TY - JOUR

T1 - Strong generators in (Formula Presented)

AU - Neeman, Amnon

N1 - Publisher Copyright:
© 2021. Department of Mathematics, Princeton University.

PY - 2021/5

Y1 - 2021/5

N2 - We solve two open problems: first we prove a conjecture of Bondal and Van den Bergh, showing that the category Dperf (X) is strongly generated whenever X is a quasicompact, separated scheme, admitting a cover by open alone subsets Spec(Ri) with each Ri of finite global dimension. We also prove that, for a noetherian scheme X of finite type over an excellent scheme of dimension ≤ 2, the derived category (Formula Presented) is strongly generated. The known results in this direction all assumed equal characteristic; we have no such restriction. The method is interesting in other contexts: our key lemmas turn out to give a simple proof that, (Formula Presented) is a separated morphism of quasi-compact, quasiseparated schemes such that (Formula Presented) takes perfect complexes to complexes of bounded-below Tor-amplitude, then f must be of finite Tor-dimension.

AB - We solve two open problems: first we prove a conjecture of Bondal and Van den Bergh, showing that the category Dperf (X) is strongly generated whenever X is a quasicompact, separated scheme, admitting a cover by open alone subsets Spec(Ri) with each Ri of finite global dimension. We also prove that, for a noetherian scheme X of finite type over an excellent scheme of dimension ≤ 2, the derived category (Formula Presented) is strongly generated. The known results in this direction all assumed equal characteristic; we have no such restriction. The method is interesting in other contexts: our key lemmas turn out to give a simple proof that, (Formula Presented) is a separated morphism of quasi-compact, quasiseparated schemes such that (Formula Presented) takes perfect complexes to complexes of bounded-below Tor-amplitude, then f must be of finite Tor-dimension.

KW - Compact generators

KW - Derived categories

KW - Schemes

UR - http://www.scopus.com/inward/record.url?scp=85129867751&partnerID=8YFLogxK

U2 - 10.4007/annals.2021.193.3.1

DO - 10.4007/annals.2021.193.3.1

M3 - Article

SN - 0003-486X

VL - 193

SP - 689

EP - 732

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 3

ER -