Homological Algebra and Sheaves

Master 2, Strasbourg, Autumn 2023

• Lecture 1 (11/09) Revision on the notion of modules.
  • Left and right modules
  • Many examples of rings and modules from linear algebra and representation theory
  • Submodule, quotient module, kernel, cokernel, image.
  • First fundemantal theorem for a morphism of modules
  • Direct sums
  • Free modules
  • Generators
  • Rank of a free module is well-defined
  Reference: Lenny Taelman's notes, Chapter 1. Schapira notes, Section 1.2.
  Exercises: Taelman's notes, Exercises to Chapter 1.
• Lecture 2 (15/09) Complexes and exact sequences.
  • Complexe of modules, morphism of complexes
  • Cohomology of a complex, induced morphism (of modules) on cohomology
  • "Functoriality": inducing morphisms on cohomology behaves well with respect to composition of morphisms of complexes
  • Examples of the philosophy of constructing invariants from cohomology of complexes: singular homology and cohomology of topological spaces, group (co)homology, cohomology of Lie algebras, Hochschild cohomology of associative algebras, etc.
  • Exact sequences
  • Short exact sequences of modules, short exact sequences of complexes of modules
  • Theorem: A short exact sequence of complexes induces a long exact sequence of cohomologies.
  • 5-Lemma
  References: Taelman's notes Chapter 2, Chapter 10.
  Exercises: Go over the proof of 5-lemma yourself! Exercises to Taelman's Chapter 2 and Chapter 10.
• Lecture 3 (18/09)
  • Snake lemma
  • Theorem: A short exact sequence of complexes induces a long exact sequence of cohomologies.
  Reference: Taelman's notes Chapter 2. Weibel's book Section 1.3.
  Excercise: go over the proof of the snake lemma yourself. Exercises to Taelman's Chapter 2 and Chapter 10.
  Homework: Proof of Theorem. Deadline 2 October.
  • Definition of categories.
  • Examples of categories: Sets, vector spaces, groups, abelian groups, rings, topological spaces, homotopy category of topological spaces.
  Reference: Taelman's notes Chapter 4, Section 1 and 2. Leinster book Section 1.1.
• Lecture 4 (22/09) Notions of categories
  • More examples of categories: differential Manifolds, Lie groups, Algebraic Varieties, Schemes, Algebraic groups, Lie algebras,
  • (Small) examples of categories: discrete categories, category with 1 object= monoid, category associted to a poset.
  • Even more examples: category of pairs of topological spaces, pointed topological spaces,...
  • Opposite category.
  • Examples from opposite monoid, from opposite poset.
  • Mono and epi
  • For Sets and A-Mod, mono=inj and epi=surj.
  • Caution: In general, mono doesn't imply inj. Example: Q\to Q/Z in the category of divisible groups, is mono.
  • Caution: In general, epi doesn't imply surj. Example: Z\to Q is epi in the category of torsionfree groups; inclusion of a dense subspace in a topological space is epi in the category of Hausdorff topological spaces.
  • Definition of isomorphism
  • For Sets and A-Mod, Isom=bij.
  • Example: In Top, Isom=Homeomorphism
  • Example: In Homotopy category of Top, Isom=(class of)homotopy equivalences
  • Isom is epi and mono.
  • Caution: epi and mono don't imply isom. Example: Z\to Q is epi and mono in Ab; inclusion of a dense subspace in a topological space is epi and mono.
  • Initial and final objects
  • Initial or final object, if exists, is unique up to a unique isomorphism.
  • Initial object is the final object in the opposite category.
  • Examples of initial and final objects: in Sets, A-modules, Top, Homotopy category of Top, Rings, etc.
  Reference: Taleman's notes Chapter 4.
  Exercices: Exercises in Taelman's Chapter 4.
• Lecture 5 (25/09)
  • Definitions: subcategories, full subcategories.
  • Examples of full subcategories: Ab in Gps; Fields in Rings.
  • Examples of non-full subcategories: Top in Sets; Grps in Sets.
  • Product of two categories.
  • Definition: Product and coprodut of two objects in a category.
  • Proposition: If exists, product is unique up to unique isomorphism. Same for coproduct. (Proof by the standard argument using the universal property.)
  • Example: In Sets, product is the cartesian product (with two natural projections), coproduct is the disjiont union (with two natural inclusions).
  • Example: In Top, product is the cartesian product equipped with product topology, coproduct is the disjiont union equipped with the natural topology.
  • Example: In A-Modules, product is given by the direct sum with two naturla projections; coproduct is also given by the direct sum with two natural inclusions.
  • Example: In Gp, product is the direct product, coproduct is the free product.
  • Product might not exist: for example in the category of integral domains.
  • Definition: Functors
  • Examples: identity functor
  • Examples: Forgetful functors: Gps -> Sets; Top -> Sets; Rings -> Ab; A-Mod ->Ab.
  • Examples: Functor of free objects: Sets -> Ab, Sets -> A-Mod, Sets -> Grps
  • Example: for an object X in a category C, we have functor Hom(X,-): C -> Sets.
  • Caution: Product and coproduct are not necessarily preserved by functors! Examples: Forgetful functor from A-Mod to Sets does not respect the coproduct. Forgetful functor from the category of algebraic varieties to Top does not respect the product.
  References and exercises: Taelman notes Chapter 9, 9.1. and Chapter 5.
• Lecture 6 (29/09)
  • Defintion: Contra-variant functors
  • Examples of contravariant functors: Hom(-,X), duality functor between k-Vect to its opposite category, Continuous functions on a topological space.
  • Definition: Morphisms of functors (natural transformations)
  • Examples of morphisms of functors: GL_n to GL_1 as functors from Rings to Groups, \pi_1 to H_1 as functors from Pointed topological spaces to Groups
  • Composition of functors
  • Isomorphism of functors: A morphism \Phi of functors is an isomorphism iff it gives an isomorphims on each object, i.e. \Phi_X is an isomorphis for all X.
  • Definition: Equivalence of categories. (! Compare the more naive notion of isomorphism of categories !)
  • Examples of equivalence of categories: k-Vect and k-Mod, G-Rep and k[G]-Mod, the duality functor between k-Vect^{f.d.} and its opposite category, the category of finite dimensional k-vector spaces is equivalent to the (small) category whose objects are natural numbers, morphisms are matrices and composition is given by multiplication of matrices.
  • Definition: Fullness, Faithfulness, essentially surjectivity of functors.
  • Theorem: A functor is an equivalence of categories if and only if it is full, faithful, and essentially surjective. (Proof next time.)
  References and exercises : Taelman notes Chapter 6. Schapira notes Chapter 1, Section 1.3.
• Lecture 7 (02/10)
  • Proof of Theorem on the criterion for equivalence of categories.
  • Example: Equivalence between the category of finite dimensional vector spaces and the category whose objects are natural numbers, morphisms are matrices.
  • Yoneda functor.
  • Yoneda Lemma, and its dual version.
  • Definition: Representable functors.
  • Definition: Presheaves on a category. Presheaves on a topological space.
  • Example of presheaves: Continuous functions on a topological space; constant presheaf.
  • Definition: Sheaves.
  References and Exercises : Taelman notes Chapter 6. Hartshorne "Algebraic Geometry", Chapter 2, Section 1.
• Lecture 8 (06/10)
  • Notion of sheaves
  • Terminologies: section, restriction.
  • Examples on a differentiable manifold: sheaf of differentiable functions, sheaf of vector fields, sheaf of differential forms, etc
  • Examples on a complex manifold: in addition to the above sheaves on the underlying differentiable manifold, we have the sheaf of holomorphic functions, sheaf of holomorphic vector fields, sheaf of holomorphic forms, etc;
  • Examples on an algebraic variety/scheme (with Zariski topology): sheaf of regular functions, sheaf of algebraic vector fields, sheaf of algebraic differential forms, etc.
  • Example: constant sheaf (given by locally constant functions).
  • Constant sheaf is not the constant presheaf.
  • Definition: stalk of a presheaf at a point.
  • Example of stalks: the stalk of the sheaf of regular functions on an algebraic variety is the local ring at that point. The stalk of the constant presheaf and the stalk of the constant sheaf are the same.
  • Sheafification.
  • Example: Sheafification of the constant presheaef is the constant sheaf.
  Referencs and Exercises: Hartshorne "Algebraic Geometry", Chapter 2, Section 1.
  • Definition: Limit
  • Example: Limit of a functor from the empty category is the final object.
  • Example: Giving a functor from the trivial 1-object category to a category C is the same giving an object in C. The limit of this functor is nothing but this object.
  Referencs and Exercises: Taelman's note: Chapter 9.
• Lecture 9 (09/10)
  • Definition: Limit and colimit
  • By the universal property, limit and colimit, if exist, are unique up to unique isomorphism.
  • Definition: Complete, cocomplete, bicomplete categories.
  • Example 1: Limit (resp. colimit) over empty category is the final (resp. initial) object.
  • Example 2: Limit (resp. colimit) over a discrete category is the product (resp. coproduct)
  • Example 3: Pull-back/fiber product is a limit; push-out is a colimit.
  • Example 4: Equalizer (resp. coequalizer) is a limit (resp. colimit). Their relation with kernal and cokernel in abelian categories.
  • Explicit construction of limit in Sets, Ab, R-Mod: the limit is given by the subset of "compatible systems" in the product, and the colimit is given by the quotient of the coproduct by certain equivalence relation.
  • In Top, product is the cartesian product with product topology; coproduct is the disjoint union with the natural topology; fiber product in Top; push-out in Top is given by gluing.
  References: Taelman notes Chapter 9, Section 1-4. Schapira notes: Section 2.1-2.4.
• Lecture 10 (13/10)
  • Characterization of limits and colimits in terms of the limits and colimits in Sets.
    More precisely, assuming existence of limits/colimits, we have isomorphism of functors: Hom(-, lim(F))=lim Hom(-, F(i))
    Hom(colim F, -)= lim Hom (F, -)
  • Limit commutes with limit; colimit commutes with colimit.
  • In general, limit does NOT commutes with colimit. Example: product and coproduct in Sets.
  • Theorem: Filtered colimit commutes with finite limit in the category Sets (or Ab, R-Mod etc).
  • Definition: Slice category.
  • Proposition ("the Yoneda functor gives a dense embedding"): Any F in the category of presheaves on C is the colimit of representables, indexed by the slice category C/F.
  • Definition: Ajoint functors.
  References: Taelman notes Chapter 9 (and the first bit of Chapter 8). Schapira notes Chapter 2. Homework: Read the proof of the Theorem on filtered colimit and finite limit, in for example, Schapira notes Theorem 2.6.6.
• Lecture 11 (16/10)
  • Adjoint functors.
  • By Yoneda, the left (resp. right) adjoint functor of a given functor, if exists, is unique up to unique isomorphism.
  • Unit, counit for a pair of adoint functors.
  • Property: for (F, G) a pair of adjoint functors, the compositions using unit and counit: F->FGF->F and G->GFG->G are both identity.
  • Left adjoint preserves colimits; right adjoint preserves limits.
  • Remark: conversely, there is a theorem ensuring the existence of right adjoint of a functor F, by checking F commutes with colimits, and that the categories are locally presentable. See the book of J.Adámek and J.Rosicky, "Locally presentable and accessible categories", Theorem 1.66. There is also a similar theorem for the exsitence of left adjoint. In fact, there are several such adjoint functor theorems in different generalities. For some more information, see nLab, Wikipedia.
  • Example: For a cocomplete category C, the functor colim: Fun(I, C)->C is the left adjoint to the constant functor C -> Fun(I, C).
    Dually, for a complete category C, the functor lim: Fun(I, C)-> C is the right adjoint to the constant functor.
  • Example: for any set X, (-xX, -^X) is a pair of adjoint functors from Sets to Sets.
  • Example: (Free, Forget) adjunctions: the followings are left adjoints of the corresponding forgetful functors:
    • Free R-module functor from Sets to R-Mod.
    • Symmetric algebra functor from k-Vect to k-Comm-Alg.
    • Free group functor from Sets to Grps
    • Universal envelopping algebra functor from k-LieAlg to k-Alg.
  • Definition: Tensor product
  • Tensor product exists.
  • By the universal property, it is unique up to unique isomorphism.
  Reference: Taelman notes Chapter 8, Chapter 8 Section 6, Chapter 7.
• Lecture 12 (20/10)
  • Construction of tensor product.
  • Examples of tensor product.
  • Proposition: the functor of tensoring with a module preserves right exact sequences. (We say that this functor is right exact.)
  • Warning: the functor of tensoring with a module does not preserve the injectivity in general.
  • Example: tensoring (over Z) with Z/2 does not preserve the injectivity of the multiplication-by-2 morphism from Z to Z.
  • Definition: a module is called flat if the functor of tensoring with this module is an exact functor, i.e. preserves injectivity.
  • Examples: free modules are flat. Q is a flat Z-module (although not free Z-module). But Z/2 is not a flat Z-module.
  • Theorem (Cartan isomorphism): we have a natural isomorphism: Hom(N\otimes M, L)= Hom(N, Hom(M, L)), which is functorial in M, N, L.
  • In particular, we have the (Tensor, Hom) adjunction: we have a pair of adjoint functors (-\otimes M, Hom(M, -)).
  • Corollary: The functor -\otimes M preserves colimits, in particular cokernel (hence surjectivity), arbitray direct sum; the functor Hom(M,-) preseves limits, in particular kernel (hence injectivity), arbitrary product.
  • The functor Hom(M, -) preserves left exact sequence. But not surjectivity in general.
  References and exercices: Taelman's notes Chapter 7.
• Lecture 13 (23/10)
  • Definition: projective modules, injective modules.
  • A module is projective iff it is a direct summand of a free module.
  • Baer criterion for injective modules.
  • Projective module is flat. The converse is not ture (for example, Q is a non projective flat Z-module). But any finitely presented flat module is projective. In fact, any flat module is a colimit of free modules.
  • Projective (injective) resolution of a module.
  • Example: A is PID, for any non zero element a, we have a free resolution 0-> A -> A -> A/a ->0, where the first arrow is the multiplication-by-a map. More generally, by the structure theorem of finitely generated modules over a PID, any finite generated A-module has a two-term free resolution.
  • Example: k[X,Y]/(X,Y) has a length-2 (= three-term) free resolution.
  • Lemma: any module admits a free resolution.
  • Definition: homotopy between two morphisms of complexes.
  • Definition: homotopy equivalence between two complexes.
  • Exercise: homotopy equivalence between morphisms is compatible with composition. We have hence the homotopy category of complexes of A-modules, denoted by Ho(A-Mod).
  • Proposition: Two homotopic morphisms between two complexes induce the same morphism on cohomology.
  • Corollary: for any integer i, we have a functor of "taking i-th cohomology" H^i: Ho(A-Mod) -> Ab.
  • Proposition: any morphism between two modules can be lifted to a morphism between their projective resolutions; the lifting is unique up to homotopy.
  • Corollary: two projective resolutions of a modules are homotopy equivalent.
  • We can define a functor P: A-Mod -> Ho(A-Mod).
  • Definition: the functor Ext^i(-, M) is defined as the composition of the following three functors:
    • P: A-Mod^{op} -> Ho(A-Mod)^{op}
    • Hom(-,M): Ho(A-Mod)^{op}-> Ho(A-Mod)
    • H^i: Ho(A-Mod) -> Ab
  • By definition, to calculate Ext^i(N, M). There are 3 steps: take a projective resolution P. of N, apply the functor Hom(-, M) to the complex P., then take the i-th cohomology.
  • Example: Ext^0(N, M)=Hom(N, M)
  • Theorem: If we have a short exact sequence of modules 0->N'->N->N''-> 0, then we have a long exact sequence of Ext-functors:
    0-> Hom(N'', M)->Hom(N, M)->Hom(N', M)-> Ext^1(N'', M)->Ext^1(N, M)->Ext^1(N', M)->Ext^2(N'', M)->...
  • The proof uses the so-called horseshoe lemma, and the theorem that a short exact sequence of complexes induces a long exact sequence of cohomology.
  • Remark: Ext^i(N, M) can be equivalently defined by using injective resolutions of M.
  References: Taelman notes: Chapters 11 and 12. For the proof of the Horseshoe lemma, see Weibel's book P.37.
• Some material that I didn't have time to cover, but now it should be easy for you to read (or invent!) by yourselves:
  • Definition of Tor functors.
  • Tor_0(M, N)=M\otimes N.
  • Theorem: If we have a short exact sequence of modules 0->N'->N->N''-> 0, then we have a long exact sequence of Tor-functors:
    ...-> Tor_1(N', M)->Tor_1(N, M)->Tor_1(N'', M) -> N'\otimes M-> N\otimes M->N''\otimes M-> 0
  • Example: Tor_1^A(A/I, A/J) is the quotient of the intersection of I and J by their product IJ.
  Reference: Chapter 3 of Weibel's book.
• Final remark: We worked on A-modules, which is a special case of an abelian category. All I said about projective/injective resolutions, Ext and Tor functors and their properties (suitably defined in order to make sense) can be generalized to an arbitrary abelian category, for example, the category of sheaves of abelian groups on a topological space.

References

• Lecture notes of Pierre Schapira: An Introduction to Categories and Sheaves.
• T. Leinster: Basic Category Theory, Cambridge Studies in Advanced Mathematics, 143
• C. Weibel: An introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics, 38.
• J. Rotman :An Introduction to Homological Algebra, Universitext
• Lecture notes of Lenny Taelman: Modules and Categories. link