DOCUMENTA MATHEMATICA, Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 565-584

Nobushige Kurokawa, Hiroyuki Ochiai, and Masato Wakayama

Absolute Derivations and Zeta Functions

Just as the function ring case we expect the existence of the coefficient field for the integer ring. Using the notion of one element field in place of such a coefficient field, we calculate absolute derivations of arithmetic rings. Notable examples are the matrix rings over the integer ring, where we obtain some absolute rigidity. Knitting up prime numbers via absolute derivations we speculate the arithmetic landscape. Our result is only a trial to a proper foundation of arithmetic.

2000 Mathematics Subject Classification: 11R27, 11R42, 14G10

Keywords and Phrases: absolute derivations, absolute mathematics, absolute schemes

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