2017/05/09 by Helmut Zöschinger, Zöschinger, Helmut
Engineering · #13C05 #13C11 #16D70 #16S90 #Advanced Numerical Analysis Techniques #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1705.03353
openalex publication_date 2017/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (R, \mathfrak m) be a commutative noetherian local ring and I an ideal of R. Let P be the class of all I-generated R-modules M (i.e. there is an epimorphism I(Λ) \twoheadrightarrow M) and let S be the class of all I∘-cogenerated R-modules N (i.e. there is a monomorphism N \hookrightarrow (I∘)Λ with I∘ = HomR(I,E)). We give a complete description of all injective and flat modules in P and S. We show that (S,P) forms a dual pair in the sense of Mehdi--Prest(2015) and that P is always closed under pure submodules. We determine all ideals I for which P is closed under submodules, S is closed under factor modules and P (resp. S) is closed under group extensions. In the last section, we examine the submodules γ(M) = ∑\U ⊂ M | U ∈ P\ and κ(M) = \bigcap \V ⊂ M | M/V ∈ S\ for all R-modules M, and we specify their explicit structure in special cases.