Cohomology Vanishing theorems over some rings containing nilpotents

2025/04/18 by Puthenpurakal, Tony J.
#13A30 #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 13D45 #Secondary 14B15

paper · doi:10.48550/arxiv.2504.13566

Abstract

(1) Let (A,\mathfrakm) be complete Noetherian local ring of dimension d and let P be a prime ideal with G_P(A) = \bigoplus_{n ≥ 0}Pⁿ/Pⁿ⁺¹ a domain. Fix r ≥ 1. If J is a homogeneous ideal of G_P^r(A) with dim G_P^r(A)/J > 0 then the local cohomology module H^d_J(G_P^r(A)) = 0. (2) Let A = K[[X₁, …,X_d]] and let \mathfrakm = (X₁, …, X_d). Assume K is separably closed. Fix r ≥ 1. Let J be a homogeneous ideal of G_\mathfrakm^r(A). We show that local cohomology modules H^j_J(G_\mathfrakm^r(A)) = 0 for j ≥ d -1 if and only if dim G_\mathfrakm^r(A)/J ≥ 2 and Proj G_\mathfrakm^r(A)/J is connected.

Citations

• Graded components of local cohomology modules
• Eulerian graded D-modules
• Twenty-Four Hours of Local Cohomology

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