2024/02/20 by Frank Lukas, Lukas, Frank
Mathematics · #16G60 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2402.13142
openalex publication_date 2024/02/20 · openalex created_date 2024/02/22 · openalex updated_date 2026/07/28
Let R be a ring with unity and X a semibrick in the module category Mod R, that is, a class of pairwise orthogonal finitely presented modules whose endomorphism rings are division rings. We study the full subcategory Filt(X) consisting of all modules admitting a filtration with factors in X. We show that Filt(X) is a wide subcategory of Mod R. For the Ext-orthogonal class X⊥ = \M ∈ Mod R | Ext1R(X,M)=0 for all X ∈ X\ we construct, for every module Y, an X⊥-envelope YX(∞) as a direct limit of iterated universal short exact sequences. Assume that every X ∈ X has projective dimension at most one and that HomR(X,R)=0 for all X ∈ X. Then the envelope RX(∞) of the regular module is isomorphic to the universal localization RX of R at X in the sense of Schofield. The X⊥-envelopes of modules in X are called Prüfer modules since they share many properties with classical Prüfer groups and with Prüfer modules over tame hereditary algebras. We prove that every injective object in Filt(X) is a direct sum of such Prüfer modules.