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SUMMARY:Using the internal language of toposes in algebraic geometry - Ing
 o Blechschmidt (University of Augsburg)
DTSTART:20160524T131500Z
DTEND:20160524T141500Z
UID:TALK66318@talks.cam.ac.uk
CONTACT:Tamara von Glehn
DESCRIPTION: We describe how the internal language of certain toposes\, th
 e associated little and big Zariski toposes of a scheme\, can be used to g
 ive simpler definitions and more conceptual proofs of the basic notions an
 d observations in algebraic geometry.\n\nThe starting point is that\, from
  the internal point of view\, sheaves of rings and sheaves of modules look
  just like plain rings and plain modules. In this way\, some concepts and 
 statements of scheme theory can be reduced to concepts and statements of i
 ntuitionistic linear algebra.\n\nFurthermore\, modal operators can be used
  to model phrases such as ``on a dense open subset it holds that'' or ``on
  an open neighbourhood of a given point it holds that''. These operators d
 efine certain subtoposes\; a generalization of the double-negation transla
 tion is useful in order to understand the internal universe of those subto
 poses from the internal point of view of the ambient topos.\n\nA particula
 rly interesting task is to find an internal characterization for a sheaf o
 f algebras to be quasicoherent\, related to an observation by Mulvey which
  Tierney called "somewhat obscure"\, and to internally construct the relat
 ive spectrum\, which\, given a quasicoherent sheaf of algebras on a scheme
  X\, yields a scheme over X. From the internal point of view\, this constr
 uction should simply reduce to an intuitionistically sensible variant of t
 he ordinary construction of the spectrum of a ring\, but it turns out that
  this expectation is too naive and that a refined approach is necessary.\n
 \nWe also discuss how the little Zariski topos can be described using the 
 internal language of the big Zariski topos\, and vice versa\; here too the
 re is a small surprise.
LOCATION:MR5\, Centre for Mathematical Sciences
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