  Regularity, complete intersection and Gorenstein properties of a local ring
can be characterized by homological conditions on the canonical homomorphism
into its residue field (Serre, Avramov, Auslander). It is also known that in
positive characteristic, the Frobenius endomorphism can also be used for these
characterizations (Kunz, ...), and more generally any contracting endomorphism.
We introduce here a class of local homomorphisms, in some sense larger than all
above, for which these characterizations still hold, providing an unified
treatment for this class of homomorphisms.
