flatness revisited

CM (Tor=0, flat extension of M)

Matsu (local property)

New CM (flat extension)

Free module over PID

Ext, Tor, pd, id

Definitions for CM, etc

CM ring - CM
CM-mod over local ring - Regular Seq.
Regular local ring - Reg. Seq.

(A,M) flat then (B,M*B) flat

new CM

Chase(Over a Noetherian ring, direct product of flat modules is flat.)

CM

M->S^-1M being injective or surjective iff s is M-regular

Post-c.i.

Extension & Contraction of scalars for modules again

Including localization

post-c.i.