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arXiv · 1604.08435

Hilbert-Kunz functions of surface rings of type ADE

Daniel Brinkmann
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Abstract

In this thesis we compute the Hilbert-Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen-Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals. For these computations we need to compute the Hilbert-series of the involved first syzygy modules as well as the fact that the surface rings of type ADE defined over the complex numbers are the rings of invariants of C[x,y] under actions defined by the finite subgroups of SL2(C). Finally, we will show that the developed tools can be used to compute Hilbert-Kunz functions of two-dimensional rings of type ADE with respect to other (monomial) ideals than the maximal. Moreover, Hilbert-Kunz functions of two-dimensional Fermat rings can be computed by the same method. There is also a paper, named "The Hilbert-Kunz functions of two-dimensional rings of type ADE", which gives a survey of main ideas and results. Moreover, the main theorem from Monskys paper "The Hilbert-Kunz multiplicity of an irreducible trinomial" will be generalized to a non-standard graded situation. This result will be used to study the behavior of the Hilbert-Kunz multiplicity in degenerations of the hypersurface rings defnied by those specific trinomials. Once more I want to thank my supervisor Holger Brenner for his guidance and friendship during the years.

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Daniel Brinkmann. 2016-04-28. Hilbert-Kunz functions of surface rings of type ADE. https://arxiv.org/abs/1604.08435

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