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Generalized Hilbert-Kunz function in graded dimension 2. (English) Zbl 1410.13003

Authors’ abstract: We prove that the generalized Hilbert-Kunz function of a graded module \(M\) over a two-dimensional standard graded normal \(K\)-domain over an algebraically closed field \(K\) of prime characteristic \(p\) has the form \(gHK(M,q)=e_{gHK}(M)q^2+\gamma(q)\), with rational generalized Hilbert-Kunz multiplicity \(e_{gHK}(M)\) and a bounded function \(\gamma(q)\). Moreover, we prove that if \(R\) is a \(\mathbb{Z}\)-algebra, the limit for \(p\rightarrow +\infty\) of the generalized Hilbert-Kunz multiplicity \(e_{gHK}^{R_{p}}(M_p)\) over the fibers \(R_p\) exists, and it is a rational number.

MSC:

13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14H60 Vector bundles on curves and their moduli
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References:

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