Characterize almost Gorensteinness via the asymptotic constant e1(U) and quantify the nn-sequence

Determine whether the almost Gorenstein property of an excellent two-dimensional normal local ring A is characterized by the vanishing of the asymptotic constant ē1(U) = 0, where U = KA/wA for a general element w in the canonical module KA and ē1(U) is defined by the asymptotic formula lengthA(U/m^n U) = n·(KX + CX)·Z − ē1(U) for sufficiently large integers n on a resolution X with canonical divisor KX, cycle CX satisfying H0(X, OX(KX + CX)) = KA, and Z representing the maximal ideal m. Additionally, clarify the possible values and behavior of the quantities {n0 − nn | n ≥ 1} and {n ≥ 0 | ē1(U) + n0 − nn}, where nn = lengthA(H0(U(−nZ))/m^n U) and n0 = lengthA(Coker H0(α0)).

Background

Section 4 develops a length formula for the module U = KA/wA tied to a resolution X of Spec(A) where OX(KX + CX) and mOX are generated and m is represented by a cycle Z. Theorem 4.3 gives LA(U/m^nU) = n(KX + CX)·Z + nn − n0, yielding e0(U) = (KX + CX)·Z and characterizing A as almost Gorenstein precisely when n0 = n1.

Question 4.6 introduces an asymptotic constant ē1(U) satisfying LA(U/m^nU) = n(KX + CX)·Z − ē1(U) for large n and notes ē1(U) ≥ 0; moreover, ē1(U) = 0 in important cases (U = H0(U) or m a pg-ideal). The question asks whether ē1(U) = 0 exactly characterizes almost Gorensteinness and requests clarification of the ranges of differences n0 − nn and related quantities.