Divergent series and Serre's intersection formula for graded rings (Q2357479)

If \(X\) is a smooth variety, Serre's intersection formula computes intersection multiplicities on \(X\). This is done via an alternating sum of the lengths of Tor groups. If \(X\) is singular, the analogous sum can be a divergent series. However, there are alternate ways coming from geometry to assign intersection multiplicities on \(X\) in many cases, although the multiplicities are often fractional. Fulton asks whether there is a connection between these alternative fractional multiplicities and Serre's formula, using analytic continuation of the divergent series. The author applies ideas of Avramov and Buchweitz to answer Fulton's question in the context of graded rings. He uses a rational function to provide an analytic continuation of the Hilbert series of the total Tor module associated to an intersection. He obtains from this a notion of intersection multiplicity that extends Serre's definition, by evaluating the rational function at \(t=1\). He shows that the multiplicity that he obtains in this way agrees with the multiplicities obtained by alternative definitions in algebraic geometry, in the cae of \(\mathbb Q\)-Cartier divisors or on normal surfaces.