2024/11/08 by Sergio Estrada, Xianping Fu, Estrada, S. +5
Mathematics · #16D90 #18E10 #18G25 #18G35 #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2411.05250
openalex publication_date 2024/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
A theory of ordinal powers of the ideal \mathfrakgS of S-ghost morphisms is developed by introducing for every ordinal λ, the λ-th inductive power J(λ) of an ideal J. The Generalized λ-Generating Hypothesis (λ-GGH) for an ideal \mathcal J of an exact category A is the proposition that the λ-th inductive power J(λ) is an object ideal. It is shown that under mild conditions every inductive power of a ghost ideal is an object-special preenveloping ideal. When λ is infinite, the proof is based on an ideal version of Eklof's Lemma. When λ is an infinite regular cardinal, the Generalized λ-Generating Hypothesis is established for the ghost ideal \mathfrakgS for the case when \mathcal A a locally λ-presentable Grothendieck category and S is a set of λ-presentable objects in \mathcal A such that ^⊥ (S^⊥) contains a generating set for \mathcal A. As a consequence of λ-GGH for the ghost ideal \mathfrakgR-mod in the category of modules R-Mod over a ring, it is shown that if the class of pure projective left R-modules is closed under extensions, then every left FP-projective module is pure projective. A restricted version n-GGH(\mathfrakg(C(R))) for the ghost ideal in C(R)) is also considered and it is shown that n-GGH(\mathfrakg(C(R))) holds for R if and only if the n-th power of the ghost ideal in the derived category D(R) is zero if and only if the global dimension of R is less than n. If R is coherent, then the Generating Hypothesis holds for R if and only if R is von Neumann regular.