On local cohomology and Hilbert function of powers of ideals (Q1411989)

The paper deals with the asymptotic behaviour of \(\text{reg} (R/I^{n})\), the Castelnuovo-Mumford regularity, where \(I\) is a homogeneous ideal in a polynomial ring \(R=K[x_1,\ldots,x_m].\) Indeed, if \(d=\dim R/I,\) \(a_i(R/I)=\max\{t; \;H_{m}^i(R/I)_t\neq 0\}\) for \(0\leq i \leq d,\) it is possible to prove that \(\lim_{n \to \infty}a_d(R/ \text{Gin}(I^n)/n\) always exists (here \(\text{Gin}(R/I)\) means the generic initial ideal of \(I\) with respect to the reverse lexicographic order; recall that \(\text{reg} (R/I)= \max \{a_i(R/I)+i\); \(0\leq i\leq d\}\)). Hence, the authors give conditions on \(\text{reg}(R/I)\) in order to have \(a_d(R/I)=a_d(R/ \text{Gin}(I));\) in particular, this happens when \(d=1.\) However an example shows that the previous equality does not always hold. The second part of the paper studies the regularity index \(\text{ri}(R/I)\) of \(R/I\), i.e. the smallest integer \(t\) for which the Hilbert function and the Hilbert polynomial coincides, i.e. \(H_{R/I}(t)=P_{R/I}(t).\) In particular, the authors prove that if \(I\) is generated in two degrees \(d_1\) and \(d_2\) then \(\text{ri}(R/I^n)\) is a linear function of \(n\) for \(n\gg 0\) whose leading coefficient is either \(d_1\) or \(d_2\)