Global Dimension and Regular Rings

In the same way that finiteness of the Gorenstein global dimension detects the Gorenstein property of a commutativeNoetherian local ring, see Sect. 19.5, finiteness of the global dimension detects the regularity property, defined below in 20.1.9. The regularity property is extended to non-local rings in the usual way, and such rings can also be characterized in terms of finiteness of homological dimensions. This theory is covered in the first two sections of the chapter; the final section discusses additional classes of local rings and compare them to regular, Gorenstein, and Cohen–Macaulay local rings.