An application of the cotangent complex to regularity and complete intersection

Abstract

Let A be a field and C a noetherian A-algebra such that C q is a quotient ring of an equicharacteristic local regular ring for each prime ideal q of C We prove that if C is geometrically reduced over A and Ω C/A is flat, then C is regular. We apply this result to prove that a reduced ring homomorphism u :A→C is regular when Ω C/A is a flat C-module and C is a complete noetherian local ring or a Nagata local ring. We also study the property of C being locally a complete intersection ring when the flat dimension of Ω C/A finite.