It is well-known that for a large class of local rings of positive characteristic,
including complete intersection rings, the Frobenius endomorphism can be
used as a test for finite projective dimension. In this paper, we exploit
this property to study the structure of such rings. One of our results
states that the Picard group of the punctured spectrum of such a ring
cannot have
-torsion.
When
is a local complete intersection, this recovers (with a purely local algebra proof) an
analogous statement for complete intersections in projective spaces first given by
Deligne in SGA and also a special case of a conjecture by Gabber. Our method also
leads to many simply constructed examples where rigidity for the Frobenius
endomorphism does not hold, even when the rings are Gorenstein with isolated
singularity. This is in stark contrast to the situation for complete intersection rings.
A related length criterion for modules of finite length and finite projective dimension
is discussed towards the end.