Length of local cohomology in positive characteristic and ordinarity

Abstract: Let $D$ be the ring of Grothendieck differential operators of the ring $R$ of polynomials in $d\geq3$ variables with coefficients in a perfect field of positive characteristic $p.$ We compute the $D$-module length of the first local cohomology module $H^1_f(R)$ of $R$ with respect to an irreducible polynomial $f$ with an isolated singularity, for $p$ large enough. The expression we give is in terms of the Frobenius action on the top coherent cohomology of the structure sheaf of the exceptional divisor of a resolution of the singularity. Our proof rests on a tight closure computation due to Hara. Since the above length is quite different from that of the corresponding local cohomology module in characteristic zero, we also consider a characteristic zero $\mathcal{D}$-module whose length is expected to equal that above, for ordinary primes.