Hemmer–Nakano Dimension in Representation Theory

• Hemmer–Nakano dimension is a homological invariant that measures how far a Schur functor preserves Ext-groups on standard-filtered modules in representation theory.
• It quantifies the quality of quasi-hereditary covers by establishing degrees of faithfulness between covering algebras and the resolved module categories.
• Its computation leverages relative dominant dimensions and Ext-vanishing conditions, offering insights into the structural and categorical properties of algebraic modules.

The Hemmer–Nakano dimension is a homological invariant in representation theory attached to a Schur functor from a split quasi-hereditary or highest-weight-type cover to the module category of another algebra. It measures the largest degree up to which the Schur functor preserves Ext-groups on a distinguished resolving subcategory, typically the standard-filtered subcategory $\mathcal F(\Delta)$. In this sense it is a numerical measure of the quality of the connection between the covering algebra and the algebra being resolved, and recent work computes it explicitly for Schur algebras and non-faithful Schur–Weyl duality situations (Cruz, 2022, Cruz et al., 6 Jul 2025).

1. Definition and categorical framework

The basic setup starts with a commutative Noetherian ring $R$, a projective Noetherian $R$-algebra $A$, and a finitely generated projective $A$-module $P$. One sets

$$B=\operatorname{End}_A(P)^{\mathrm{op}}, \qquad F=\operatorname{Hom}_A(P,-)\colon A\text{-mod}\to B\text{-mod}.$$

The exact functor $F$ is the Schur functor. In the highest-weight setting, $A$ is a split quasi-hereditary $R$-algebra with standard modules $R$0, and the standard-filtered subcategory is

$R$1

The Hemmer–Nakano dimension is then defined for a resolving subcategory $R$2 as

$R$3

If $R$4 is not even a $R$5-$R$6-cover, the convention is

$R$7

For split quasi-hereditary covers, taking $R$8, this is the degree of faithfulness of standard modules in Rouquier’s sense (Cruz, 2022).

Object	Definition	Role
$R$9	projective Noetherian $R$0-algebra	covering algebra
$R$1	finitely generated projective $R$2-module	defines the cover
$R$3	$R$4	target algebra
$R$5	$R$6	Schur functor
$R$7	standard-filtered subcategory	main test class
$R$8	maximal Ext-degree preserved by $R$9	homological quality measure

A cover $A$0 of $A$1 means that $A$2 is fully faithful on $A$3-projectives, equivalently that the canonical map

$A$4

is an isomorphism. The Hemmer–Nakano dimension refines this double-centralizer property by asking how far Ext-comparison persists beyond degree $A$5 (Cruz, 2022).

2. Faithfulness levels and Ext-comparison

The language of $A$6-faithfulness organizes the invariant. For modules $A$7 with a standard filtration, $A$8 is $A$9-faithful if

$A$0

This yields a hierarchy. A $A$1-faithful cover means that the Schur functor is faithful on the chosen class. A $A$2-faithful cover means that $A$3 is full and faithful on $A$4. A $A$5-faithful cover means that $A$6 induces an exact equivalence

$A$7

with inverse given by the right adjoint $A$8 restricted appropriately. Higher values record preservation of increasingly long Ext-patterns (Cruz, 2022).

The right adjoint is

$A$9

In the split quasi-hereditary case the unit

$P$0

detects low-degree faithfulness: $P$1 is $P$2-faithful iff $P$3 is monic for all $P$4, and $P$5-faithful iff $P$6 is an isomorphism for all such $P$7. For $P$8, $P$9 is $$B=\operatorname{End}_A(P)^{\mathrm{op}}, \qquad F=\operatorname{Hom}_A(P,-)\colon A\text{-mod}\to B\text{-mod}.$$0-faithful iff

$$B=\operatorname{End}_A(P)^{\mathrm{op}}, \qquad F=\operatorname{Hom}_A(P,-)\colon A\text{-mod}\to B\text{-mod}.$$1

This formulation turns Hemmer–Nakano dimension into a vanishing problem for derived adjoints (Cruz, 2022).

A related formulation appears for quasi-hereditary covers arising from Schur–Weyl duality. For a quasi-hereditary algebra $$B=\operatorname{End}_A(P)^{\mathrm{op}}, \qquad F=\operatorname{Hom}_A(P,-)\colon A\text{-mod}\to B\text{-mod}.$$2 with Schur functor

$$B=\operatorname{End}_A(P)^{\mathrm{op}}, \qquad F=\operatorname{Hom}_A(P,-)\colon A\text{-mod}\to B\text{-mod}.$$3

the Hemmer–Nakano dimension of $$B=\operatorname{End}_A(P)^{\mathrm{op}}, \qquad F=\operatorname{Hom}_A(P,-)\colon A\text{-mod}\to B\text{-mod}.$$4 is the maximal value $$B=\operatorname{End}_A(P)^{\mathrm{op}}, \qquad F=\operatorname{Hom}_A(P,-)\colon A\text{-mod}\to B\text{-mod}.$$5 such that for all $$B=\operatorname{End}_A(P)^{\mathrm{op}}, \qquad F=\operatorname{Hom}_A(P,-)\colon A\text{-mod}\to B\text{-mod}.$$6,

$$B=\operatorname{End}_A(P)^{\mathrm{op}}, \qquad F=\operatorname{Hom}_A(P,-)\colon A\text{-mod}\to B\text{-mod}.$$7

In this form, the invariant measures the range of cohomological degrees for which the Schur functor induces Ext-isomorphisms between the quasi-hereditary cover and the quotient algebra (Cruz et al., 6 Jul 2025).

3. Structural bounds and computational principles

A central structural theorem gives an upper bound by the size of the simple spectrum of $$B=\operatorname{End}_A(P)^{\mathrm{op}}, \qquad F=\operatorname{Hom}_A(P,-)\colon A\text{-mod}\to B\text{-mod}.$$8. If $$B=\operatorname{End}_A(P)^{\mathrm{op}}, \qquad F=\operatorname{Hom}_A(P,-)\colon A\text{-mod}\to B\text{-mod}.$$9 is a field and $F$0 is split quasi-hereditary, with

$F$1

then sufficiently high faithfulness forces $F$2 to become an equivalence. More precisely, if $F$3 is a split quasi-hereditary $F$4-faithful cover of $F$5, then

$F$6

is an equivalence, and in fact $F$7 is an equivalence of categories. Since $F$8, this implies

$F$9

Thus the invariant cannot grow beyond the point at which the cover collapses to Morita-type equivalence (Cruz, 2022).

The main computational mechanism is relative dominant dimension. For suitable resolving subcategories $A$0, the paper proves the inequalities

$A$1

and

$A$2

This places Hemmer–Nakano dimension between two dominant-dimension expressions, with the ambient ring dimension $A$3 supplying the correction term in the integral setting (Cruz, 2022).

The invariant is also local on the ground ring. If $A$4 is an $A$5-$A$6-cover, then $A$7 is the corresponding cover over any flat Noetherian $A$8-algebra $A$9. Conversely, for regular rings, cover properties can be recovered from localizations and residue fields. A particularly useful criterion states that for a regular local $R$0 with quotient field $R$1, if

1. $R$2 is $R$3-faithful, and
2. for every height-one prime $R$4, the reduction modulo $R$5 is $R$6-faithful,

then $R$7 is already $R$8-faithful. Dually, quotienting by a projective ideal lowers faithfulness by at most one, and reduction modulo a prime $R$9 of height $R$00 lowers it by at most $R$01 (Cruz, 2022).

4. Schur algebras and integral formulas

For $R$02, the Schur algebra

$R$03

is split quasi-hereditary, with standard modules given by Weyl modules. The Schur functor

$R$04

links $R$05-modules to $R$06-modules and sends Weyl modules to dual Specht modules. In this setting the Hemmer–Nakano dimension can be computed from dominant-dimension formulas (Cruz, 2022).

The relevant dominant-dimension identities are

$R$07

and, for the characteristic tilting module $R$08,

$R$09

These formulas feed directly into exact values of the Hemmer–Nakano dimension (Cruz, 2022).

In equal characteristic local rings containing a field, the formulas are

$R$10

and

$R$11

In unequal characteristic local rings the values shift by one: $R$12 and

$R$13

These identities show that the integral setup can exhibit better homological behavior than residue-field specializations, and they supply a model for computing Hemmer–Nakano dimension from relative dominant dimension rather than from direct Ext-computation alone (Cruz, 2022).

5. The exact value for the quasi-hereditary cover of $R$14

A recent explicit computation concerns the Schur algebra $R$15 over a field $R$16 of characteristic $R$17, its Ringel dual $R$18, and the centralizer algebra

$R$19

where $R$20 is the annihilator of $R$21. The relevant quasi-hereditary cover comes from Schur–Weyl duality and the Schur functor from the Ringel dual side to $R$22-modules. In this non-faithful case $R$23, the Hemmer–Nakano dimension captures the “homological depth” of Schur–Weyl duality (Cruz et al., 6 Jul 2025).

The computation depends on the relative dominant dimension

$R$24

valid for any field $R$25 of characteristic $R$26. The paper also derives

$R$27

These vanishings feed into the quasi-hereditary cover theorem, but the paper distinguishes this larger Ext-vanishing range from the final Hemmer–Nakano dimension of the cover itself (Cruz et al., 6 Jul 2025).

The main numerical result is that for $R$28,

$R$29

Equivalently, the Schur functor induces isomorphisms

$R$30

for all $R$31 and all

$R$32

while the isomorphism fails in general for $R$33. The formula $R$34 is verified separately for $R$35, while $R$36 is excluded from the new computation and had already been handled in earlier work (Cruz et al., 6 Jul 2025).

This calculation is embedded in a broader higher Auslander-theoretic picture. The paper proves that

$R$37

forms a relative $R$38-Auslander pair, and more precisely that in the principal block

$R$39

On the $R$40-side, $R$41 is a $R$42-quasi-precluster tilting module. Consequences include that the Young module $R$43 has infinite projective dimension, every other non-projective Young module in the principal block has finite projective dimension $R$44, and

$R$45

is a tilting $R$46-module (Cruz et al., 6 Jul 2025).

6. Terminological scope and distinction from geometric “Nakano” theories

The term Hemmer–Nakano dimension belongs to representation theory, Schur functors, quasi-hereditary covers, and Ext-comparison. It should be distinguished from the much older and much broader uses of Nakano in complex geometry and vector-lattice theory. Several recent arXiv papers with “Nakano” in the title do not define or use any notion called Hemmer–Nakano dimension. Instead, they concern Nakano curvature positivity, Nakano vanishing, Demailly–Nadel-type vanishing for holomorphic vector bundles, augmented-base-locus refinements of Nakano vanishing, Monge–Ampère-type equations for Nakano positive curvature tensors, Nakano semipositivity of direct image bundles, or Nakano carrier theorems in Riesz spaces (Raufi, 2012, Wu, 2020, Pan, 9 Jan 2025, Zou, 2021, Chil et al., 3 Mar 2025).

This terminological separation matters because the shared word “Nakano” does not signal a shared invariant. In the geometric papers, Nakano refers to positivity or vanishing in complex differential or algebraic geometry, such as the curvature condition

$R$47

or variants of Nakano semipositivity and Nakano carrier theory. In the representation-theoretic papers, by contrast, the Hemmer–Nakano dimension is an Ext-theoretic degree bound for a Schur functor. A plausible implication is that the name records historical influence rather than a direct conceptual overlap between these literatures (Raufi, 2012, Cruz, 2022).