Mathematics Notes

I have tried to make the notes as readable as possible, but no doubt there are some idiosyncracies. Many notes are heavily cross-referenced with other notes. I use an acronym system, where a reference of the form (MRS, Proposition 6) refers to my “Modules over Ringed Spaces” notes, for example. I have tried to document the relevant acronyms here, but there may be references to notes that I haven’t published online.

References

It goes without saying that most of the results in my notes are from one book or another. I’ve tried to list some of the main sources below, together with my shorthand for each book.

  • Weibel: Weibel’s “An Introduction to Homological Algebra”.
  • H & S: Hilton & Stammbach’s “A Course in Homological Algebra”.
  • A & M: Atiyah & Macdonald’s “Introduction to Commutative Algebra”.
  • Z & S: Zariski and Samuel’s books on commutative algebra.
  • EFT: My Elementary Field Theory notes, based on Z & S’s chapter on field theory.
  • Mitchell: B. Mitchell’s “Category Theory”.
  • H or Hartshorne: Hartshorne’s “Algebraic Geometry”.

Corrections / Updates

  • The definition of a perfect pairing in TES was not correct. This has now been fixed (thanks for Laurent Busé for noticing this one).

Algebraic Geometry

  • Sheaves of Groups and Rings: (SGR) Sheaves of sets (incomplete), sheaves of abelian groups, stalks, sheaf Hom, tensor products, inverse and direct image, extension by zero.
  • Modules over a Ringed Space: (MRS) Inverse and direct image, tensor products, ideals, locally free sheaves, exponential tensor products, sheaf Hom, coinverse image and extension by zero. Sheaves of graded modules over sheaves of graded rings, quasi-structures, modules over schemes, sheaves of algebras and sheaves of graded algebras (Quite rough in places, I’m in the process of typing written notes).
  • Sheaves of Algebras: (SOA) Direct and inverse image for algebras, modules over sheaves of algebras, ideals, generating algebras, tensor products. Sheaves of graded algebras, their modules, and generating graded algebras. Sheaves of super algebras and tensor products.
  • Special Sheaves of Algebras: (SSA) Sheaves of tensor algebras, symmetric algebras, exterior algebras, polynomial algebras and ideal products. Complete study of these constructions as adjoints.
  • Modules over a Scheme: (MOS) Ideals, special functors (extension by zero and coextension of scalars), locally free sheaves, sheaf Hom, extension of coherent sheaves.
  • The Proj Construction: (TPC) Functorial properties, products, linear morphisms, projective morphisms, dimensions of some schemes, points of projective space.
  • Modules over Projective Schemes: (MPS) Properties of the functor associating a graded module with a quasi-coherent sheaf on Proj.
  • An Adjunction for Modules over Projective Schemes: (AAMPS) The adjoint for taking the associated sheaf of a graded module, the quasi-coherent case.
  • An Equivalence for Modules over Projective Schemes: (AEMPS) The projective version of some important theorems in the affine case.
  • Relative Affine Schemes: (RAS) Affine morphisms, the Spec construction, the sheaf associated to a sheaf of quasi-coherent modules over an algebra.
  • The Segre Embedding: (SEM) Pullback of Proj schemes, properties of projective morphisms.
  • Concentrated Schemes: (CON) Basic properties of quasi-compact and quasi-separated schemes and morphisms. In our notes a quasi-compact quasi-separated morphism (or scheme) is called concentrated.
  • Section 2.6 – Divisors: (DIV) Weil divisors, divisors on curves, cartier divisors, invertible sheaves.
  • Section 2.7 – Projective Morphisms: (PM) Morphisms to Pn, the duple embedding, ample invertible sheaves, linear systems.
  • Section 2.7.1 – Blowing Up: (BU) Definition of the blow-up, blowing up of varieties.
  • Section 2.8 – Differentials: (DIFF) Kahler differentials, sheaves of differentials, nonsingular varieties, rational maps, applications, some local algebra.
  • Section 2.9 – Formal Schemes: (FS) Inverse limits, completion, adic rings (complete rings of fractions, local completion), affine formal schemes (this note is not yet complete).
  • Section 3.2 – Cohomology of Sheaves: (COS) Definition of cohomology, the module structure and the presheaf of cohomology. A vanishing theorem of grothendieck, cohomology of noetherian schemes, Cech cohomology, the cohomology of projective space, Ext groups and sheaves.
  • Section 3.8 – Higher Direct Images of Sheaves: (HDIS) Definition of higher direct image functors, module structure and properties for schemes. Definition of the higher coinverse image functors, and their properties. Direct image and quasi-coherent sheaves, uniqueness of cohomology.
  • Section 3.7 – Serre Duality: (SDT) Notes on Serre Duality and dualising sheaves as presented in Hartshorne.
  • The Relative Proj Construction: (TRPC) Associating a Proj with a sheaf of graded algebras. The sheaf associated to a sheaf of graded modules, the graded module associated to a quasi-coherent sheaf, functorial properties, ideal sheaves and closed subschemes, the duple embedding, twisting with invertible sheaves.
  • Ample Families: (AMF) Ample sheaves and ample families of sheaves on arbitrary schemes, as described in EGA and later SGA.

Commutative Algebra

  • Matsumura: (MAT) General rings, flatness, depth, Cohen-Macaulay rings, normal and regular rings, koszul complexes, unique factorisation.
  • Matsumura Part II: (MAT2) Extension of a ring by a module, derivations and differentials, separability.
  • Noether Normalisation: First introduction to various versions of Noether normalisation.
  • More Noether Normalisation: A version of noether normalisation involving separability.
  • Hensel’s Lemma: Hensel’s Lemma and a few small examples.
  • Cohen’s Theorem.
  • Graded Rings and Modules: (GRM) Definitions and basic properties, the category of graded modules, quasi-structures, grading tensor products.
  • Tensor, Exterior and Symmetric algebras: (TES) The tensor algebra and properties, exterior algebra and properties, including: dimension theorems, the determinant formula (i.e. highest exterior powers), and duality properties, the symmetric algebra and properties.
  • Automorphisms of Power Series Rings: (APSR) Constructing automorphisms of power series rings from an independent family of power series.

Category Theory and Noncommutative Algebra

  • Foundations for Category Theory: (FCT) Outline of the problem of foundations in category theory, first order theories, NBG and associated problems, review of ZFC and grothendieck universes. This forms the logical background for all my notes.
  • Abelian Categories: (AC) Definition of categories, limits and colimits, functor categories, pointwise limits and colimits, adjoint functors, abelian categories, grothendieck abelian categories and reflective subcategories (mostly just to fix notation. These notes are not a complete reference on category theory).
  • Diagram Chasing in Abelian Categories: (DCAC) Proving the Five Lemma in an abelian category, using an embedding to establish diagram chasing in arbitrary abelian categories (we use only the “first” embedding theorem into the category of abelian groups, since its proof is more accessible, and you probably have to do the same amount of work to avoid well-definedness issues for the connecting morphism even with the better embeddings).
  • Derived Functors: (DF) (co)chain complexes in an abelian category, (co)homology, projective and injective resolutions, left and right derived functors of additive functors between abelian categories, long exact (co)homology sequences, long exact sequences of derived functors, dimension shifting and acyclic resolutions, change of base, homology and colimits, cohomology and limits, delta functors.
  • Triangulated Categories Part I: (TRC) [Verdier quotients] Triangulated categories, triangulated functors, homotopy colimits, localising subcategories, right derived functors, left derived functors, portly considerations.
  • Triangulated Categories Part II: (TRC2) [Thomason localisation] Finer localising subcategories, perfect classes, small objects, compact objects, portly considerations, morphisms in the quotient.
  • Triangulated Categories Part III: (TRC3) [Brown representability] Representability theorems, compactly generated triangulated categories.
  • Derived Categories Part I: (DTC) Homotopy categories, derived categories (extending functors, introduction to hearts, bounded derived categories), homotopy resolutions, homotopy direct limits, bousfield subcategories, existence of resolutions.
  • Derived Categories Part II: (DTC2) Derived functors, derived Hom, derived Tensor, brown representability.
  • Derived Categories Of Sheaves: (DCOS) Representing cohomology, derived direct image, derived sheaf Hom, derived Tensor, derived inverse image.
  • Derived Categories Of Quasi-coherent Sheaves: (DCOQS) Derived direct image, Derived inverse image, Sheaves with quasi-coherent cohomology, local cohomology triangle, resolutions by Cech sheaves and the Cech triangles, Neeman’s unbounded Grothendieck duality, comparing the derived category of quasi-coherent sheaves with the derived category of sheaves with quasi-coherent cohomology, quasi-coherent hypercohomology, perfect complexes and compactness, projection formula and friends.
  • Ext: (EXT) Ext in general abelian categories, using injectives and projectives and balancing the two, Ext for linear categories, dimension shifting, Ext and coproducts, Ext for commutative rings, another characterisation of derived functors.
  • Tor: (TOR) Tor on the left and right and balancing the two, dimension shifting, Tor and colimits, Tor for commutative rings and bimodules, criteria for flatness.
  • Dimensions: (DIM) Projective dimension, injective dimension, global dimension, flat dimension and change of rings.
  • Spectral Sequences: (SS) Definition, basic convergence properties, the spectral sequence of a complex filtration, the two spectral sequences of a bicomplex, the Grothendieck spectral sequence.
  • Algebra in a Category: (ALCAT) Groups, rings and modules in an arbitrary category. Sheaves of groups, rings and modules and graded versions.
  • Linearised Categories: (LC) Generalise the group ring construction to the linearisation of any small category with respect to a sheaf of rings, the graded version of this construction. Includes proof that the category of graded sheaves of modules is grothendieck abelian.

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