# Adeel A. Khan

Home Papers

2019

Absolute purity for the rational motivic sphere spectrum.
Joint with Frédéric Déglise, Jean Fasel, and Fangzhou Jin.
Preprint, 12 pages, February 2019.

A gluing result for weight structures along semi-orthogonal decompositions, which is applied to give some new examples of regular ring spectra.
Joint with Vladimir Sosnilo.
Preprint, 12 pages, January 2019.

2018

Proves topological invariance in SH[1/p].
Joint with Elden Elmanto.
Preprint, 10 pages, last updated January 2019 (changelog).

Proves projective bundle and blow-up formulas in K-theory for derived stacks.
Also gives a direct new proof of cdh descent for homotopy invariant K-theory.
This version extends the cdh descent criterion to stacks.
Preprint, 18 pages, last updated November 2018 (changelog).

Relates the theory of motivic fundamental classes with the recognition principle for infinite **P**^{1}-loop spaces.
Joint with Elden Elmanto, Marc Hoyois, Vladimir Sosnilo, and Maria Yakerson.
Preprint, 38 pages, September 2018.

Intersection theory à la Fulton, in the motivic homotopy category.
Joint with Frédéric Déglise and Fangzhou Jin.
Preprint, 43 pages, last updated December 2018 (changelog).

A complete description of the functor of points of a blow-up in regularly immersed centre, using the language of derived algebraic geometry.
Joint with David Rydh.
Preprint, 19 pages, last updated May 2019 (changelog).

2017

Develops a motivic version of the theory of (grouplike) E_{∞}-spaces.
Joint work with Elden Elmanto, Marc Hoyois, Vladimir Sosnilo, and Maria Yakerson.
Preprint, 77 pages, last updated September 2018.

A revised version of an older preprint called Brave new motivic homotopy theory II.
Compares **A**^{1}-homotopy theory over a spectral scheme with classical **A**^{1}-homotopy theory over the underlying classical scheme, and deduces some consequences for the homotopy invariant K-theory of commutative ring spectra.
Joint work with Denis-Charles Cisinski, in preparation.

2016

An analogue of Kashiwara's lemma for **A**^{1}-homotopy invariant Nisnevich sheaves over spectral algebraic spaces.
This is a rewritten version of an older preprint called Brave new motivic homotopy theory I.
27 pages, last updated April 2019 (changelog).

My Ph.D. thesis (2016).
The foundational results of motivic homotopy theory are generalized to the setting of derived algebraic geometry.
Notably, the formalism of Grothendieck’s six operations is extended to this setting.
122 pages, minor update in September 2018.

2014