Annals of K-Theory

On derived categories of arithmetic toric varieties

Matthew Ballard, Alexander Duncan, and Patrick McFaddin

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

derived categories exceptional collections Galois descent toric varieties


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.

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