“Infinite” Properties of Certain Local Cohomology Modules of Determinantal Rings

Abstract

For given integers m,n ≥ 2 there are examples of ideals I of complete determinantal local rings \((R,\mathfrak {m}), \dim R = m+n-1, \text {grade}~I = n-1,\) with the canonical module ω_R and the property that the socle dimensions of \(H^{m+n-2}_{I}(\omega _{R})\) and \(H^{m}_{\mathfrak {m}}\left (H^{n-1}_{I}(\omega _{R})\right )\) are not finite. In the case of m = n, i.e., a Gorenstein ring, the socle dimensions provide further information about the τ-numbers as studied in Mahmood and Schenzel (J. Algebra372, 56–67, 10). Moreover, the endomorphism ring of \(H^{n-1}_{I}(\omega _{R})\) is studied and shown to be an R-algebra of finite type but not finitely generated as R-module generalizing an example of Schenzel (J. Algebra 344, 229–245, 15).