Homological algebra and sheaves

• Introduction to the second half of the course

• Definition of Tor groups.

• Proposition : an A-module M is flat if and only if the functors Tor^A_i(M,-) vanish for every i > 0.

• Freyd's theorem : a category admitting too large coproducts is a poset.

• Regular cardinals, \kappa-filtered and \kappa-compact objects.

• Generators for a category, presentable categories, different characterizations.

• Adjoint functor theorem.

• Example : sheaves are presentable.

References

• Weibel, Introduction to Homological Algebra, §§3.1-3.2.

• Adamek, Rosicky, Locally Presentable and Accessible Categories (LNM series). Especially §0, §1.A, §1.B.

Some incomplete notes.

A set of exercises.

• Additive categories.

• Examples (Ab, Mod_R, Rep_G, Rep_g) and non-examples (Set, Grp, CRing).

• Additive functors.

• Abelian categories and their basic properties.

• Definition of image and coimage. Theorem : in an abelian category, for every f : X -> Y, Im(f) = CoIm(f).

References.

• Weibel, Introduction to Homological Algebra, §§A.4.

• Borceaux, Handbook of Category Theory, Vol. II, §§1.1-1.6.

Some incomplete notes.

• Reflective subcategories. Examples. Theorem : if D is reflective in C and C is (co)complete, then the same goes for D. If C is abelian and the left adjoint C -> D to the inclusion commutes with finite limits, then D is abelian.

• Grothendieck's axioms AB3, AB4, AB5 and the notion of Grothendieck abelian category.

• Theorem: if A is a Grothendieck abelian category, then for every topological space X, Sh(X;A) is a Grothendieck abelian category as well.

• Definition of pushforward and pullback for sheaves.

References

• Borceaux, Handbook of categorical algebra, Volume I, §3.5.

• Grothendieck, Some aspects of homological algebra (Tohoku paper). Available here, §§1.6-1.10.

• Stacks Project, Tag 008C.

Some incomplete notes.

• Characterization of the inverse image for sheaves.

• Gabriel-Popescu embedding theorem for Grothendieck abelian categories.

• Definitions: having enough injectives and having enough projectives.

• An object in a Grothendieck abelian category is injective if and only if it has the lifting property against subobjects of the generator.

• Theorem: a Grothendieck abelian category always has enough injectives.

References

• Stacks Project, Tag 05AB

Some incomplete notes.

A set of exercises.

• The notion of cohomological and homological \delta-functor. Examples (Tor and Exts). Universality. Uniqueness of universal \delta-functors.

• Let A be an abelian category with enough injectives. Then an object I is injective if and only if every short exact sequence 0 -> I -> M -> N -> 0 splits after application of any additive functor.

• Let A be an abelian category with enough injectives and let F : A -> B be a left exact functor. Construction of the right derived \delta-functor {R^i F, \delta^i}.

• Theorem : right derived \delta-functors are universal.

• Notion of acyclic object. Example (flat objects for - \otimes_A M).

• Theorem : acyclic resolutions can be used to compute derived \delta-functors.

References

• Grothendieck, Some aspects of homological algebra. Available here, §§2.1-2.3.

• Weibel, Introduction to homological algebra, §§2.1-2.5.

Some incomplete notes.

• Characterization of constant sheaves

• Definition of sheaf cohomology

• Sheaf cohomology of a point

• Notion of flasque sheaf

• Theorem : every injective sheaf is flasque, and flasque sheaves are acyclic.

References

• Iversen, Cohomology of sheaves

• Schapira, An Introduction to Categories and Sheaves

• Concerning flasque sheaves : see Stacks Project, Tag 09SV

Warning. Notice however that I didn't follow any of the above references, and that the order the material is treated can change dramatically from one presentation to the other. For instance, Schapira's notes treat first derived categories, and then sheaf cohomology, while in this course I decided to do the opposite. This means that he is allowed to use some machinery that we still didn't develop at this point of the course. Nevertheless, they are excellent notes, and I encourage all of you to go through them.

Some incomplete notes.

• Functoriality of sheaf cohomology

• Homotopy invariance : statement and easy parts of the proof

• Mayer-Vietoris sequence

• Computation of the cohomology of spheres S^n

• Derived functor of inverse limit, lim^1.

References

Same as for the previous lecture, with the same warning.

For Mayer Vietoris, see Stacks Project, Tag O1E9.

Some incomplete notes.

• Excision sequence.

• Sheaf cohomology of the complex projective space.

• Soft sheaves and the sheaf-de Rham comparison theorem.

References

Same as for the previous lectures, with the same warning.

For the excision and the de Rham cohomology of the complex space, see also §6.2 in these notes by Arapura.

For the cellular decomposition of the complex projective space, see for instance

• Hatcher, Algebraic Topology, Examples 0.4 and 0.6.

Some incomplete notes.

• Problèmes d'existences de fonctions méromorphes sur C : (i) avec parties principales données ; (ii) avec diviseur des pôles et des zéros donné.

• Faisceau des fonctions méromorphes sur les variétés complexes.

• Distributions additives de Cousin. Exemples. Reformulation faisceautique et lien avec l'annulation H^1(X;O_X) = 0.

• Notion de faisceau cohérent, énoncé du théorème de cohérence de Oka (cas des variétés complexes), théorème B de Cartan et définition de variété Stein. Différentes caractérisations et exemples (sans preuves).

• Distributions multiplicatives de Cousin. Exemples. Reformulation faisceautique et lien avec l'annulation H^1(X,O_X^*) Suite exacte exponentielle et lien avec les annulations H^1(X,O_X) = H^2(X,Z) = 0.

• Complexe de Cech associé à un recouvrement et définition de cohomologie de Cech.

References

• Grauert, Remmert, Theory of Stein spaces, §V.2.

• Grauert, Remmert, Coherent analytic sheaves, §2.1 - 2.2, §2.5.

Some incomplete notes.