abstract-colloquium-logicum2018.txt

New reduction techniques in commutative algebra driven by logical methods

Commutative algebra abounds with techniques to reduce given situations to
simpler ones, for instance passing to a quotient or passing to a
localization. These techniques switch a given ring with another.

But there are also reduction techniques driven by logical methods, whose effects
cannot be mimicked by classical techniques. They arise from certain topological
models and switch a given ring not with a further ring, but with a suitable
forcing model. For instance, there is a way of forcing any ring to be a local
ring.

These forcing models make an interesting trade-off: On the one hand, they enjoy
better properties than the original ring while being simultaneously sufficiently
close to the original ring in order to be useful, but on the other hand, they
only validate intuitionistic logic, not classical logic.

They were already discovered in the 1970s, but their applicability to
commutative algebra was only properly recognized and studied in the 2000s, under
the umbrella term *dynamical methods in algebra*. The talk presents this
topic from the point of view of new results [1,2]: It's possible to force any
reduced ring to be a field, thereby importing any intuitionistic theorem about
fields into the realm of reduced rings. This technique has been used to give an
almost trivial and even constructive proof of *Grothendieck's generic freeness
lemma*, an important theorem in algebraic geometry, which substantially
improved on the longer, somewhat convoluted and unconstructive previously-known
proofs.

[1] I. Blechschmidt. An elementary and constructive proof of Grothendieck's
generic freeness lemma. 2018. Preprint available at
https://rawgit.com/iblech/internal-methods/master/paper-generic-freeness.pdf.

[2] I. Blechschmidt. Using the internal language of toposes in algebraic
geometry. PhD thesis, 2018. Available at
https://rawgit.com/iblech/internal-methods/master/notes.pdf.