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CATEGORIES:Category Theory Seminar
SUMMARY:Using the internal language of toposes in algebrai
c geometry - Ingo Blechschmidt (University of Augs
burg)
DTSTART;TZID=Europe/London:20160524T141500
DTEND;TZID=Europe/London:20160524T151500
UID:TALK66318AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/66318
DESCRIPTION: We describe how the internal language of certain
toposes\, the associated little and big Zariski to
poses of a scheme\, can be used to give simpler de
finitions and more conceptual proofs of the basic
notions and observations in algebraic geometry.\n\
nThe starting point is that\, from the internal po
int of view\, sheaves of rings and sheaves of modu
les look just like plain rings and plain modules.
In this way\, some concepts and statements of sche
me theory can be reduced to concepts and statement
s of intuitionistic linear algebra.\n\nFurthermore
\, modal operators can be used to model phrases su
ch as ``on a dense open subset it holds that'' or
``on an open neighbourhood of a given point it hol
ds that''. These operators define certain subtopos
es\; a generalization of the double-negation trans
lation is useful in order to understand the intern
al universe of those subtoposes from the internal
point of view of the ambient topos.\n\nA particula
rly interesting task is to find an internal charac
terization for a sheaf of algebras to be quasicohe
rent\, related to an observation by Mulvey which T
ierney called "somewhat obscure"\, and to internal
ly construct the relative spectrum\, which\, given
a quasicoherent sheaf of algebras on a scheme X\,
yields a scheme over X. From the internal point o
f view\, this construction should simply reduce to
an intuitionistically sensible variant of the ord
inary construction of the spectrum of a ring\, but
it turns out that this expectation is too naive a
nd that a refined approach is necessary.\n\nWe als
o discuss how the little Zariski topos can be desc
ribed using the internal language of the big Zaris
ki topos\, and vice versa\; here too there is a sm
all surprise.
LOCATION:MR5\, Centre for Mathematical Sciences
CONTACT:Tamara von Glehn
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