Multiplicity bounds in graded rings

Original · EN

The F-threshold cʲ() of an ideal with respect to an ideal J is a positive characteristic invariant obtained by comparing the powers of with the Frobenius powers of J. We study a conjecture formulated in an earlier paper HMTW by the same authors together with M. Mustaţă, which bounds cʲ() in terms of the multiplicities e() and e(J), when and J are zero-dimensional ideals and J is generated by a system of parameters. We prove the conjecture when and J are generated by homogeneous systems of parameters in a Noetherian graded k-algebra. We also prove a similar inequality involving, instead of the F-threshold, the jumping number for the generalized parameter test submodules introduced in ST.