2025/07/08 by Abdolnaser Bahlekeh, Bahlekeh, Abdolnaser, Shokrollah Salarian +1
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2507.05825
openalex publication_date 2025/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper contains two theorems concerning the vanishing of natural transformations of (co)homology functors. Precisely, assume that R is a right noetherian ring and f: M\rt N is a morphism of finitely generated right R-modules. The first theorem proves that the natural transformation \Ext1(f, -) vanishes over the category of finitely generated right R-modules if and only if \Tor1(f, -) vanishes over the category of finitely generated left R-modules. As a corollary of this result, we establish that \Ext1(f, -) is epic if and only if \Tor1(f, -) is monic. The second theorem shows that if R is left and right noetherian and M, N are Gorenstein projective, then the natural transformations \Tor1(f, -), \Ext1(-, f) and \Ext1(f, -) vanish over the category of finitely generated Gorenstein projective modules, simultaneously. This, in particular, yields that over Gorenstein projective modules, the notions of phantom morphisms and \Ext-phantom morphisms coincide. Also, it is proved that if R is n-Gorenstein, then for any integer i>n, the natural transformations \Exti(f, -), \Exti(-, f) and \Tori(f, -) vanish over finitely generated modules, simultaneously. As an interesting consequence, we show that under the same assumptions, \Exti(-, f) is epic (resp. monic) if and only if \Exti(f, -) is monic (resp. epic) if and only if \Tori(f, -) is epic (resp. monic).