Università degli studi dell'Insubria


A.A. di erogazione 2019/2020

Laurea Magistrale in MATEMATICA
 (A.A. 2019/2020)


Anno di corso: 
Tipologia di insegnamento: 
Settore disciplinare: 
Primo Semestre
Ore di attivita' frontale: 
Dettaglio ore: 
Lezione (64 ore)

The aim of the course is to understand why toposes constitute the natural theoretical framework for defining cohomological invariants, and to learn about the essential properties of cohomology as reformulated in the topos-theoretic setting.


A basic familiarity with the language of category theory would be useful, but is not essential.

Abelian categories and their properties. Complexes and exact sequences. The five lemma and the snake lemma. Homology of a complex and the Mayer-Vietoris long exact sequence. Homotopy of chain complexes and invariance of homology with respect to it. Projective and injective resolutions. Derived categories. Derived functors. Spectral sequences. Grothendieck topologies. Categories of sheaves on a site. The category of internal abelian groups in a topos, and its properties. Cohomology of a topos. Derived functors of direct image, tensor product and hom functors. Čech cohomology (hypercoverings). Torsors.
Examples of cohomologies of toposes coming from Algebraic Geometry. Étale cohomology: definition and basic properties. Computation of the cohomology of curves with constant coefficients. The six operations formalism (Poincaré duality for toposes). The notion of constructible sheaf. Base change for proper morphisms. Base change by smooth morphisms. Künneth formula. Cohomology class of a subscheme; in particular, computation of the cohomology class of the diagonal. The Grothendieck-Lefschetz fixed-point formula.

Theoretical lectures and exercise sessions.

Modalita' di verifica dell'apprendimento: 

Oral examination based on a seminar on a topic extending the contents of the course chosen in agreement with the Lecturers.

M. Artin, A. Grothendieck, J.-L. Verdier, Théorie des topos et cohomologie étale des schémas. Tomes 1 à 3 (SGA4), second edition published as Lecture Notes in Math., vols. 269, 270 and 305 (Springer-Verlag, 1972), reedition available at http://www.normalesup.org/~forgogozo/SGA4/tomes/SGA4.pdf .
A. Grothendieck, Sur quelques points d'algèbre homologique, Tohoku Math. J. Volume 9 (2-3), 1957, 119-221.
J. S. Milne, Étale Cohomology (Princeton University Press, 1980).
J. S. Milne, Lectures on Étale Cohomology, available at https://www.jmilne.org/math/CourseNotes/LEC.pdf .
G. Tamme, Introduction to Etale Cohomology (Springer-Verlag, 1994).
C. Weibel, An Introduction to Homological Algebra (Cambridge University Press, 2008).

Please contact the Lecturers by e-mail for obtaining an appointment.