Some Homological Conjectures Over Idealization Rings

Let (R,m,k) be a Noetherian local ring and let M be a finitely generated R-module. The main focus of this paper is to give positive answers for some long-standing homological conjectures over the idealization ring R M. First, if N is a R k-module, we show that the vanishing of ExtR kⁱ(N,N⊕ (R k)) for $i=1,2,3$ gives that N is free, and this provides a sharpened version of the Auslander-Reiten conjecture over R k. Also, we give a characterization of the Betti numbers of an R-module over the idealization ring R M and, as a biproduct, we derive that the Jorgensen-Leuschke conjecture holds true for R M. Further, we show that the true of Buchsbaum-Eisenbud-Horrocks and Total Rank conjectures over R implies the true over R M. This establishes particular answers for both conjectures for modules with infinite projective dimension, especially when R is regular or a complete intersection ring. As applications of the idealization ring theory, we show that the Zariski-Lipman conjecture holds for any ring R provided the Betti numbers of the R-derivation module Derₖ(R), seen as R k-module, satisfy the inequality βₙR k(Derₖ(R))≤βₙ₋₁R k(Derₖ(R)) for some $n>0$. Some implications regarding the Herzog-Vasconcelos conjecture are also provided.