Spectra of Modules

AbstractThis manuscript solves the problem that the so-called “stable category” Mod Λ of an Artin algebra is in general not triangulated. The method is to mimick topology and hence first form the Spanier–Whitehead category Stab(Mod Λ) and then construct a category Spectra of “spectra of modules” which completes the compact part of Stab(Mod Λ) under small coproducts.Spectra is then a triangulated substitute for Mod Λ.The main results are that Spectra is a compactly generated triangulated category which contains the compact part of Stab(Mod Λ) as a full subcategory and even admits a precise description of its compact objects, which only form a small set of isomorphism classes.As an application, it is proved that over an Artin algebra, the Gorenstein projective modules form a pre-covering class. This was previously only known for rings satisfying strong homological conditions