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The Albert-Brauer-Hasse-Noether Theorem and Global Dimension of Crossed Products

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The Albert-Brauer-Hasse-Noether Theorem and Global Dimension of Crossed Products Aristotle University of Thessaloniki Faculty of Natural Sciences School of Mathematics The Albert-Brauer-Hasse-Noether Theorem and Global Dimension of Crossed Products Palaisti, Marina A thesis presented to The Department of Mathematics in partial fulfillment of the requirements for the degree of Master in Science in the subject of Pure Mathematics. Thessaloniki, 2014 ii Aristotle University of Thessaloniki Faculty of Natural Sciences School of Mathematics The Albert-Brauer-Hasse-Noether Theorem and Global Dimension of Crossed Products Supervisor: Author: Prof. Palaisti, Marina Theohari-Apostolidi, Theodora Committee Members: Prof. Theohari-Apostolidi, Theodora Prof. Haralambous, Hara Prof. Tzouvaras, Athanasios Thessaloniki, 2014 Παλαιστή Μαρίνα, Πτυχιούχος Μαθηµατικός Α.Π.Θ. Copyright ©Παλαιστή Μαρίνα Με επιφύλαξη παντός δικαιώµατος. All rights reserved Απαγορεύεται η αντιγραφή, αποθήκευση και διανοµή της παρούσης ερ- γασίας, εξ΄ ολοκλήρου ή τµήµατος αυτής, για εµπορικό σκοπό. Επιτρέπεται η ανατύπωση, αποθήκευση και διανοµή για σκοπό µη κερδοσκοπικό, εκ- παιδευτικής ή ερευνητικής ϕύσης, υπό την προϋπόθεση να αναφέρεται η πηγή προέλευσης και να διατηρείται το παρόν µήνυµα. Ερωτήµατα που αφορούν τη χρήση της εργασίας για κερδοσκοπικό σκοπό πρέπει να απευθύνονται στη συγγραφέα. Οι απόψεις και τα συµπεράσµατα που περιέχονται σε αυτό το έγγραφο εκφράζουν τη συγγραφέα και δεν πρέπει να ερµηνευθεί ότι εκφράζουν το Α.Π.Θ. iii iv "... he wrote to me that algebraic number theory was the most beautiful topic he had ever come across and that the sole consolation in misery was his lecturing on class field theory... This was indeed the kind of mathematics he had admired most: the main results are of great scope, of great aesthetic beauty, but the proofs are technically extremely hard." A.Borel about Harish-Chandra, 1995. vi Introduction This thesis is concerned with the Albert-Brauer-Hasse-Noether Theorem, and also with global dimensions of crossed products, which are very im- portant not only in the theory of structure of algebras, but also in algebraic number theory. The Albert-Brauer-Hasse-Noether Theorem was established in Novem- ber 9, 1931, by Abraham Adrian Albert(1905-1972), Richard Brauer(1901- 1977), Helmut Hasse(1898-1979) and Amalie Emmy Noether(1882-1935). It was published in 1932, and dedicated to Kurt Hensel(1861-1941), on the occasion of his 70th birthday. From the survey article of P. Roquette [Roq] we get the following historical elements of the collaboration of Albert, Brauer, Hasse and Noether, to prove their celebrating theorem. Brauer became interested in class field theory, because he believed that its results would be important when applied to representation theory of groups. Hasse became interested in the theory of algebras because he had noticed that class field theory and the local p-adic theory could be used there profitably. Finally, Noether, who brought Brauer and Hasse toghether, was motivated by her belief that the non-commutative theory of algebras should be used for a better understanding of class field theory. At that time the results of Dickson(1874-1954) and his disciples were noted with lots of interest by the mathematicians around Noether. There- fore, during the collaboration between Brauer, Hasse and Noether, Hasse wrote a letter to Dickson, describing his work in it, and asking about the existence of non-cyclic division algebras of index 4 over a number field. Dickson forwarded it to a disciple of him, Albert. The last replied to Hasse that he was very interested in it, and introduced himself, so they started communicating and exchanging ideas about the steps of the proof that were to handle. Albert had already in 1931 developed different tools from the other three, in order to approach a part of this theorem. Thus, in vii viii INTRODUCTION their paper, Brauer, Hasse and Noether inserted a footnote, aknowledging Albert and giving him "an independent share of the proof". Later, in 1932, in a joint paper between Albert and Hasse, the theorem had been proved again, using Albert’s methods this time. The Albert-Brauer-Hasse-Noether Theorem is one of the most profound results in the theory of central simple algebras. The paper starts with the following sentence. "At last our joint endeavours have finally been successful, to prove the fol- lowing theorem, which is of fundamental importance for the structure theory of algebras, and also beyond..." The Albert-Brauer-Hasse-Noether Theorem is a very critical step in the view of algebraic number theory. It allows a complete classification of division algebras over a number field, by means of Hasse invariants. Therefore, one can determine the Brauer group of an algebraic number field, and also can describe the splitting fields of a division algebra, by describing their local behavior, which is extremely useful in representation theory. Fur- thermore, the Albert-Brauer-Hasse-Noether Theorem opened new paths, giving the ability to understand class field theory via the structure of alge- bras. In this thesis we discuss several topics of the theory of structure and properties of central simple algebras, from an algebraic number-theoretic point of view. We examine the Albert-Brauer-Hasse-Noether Theorem, and the required theories over algebraic number fields, and not exactly on global fields. In chapter 1 we introduce the central simple algebras over an algebraic number field K and their fundamental properties, and also we introduce the Brauer group of a field. Central simple K-algebras are really useful algebraic structures, not only because of their simplicity as algebras, but also due to the fact that their center is precisely the field K. An equivalence relation is defined on central simple algebras, and their classes form a group, the Brauer group. The elements of the last gives a classification of all division algebras over the field K. In chapter 2 we are concerned with the extremely important concept of crossed products algebras and the formalistic theory of group cohomology. Crossed products are a specific kind of central simple algebras, having especially beautiful properties, such as their correspondence to cocycles and their ability to preserve their natural properties. Furthermore, we discuss the basic stuff about cyclic algebras and we prove that a division ix algebra of degree 3 over an algebraic number field is cyclic. In chapter 3 we provide the needed machinery of the theory of valua- tions on algebraic number fields and the P-adic completions of them, in order to present and prove the main theorem of this thesis, the Albert- Brauer-Hasse-Noether Theorem. One of the main applications of the The- orem is that every central simple algebra is cyclic. Finally, in chapter 4 we discuss the generalization of our previous con- cepts. We give a little information about the cohomology modules, from an arbitrary algebra A to an (A, A)-bimodule M, and we compute the coho- mology modules H 0,H 1 and H 2. Moreover, we introduce the construction of the general crossed products, and we view them from a homological point of view, giving relations between global and cohomological dimen- sion for several crossed products. Finally, we concetrate to the case of zero global dimension, in which we prove that the group in this case is a tor- sion group, although it is not necessarily finite and we construct a crossed product which is a division ring. We refer to [Re] and [Pi] for the theory of central simple algebras and the cohomology theory, and we follow mainly [Re] for the proofs of the presented theorems. We refer to [AR] and [Yi] for the context of chapter 4. For many years, all division algebras were constructed as crossed prod- ucts, starting in 1843 with Hamilton’s(1805-1865) real quaternions. Even- tually, Albert, Dickson and Wedderburn(1882-1948) proved that all divi- sion algebras of degree 2, 3, 4, 6 or 12 are cyclic, and hence crossed prod- ucts, and for much of the 20th century, it was conjectured that all division algebras are crossed products. But, in 1972, Amitsur(1921-1994) pro- duced a counterexample. He constructed non-crossed product division algebras of any degree n divisible by 23 or p2, where p is a prime number, provided p is prime to the residue characteristic. This construction led to a number of questions about the properties of division algebras. One of the most important open problems regarding central simple K-algebras, and the motivation of the study of this thesis project, is to construct a non-cyclic division algebra of degree a prime p over a field K [ABGV, Problem 1.1]. It is trivial that a division algebra of degree 2 over K is cyclic. Also, a division algebra of degree 3 over K is cyclic, as shown by Wedderburn in 1921. For p > 3 this problem is completely open. If we specialize this problem to the case where K is a number field, then the answer is given by the Albert-Brauer-Hasse-Noether Theorem. This asserts that every division algebra of degree p over a number field is cyclic. x INTRODUCTION From the Albert-Brauer-Hasse-Noether Theorem and some theorems of Frobenius(1849-1917), Hasse, Tsen(1898-1940) Wedderurn and Witt(1911- 1991), today we know that all division algebras over the fields K, KP ,KP ((t)), where K is any number field, Fq, Fq (t) , Fq ((t)) , R, R (t) , R ((t)) , C, C (t) , C ((t)) are crossed products [B]. I would wholeheartedly like to thank my supervisor, professor Theodora Theohari-Apostolidi, who has been an invaluable inspiration for me all these years, for the effort she put to develop this thesis, for the full under- standing and endless patience, and for the detailed guidance in my every step. I would also like to thank the other members of my thesis commit- tee, professor Hara Charalambous and professor Athanasios Tzouvaras, for their accurate and helpful comments, suggestions and corrections, which contributed to improve the present work.
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