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# Hilbert–Kunz function

In algebra, the Hilbert–Kunz function of a local ring (R, m) of prime characteristic p is the function

\( f(q) = \operatorname{length}_R(R/m^{[q]}) \)

where q is a power of p and m[q] is the ideal generated by the q-th powers of elements of the maximal ideal m. [1]

The notion was introduced by Ernst Kunz, who used it to characterize a regular ring as a Noetherian ring in which the Frobenius morphism is flat.

References

Conca, Aldo (1996). "Hilbert-Kunz function of monomial ideals and binomial hypersurfaces" (PDF). http://www.dima.unige.it. Springer Verlag 90, 287 - 300. Retrieved 23 August 2014.

Bibliography

E. Kunz, "On noetherian rings of characteristic p," Am. J. Math, 98, (1976), 999–1013. 1

Edward Miller, Lance; Swanson, Irena (2012). "Hilbert-Kunz functions of 2 x 2 determinantal rings". arXiv:1206.1015.

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