In 1985, P. C. Roberts  proved the vanishing theorem of intersection multiplicities for a local ring that satisfies , where is the Riemann-Roch map for with regular base scheme . 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 shall give basic properties and examples of Roberts rings in the paper.
"On Roberts rings." J. Math. Soc. Japan 53 (2) 333 - 355, April, 2001. https://doi.org/10.2969/jmsj/05320333