Annals of K-Theory

On derived categories of arithmetic toric varieties

Matthew Ballard, Alexander Duncan, and Patrick McFaddin

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Abstract

We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections, making it possible to give concrete descriptions of their derived categories. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskiĭ and Klyachko, and toric varieties associated to Weyl fans of type A. Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly nontoric) varieties over nonclosed fields.

Article information

Source
Ann. K-Theory, Volume 4, Number 2 (2019), 211-242.

Dates
Received: 13 April 2018
Revised: 3 January 2019
Accepted: 18 January 2019
First available in Project Euclid: 13 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.akt/1565661792

Digital Object Identifier
doi:10.2140/akt.2019.4.211

Mathematical Reviews number (MathSciNet)
MR3990785

Zentralblatt MATH identifier
07102033

Subjects
Primary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 14M25: Toric varieties, Newton polyhedra [See also 52B20]
Secondary: 14G27: Other nonalgebraically closed ground fields 19E08: $K$-theory of schemes [See also 14C35]

Keywords
derived categories exceptional collections Galois descent toric varieties

Citation

Ballard, Matthew; Duncan, Alexander; McFaddin, Patrick. On derived categories of arithmetic toric varieties. Ann. K-Theory 4 (2019), no. 2, 211--242. doi:10.2140/akt.2019.4.211. https://projecteuclid.org/euclid.akt/1565661792


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