Dimension and singularity theory for local rings of finite embedding dimension

In this paper, we develop an algebraic theory for local rings of finite embedding dimension. 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).