Student seminar


The goal of the seminar is to come to an understanding of the proof by Deligne and Illusie of the degeneration of the Hodge-to-de Rham spectral sequence. The setup is as follows: let $k$ be a field, $X$ a smooth and proper scheme over $k$, then one can compute algebraic de Rham cohomology on one hand, and sheaf cohomology of the sheaves $\Omega_{X/k}^p$ on the other. These are related by means of a spectral sequence \begin{equation} \mathrm{E}_1^{p,q}=\mathrm{H}^q(X,\Omega_{X/k}^p)\Rightarrow\mathrm{H}_{\mathrm{dR}}^{p+q}(X/k). \end{equation}

When $k=\mathbb{C}$ one can use techniques from complex analytic geometry, such as the Poincaré lemma and Hodge theory. It took until 1987 to find a purely algebraic proof. This proof is very interesting, because it combines techniques from characteristic $p$, deformation theory, algebraic geometry and derived categories to come to an important result.


February 25
Introductory lecture and organisation of the seminar
by Pieter Belmans, in G.016, at 13h00
March 4
Differentials and smoothness (notes)
by Dennis Presotto, in G.016, at 13h00
March 11
Additional properties of smoothness (notes)
by Julia Ramos Gonzalez, in G.016, at 13h00
March 25
Lifting smooth morphisms
Pieter Belmans, in G.016, at 13h00
April 1
Frobenius morphisms and Cartier isomorphism
by Theo Raedschelders, in G.016, at 13h00
A short primer on non-commutative Hodge-to-de Rham degeneration
by Pieter Belmans, in G.016, at 15h00
April 29
Degeneration in positive characteristic (notes)
Jens Hemelaer, in G.016, at 13h00
May 6
Limits of schemes (notes)
Sebastian Klein, in G.016, at 13h00
May 13
From characteristic zero to characteristic $p$
Liran Shaul, in G.016, at 13h00


In due time I might add these.