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 Ext_R⋉ k^i(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 Der_k(R), seen as R⋉ k-module, satisfy the inequality β_n^R⋉ k(Der_k(R))≤β_n-1^R⋉ k(Der_k(R)) for some n>0. Some implications regarding the Herzog-Vasconcelos conjecture are also provided.