B. Johnston et J. Verma, LOCAL COHOMOLOGY OF REES-ALGEBRAS AND HILBERT-FUNCTIONS, Proceedings of the American Mathematical Society, 123(1), 1995, pp. 1-10
Let I be an ideal primary to the maximal ideal in a local ring. We uti
lize two well-known theorems due to J.-P. Serre to prove that the diff
erence between the Hilbert function and the Hilbert polynomial of I is
the alternating sum of the graded pieces of the graded local cohomolo
gy (with respect to its positively-graded ideal) of the Pees ring Of I
. This gives new insight into the higher Hilbert coefficients of I. Th
e result is inspired by one due to J. D. Sally in dimension two and is
implicit in a paper by D. Kirby and H. A. Mehran, where very differen
t methods are used.