We establish new results on (co)homology vanishing and freeness criteria for finite modules. In one of the applications, we provide a (spectral sequence) proof of the long-standing Auslander-Reiten conjecture for the class of Cohen-Macaulay Burch rings, and we also address related problems such as the Tachikawa and the Huneke-Wiegand conjectures, among further topics of interest (e.g., tensor products of modules). Inspired by a paper of Huneke and Leuschke, we obtain characterizations of when a local ring is regular, or a complete intersection, or Gorenstein. For the regular case, we describe progress on certain classical differential problems, namely, the strong version of the Zariski-Lipman conjecture and Berger's conjecture. Along the way, we generalize several results from the literature and propose some questions.
Let m≤n and r≥1. We study one-parameter deformations F:[0,1]×(Rn,0)→(Rm,0) of class C1,r+1 relative to a fixed closed germ Z⊂(Rn,0) such that SingFt⊂Z for all t∈[0,1]. Writing P=∂tF and denoting by ν(dxF) the minimal singular value of the spatial differential, we prove that if|∂xsP(t,x)|≤Cν(dxF(t,x))r+1−|s|dist(x,Z)1+α,|s|≤r, for some α>0, then F is Cr-trivial near Z along [0,1]; in fact, dist(x,Z)1+α may be replaced by dist(x,Z)θ with any θ>0, so the borderline case α=0 is included. No regularity of Z is assumed. The trivializing isotopy is the flow of the Thom–Levine field W=−RP (R being the Moore–Penrose right inverse of dxF), extended by zero across Z; it fixes Z, has the r-jet of the identity along Z and is bi-Lipschitz with explicit distortion bounds. We also give a minor-based reformulation of the hypothesis and a pathwise multiparameter variant.
The purpose of this paper is to find coefficient bounds for the class Gϑκ(L) of the parametrically-augmented Bazilevič-type normalized analytic functions in the open unit disk, which are associated with a lung-shaped domain. For functions belonging to this general class, we derive estimates for the initial Taylor-Maclaurin coefficients a2 and a3. We also find some realistic upper bound for the Fekete-Szegö functional for f∈Gϑκ(L). Similar outcomes are determined for the function log(f(z)z) and the inverse function f−1(z). Moreover, we evaluate the second Vandermonde determinants whose entries involve both Taylor-Maclaurin initial coefficients and the logarithmic coefficients.
