A vanishing theorem for finitely supported ideals in regular local rings
A cohomological vanishing property is proved for finitely supported ideals in an arbitrary d-dimensional regular local ring. (Such vanishing implies some refined Briancon-Skoda-type results, not otherwise known in mixed char- acteristic.) It follows that the adjoint e I of a finitely supported ideal I has order ord e I = sup(ordI +1 −d, 0), and that taking adjoints of finitely supported ideals commutes with taking strict transforms at infinitely near points. In particular, e I is also finitely supported.
Adjoints of ideals in regular local rings
The adjoint of an ideal I in a regular local ring R is the R-ideal adj(I):=H^0(Y, I\omega_Y), where f:Y -> Spec(R) is a proper birational map with Y nonsingular and IO_Y invertible, and \omega_f is a canonical relative dualizing sheaf. (Such an f is supposed to exist.) The basic conjecture is that I.adj(I^n)=adj(I^{n+1}) whenever n is >= the analytic spread of I. This is a strong version of the Briancon-Skoda theorem, implying all other known versions. It follows from the (conjectural) vanishing of H^i(Y,\omega_Y) for all i>0. When R is essentially of finite type over a char. 0 field, that vanishing has been deduced by Cutkosky from Kodaira vanishing. It also holds whenever dim.R = 2; and in this case we can say considerably more about adjoints, tying in e.g., with classical material on adjoint curves and conductors.
A new class of numerical sequences and its applications to sine and cosine series
First, it is proved that the class of classical quasi-monotone sequences is not comparable to the newly defined class of sequences of ‘rest bounded variation’. Considering this result, we prove three sample theorems for sequences of rest bounded variation, being analogues of the theorems proved earlier for monotone or quasi-monotone sequences. One of them gives a partial answer to a question raised by R. P. Boas Jr.