BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Roberto Pirisi (La Sapienza Università di Roma) DTSTART:20210630T150000Z DTEND:20210630T155000Z DTSTAMP:20260413T061457Z UID:zoomnovy/9 DESCRIPTION:Title: Unramified cohomology\, cohomological invariants and root stacks\nby Roberto Pirisi (La Sapienza Università di Roma) as part of Algebraic grou ps and algebraic geometry: in honor of Zinovy Reichstein's 60th birthday\n \n\nAbstract\nGiven 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 codim ension one point is zero. When X is proper\, this group is a birational in variant and it being non-trivial disproves stable rationality. It was firs t used in this way\, in the form of the unramified Brauer group\, by Artin ⁠–Mumford and Saltman\, and then in higher degrees by Colliot-Thélèn e⁠–Ojanguren. \nGiven an algebraic group G/k\, the cohomological invar iants Inv(G\,M) are the natural transformations between the functor of G-t orsors 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-repres entation V such that the subset U where G acts freely has complement of co dimension 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 cl assifying stack BG\, extended them to invariants of general algebraic stac ks\, 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 unra mified cohomology to algebraic stacks. Moreover\, cohomological invariants can be used to compute Brauer groups\, which I and Andrea di Lorenzo rece ntly did for the moduli stacks of smooth Hyperelliptic curves.\nUnfortunat ely\, it's easy to see that even for smooth\, projective Deligne Mumford s tacks 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 t o add root stacks\, which can modify cohomological invariants rather drast ically. I will describe recent work with Andrea Di Lorenzo in which we fin d a formula to describe the cohomological invariants of a root stack and u se it to show that different natural compactifications of the moduli stack s of Hyperelliptic curves\, while being seemingly almost identical\, have vastly different cohomological invariants.\n LOCATION:https://researchseminars.org/talk/zoomnovy/9/ END:VEVENT END:VCALENDAR