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SUMMARY:Federico Binda
DTSTART:20210525T160000Z
DTEND:20210525T170000Z
DTSTAMP:20260423T035731Z
UID:eAKTS/21
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/eAKTS/21/">G
 AGA type conjecture for the Brauer group via derived geometry</a>\nby Fede
 rico Binda as part of electronic Algebraic K-theory Seminar\n\nLecture hel
 d in 979 0634 7355.\n\nAbstract\nIn Brauer III\, Grothendieck considered t
 he problem of comparing the cohomological Brauer group $Br(X) = H^2_{et}(X
 \,G_m)$ of a scheme $X$\, proper and flat over a henselian DVR $R$\, and t
 he inverse limit of the Brauer groups $\\lim_nBr(X_n)$\, where $X_n = X\\o
 times_R R/m^n$. He proved that the canonical map $Br(X) \\to \\lim_n Br(X_
 n)$ is injective under a number of restrictions\, and left as an open prob
 lem the question on whether the formal injectivity holds in a fairly gener
 al setting.\nThanks to the machinery of derived algebraic geometry and the
  results of To\\"en on derived Azumaya algebras and derived Morita theory\
 , we are able to rephrase Grothendieck’s question in terms of a formal G
 AGA-type problem for smooth and proper categories\, enriched over the $\\i
 nfty$-category $QCoh(X)$ of quasi-coherent $O_X$-modules. In this framewor
 k we can show that Grothendieck’s injectivity conjecture always holds fo
 r a proper derived scheme $X \\to S$ where S is the spectrum of any comple
 te Noetherian local ring\, if we are willing to replace the inverse limit 
 $\\lim_n Br(X_n)$ with the Brauer group $Br(X)$ of the formal scheme $\\ma
 thfrak{X}$ given by the colimit of the thickenings $X_n$. The obstruction 
 involving the inverse system $Pic(X_n)$ considered by Grothendieck appears
  naturally in the Milnor sequence for a certain tower of spaces. This is a
  joint work in progress with Mauro Porta (IRMA\, Strasbourg).\n
LOCATION:https://researchseminars.org/talk/eAKTS/21/
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