Dominance and common deformations of short local rings
Henry Zweiman
September 9, 2026
Abstract
Let be a commutative Artinian local ring with . We construct a one-dimensional complete Cohen–Macaulay local ring whose quotients by two regular elements outside the square of its maximal ideal are and . Consequently, dominance and uniform dominance are each equivalent for and its associated graded ring. Their dominant indices differ by at most the bounds and the reverse inequality when finite. Combining this construction with Kimura’s equicharacteristic dominance theorem and known deformation and flat-descent results, we prove that every non-complete-intersection Gorenstein local ring satisfying either or is dominant. This answers both clauses of Kobayashi–Takahashi Question 7.1 without restrictions on characteristic or residue-field cardinality.
1. Introduction
All rings in this paper are commutative, Noetherian, and have identity. A local ring is dominant if
Here is the quotient of the bounded derived category of finitely generated modules by the perfect complexes. A thick subcategory is closed under shifts, cones, and direct summands. Dominance was introduced by Takahashi in connection with subcategory classification [6]. For an Artinian ring, generates , so dominance says that this category has no nonzero proper thick subcategory.
Short local rings, here meaning those with , have long provided examples and tests for homological conjectures; see Avramov–Iyengar–Şega [1]. In the Gorenstein case, Kobayashi and Takahashi ask whether every non-complete-intersection ring with , or more generally with , is dominant [3, Question 7.1]. The multiplicity is Hilbert–Samuel multiplicity, and .
Kimura recently proved the short case when contains a field, and the higher-dimensional case in equal characteristic with infinite residue field [2, Theorem 7.3]. His theorem supplies the essential representation-theoretic input here. In particular, the first non-complete-intersection Gorenstein dominant examples already belong to that work. Our contribution is a common deformation that transfers dominance from the associated graded ring to an arbitrary short local ring, including rings without a coefficient field. The construction does not require the Gorenstein hypothesis.
Theorem 1.1.
Let be an Artinian local ring with , and put . There exist a one-dimensional complete Cohen–Macaulay local ring and -regular elements such that
If has residue characteristic and in , one can take finite free of rank over a complete Cohen discrete valuation ring with uniformizer , with and for an element satisfying .
The elements and need not be distinct. The condition that both lie outside is essential for the categorical application: dominance need not descend through a regular element in ; see [6, Corollary 6.12].
Theorem 1.2.
With and as above, is dominant if and only if is dominant. Also, is uniformly dominant if and only if is uniformly dominant. If their dominant indices are finite, then
Theorem 1.3.
Let be a Gorenstein local ring that is not a complete intersection. If either or , then is dominant.
Thus Theorem 1.3 answers Question 7.1 of [3]. The removal of the characteristic restriction uses Theorem 1.1; the removal of the finite-residue-field obstruction in positive dimension uses Liu’s flat descent [4]. We make no assertion that the rings in Theorem 1.3 are uniformly dominant. Theorem 1.2 transfers that stronger question to their graded Artinian counterparts without answering it.
2. The common deformation
Write
Since , multiplication gives a well-defined symmetric bilinear map
Its values span . The graded ring is precisely the vector space with multiplication given by in positive degree and all products of total degree at least three zero.
Lemma 2.1.
If contains a field, then as local rings.
Proof.
The Artinian ring is complete, so the equicharacteristic Cohen structure theorem gives a coefficient field identified with ; see [5, Tag 032A]. Choose a -linear complement of in . The decomposition has exactly the multiplication described in (3), because . Identifying with gives the desired isomorphism.
We now treat rings that do not contain a field. Their residue field has characteristic , with in . By the Cohen structure theorem there is a complete discrete valuation ring of characteristic zero, with maximal ideal and residue field , and a local homomorphism
inducing the chosen residue-field isomorphism [5, Tag 032A and its proof]. This statement holds for imperfect as well. The map (4) is not asserted to be injective. All uses of scalars from in are through this map.
Choose elements whose classes are a basis of , and a -basis of . Define by
This equality uses the canonical -module structure on , annihilated by . Choose symmetric lifts of these coefficients. Empty lists and empty sums are allowed when .
Define a free -module
with a -bilinear multiplication, unit , and rules
These rules define an associative commutative algebra: on triples of nonunit basis elements, both associated products are zero; triples containing satisfy associativity automatically. Put
Then is an ideal with and . Therefore is local, with maximal ideal , and has dimension one. It is Noetherian and complete: it is finite free over , and the -adic and -adic topologies agree since and . Multiplication by is injective, so is Cohen–Macaulay. Reducing (6) modulo gives
The coefficient map and the assignments , define a ring homomorphism . Indeed, the differences between the lifted coefficients and their residues annihilate , and all products involving vanish. It is surjective: first represent an element of modulo by an element of , then represent the remaining class modulo by a -linear combination of the , and finally represent the remainder by the .
Adapt the bases to the nonzero element . If , choose and set . If , choose and set . In both cases and .
Lemma 2.2.
The element is -regular, the map induces an isomorphism , and both and belong to .
Proof.
Let be the fraction field of . In , the element is invertible, with inverse
Since is -free, it embeds in this algebra, proving that is regular.
In the ordered basis , multiplication by is strictly triangular. Thus multiplication by has determinant . The length formula for the cokernel of a square matrix of nonzero determinant over a discrete valuation ring gives
The -composition factors of are all , which also has -length one. Hence . The surjection is an isomorphism by additivity of length.
Finally, the projection with kernel satisfies . Both and project to , so neither lies in .
When does not contain a field, the algebra (5) and Lemma 2.2 give the result with and . In the remaining case, Lemma 2.1 allows and . This ring is complete of dimension one, is regular and outside its maximal-ideal square, and an Artinian ring is Cohen–Macaulay.
3. Transfer of categorical generation
We recall the exact deformation properties needed. If is local and is -regular, then
For quantitative generation, let be the subcategory obtained from shifts, finite direct sums, and direct summands of using at most successive extensions, with . Takahashi’s dominant index is
A ring is uniformly dominant when this index is finite. A regular ring has index . For a regular element , the known bounds are
Interchanging and gives the other inequality and the equivalence of finiteness. No uniform bound is inferred from dominance alone.
4. Gorenstein rings of almost minimal multiplicity
We first verify the hypotheses needed to apply Kimura’s theorem to .
Lemma 4.1.
Let be Gorenstein, with , and suppose is not a complete intersection. Then is a Gorenstein local -algebra, is not a complete intersection, and has embedding dimension at least three.
Proof.
By [3, Lemma 7.2(1)], and . Choose a generator of . The multiplication pairing on with values in is nonsingular. Indeed, a vector in its radical lifts to an element of annihilating , which lies in and hence has zero degree-one class. Thus has socle , and is Gorenstein with Hilbert function .
Write in its graded presentation. The space of quadratic relations has dimension . As contains no linear form, these relations give linearly independent classes in . Therefore
An Artinian complete intersection in this regular local ring has a defining ideal minimally generated by elements. Hence is not a complete intersection.
If , Lemma 4.1 makes a non-complete-intersection Gorenstein local ring containing a field. Kimura’s theorem [2, Theorem 7.3(1)] says that is dominant. Theorem 1.2 now implies that is dominant.
Suppose and the residue field is infinite. Set . Choose whose initial linear forms give a homogeneous system of parameters for . This is possible over an infinite field. Their ideal is a minimal reduction of , and their classes in are linearly independent. Since is Cohen–Macaulay, this is a regular parameter sequence. The Artinian quotient therefore satisfies
These are the usual minimal-reduction equalities, also used in [2, proof of Theorem 7.3(2)]. The ring is Gorenstein and not a complete intersection: each of these properties is equivalent across quotient by a regular sequence. If is its maximal ideal, then
Thus is zero or a simple -module, and . The Artinian case proves that is dominant. Applying (8) successively to the parameter sequence proves dominance of .
For a finite residue field, put . The map is faithfully flat and local, its maximal ideal is , and its residue field is the infinite field . Flatness gives
Consequently the Hilbert functions, embedding dimensions, dimensions, and multiplicities agree. The closed fiber is a field, so is Gorenstein, and is a complete intersection if and only if is; see [5, Tags 0BJL and 09Q7]. The infinite-field case applies to . Liu’s flat-descent theorem [4, Theorem 3.8] then gives dominance of .
5. Examples, verification, and scope
The two positions of the residue characteristic in the maximal-ideal filtration both occur. Let , let , and form the algebra with basis and rules
The quotients and have Hilbert function and nonsingular multiplication pairing. They are Gorenstein and dominant. In the first, generates the socle and the characteristic is . In the second, has nonzero degree-one class, , and the characteristic is . Their common graded counterpart has characteristic . These examples illustrate the transfer; the theorem covers arbitrary residue fields and arbitrary symmetric multiplication tensors.
The accompanying standard-library Python program constructs exact finite quotients using the lattice generated by the columns of multiplication by . Its triangular matrix has diagonal entries , giving unique representatives with every coordinate in . The program checks associativity on the integral basis, stability of the relation lattice under multiplication, the quotient cardinality, the first two maximal-ideal layers, the socle, and the position and order of .
The recorded run covers 258 quotient rings, including 86 Gorenstein and 172 non-Gorenstein cases. It enumerates 16,860 elements and checks 31,698 basis associativity identities. The examples have characteristics . These are diagnostics for the construction; the proof over an arbitrary Cohen ring is the argument of Section 2.
The construction uses at a precise point: every triple product of positive-degree basis elements vanishes, so arbitrary lifts of the multiplication coefficients remain associative. For longer local rings, lifting a multiplication table introduces additional associativity conditions. No corresponding arbitrary-length lifting theorem is asserted. Nor does Theorem 1.2 imply that every non-Gorenstein short ring is dominant.
The quantitative transfer leaves a focused further problem: determine whether the non-complete-intersection Gorenstein graded rings with Hilbert function are uniformly dominant, and, if so, bound their dominant indices in terms of . Any such bound transfers immediately to short Gorenstein rings without a coefficient field by (2).
Preparation.
This manuscript was developed with AI assistance. The proof and accompanying computations have undergone internal checks; independent expert review has not been completed.
References
[1] L. L. Avramov, S. B. Iyengar, and L. M. Şega, Free resolutions over short local rings, J. Lond. Math. Soc. (2) 78 (2008), 459–476. https://arxiv.org/abs/0707.4451.
[2] K. Kimura, Thick subcategories over weakly symmetric algebras with radical cube zero, preprint, 2026, version 1. https://arxiv.org/abs/2609.05781v1.
[3] T. Kobayashi and R. Takahashi, On the ubiquity of uniformly dominant local rings, preprint, 2026, version 1. https://arxiv.org/abs/2603.10810v1.