# Projekt:Computeralgebra-Berechnungen/Symmetrische Hilbert-Kunz Theorie/Syz2/Fermat-Quartik (vier Variaben)/x^3,y^3,z^3,w^3

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Für die Fermathyperfläche

${\displaystyle X^{4}+Y^{4}+Z^{4}+W^{4}=0}$

ist der Anfang der Idealauflösung des Ideals ${\displaystyle {}(x^{3},y^{3},z^{3},w^{3})}$ gleich

${\displaystyle {}\ldots \longrightarrow R(-7)^{4}\oplus R(-9)^{4}\longrightarrow R(-6)^{6}\oplus R(-4)\longrightarrow R(-3)^{4}\longrightarrow R\longrightarrow R/I\longrightarrow 0\,.}$

Der Rang des zweiten Syzygienbündels ist vier.

q ${\displaystyle {}\sum h^{2}(S^{q}(Syz_{2})(m))}$ durch Rang durch Rang ${\displaystyle {}q^{3}}$ 1 1181 295,25 295,25 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 2 20601 2060,1 257,5125 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 3 133361 6668.05 246,9648 4 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 5 6 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 7 8 9 10 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 11 12 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 13 14 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 15 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 16 17 18 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$ 19 20 ${\displaystyle {}-}$ ${\displaystyle {}-}$ ${\displaystyle {}-}$