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\title{Using the internal language of toposes in algebraic geometry}
\author{Ingo Blechschmidt}
\email{ingo.blechschmidt@math.uni-augsburg.de}

\begin{document}

\begin{abstract}
  We describe how the internal language of certain
  toposes, the associated petit and gros Zariski toposes of a scheme, can be
  used to give simpler definitions and more conceptual proofs of the basic
  notions and observations in algebraic geometry.
  % This is useful for studying schemes from a local or relative point of view.

  The 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 intuitionistic linear algebra.

  Furthermore, 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 define certain subtoposes; a
  generalization of the double-negation translation is useful in order to
  understand the internal universe of those subtoposes from the internal point
  of view of the ambient topos.

  A particularly interesting task is to internalize the
  construction of the relative spectrum, which, given a quasicoherent sheaf of algebras
  on a scheme~$X$, yields a scheme over~$X$. From the internal point of
  view, this construction should simply reduce to an intuitionistically sensible
  variant of the 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.
\end{abstract}

\maketitle
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\end{document}

* Krull dimension
* Cartier divisors?
* Box translation
* internal spectrum
* big Zariski topos: criterion for affineness; étale topology