Motives of Algebraic Stacks

Abstract

The study of motives of algebriac stacks was initiated by Bertrand To\"{e}n for Deligne-Mumford stack, and subsequently, developed and extended by various authors to more general alegbraic stacks. There exists considerable literature on motives of algebraic stacks. However, there are often two limitations to the treatments found in literature: firstly, that they almost exclusively work with the \'{e}tale topology as opposed to the Nisnevich topology; and secondly, that they primarily focus on Deligne-Mumford stacks or quotient stacks.

In this thesis, we further study the theory of motives for algebraic stacks and attempt to improve upon the above shortcomings. The main thrust of this thesis is to show that a reasonable theory of motivic cohomology for algebraic stacks exists in the Nisnevich topology.

We show that the construction of the \'{e}tale motive due to Utsav Choudhury can also carried out in the Nisnevich topology. This gives us a canonical definition of the motive of an algebraic stack which has many functorial properties.

Using this definition, we show that for stacks that are Nisnevich locally global quotient stacks many of the standard exact triangle of motivic cohomology of schemes, viz., projective bundle formula, Meyer-Vietoris triangle, Gysin triangle, and blow-up triangle, continue to hold. We also observe that \'{e}tale sheafification of this motive recovers the \'{e}tale motive usually found in literature.

Further, we show that for quotients of smooth schemes by linear algebraic groups, the Edidin-Graham-Totaro (higher) Chow groups and the motivic cohomology groups agree integrally. This was earlier known only with rational coefficients by the work of Roy Joshua.