Snopovi modula na shemama

master's thesis

• Title (english): Sheafs of modules on schemas
• Author: Lovro Greganić
• Mentor: Goran Muić (mentor)
• Committee member: Goran Muić (predsjednik povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Committee member: Nikola Sandrić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2024-07-17, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu govorimo o shemama, koje Alexander Grothendieck 1960. uvodi u djelu EGA. One lokalno odgovaraju komutativnim prstenima s jedinicom i generaliziraju pojmove kvaziprojektivnih mnogostrukosti. Također govorimo o kvazikoherentnim snopovima koji lokalno odgovaraju modulima nad prstenom. U prvom poglavlju dajemo osnove iz teorije kategorija, naime pojmove ekvivalencije kategorija i limesa. Također dajemo kratak osvrt na osnovne pojmove iz klasične algebarske geometrije; pojmove (kvazi)projektivnih, (kvazi)afinih mnogostrukosti i regularnih funkcija. U drugom poglavlju, opisujemo definiciju snopa te dajemo osnovna svojstva. Potom definiramo spektar prstena, strukturni snop i (afine) sheme te pokazujemo ekvivalenciju kategorija komutativnih prstena s jedinicom i afinih shema. Također pokazujemo kako afine mnogostrukosti shvaćamo kao sheme te dajemo primjere shema (posebno projektivnih prostora). Nadalje, dajemo kratak pregled svojstva shema i definiramo pojam afinih lokalnih svojstava te neke rezultate. U trećem poglavlju, najprije definiramo još neka potrebna svojstva iz teorija snopova. Definiramo \(O_X\)-module, (kvazi)koherentne snopove te dokazujemo ekvivalenciju kategorija kvazikoherentnih snopova nad afinom shemom i modula nad prstenom. Nadalje, pokazujemo još neke primjere i svojstva kvazikoherentnim snopovima te završavamo rad s nekim rezultatima o njima na projektivnim prostorima.

Abstract (english)

In this paper, we talk about schemes, which Alexander Grothendieck 1960. introduces in work EGA. Schemes locally correspond to commutative rings with identity and they generalize quasiprojective varieties. Also, we talk about quasicoherent sheaves which locally correspond to modules over a ring. In the first chapter, we give preliminaries in category theory, namely concepts of equivalence of categories and limits. Also, we give a short discussion about concepts in classical algebraic geometry such as definitions of (quasi)projective, (quasi)affine varieties and regular functions. In the second chapter, we define sheaves and some basic properties. Then, we define the spectrum of a ring, structure sheaf, (affine) schemes and give equivalence of categories of commutative rings and affine schemes. Also, we show how affine varieties can be understood as schemes and give examples of schemes (in particular, projective spaces). Furthermore, we give a short presentation of the properties of schemes and define the concept of affine local property and some results. In the third chapter, firstly we introduce more properties of sheaves. We define \(O_X\)-modules, (quasi)coherent sheaves and we prove the equivalence of categories of quasicoherent sheaves and modules over a ring. Furthermore, we give examples and properties of quasicoherent sheaves and we conclude the paper with some results on quasicoherent sheaves on projective spaces.

• Keywords: kvaziprojektivne mnogostrukosti; ekvivalencije kategorija; limes; regularne funkcije
• Language: croatian
• URN:NBN: urn:nbn:hr:217:336399
• Study programme: Title: Theoretical Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-07-24 11:34:12