The Injective Spectrum of a Right Noetherian Ring

  • Harry Gulliver

Student thesis: Phd


The injective spectrum was introduced briefly by Gabriel, building on work of Matlis. It is a ringed space associated to a ring R that agrees with the usual Zariski spectrum when R is commutative noetherian, and allows the Zariski spectrum to be extended to the case of noncommutative rings (or even Grothendieck categories, as the injective spectrum depends only on the category of (right) modules, and not on the ring itself). In this thesis, we study the injective spectra of right noetherian rings, establishing a number of basic topological properties, and relating the topological dimension of the spectrum to the Krull dimension of the ring (in the sense of deviation of the poset of right ideals). We also compute a number of examples, illustrating both the geometrically nice behaviour possible, and the more unpleasant behaviour that the injective spectrum can exhibit. We further establish partial results on functoriality of the injective spectrum, and explore links with the torsion spectrum developed by Golan, and consider sheaves of modules over the injective spectrum.
Date of Award31 Dec 2019
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorMichael Prest (Supervisor) & John Stafford (Supervisor)


  • Noncommutative geometry
  • Injective spectrum
  • Torsion theories
  • mathematics
  • Rings and modules
  • Zariski spectrum

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