On Roberts rings. (English) Zbl 1050.13012

Kurano, Kazuhiko

J. Math. Soc. Japan 53, No. 2, 333-355 (2001).

Summary: P. C. Roberts [Bull. Am. Math. Math. Soc., New Ser. 13, 127–130 (1985; Zbl 0585.13004)] proved the vanishing theorem of intersection multiplicities for a local ring that satisfies \(\tau_{A/S}([A])= [\text{Spec}\, A]_{\dim A}\), where \(\tau_{A/S}\) is the Riemann-Roch map for \(\text{Spec}\,A\) with regular base scheme \(\text{Spec}\,S\). We refer such rings as Roberts rings. For rings of positive characteristic, we can characterize Roberts rings by the Frobenius maps. For rings with field of fractions of characteristic 0, we can characterize Roberts rings by some Galois extensions.
We give basic properties and examples of Roberts rings.

MSC:

13H15

Multiplicity theory and related topics

Keywords:

vanishing of intersection multiplicities; Roberts rings; Frobenius maps; Galois extensions

Citations:

Zbl 0585.13004