arXiv subjects

Luc Illusie

Publications and source records attributed to Luc Illusie.

4 recordsLinked to original sources

arXiv2016

Around the Thom-Sebastiani theorem

For germs of holomorphic functions $f : (\mathbf{C}^{m+1},0) \to (\mathbf{C},0)$, $g : (\mathbf{C}^{n+1},0) \to (\mathbf{C},0)$ having an isolated critical point at 0 with value 0, the classical Thom-Sebastiani theorem describes the vanishing cycles group $Φ^{m+n+1}(f \oplus g)$ (and its monodromy) as a tensor product $Φ^m(f) \otimes Φ^n(g)$, where $(f \oplus g)(x,y) = f(x) + g(y), x = (x_0,...,x_m), y = (y_0,...,y_n)$. We prove algebraic variants and generalizations of this result in étale cohomology over fields of any characteristic, where the tensor product is replaced by a certain local convolution product, as suggested by Deligne. They generalize arXiv:1105.5210. The main ingredient is a Künneth formula for $RΨ$ in the framework of Deligne's theory of nearby cycles over general bases. In the last section, we study the tame case, and the relations between tensor and convolution products, in both global and local situations.

math.AG↗

arXiv2016

Quotient stacks and equivariant étale cohomology algebras: Quillen's theory revisited

Let $k$ be an algebraically closed field. Let $Λ$ be a noetherian commutative ring annihilated by an integer invertible in $k$ and let $\ell$ be a prime number different from the characteristic of $k$. We prove that if $X$ is a separated algebraic space of finite type over $k$ endowed with an action of a $k$-algebraic group $G$, the equivariant étale cohomology algebra $H^*([X/G],Λ)$, where $[X/G]$ is the quotient stack of $X$ by $G$, is finitely generated over $Λ$. Moreover, for coefficients $K \in D^+_c([X/G],\mathbb{F}_{\ell})$ endowed with a commutative multiplicative structure, we establish a structure theorem for $H^*([X/G],K)$, involving fixed points of elementary abelian $\ell$-subgroups of $G$, which is similar to Quillen's theorem in the case $K = \mathbb{F}_{\ell}$. One key ingredient in our proof of the structure theorem is an analysis of specialization of points of the quotient stack. We also discuss variants and generalizations for certain Artin stacks.

math.AG↗

arXiv2012

Travaux de Gabber sur l'uniformisation locale et la cohomologie etale des schemas quasi-excellents. Seminaire a l'Ecole polytechnique 2006--2008

This book contains notes of a seminar on Ofer Gabber's work on the etale cohomology and uniformization of quasi-excellent schemes. His main results include (cf. introduction) constructibility theorems (for abelian or non-abelian coefficients), vanishing theorems (e.g. affine Lefschetz), uniformization for the "prime-to-l alteration topology", rigidity for non-abelian coefficients, a new proof of the absolute purity conjecture, duality, etc.

math.AG↗

arXiv2011

Odds and ends on finite group actions and traces

In this article, we study several problems related to virtual traces for finite group actions on schemes of finite type over an algebraically closed field. We also discuss applications to fixed point sets. Our results generalize previous results obtained by Deligne, Laumon, Serre and others.

math.AG↗