Let be a commutative graded Artinian Gorenstein algebra over an arbitrary field, with Hilbert function and . We prove that its dominant index is exactly one: the residue field belongs to level two from every nonzero object of the stable module category. The proof combines Ringel’s eventual bristle-generation theorem for Kronecker representations with an explicit identity between normalized syzygies and kernel reflections. Distant cosyzygies map onto every bristle, producing a single exact sequence between sums of shifts of the original module whose kernel has a simple summand after a shift. Every nonzero stable object consequently has generation time at most three. The common deformation of a short local ring and its associated graded ring also gives dominant index at most seven without a coefficient-field assumption. These results establish uniform dominance for this class; they do not decide whether every dominant local ring is uniformly dominant.
1. Introduction and statements
Let be an Artinian Gorenstein local ring. Its stable category has finitely generated modules as objects and homomorphisms modulo those factoring through free modules as morphisms. It is triangulated, with suspension , and agrees with the singularity category. For an object , let consist of finite direct sums and direct summands of shifts of . Inductively, is obtained by taking summands of extensions of objects in by objects in .
Following Takahashi [5], the dominant index is the least integer such that for every nonzero . Finiteness of this invariant is uniform dominance. Dominance alone asks only that belong to the thick closure of each nonzero object, with no uniform bound on the number of extensions.
Kobayashi and Takahashi study uniform dominance for several classes of local rings and ask about dominance of non-complete-intersection short Gorenstein rings [2, Question 7.1]. Kimura proves dominance in the coefficient-field case through a classification of thick subcategories over weakly symmetric algebras [1, Theorem 7.3]. That qualitative result is already known and is not claimed as new here. The present paper establishes a sharp uniform bound for the commutative short graded case.
Theorem 1.1.
Let be any field and let be a commutative graded Artinian Gorenstein -algebra, generated in degree one, with
Then . More precisely, for every nonzero ,
The first bound is sharp for every such algebra.
The essential existing input is Ringel’s theorem that sufficiently high kernel reflections of a matrix pencil have enough eigenvectors to span the source space [4, Section 5, Theorem 2]. This is the matrix formulation of his eventual bristle-generation theorem [3, Main Theorem 1.3]. We show that even normalized syzygies realize these reflections. We then use bristles only as intermediate factors of a map between shifts of the original module. They need not themselves have been constructed in its thick closure.
Corollary 1.2.
Let be an Artinian Gorenstein local ring with and . If contains a field, then . In general, is uniformly dominant and .
The second assertion uses the common-deformation construction in [6]; its role and proof are recalled in Section 5. Theorem 1.1 is independent of that construction. The global question whether a dominant local ring can fail to be uniformly dominant [5, Question 6.12] remains open beyond the classes treated here.
2. Two-layer modules and normalized syzygies
Throughout Sections 2–5 let satisfy Theorem 1.1. Choose in and a basis of . There is an invertible symmetric matrix with
Indeed the multiplication pairing on is nondegenerate because the socle of is . Symmetry follows from commutativity. No diagonalization of this pairing will be used.
Lemma 2.1.
Every finite -module with no free summand is annihilated by and admits a grading concentrated in degrees zero and one. If it also has no simple summand, then its radical and socle coincide.
Proof.
If , the map sending to is injective: every nonzero ideal contains the one-dimensional socle, whereas its kernel does not. Since is self-injective, this injection splits. This contradicts the absence of free summands. Thus . Choose a -linear complement to . Multiplication by sends into and kills , giving the grading.
Finally, any vector-space complement to in splits off as a semisimple module: choose the complement to in to contain it, and split the corresponding top vector subspace. Thus absence of simple summands implies .
Call a nonzero module with no free summands persistent if none of its minimal syzygies or cosyzygies, indexed by all integers, has a simple summand. This terminology is used only within this paper. If is not persistent, then , so such modules already satisfy the theorem.
A two-layer module is specified by vector spaces , and arrows , giving an -Kronecker representation. For a persistent module the arrows jointly span and their kernels have zero intersection. Normalize every syzygy to degrees zero and one, and denote the resulting operation by .
Lemma 2.2.
Let be persistent, with and . Put
Then has degree-zero space , degree-one space , and arrows
where is the th projection. In particular,
Moreover, normalized syzygy is exact on any short exact sequence of persistent two-layer modules, and on all its iterates.
Proof.
The natural minimal cover is an isomorphism in degree zero. Its kernel has degree-one part and degree-two part . Multiplication in the cover gives (2). After shifting degrees, persistence and Lemma 2.1 ensure that the degree-one component is the radical. This proves the first dimension formula. A graded injective envelope embeds into , since the socle of is ; the quotient, after normalization, has dimensions .
For exactness, take degree-zero components of a graded short exact sequence and tensor them over with . These give an exact sequence of the natural covers. The snake lemma gives an exact sequence of their kernels. Normalize the degrees. Persistence supplies the same hypotheses at the next step, so the argument iterates. This is also the exactness construction appearing in [1, Lemma 3.11]; the local argument given here requires no ambient proper thick subcategory.
Lemma 2.3.
Put . For a persistent module , write . Then
Also as .
Proof.
The matrix for cosyzygy in (3) has eigenvectors and , with eigenvalues and respectively. Write the initial vector as their linear combination. Since the dimension vector remains strictly positive for all positive and negative iterates, both coefficients are nonnegative. Neither coefficient can be zero, because are positive integers and the slopes are irrational. Irrationality follows from for . Both coefficients are therefore positive. The conclusions follow by taking positive and negative iterates. Finally, and , so .
3. Kernel reflections and bristles
For a pencil , let have source , target , and arrows . This is precisely the kernel reflection used in [4, Section 5]. For let replace the column of arrows by times that column.
Lemma 3.1.
On persistent modules there are natural identifications
Consequently for every nonnegative integer .
Proof.
The first identity is (2). The row map of the changed pencil is . Its kernel is . Under this identification, its projection arrows are times the old projections. Hence
It follows that , since is symmetric. These are identifications of the arrow maps, not just the dimensions. Iteration is legitimate by persistence.
A bristle is the pencil , where . It is also a cyclic -module of length two. We use the following consequence of Ringel’s theorem.
Theorem 3.2 (Ringel).
For and any pencil , the source of is spanned by vectors whose arrow images span a one-dimensional space, for all sufficiently large .
This is [4, Section 5, Theorem 2], deduced there from [3]. It applies over an arbitrary field. In particular, if is generated by its source space, a basis chosen from these vectors gives a surjection from a direct sum of bristles onto .
Lemma 3.3.
For every persistent , there is an integer such that maps onto every bristle whenever .
Proof.
Let be generated in degree zero, with dimensions . A map is a pair of functionals and satisfying
These are at most linear equations on unknowns. Thus the space of maps has dimension at least . If a map is nonzero, then , since . Its top map is consequently surjective, and the images of the arrows then give surjectivity on the target degree as well. Lemma 2.3 makes the displayed lower bound positive for and all sufficiently large , independently of .
Remove free summands from a representative of . If it is not persistent, then . Otherwise choose large enough that Theorem 3.2 applies to the even reflection . By Lemma 3.1, the normalized module
is generated by bristles. Put . A basis of eigenvectors in gives a fixed epimorphism
For every sufficiently large , Lemma 3.3 supplies epimorphisms for all . Compose their direct sum with (5) to obtain
Both the middle and right modules belong to , so . Choose still larger, if necessary, so that
Here and have already been fixed, and Lemma 2.3 guarantees such .
The kernel has no free summand, since it is annihilated by . If it were persistent, applying normalized syzygy times to (6), using Lemma 2.2, would give an epimorphism
This contradicts (7). A zero kernel gives the same contradiction. Thus some stable shift of has as a summand. Consequently .
To prove sharpness, take a bristle as an -module. It is indecomposable and nonprojective. Minimal stable syzygies and cosyzygies preserve nonprojective indecomposability, since is self-injective and its stable category is Krull–Schmidt. The two dimension transformations in (3) preserve
Starting from gives . Suppose a simple module first occurs in either direction. Up to that step the preceding modules are nonsimple indecomposables, so the minimal-cover or injective-envelope calculation in Lemma 2.2 gives the same dimension transformation at that step. A one-dimensional graded module has vector or and hence , a contradiction. Thus every shift of is nonsimple. Krull–Schmidt and cancellation of free summands imply that is not a summand of a finite sum of those shifts. Hence , proving .
Finally Lemma 2.1 shows that every nonfree module is an extension of two semisimple modules. Therefore . The usual composition rule for levels gives for every nonzero .
Remark 4.1.
The proof does not bound the shift in Ringel’s theorem or the number of bristle generators. Such bounds are unnecessary: shifts and finite sums are level-one operations. The crucial order is to fix and first, and only then choose in (7). Allowing to grow with would not give the stated contradiction.
5. Consequences, transfer, and an example
The generation time of an object is the least for which the whole category is its level . Thus Theorem 1.1 gives an upper bound of three for the generation time of every nonzero object, and in particular for the ultimate dimension of . This does not assert that every integer up to three occurs in its Orlov spectrum.
For completeness, we recall the specific deformation input used to remove the coefficient-field assumption. The construction below is the short Gorenstein case of [6, Theorems 1.1 and 1.2].
Proposition 5.1.
If is an Artinian Gorenstein local ring with and , and , then
Proof.
The socle is , and the multiplication pairing on is nondegenerate. If has a coefficient field, a complement to in identifies with .
Otherwise its residue characteristic is nonzero in . Choose a complete Cohen discrete valuation ring , with uniformizer and residue field , mapping to . Such a map need not be injective. Choose lifts of a basis of and a socle generator . Lift their symmetric multiplication coefficients arbitrarily to , and define
This associative commutative ring is finite free over , complete local of dimension one, and Cohen–Macaulay. The assignment , is a surjection , as can be checked on the three layers of the maximal-ideal filtration.
If , choose and set . If , choose and set . Then and lies in the kernel. Multiplication by on the free -module has determinant , so it is injective and its cokernel has -length . The surjection is therefore an isomorphism, since has the same length. Also .
If is the maximal ideal of , the augmentation sends into , whereas it sends and to . Thus both regular elements belong to . Takahashi’s regular-element inequalities [5, Theorem 6.2] give
The associated graded ring satisfies Theorem 1.1. If contains a field, its completeness supplies a coefficient field and . Otherwise apply Proposition 5.1 and .
Here is an explicit instance of (6). Over , take , , and . Then
The accompanying exact computation selects five bristle generators of , factors each through , and obtains an epimorphism . Its kernel has graded dimensions and radical dimension . A complement to its radical in the degree-one space therefore gives eight simple direct summands. This is a concrete one-cone witness for .
The file check.py also verifies the reflection identity on 288 finite-field matrix examples and on 24 additional alternating pairings in characteristic two. These computations check the matrix identifications and the displayed example; the general theorem rests on the proof and Ringel’s cited result, not on finite verification.
6. Further questions
The exact Orlov spectrum of the stable category in Theorem 1.1 is not determined here. The bound seven in Proposition 5.1 is likewise not asserted to be sharp. A direct mixed-characteristic construction could improve it. More generally, the one-cone argument suggests looking for generation theorems by uniformly small intermediate modules over other algebras. Such a theorem would have to be established for the relevant module category; the present Kronecker result cannot simply be assumed outside the short case.
References
[1] K. Kimura, Thick subcategories over weakly symmetric algebras with radical cube zero, preprint, 2026, arXiv:2609.05781v1.
[2] T. Kobayashi and R. Takahashi, On the ubiquity of uniformly dominant local rings, preprint, 2026, arXiv:2603.10810v1.
[3] C. M. Ringel, Kronecker modules generated by modules of length 2, Representations of Algebras, Contemporary Mathematics 705 (2018), 189–214; arXiv:1612.07679; doi:10.1090/conm/705/14194.
[4] C. M. Ringel, The eigenvector variety of a matrix pencil, 2017, arXiv:1703.04097v2, especially Section 5, Theorem 2.
[5] R. Takahashi, Uniformly dominant local rings and Orlov spectra of singularity categories, arXiv:2412.00669; author version udim8.pdf, Theorem 6.2 and Question 6.12.
[6] H. Zweiman, Dominance and common deformations of short local rings, 2026, public manuscript P36, Theorems 1.1 and 1.2.