Unramified cohomology, cohomological invariants and root stacks

Abstract: Given a smooth algebraic variety X/k, its (Bloch-Ogus-Rost) unramified cohomology H_nr(X,M) with coefficients in a cycle module M is the subgroup of M(k(X)) given by the elements whose residue at each codimension one point is zero. When X is proper, this group is a birational invariant and it being non-trivial disproves stable rationality. It was first used in this way, in the form of the unramified Brauer group, by Artin⁠–Mumford and Saltman, and then in higher degrees by Colliot-Thélène⁠–Ojanguren. Given an algebraic group G/k, the cohomological invariants Inv(G,M) are the natural transformations between the functor of G-torsors over fields and the cycle module M. There are many examples dating back up to the beginning of the 20th century, but they were introduced in the present form by Serre. A result by Totaro shows that given a G-representation V such that the subset U where G acts freely has complement of codimension 2 or more, we have Inv(G,M)=H_nr(U/G,M). A few years ago, I reinterpreted the idea of cohomological invariants as invariants of the classifying stack BG, extended them to invariants of general algebraic stacks, and showed that on schemes we have Inv(X,M) = H_nr(X,M) and in fact they can be seen as the "only possible" extension of Bloch-Ogus-Rost unramified cohomology to algebraic stacks. Moreover, cohomological invariants can be used to compute Brauer groups, which I and Andrea di Lorenzo recently did for the moduli stacks of smooth Hyperelliptic curves. Unfortunately, it's easy to see that even for smooth, projective Deligne Mumford stacks cohomological invariants are not a birational invariant. One way to see this is that while any birational map between smooth proper schemes is given, at least in char(k)=0, by a sequence of blow-ups and blow-downs, which leave cohomological invariants unchanged, for DM stacks we have to add root stacks, which can modify cohomological invariants rather drastically. I will describe recent work with Andrea Di Lorenzo in which we find a formula to describe the cohomological invariants of a root stack and use it to show that different natural compactifications of the moduli stacks of Hyperelliptic curves, while being seemingly almost identical, have vastly different cohomological invariants.

algebraic geometrygroup theorynumber theoryrepresentation theory

Audience: researchers in the topic

Algebraic groups and algebraic geometry: in honor of Zinovy Reichstein's 60th birthday