Auslander–Buchsbaum–Serre Criterion

• Auslander–Buchsbaum–Serre criterion is a pivotal theorem in commutative algebra that equates ring regularity with the finite projective or flat dimension of its residue field.
• It bridges key homological invariants—such as projective dimension, flat dimension, and global dimension—to provide a unified test for the regularity of Noetherian local rings.
• The approach underpins formal verification in systems like Lean4 and extends to broader contexts, including the study of complete intersections and Gorenstein properties.

The Auslander–Buchsbaum–Serre criterion is a foundational result in commutative algebra that provides a precise homological characterization of regular local rings. It asserts that the regularity of a Noetherian local ring $(R,\m,k)$ is equivalent to the finiteness of the projective dimension of its residue field $k$, or, equivalently, to the finiteness of the flat dimension of the canonical surjection $R \to k$. This result sits at the intersection of homological algebra, local algebra, and singularity theory and links the structure of a ring to the properties of its residue field via derived functors.

1. Formal Statement and Equivalent Conditions

The Auslander–Buchsbaum–Serre criterion, in its classical form for a commutative Noetherian local ring $(R,\m,k)$, states the following equivalences:

• $R$ is regular
• The projective dimension of $k$ as an $R$-module is finite; that is,

$\pd_R(k) < \infty$

• The flat dimension of the canonical surjection $\varphi: R \to k$ is finite:

$\fd_R(\varphi) < \infty$

Furthermore, these invariants are equal when any of the conditions hold,

$k$0

where $k$1 denotes the Krull dimension of $k$2 (Majadas, 2012).

This is equivalently formulated in terms of global dimension: $k$3 as formalized in Lean4 as $k$9 (Guan et al., 28 Oct 2025).

2. Homological Invariants: Definitions and Structures

• Projective Dimension ($k$4): The minimal length of a projective resolution of a finitely generated $k$5-module $k$6:

$k$7

Alternatively,

$k$8

• Flat Dimension ($k$9): For a local homomorphism $R \to k$0, the flat dimension of $R \to k$1 as an $R \to k$2-module.
• Depth ($R \to k$3): The infimum of $R \to k$4 with $R \to k$5.
• Embedding Dimension ($R \to k$6): The dimension of $R \to k$7 as a vector space over $R \to k$8.
• Krull Dimension ($R \to k$9): The supremum of the lengths of chains of prime ideals in $(R,\m,k)$0.
• André–Quillen Homology ($(R,\m,k)$1): For $(R,\m,k)$2,

$(R,\m,k)$3

This arises by considering the generation of the maximal ideal $(R,\m,k)$4 by a regular sequence (Majadas, 2012).

• Global Dimension ($(R,\m,k)$5): The supremum of projective dimensions over all $(R,\m,k)$6-modules:

$(R,\m,k)$7

(Guan et al., 28 Oct 2025).

3. Proof Outline and Methodological Approaches

Two main approaches are highlighted in the literature:

Classical Koszul/Ext Argument:

• With $(R,\m,k)$8 a minimal set of generators for $(R,\m,k)$9, the minimal free resolution of $R$0 starts as:

$R$1

Under finiteness of $R$2, this forces the minimal number of generators $R$3 of $R$4 to equal $R$5, characterizing $R$6 as regular. The entirety of the resolution relies on syzygies and vanishing properties of Ext in high degrees, ensuring that $R$7 is generated by a regular sequence of length equal to the dimension.

Avramov’s Injectivity and André–Quillen Approach:

• The canonical surjection $R$8 possesses the $R$9-vanishing property, i.e., the map

$k$0

is the zero map. Avramov’s theorem asserts that under finite flat dimension (specifically, $k$1), this map is injective. As it is also zero, one concludes $k$2, which is the defining property of regularity for $k$3 (Majadas, 2012).

Formalization Approach (Lean4/Mathlib4):

• The criterion is mechanized in Lean4 not via the Koszul complex, but by leveraging the theory of maximal Cohen–Macaulay modules and a special case of the Ferrand–Vasconcelos theorem: if the maximal ideal has finite projective dimension, it is generated by a regular sequence. This formalization also requires auxiliary results: the depth of modules, Cohen–Macaulay property, unmixedness, associated primes, and Hilbert's Syzygy theorem (Guan et al., 28 Oct 2025).

4. Unification with Broader Homological Criteria

The Auslander–Buchsbaum–Serre criterion exemplifies a unified homological approach to structural properties of rings. Majadas’s framework extends the logic of the criterion to complete intersection and Gorenstein properties by studying analogous vanishing of André–Quillen homology in higher degrees or through Ext vanishing and upper-complete-intersection dimension. The approach is as follows:

• Identify local homomorphisms with $k$4-vanishing,
• Prove that finite flat dimension together with $k$5-vanishing implies vanishing of André–Quillen homology in degree $k$6,
• Interpret this vanishing as equivalent to the structural property (regularity, complete intersection, or Gorenstein) (Majadas, 2012).

A plausible implication is that this philosophy generalizes regularity tests and connects them deeply with the homological algebraic structure of local homomorphisms.

5. Rigorous Computational and Formal Tools

The formalization of the criterion in Lean4/mathlib4 involves:

• Explicit predicates and definitions for projective/global dimension, depth, Cohen–Macaulay property,
• Rees’s theorem and precise Ext-vanishing equivalences,
• Rigorous category-theoretic treatment of derived functors,
• The weakened Ferrand–Vasconcelos theorem specialized to the maximal ideal,
• Hilbert's Syzygy theorem for polynomial rings,
• Deep use of ideal theoretic and homological lemmas, such as behavior of associated primes, Krull dimension, regular sequences, and specialization/localization principles (Guan et al., 28 Oct 2025).

In the formalized system, the statement appears as: $R$0 and modules with $k$7 (maximal Cohen–Macaulay) are shown to be free in the regular case.

6. Auxiliary Theorems and Interconnections

The development of the criterion and its formalization interlocks with several key results:

• Hilbert’s Syzygy theorem ($k$8),
• Unmixedness theorem for Cohen–Macaulay rings,
• Rees theorem for regular sequences and Ext vanishing,
• Depth lemmas—Auslander–Buchsbaum and Ischebeck,
• Nakayama’s lemma, prime avoidance, and properties of regular rings localized at primes.

These results collectively establish a robust infrastructure for connecting module-theoretic properties to ring-theoretic regularity.

7. Significance and Broader Impact

The Auslander–Buchsbaum–Serre criterion not only provides a practical test for regularity of Noetherian local rings, but also offers a prototype for homological characterizations of more subtle singularity classes. Its formalization, as achieved in Lean4, demonstrates the maturity of contemporary proof assistant ecosystems for sophisticated homological theorems (Guan et al., 28 Oct 2025). The approach of characterizing structural ring properties via vanishing of homological invariants plays a fundamental role in both classical algebraic geometry and modern computational methods. The criterion’s central role is maintained in both theoretical developments and computational verification systems.