Student seminar

## Goal

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.

## Schedule

- 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

## References

In due time I might add these.

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