  In this paper, an algebraic theory for local rings of finite embedding
dimension is developed. Several extensions of (Krull) dimension are proposed,
which are then used to generalize singularity notions from commutative algebra.
Finally, variants of the homological theorems are shown to hold in equal
characteristic.
  This theory is then applied to Noetherian local rings in order to get: (i)
over a Cohen-Macaulay local ring, uniform bounds on the Betti numbers of a
Cohen-Macaulay module in terms of dimension and multiplicity, and similar
bounds for the Bass numbers of a finitely generated module; (ii) a
characterization for being respectively analytically unramified, analytically
irreducible, unmixed, quasi-unmixed, normal, Cohen-Macaulay, pseudo-rational,
or weakly F-regular in terms of certain uniform arithmetic behavior; (iii) in
mixed characteristic, the Improved New Intersection Theorem when the residual
characteristic or ramification index is large with respect to dimension (and
some other numerical invariants).
