This bachelor’s thesis constitutes a (relatively) self-contained exposition of a snippet of algebraic geometry, sufficent for the formulation of étale cohomology (in terms of the derived-functors approach, á la Grothendieck). We start by giving an account of a few fundamental notions from classical algebraic geometry, before generalizing our discussion to locally ringed spaces to then move on to Grothendieck topologies and sheaves. We then specify to a discussion on étale topology, where we provide examples of étale sheaves and short exact sequences of such. We show that the category of étale sheaves (of R-modules) has enough injectives, and then segue to the construction of the étale cohomology. We also explain why this cohomology theory allows short exact sequences of sheaves to induce long exact sequences of cohomologies.
In this thesis we will explore how to study finite dimensional complex representations of the symmetric groups S_n with the use of so called PSH-algebras. In chapter 1 the PSH-algebra and some of its most important related objects are defined, and then the rest of chapter 1 is dedicated to develop the important theory around PSH-algebras which is to be applied later in chapter 2. In chapter 2 a specific PSH-algebra is then constructed using objects related to representation theory of the symmetric groups, and the general theorems and objects proved and defined in chapter 1, are interpreted in the setting of this specific PSH-algebra as statements about representation theory of the symmetric groups.
The goal of this thesis is to provide a (well known) Morita theoretic proof of the Dold–Kan correspondence while also introducing the prerequisites needed to understand it. In particular, we introduce basic category theory, simplicial sets, and Morita theory for both rings and categories.
Inom linja ̈r algebra har varje vektorrum ett s ̊a kallat dualrum, vilket är ett vektorrum bestående av alla linjära funktioner från det ursprungliga rummet till sin kropp. Att beräkna dimensionen av ett dualrum tillhörande ett ändlig-dimensionellt vektorrum är relativt enkelt, för oändlig-dimensionella vektorrum är det mer komplicerat. Den sats vi ska diskutera, Erdős–Kaplansky Satsen, ämnar lösa den frågan med påståendet att ett dualrum tillhörande ett oändlig-dimensionellt vektorrum har dimension lika med sin kardinalitet.
This thesis looks at characterising countably infinitely categorical theories. That is theories for which every countably infinite model is isomorphic to every other countably infinite model. The thesis looks at the Lindenbaum-Tarski algebra, Henkin theories, types and then ends with the Ryll-Nardzewski theorem which provides several equivalences to a theory being countably infinitely categorical.
This thesis will cover an elementary proof of the Riemann–Roch Theorem for planecurves. We will introduce the notions of divisors, which is a convenient way of com-puting multiplicities of rational function, then continuing by introducing differentials.Furthermore we will introduce the K-vector space L(D), consisting of rational func-tions which are controlled by a divisor D. This is followed by presenting some moreresults before we arrive at an elementary proof of the Riemann–Roch Theorem.
A definition of abstract logic is presented. This is used to explore and compare some abstract logics, such as logics with generalised quantifiers and infinitary logics, and their properties. Special focus is given to the properties of completeness, compactness, and the Löwenheim-Skolem property. A method of comparing different logics is presented and the concept of equivalent logics introduced. Lastly a proof is given for Lindström's theorem, which provides a characterization of elementary logic, also known as first-order logic, as the strongest logic for which both the compactness property and the Löwenheim-Skolem property, holds.
This paper examines the Rado graph, the unique, countably infinite, universalgraph. Many of the central properties are covered in detail, and various constructionsare provided, using results from a variety of fields of mathematics. A variantof the Rado graph was initially constructed by Ackermann. The actual Rado graphwas studied later, by Erdős and Rényi, before Rado rediscovered it from a differentperspective. A multitude of other authors have since then contributed to the subject.
One of the most important mathematicians of all time was Euclid. Even though his books laid thegroundwork for plane geometry, they did have some limitations. In this paper we show why some importantconstructions are impossible, as well as giving Swedish mathematics teachers some ideas on how to implementthis in their lessons.
In this thesis we study and classify all isolated and completely isolatedsubsemigroups and congruences in classical finite transformationsemigroups.