  Broadening existing results in the literature to much wider classes of rings,
we prove among other things:
  1. Reduced quotients of excellent regular rings of characteristic $p$ admit
big test elements,
  2. The set of F-jumping numbers of a principal ideal in a locally excellent
regular ring is a discrete subset of $\mathbb Q$, and
  3. If $R$ is a quotient of a locally excellent regular ring of prime
characteristic, then there is a uniform upper bound on the
Hartshorne-Speiser-Lyubeznik numbers of the injective hulls of the residue
fields of $R$.
  To do so, we develop the parallel theories of Ohm-Rush and intersection flat
algebras. We show that both properties can be checked locally in flat maps of
Noetherian rings. We show that intersection-flatness admits a content theory
parallel to that of Ohm-Rush content for Ohm-Rush algebras. We develop descent
results for these properties. Using the descent result for intersection
flatness, we obtain a local condition under which a faithfully flat map of
Noetherian rings must be intersection-flat. The local condition for
intersection-flatness allows us to conclude that finitely generated faithfully
flat algebras over a Noetherian ring are intersection-flat. Combining the local
condition for intersection flatness with results of Kunz and Radu yields the
conclusion that the Frobenius endomorphism associated to a locally excellent
regular ring of prime characteristic is intersection-flat, thus answering a
question of Sharp. Applications of the latter result include the three
enumerated results above. We also get applications to tight closure and
parameter test ideals.

Commutative Algebra