Külshammer–Reynolds Ideals in Finite-Dimensional Algebras

• Külshammer–Reynolds ideals are central invariants defined through Frobenius powers and commutator subspaces in finite-dimensional algebras with a non-degenerate symmetric bilinear form.
• They form a descending chain from the full center to the Reynolds ideal, providing a refined filtration that aids in distinguishing derived and stable equivalence classes.
• The theory extends to non-symmetric algebras via trivial extensions and further generalizes to higher Hochschild (co)homology and graded settings, underscoring its broad applicability in representation theory.

Searching arXiv for the cited papers and closely related work on Külshammer–Reynolds ideals. Külshammer–Reynolds ideals are central or center-adjacent invariants attached to finite-dimensional algebras in positive characteristic, originating in the study of symmetric algebras and later extended to non-symmetric algebras, stable settings, graded categories, triangulated categories, and higher Hochschild theory. In the classical symmetric case, they arise from the commutator subspace and Frobenius powers, producing a descending chain of ideals

$$Z(A)=T_0(A)^\perp \supseteq T_1(A)^\perp \supseteq T_2(A)^\perp \supseteq \cdots,$$

where

$T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$

Within the historical terminology recorded by Zimmermann, the Reynolds ideal is the terminal term

$R(A)=Z(A)\cap \soc(A),$

while Külshammer called the ideals $T_n(A)^\perp$ the generalised Reynolds ideals (Zimmermann, 2010). Subsequent work recast the theory in terms of Hochschild homology, trivial extensions, graded centers, and derived invariance (Zimmermann, 2010, Volkov et al., 2017).

1. Classical symmetric-algebra construction

The basic setting is a finite-dimensional algebra $A$ over a field $K$ or $k$ of characteristic $p>0$, with the symmetric case distinguished by the existence of a non-degenerate associative symmetric bilinear form

$\langle-,-\rangle : A\times A \to K.$

A key structural identity is

$[A,A]^\perp = Z(A),$

equivalently $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$0 when $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$1 denotes the commutator space (Zimmermann, 2010, Brenner et al., 2022). This is the reason the construction lands in the center.

For each $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$2, one defines

$T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$3

If $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$4 is symmetric, the orthogonal complement

$T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$5

is an ideal of the center, and these ideals form a descending chain

$T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$6

(Zimmermann, 2010, Zimmermann, 2010). Zimmermann explicitly records this as the standard ideal-theoretic form of the classical Külshammer construction (Zimmermann, 2010).

The adjoint-map formulation is equally central. In the symmetric case, one has a non-degenerate pairing between $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$7 and $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$8, and the Frobenius-semilinear maps $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$9 and $R(A)=Z(A)\cap \soc(A),$0 are characterized by

$R(A)=Z(A)\cap \soc(A),$1

and

$R(A)=Z(A)\cap \soc(A),$2

The image of $R(A)=Z(A)\cap \soc(A),$3 is $R(A)=Z(A)\cap \soc(A),$4, so the central ideals may be viewed equivalently as images of Frobenius-adjoint operators (Zimmermann, 2010).

2. Reynolds ideal and its relation to the Külshammer filtration

The Reynolds ideal occupies the terminal position in the classical filtration. In Zimmermann’s survey, the notation is

$R(A)=Z(A)\cap \soc(A),$5

and the chain is described as interpolating between the whole center and this final ideal (Zimmermann, 2010). In the historical terminology recorded there, the Reynolds ideal is the bottom term, while the full family $R(A)=Z(A)\cap \soc(A),$6 consists of generalized Reynolds ideals (Zimmermann, 2010).

For group algebras $R(A)=Z(A)\cap \soc(A),$7, the Reynolds ideal is described as

$R(A)=Z(A)\cap \soc(A),$8

and, using symmetric-algebra language,

$R(A)=Z(A)\cap \soc(A),$9

(Zimmermann, 2010). This places the Reynolds ideal at the intersection of centrality and the socle, a theme that persists across later structural work.

The paper “Ideals in the center of symmetric algebras” studies precisely the Reynolds side of the theory. It takes

$T_n(A)^\perp$0

as a basic central subspace and proves a sharp criterion: if $T_n(A)^\perp$1 is symmetric, then

$T_n(A)^\perp$2

and in this case

$T_n(A)^\perp$3

(Brenner et al., 2022). In particular, every symmetric local algebra is basic, so for every symmetric local algebra,

$T_n(A)^\perp$4

(Brenner et al., 2022).

A broader commutator-theoretic criterion also appears there. For $T_n(A)^\perp$5,

$T_n(A)^\perp$6

and consequently

$T_n(A)^\perp$7

(Brenner et al., 2022). This places the Reynolds ideal squarely within the same commutator-centered framework as the Külshammer filtration.

3. Hochschild-theoretic reformulation and higher operations

A major reinterpretation of the theory is Hochschild-theoretic. For any finite-dimensional algebra $T_n(A)^\perp$8,

$T_n(A)^\perp$9

with

$A$0

(Zimmermann, 2010). In degree $A$1, the Külshammer construction is therefore governed by the $A$2-power map on $A$3, and the center appears as the dual object in the symmetric case.

Zimmermann’s paper on Hochschild homology invariants generalizes this further. For a finite-dimensional symmetric algebra over a perfect field of characteristic $A$4, there are higher maps

$A$5

defined using the non-degenerate pairing

$A$6

and characterized by

$A$7

for all

$A$8

(Zimmermann, 2010). The degree-$A$9 case recovers the classical map: $K$0 (Zimmermann, 2010).

This higher-homological viewpoint preserves the core logic of the original construction: Frobenius powers on homology are adjoint to operators defined using duality. A plausible implication is that the classical Külshammer filtration should be understood as the degree-$K$1 shadow of a broader Hochschild-theoretic structure, rather than as an isolated phenomenon.

Zimmermann’s survey makes the same point from a different angle. For symmetric algebras, the cup product induces $K$2-power maps on Hochschild cohomology, and the corresponding right adjoints yield operations

$K$3

satisfying

$K$4

(Zimmermann, 2010). The paper presents these as higher Hochschild analogues of the classical degree-$K$5 theory.

4. Non-symmetric algebras and generalized Reynolds ideals

The classical orthogonal-complement definition depends on symmetry. For arbitrary finite-dimensional algebras, the standard workaround is the trivial extension algebra

$K$6

or, in Zimmermann’s notation,

$K$7

(Zimmermann, 2010, Zimmermann, 2010). Its multiplication is

$K$8

and it carries the symmetric bilinear form

$K$9

hence $k$0 is always symmetric (Zimmermann, 2010).

This construction lets one transport Külshammer-type data from $k$1 back to $k$2. The key identity recalled in both Zimmermann papers is

$k$3

or equivalently, in the survey’s ordering convention,

$k$4

(Zimmermann, 2010, Zimmermann, 2010). These annihilator spaces are exactly the objects identified there as the standard Reynolds-type reformulation in the non-symmetric setting (Zimmermann, 2010).

The survey explicitly connects this to the phrase “generalized Reynolds ideals for non-symmetric algebras,” citing the earlier work of Bessenrodt, Holm, and Zimmermann (Zimmermann, 2010). Thus, in the non-symmetric case, the analogue of the classical chain is no longer a family of ideals directly inside $k$5, but rather a family obtained through annihilator subspaces inside the symmetric trivial extension.

The same strategy extends to higher degree. For arbitrary finite-dimensional $k$6, the higher Külshammer-type maps are defined by

$k$7

where

$k$8

(Zimmermann, 2010). This uses the split injection and projection on Hochschild homology induced by the trivial extension. The construction shows that the non-symmetric generalization is not ad hoc; it is mediated by a canonical symmetric envelope.

5. Invariance under Morita, derived, and stable equivalences

One of the main reasons Külshammer–Reynolds ideals became important in representation theory is that they are preserved under strong equivalence relations. Zimmermann’s survey states Morita invariance in the following form: if

$k$9

is a Morita equivalence, then the induced algebra isomorphism

$p>0$0

satisfies

$p>0$1

under the hypotheses that the field is perfect of characteristic $p>0$2 and $p>0$3 are finite-dimensional (Zimmermann, 2010).

Derived invariance is a more refined statement. For finite-dimensional symmetric algebras over a perfect field of characteristic $p>0$4, if

$p>0$5

then any two-sided tilting complex inducing the equivalence yields an algebra isomorphism

$p>0$6

such that

$p>0$7

(Zimmermann, 2010). In the higher Hochschild-homological formulation, Zimmermann states that for symmetric algebras,

$p>0$8

and for arbitrary finite-dimensional algebras,

$p>0$9

(Zimmermann, 2010). These equalities express derived invariance directly at the level of Hochschild homology operations.

Stable equivalence of Morita type requires modification because the ordinary center is not, in general, a stable invariant. The survey therefore replaces it with the stable center

$\langle-,-\rangle : A\times A \to K.$0

and records that, for symmetric algebras,

$\langle-,-\rangle : A\times A \to K.$1

under stable equivalence of Morita type (Zimmermann, 2010). This shows that the filtration survives even when the ambient center itself does not.

A plausible implication is that Külshammer–Reynolds ideals are best regarded as a hierarchy of equivalence-sensitive invariants: the ideals themselves in the Morita and derived settings, and their quotient data in the stable setting.

6. Graded categories, triangulated categories, and Hochschild cohomology ideals

Volkov and Zvonareva generalize the theory from algebras to graded categories. For a graded category $\langle-,-\rangle : A\times A \to K.$2 over a field $\langle-,-\rangle : A\times A \to K.$3 of characteristic $\langle-,-\rangle : A\times A \to K.$4, they define a $\langle-,-\rangle : A\times A \to K.$5-power map

$\langle-,-\rangle : A\times A \to K.$6

on the graded abelianization, and then

$\langle-,-\rangle : A\times A \to K.$7

(Volkov et al., 2017). The corresponding Külshammer-type ideals are annihilators in the graded center: $\langle-,-\rangle : A\times A \to K.$8 with

$\langle-,-\rangle : A\times A \to K.$9

(Volkov et al., 2017). The Reynolds ideals are defined by intersection: $[A,A]^\perp = Z(A),$0 (Volkov et al., 2017).

The resulting chains

$[A,A]^\perp = Z(A),$1

and

$[A,A]^\perp = Z(A),$2

are decreasing sequences of graded ideals and are invariant under degree-preserving equivalences of graded categories (Volkov et al., 2017).

For a category with automorphism $[A,A]^\perp = Z(A),$3, the same construction applies to the orbit category $[A,A]^\perp = Z(A),$4, giving ideals in the graded center

$[A,A]^\perp = Z(A),$5

(Volkov et al., 2017). For an algebra $[A,A]^\perp = Z(A),$6, these ideals can be pulled back to Hochschild cohomology by the characteristic homomorphism

$[A,A]^\perp = Z(A),$7

leading to

$[A,A]^\perp = Z(A),$8

and

$[A,A]^\perp = Z(A),$9

(Volkov et al., 2017). These form decreasing chains of graded ideals in $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$00 and are derived invariant (Volkov et al., 2017).

In the $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$01-Calabi–Yau case, the theory regains an orthogonality form parallel to the classical symmetric case. Volkov and Zvonareva prove

$T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$02

and, over a perfect field, also

$T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$03

for a Frobenius-adjoint operator $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$04 (Volkov et al., 2017). This shows that the annihilator description and the orthogonal-complement description are not competing viewpoints but manifestations of the same structure under suitable duality hypotheses.

7. Scope, applications, and nearby but distinct uses of “Külshammer”

The survey literature emphasizes that Külshammer-type invariants have been used to distinguish derived and stable equivalence classes in subtle representation-theoretic situations (Zimmermann, 2010). Zimmermann highlights applications to dihedral type, semidihedral type, deformed preprojective algebras of type $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$05, and classification problems for domestic and polynomial-growth symmetric or weakly symmetric algebras (Zimmermann, 2010). In these contexts, even the quotient rings

$T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$06

may detect scalar parameters when dimensions alone do not (Zimmermann, 2010).

Structural work on the center also feeds back into the theory. The study of $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$07, $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$08, and $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$09 in symmetric algebras supplies exact orthogonality identities such as

$T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$10

and

$T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$11

as well as tensor-product and trivial-extension formulas including

$T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$12

and

$T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$13

(Brenner et al., 2022). These results do not define the Külshammer ideals $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$14 directly, but they clarify the central and commutator-theoretic environment in which those ideals live.

By contrast, the paper “The Picard group of an order and Külshammer reduction” is not about Külshammer–Reynolds ideals in the standard sense. It does not define

$T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$15

nor Reynolds ideals, generalized Reynolds ideals, symmetrizing forms, or the Frobenius-power construction

$T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$16

(Eisele, 2018). Its relevance is indirect: it proves a different result called “Külshammer reduction” in the context of Donovan’s conjecture and develops structural control of outer automorphism groups of $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$17-orders, which is potentially useful when studying invariants preserved under Morita equivalence (Eisele, 2018). This distinction is necessary because the term “Külshammer” occurs in more than one representation-theoretic lineage.

Taken together, the cited works present Külshammer–Reynolds ideals as a layered theory. In the classical symmetric case, they are the central ideals $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$18 descending from $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$19 to the Reynolds ideal $T_n(A)=\{x\in A\mid x^{p^n}\in [A,A]\}.$20. In the non-symmetric case, they reappear through trivial extensions and annihilator spaces. In higher degree, they become operators on Hochschild homology and ideals in Hochschild cohomology. In graded and triangulated settings, they become derived invariants in graded centers and orbit categories. Across these forms, the stable themes are commutators, Frobenius powers, orthogonality or annihilation, and the persistent role of the center (Zimmermann, 2010, Zimmermann, 2010, Volkov et al., 2017, Brenner et al., 2022).