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Κατεύθυνση Θεωρητικά Μαθηματικά

Library of the School of Science

Library of the School of Science

2021-05-02

2021

Karakikes Miltiadis

Αριστείδης Κοντογεώργης, Καθηγητής, Τμήμα Μαθηματικών, Σχολή Θετικών Επιστημών του Εθνικού και Καποδιστριακού Πανεπιστημίου Αθηνών

Introduction to Derived Categories

English

Introduction to Derived Categories

The theory involving derived functors on abelian categories is a fundamental concenpt of homological algebra that has a lot of applications on modern mathematics, especially on algebraic geometry, and even on theoretical physics. Derived functors, being an important tool, it was necessary to take the next step and extend our theory, introducing ourselves to derived categories, which simplify a lot of homological algebra. Instead of looking at the objects of a category, we study their chain complexes which are equipped with a stronger concept of equivalence (that of quasi-isomorphisms). All this has its roots in [Tohoku], the work of Alexander Grothendieck and, his student, Jean-Louis Verdier, which provided us with essential tools to avoid using the, more complex, spectral sequences.

The motto - idea, one will find in every paper concerning the derived categories, is “Complexes Good, Homology of Complexes Bad”. As stated above, we needed chain complexes as a natural invariant because they have all the information we want about the homotopy of a space - homology holds little information about that - thus the motto. With this, quasi-isomorphisms, which are isomorphisms of the induced maps on homology, come in to play a crucial role into the construction of derived categories, identifying the isomorphic complexes in our new category, which is the stronger equivalence relation we needed. Namely, homotopic complexes are also quasi-isomorphic.

The main goal of this thesis is to prove Grothendieck’s Spectral Sequence Theorem 5.2.2 which computes the derived functors of the composition of two functors, by knowing the derived functors of each functor. The Leray spectral sequence 5.5.8 and the Lyndon-Hochschild-Serre spectral sequence are just a couple out of the many special cases of Grothendieck’s spectral sequence.

We will see two proofs of this result. The first time using some basic knowledge about spectral sequences and hypercohomology and the second time, a more direct - simplified proof, using derived category language. It wouldn’t be false to assume that the derived category has provided us with a simpler way of doing calculations which are rather complicated if done using the spectral sequence formula.

More precisely, in Chapter 1 we are reminded of some basic definitions and structures such as (co)homology, homotopy, exact triangles and triangulated categories, thus setting the stage for the following chapters. In Chapter 2 we construct the left derived functors (and dually the right ones), we show that they are well defined and equip ourselves with the needed propositions. In Chapter 3 we take a quick glance into the spectral sequences world, understanding how they are defined, how they converge as well as seeing some examples of already known results (like the Snake Lemma) being easily proven using our new tool. Next, we localize the homotopy category of an abelian category with respect to quasi-isomorphisms, thus obtaining the Derived Category in Chapter 4, where we also explain how this works (object, morphism, composition - wise). Finally, in Chapter 5, we have all the tools we need to prove Grothendieck’s Spectral Sequence Theorem, using both spectral sequences and derived categories, the later making the proof as easy as a corollary. As a direct application of this, we take a quick look at Leray’s Spectral Sequence.

The motto - idea, one will find in every paper concerning the derived categories, is “Complexes Good, Homology of Complexes Bad”. As stated above, we needed chain complexes as a natural invariant because they have all the information we want about the homotopy of a space - homology holds little information about that - thus the motto. With this, quasi-isomorphisms, which are isomorphisms of the induced maps on homology, come in to play a crucial role into the construction of derived categories, identifying the isomorphic complexes in our new category, which is the stronger equivalence relation we needed. Namely, homotopic complexes are also quasi-isomorphic.

The main goal of this thesis is to prove Grothendieck’s Spectral Sequence Theorem 5.2.2 which computes the derived functors of the composition of two functors, by knowing the derived functors of each functor. The Leray spectral sequence 5.5.8 and the Lyndon-Hochschild-Serre spectral sequence are just a couple out of the many special cases of Grothendieck’s spectral sequence.

We will see two proofs of this result. The first time using some basic knowledge about spectral sequences and hypercohomology and the second time, a more direct - simplified proof, using derived category language. It wouldn’t be false to assume that the derived category has provided us with a simpler way of doing calculations which are rather complicated if done using the spectral sequence formula.

More precisely, in Chapter 1 we are reminded of some basic definitions and structures such as (co)homology, homotopy, exact triangles and triangulated categories, thus setting the stage for the following chapters. In Chapter 2 we construct the left derived functors (and dually the right ones), we show that they are well defined and equip ourselves with the needed propositions. In Chapter 3 we take a quick glance into the spectral sequences world, understanding how they are defined, how they converge as well as seeing some examples of already known results (like the Snake Lemma) being easily proven using our new tool. Next, we localize the homotopy category of an abelian category with respect to quasi-isomorphisms, thus obtaining the Derived Category in Chapter 4, where we also explain how this works (object, morphism, composition - wise). Finally, in Chapter 5, we have all the tools we need to prove Grothendieck’s Spectral Sequence Theorem, using both spectral sequences and derived categories, the later making the proof as easy as a corollary. As a direct application of this, we take a quick look at Leray’s Spectral Sequence.

Science

Derived Categories, Derived Functors, Composition of Derived Functors, Category Theory, Grothendieck, Alexander Grothendieck, Verdier, Tohoku, Triangulated Categories, Spectral Sequences, Right Derived Functors, Left Derived Functors, Total Derived Functors, Hyper Derived Functors, Left Hyper Derived Functors, Right Hyper Derived Functors, Hypercohomology, Grothendieck's Spectral Sequence Theorem, Grothendieck's Spectral Sequence, Leray Spectral Sequence

Yes

2

No

8

85