Homological Algebra II [Lecture notes]

Homological Algebra (TCC 2016)

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Download Homological Algebra Ii Lecture Notes

Harvard undergraduate theses an excellent collection of the mathematics undergraduate theses completed in the last few years at Harvard. Bas Edixhoven this is a list of notes from talks that Edixhoven has given over the years. Algebraic Number Theory-Holden. The Theory of Witt Vectors-Rabinoff.

Useful Link

Complex Analytic and Differential Geometry-Demailly. Uniformization Theorem-Chan. Analytic Vector Bundles-Andrew These notes are truly amazing. Complex Manifolds-Koppensteiner. Kahler Geometry-Speyer. Differential Topology-Dundas. Spaces and Questions-Gromov. Introduction to Cobordism-Weston. Groups Acting on the Circle-Ghys.

Elliptic Curves

Lie Groups-Ban comes with accompanying lecture videos. Very Basic Lie Theory-Howe. Geometric Group Theory-Bowditch. Categories and Homological Algebra-Schapira. Representation Theory-Etingof. Commutative Algebra-Haines. Geometric Commutative Algebra-Arrondo. Examples in Category Theory-Calugereanu and Purdea. Foundations of Algebraic Geometry-Vakil. Analytic Techniques in Algebraic Geometry-Demailly. Algebraic Geometry-Gathmann Contributed by Mohan. Galois Theory for Schemes-Lenstra.

Rational Points on Varieties-Poonen. Teaching Schemes-Mazur. If we're going to mention Hatcher famous to me for the algebraic topology notes , we might as well also mention a few other books that are online, like Algebra chapter 0 , Stanley's insane first volume of Enumerative Combinatorics which reminds me: generatingfunctionology. Also I don't see topology without tears mentioned. The sheer number of books and notes on differential geometry and lie theory is mind-boggling, so I'll have to update later with the juicier ones.

Let's not forget the AMS notes online back through - they're very nice reading as well. James Binney Oxford - Great quantum mechanics videos and a free accompanying downloadable book. Not strictly speaking "math," but plenty of it. Gathmann's notes on algebraic geometry. I think they are one of the best places from where one can start learning algebraic geometry. Lecture notes on abstract algebra, galois theory and introductory algebraic number theory by Frederique Oggier.

For Spanish speaking users, I can link to Carlos Ivorra's website.

He has the following material:. Zev Chonoles, a graduate student at the University of Chicago and sometime poster here, has several wonderful sets of lecture notes of the first year graduate courses at U of Chicago. They're terrific and strongly recommended. Kowalski of ETH Zurich in Germany has some very good,substantial notes at his webpage on analysis, representation theory and graph theory research.

Worth a look. Alex, you'll be happy to know I've compiled a vast list of lecture notes,with my commentary on each. This list contains just about all the notes at this thread and many more.

Lecture Notes

It will be part of a fully functional website affiliated with my blog, "Tables, Chairs and Beermugs" and will attempt to catalog and comment on all currently available sources for mathematics students under 30 dollars U. I'll post a link to the site here when the first version is ready and online. In a way,I'm kind of upset that you posted this question and the links beating me to it.

Oh well, I'm putting mine up anyway. Notes taken by Richard Garner on a course by Eugenia Cheng. They don't cover exactly the same topics.

They are quite different: the first one is more to-the-point, with less details but with some great explanations and insights. It covers a bit more material, if I'm not mistaken.

For example, it has proofs for both adjoint functor theorems, and they are very nice, especially the one for the GAFT, in my opinion. In this field, one of my contributions is a characterisation for n-coherent rings using closure properties and constructing hereditary cotorsion pairs which involve the classes of modules of type FPn, the FPn-injective modules and the FPn-flat modules. These concepts are in turn generalisations of finitely presented, injective and flat modules, respectively.

I have also used these classes of modules in the process of obtaining new model structures on the category of chain complexes over a ring. One important aspect on this matter is looking for conditions under which a ring homomorphism yields Quillen adjunctions between these structures. Motivated by some properties satisfied by Gorenstein projective and Gorenstein injective modules over an Iwanaga-Gorenstein ring, we present the concept of left and right n -cotorsion pairs in an abelian category C.

This concept and its dual generalise the notion of complete cotorsion pairs, and has an appealing relation with left and right approximations, especially with those having the so called unique mapping property. The main purpose of this paper is to describe several properties of n -cotorsion pairs and to establish a relation with complete cotorsion pairs. We also give some applications in relative homological algebra, that will cover the study of approximations associated to Gorenstein projective, Gorenstein injective and Gorenstein flat modules and chain complexes, as well as m -cluster tilting subcategories.

A recent result by J. In this paper, we generalise this result to a certain relativisation of Gorenstein flat modules, which we call Gorenstein B -flat modules, where B is a class of right R -modules.

Reviews of Michael Atiyah's books

This will allow us to show approximation properties concerning these modules, and also to obtain a relative version of the model structure described before. Moreover, we also present and prove our results in the category of complexes of left R-modules, study other model structures on complexes constructed from relative Gorenstein flat modules, and compare these models via computing their homotopy categories.

A natural source to get such pairs is by means of cotorsion triplets. In this paper we study the connection between balanced pairs and cotorsion triplets by using recent quiver representation techniques. In doing so, we find a new characterization of abelian categories having enough projectives and injectives in terms of the existence of complete hereditary cotorsion triplets. We show that some characterizations analogous to that of injective, FP-injective and flat complexes exist for FPn-injective and FPn-flat complexes.

We also introduce and study FPn-injective and FPn-flat dimensions of modules and complexes, and give a relation between them in terms of Pontrjagin duality. Finally, we construct model structures on the category of complexes from the classes of modules with bounded FPn-injective and FPn-flat dimensions, and analyze several conditions under which it is possible to connect these model structures via Quillen functors and Quillen equivalences. We revisit Auslander—Buchweitz approximation theory and find some relations with cotorsion pairs and model category structures. We also establish correspondences between these model structures, relative cotorsion pairs, Frobenius pairs, and Auslander—Buchweitz contexts.

Some applications of this theory are given in the context of Gorenstein homological algebra, and connections with perfect cotorsion pairs, covering subcategories and cotilting modules are also presented and described. We study the interplay between the notions of n -coherent rings and finitely n -presented modules, and also study the relative homological algebra associated to them. We show that the n -coherency of a ring is equivalent to the thickness of the class of finitely n -presented modules.

The relative homological algebra part comes from the study of orthogonal complements to this class of modules with respect to the Ext and Tor functors. We also construct cotorsion pairs from these orthogonal complements, allowing us to provide further characterizations of n -coherent rings. We construct Abelian model structures on the category of chain complexes over a ring R , from the notion of homological dimensions of modules.

Using this result, we prove that there is a unique Abelian model structure on the category of chain complexes over R , where the exact complexes are the trivial objects and the complexes with projective dimension at most n form the class of trivially cofibrant objects. In a previous work by D.

Bravo et. We extend this result by finding a new Abelian model structure with the same trivial objects where the cofibrant objects are given by the class of complexes whose terms are modules with projective dimension at most n.

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