png (Campbell et al., 2019, p. 18)
File:CWZ19_ZUX3SCDW.png (Campbell et al., 2019, p. 18)
File:CWZ19_DWL3RXHA.png (Campbell et al., 2019, p. 18)
File:CWZ19_5Q3R8IKH.png (Campbell et al., 2019, p. 18)
File:CWZ19_TLVY5Y4M.png (Campbell et al., 2019, p. 19)
File:CWZ19_VMPRCGPM.png (Campbell et al., 2019, p. 19)
File:CWZ19_H3A38XA4.png (Campbell et al., 2019, p. 24)
File:CWZ19_3FDUC7LE.png (Campbell et al., 2019, p. 25)
File:CWZ19_YYC6CTCF.png (Campbell et al., 2019, p. 29)
File:CWZ19_V2BZL9RD.png (Campbell et al., 2019, p. 29)
File:CWZ19_DY4S6BVY.png (Campbell et al., 2019, p. 30)
File:CWZ19_TIBADXKJ.png (Campbell et al., 2019, p. 30)
File:CWZ19_4UVGXUMQ.png (Campbell et al., 2019, p. 31)
- Main result
File:CWZ19_5CPP9LRW.png (Campbell et al., 2019, p. 31)
File:CWZ19_EF7KGDBC.png (Campbell et al., 2019, p. 31)
File:CWZ19_NCAG38BI.png (Campbell et al., 2019, p. 32)
File:CWZ19_HJSGSGKU.png (Campbell et al., 2019, p. 33)
File:CWZ19_FTDGDU88.png (Campbell et al., 2019, p. 34)
File:CWZ19_79DLHM69.png (Campbell et al., 2019, p. 34)
File:CWZ19_7GJH8RN8.png (Campbell et al., 2019, p. 34)
File:CWZ19_F5G3S88D.png (Campbell et al., 2019, p. 35)
File:CWZ19_BDU2MIQ3.png (Campbell et al., 2019, p. 35)
File:CWZ19_IMLYC76N.png (Campbell et al., 2019, p. 35)
