BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Uriya First (University of Haifa) DTSTART:20200428T203000Z DTEND:20200428T213000Z DTSTAMP:20260423T124906Z UID:MITNT/3 DESCRIPTION:Title: Ge neration of algebras and versality of torsors\nby Uriya First (Univers ity of Haifa) as part of MIT number theory seminar\n\n\nAbstract\nThe prim itive element theorem states that every finite separable field\nextension L/K is generated by a single element. An almost equally known\nfolklore fa ct states that every central simple algebra over a field can be\ngenerated by 2-elements.\n\nI will discuss two recent works with Zinovy Reichstein (one is forthcoming)\nwhere we establish global analogues of these results . In more detail\, over\na ring R (or a scheme X)\, separable field extens ions and central simple\nalgebras globalize to finite etale algebras and A zumaya algebras\,\nrespectively. We show that if R is of finite type over an infinite field K\nand has Krull dimension d\, then every finite etale R -algebra is generated\nby d+1 elements and every Azumaya R-algebra of degr ee n is generated by\n2+floor(d/[n-1]) elements. The case d=0 recovers the well-known facts\nstated above. Recent works of B. Williams\, A.K. Shukla and M. Ojanguren\nshow that these bounds are tight in the etale case and suggest that they\nshould also be tight in the Azumaya case.\n\nThe proof makes use of principal homogeneous G-bundles T-->X (G is an\naffine algebr aic group over K) which can specialize to any principal\nhomogeneous G-bun dle over an affine K-variety of dimension at most d. In\nparticular\, such G-bundles exist for all G and d.\n LOCATION:https://researchseminars.org/talk/MITNT/3/ END:VEVENT END:VCALENDAR