Анализа компутативних прстена придруживањем симплицијалних комплекса

Abstract

Predmet izuqavaa doktorske disertacije su simplicijalni kompleksi pridrueni komutativnim prstenima sa jedinicom. Generalno, kombi- natorni objekti mogu biti pridrueni prstenima na razliqite naqine, i u ovoj disertaciji izuqavamo vixe simplicijalnih kompleksa koji daju interesantne rezultate. Fokus rada je odreivae homotopskog tipa geometrijske realizacije takvih simplicijalnih kompleksa u slu- qajevima kada je to mogue. Za djelimiqno ureen skup netrivijalnih ideala u komutativnom prstenu, definixe se ureajni kompleks i odreuje egov homotopski tip u generalnom sluqaju. Simplicijalni kompleks moe biti i indirektno pridruen prstenu, kao kompleks nezavisnosti nekog grafa ili hipergrafa koji je pridru- en prstenu. Za komaksimalan graf definixemo egov kompleks neza- visnosti i odreujemo homotopski tip za generalne komutativne prstene sa jedinicom. Da e, ova teza se bavi i izuqavaem nula djelite a tako xto se po- smatraju ideali koji su nula djelite i i definixe se kompleks ideala nula djelite a. Homotopski tip ovog simplicijalnog kompleksa odre- uje se za konaqne prstene kao i za prstene sa beskonaqno mnogo mak- simalnih ideala. U ovom dijelu koristi se diskretna teorija Morsa za simplicijalne komplekse. Teoreme dokazane u disertaciji primje- ujemo na neke klase komutativnih prstena qime dolazimo do intere- santnih kombinatornih rezultata.

This dissertation examines simplicial complexes associated with commutative rings with unity. In general, a combinatorial object can be attached to a ring in many dierent ways, and in this dissertation we examine several simplicial complexes attached to rings which give interesting results. Focus of this thesis is determining the homotopy type of geometric realization of these simplicial complexes, when it is possible. For a partially ordered set of nontrivial ideals in a commutative ring with identity, we investigate order complex and determine its homotopy type for the general case. Simplicial complex can also be associated to a ring indirectly, as an independence complex of some graph or hypergraph which is associated to that ring. For the comaximal graph of commutative ring with identity we dene its independence complex and determine its homotopy type for general commutative rings with identity. This thesis also focuses on the study of zero-divisors, by investigating ideals which are zero-divisors and dening zero-divisor ideal complex. The homotopy type of geometric realization of this simplicial complex is determined for rings that are nite and for rings that have innitely many maximal ideals. In this part of the thesis, we use the discrete Morse theory for simplicial complexes. The theorems proven in this dissertation are then applied to certain classes of commutative rings, which gives us some interesting combinatorial results.