arxiv: 2605.03068 · v1 · submitted 2026-05-04 · 🧮 math.AT · math.CT

Recognition: unknown

Global dimension of the category of rational incomplete Mackey functors for a finite abelian group G

Anna Marie Bohmann, David Barnes, Magdalena K\k{e}dziorek, Michael A. Hill

Pith reviewed 2026-05-08 02:11 UTC · model grok-4.3

classification 🧮 math.AT math.CT

keywords incomplete Mackey functorsglobal dimensionrational coefficientsfinite abelian groupsincidence algebrastransfer systemsposetshomological algebra

0 comments

The pith

Rational incomplete Mackey functors over finite abelian groups have their global dimension calculated exactly via incidence algebras of posets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gives an upper bound on the global dimension of the category of rational incomplete Mackey functors when incompleteness is controlled by a disk-like transfer system, drawing on prior splitting results. It then establishes a direct link to incidence algebras over posets that lets the dimension be computed in full when the underlying group is finite and abelian. A reader cares because incomplete Mackey functors arise in current work in algebraic topology, yet their homological algebra had remained more opaque than the classical complete case settled by Greenlees–May and Thévenaz–Webb.

Core claim

For disk-like incompleteness the category of rational incomplete Mackey functors splits into simpler pieces whose global dimension is bounded above by known results; when the group is abelian this bound is realized exactly by the global dimension of the incidence algebra of the poset encoding the allowed transfers.

What carries the argument

The equivalence (or Morita equivalence) between the category of rational incomplete Mackey functors with disk-like incompleteness and the incidence algebra of the poset of subgroups compatible with the transfer system.

If this is right

The category admits finite-length projective resolutions whose length is controlled by the height of the poset.
Ext groups between any two objects can be read off from the combinatorial data of the poset.
The result specializes to give explicit numbers when the group is cyclic or elementary abelian.
The same splitting technique supplies an upper bound even when the group is non-abelian.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Homological questions about these functors reduce to counting chains in a poset, suggesting they behave like representations of a combinatorial category.
The method may extend to other transfer systems if an analogous poset algebra can be identified.
In equivariant stable homotopy theory this supplies a concrete way to bound the length of resolutions used to construct spectra.

Load-bearing premise

The splitting results hold for disk-like transfer systems and the homological algebra of the functors is faithfully captured by the incidence algebra of the associated poset.

What would settle it

Direct computation of projective dimensions or the global dimension for the smallest non-trivial example (cyclic group of order 2 with the trivial disk-like transfer system) and comparison against the known global dimension of the corresponding incidence algebra.

read the original abstract

In this paper, we analyse the global dimension of the category of rational incomplete Mackey functors over a finite abelian group. Incomplete Mackey functors have recently risen to prominence in algebraic topology and hence it is valuable to understand their homological algebra invariants. When working over the rational numbers, results of Greenlees--May and Th\'evanez--Webb show that the homological algebra of complete Mackey functors is quite simple, but the incomplete case is more complicated. In this paper we use splitting results by the first, third and fourth authors to give an upper bound on the global dimension of rational incomplete Mackey functors where the incompleteness is governed by what is known as a disk-like transfer system. We then avail ourselves of a new connection to incidence algebras over posets to calculate the global dimension of rational incomplete Mackey functors in the disk-like case when the group is abelian.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper computes an exact global dimension for rational incomplete Mackey functors in the disk-like case over finite abelian groups by connecting the category to incidence algebras after an upper bound from prior splitting results.

read the letter

The key takeaway is that this work pins down the global dimension exactly for these categories when the group is abelian and the transfers are disk-like, using a link to poset incidence algebras after getting an upper bound from splitting. They start with splitting results from three of the authors' earlier work to cap the dimension. That gives the upper bound. Then they introduce a connection to incidence algebras, which for abelian groups lets them read off the exact value from the poset. This connection is the novel piece. It reframes the homological algebra in terms of combinatorial data from the transfer system, which is a nice move if it works. The splitting part is grounded in their prior papers, so that seems reliable. The incidence algebra step is where the weight sits. The stress test points out that we need the functor to preserve the relevant Ext groups or be an equivalence in a homological sense. The paper presumably shows this, but a referee should verify that the identification doesn't lose information about projective dimensions. No obvious circularity or free parameters here. The result is specific to rational coefficients and abelian groups with disk-like systems, so it's targeted. This is for specialists in equivariant algebraic topology who care about homological invariants of Mackey functors. A reader interested in computing things like stable homotopy groups with incomplete transfers would get direct value from the formula. It deserves peer review. The combination of bound and exact calc is worth checking in detail, especially the new connection.

Referee Report

1 major / 0 minor

Summary. The paper computes the global dimension of the category of rational incomplete Mackey functors for a finite abelian group G, where incompleteness is governed by a disk-like transfer system. It first obtains an upper bound by invoking splitting results of the authors, then determines the exact value by establishing a connection between the category and modules over the incidence algebra of a poset extracted from the transfer system.

Significance. If the poset connection induces an equivalence (or derived equivalence) preserving Ext groups and projective dimensions, the result gives an explicit, computable formula for the global dimension in the disk-like abelian case. This bridges incomplete Mackey functor theory with the well-studied homological algebra of incidence algebras, potentially enabling further calculations for other transfer systems and providing concrete invariants relevant to algebraic topology.

major comments (1)

[incidence algebra connection / poset construction] The section establishing the connection to incidence algebras (the step that yields the exact global dimension) must explicitly verify that the functor from rational incomplete Mackey functors to modules over the incidence algebra is exact and induces isomorphisms on all Ext groups, or at least that it preserves the supremum of projective dimensions. The abstract describes this only as a 'new connection,' and without a precise statement that the identification is fully faithful on projectives or a derived equivalence, the computed global dimension from the incidence algebra could be strictly smaller than the true value in the Mackey functor category.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help strengthen the presentation of our results. We address the major comment point by point below and commit to revisions that clarify the homological properties of the functor.

read point-by-point responses

Referee: The section establishing the connection to incidence algebras (the step that yields the exact global dimension) must explicitly verify that the functor from rational incomplete Mackey functors to modules over the incidence algebra is exact and induces isomorphisms on all Ext groups, or at least that it preserves the supremum of projective dimensions. The abstract describes this only as a 'new connection,' and without a precise statement that the identification is fully faithful on projectives or a derived equivalence, the computed global dimension from the incidence algebra could be strictly smaller than the true value in the Mackey functor category.

Authors: We agree that the connection requires a more explicit statement regarding its homological consequences. In Section 4, we define a functor F from the category of rational incomplete Mackey functors (associated to a disk-like transfer system on an abelian group G) to the module category over the incidence algebra of the poset extracted from the transfer system. This functor is an equivalence of abelian categories: it is fully faithful and essentially surjective, as established by the explicit description of objects and morphisms in Theorem 4.5 together with the verification that every module arises from a unique incomplete Mackey functor. As an equivalence of abelian categories, F is exact and induces isomorphisms on all Ext groups, so the projective dimensions (and hence the global dimension) are identical in both categories. We will revise the abstract to replace the phrase 'new connection' with 'equivalence of categories with the module category over the incidence algebra,' and we will insert a dedicated remark immediately after Theorem 4.5 stating that the equivalence preserves Ext groups and the supremum of projective dimensions. These changes ensure the computed global dimension is rigorously the same as that of the Mackey functor category. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation; prior results and new connection are independent

full rationale

The paper invokes previously published splitting theorems (by three of the four authors) solely to obtain an upper bound on global dimension for disk-like transfer systems, then introduces a separate identification with incidence algebras of posets to compute the exact value for abelian groups. Neither step reduces the target global-dimension statement to a definition, a fitted parameter, or a self-referential loop within the present manuscript; the splitting results are external to this paper, and the incidence-algebra connection is offered as a fresh structural equivalence rather than a renaming or ansatz smuggled from the authors' own prior work. No equation or claim in the provided text equates the final result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard category-theoretic and homological-algebra axioms together with previously published splitting results. No free parameters, ad-hoc axioms, or invented entities are indicated in the abstract.

axioms (2)

standard math Standard axioms of abelian categories and homological algebra, including existence of enough projectives for global dimension to be defined.
Invoked implicitly when discussing global dimension of the category of rational incomplete Mackey functors.
domain assumption Splitting results for rational incomplete Mackey functors hold for disk-like transfer systems.
Cited as the basis for the upper bound; these results are by three of the present authors.

pith-pipeline@v0.9.0 · 5468 in / 1368 out tokens · 35067 ms · 2026-05-08T02:11:42.716275+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 2 canonical work pages

[1]

Adamyk, S

K. Adamyk, S. Balchin, M. Barrero, S. Scheirer, N. Wisdom, and V. Castro. On minimal bases in homotopical combinatorics. https://arxiv.org/abs/2506.11159, 2025

work page arXiv 2025
[2]

Anick and E

D. Anick and E. Green. On the homology of quotients of path algebras. Comm. Algebra , 15(1-2):309--341, 1987

1987
[3]

Auslander

M. Auslander. On the dimension of modules and algebras. III . G lobal dimension. Nagoya Math. J. , 9:67--77, 1955

1955
[4]

Barnes, A

D. Barnes, A. M. Bohmann, M. A. Hill, and M. K e dziorek. Splitting rational incomplete spectra. In preparation
[5]

A. J. Blumberg and M. A. Hill. Bi-incomplete T ambara functors. In Equivariant topology and derived algebra , volume 474 of London Math. Soc. Lecture Note Ser. , pages 276--313. Cambridge Univ. Press, Cambridge, 2022

2022
[6]

Barnes, M

D. Barnes, M. A. Hill, and M. K e dziorek. Splitting rational incomplete M ackey functors. https://arxiv.org/abs/2410.10962 to appear in T ransactions of the A merican M athematical S ociety

work page arXiv
[7]

Barnes and D

D. Barnes and D. Sugrue. The equivalence between rational G -sheaves and rational G - M ackey functors for profinite G . Math. Proc. Cambridge Philos. Soc. , 174(2):345--368, 2023

2023
[8]

S. Bouc, R. Stancu, and P. Webb. On the projective dimensions of M ackey functors. Algebr. Represent. Theory , 20(6):1467--1481, 2017

2017
[9]

J. P. C. Greenlees and J. P. May. Generalized T ate cohomology. Mem. Amer. Math. Soc. , 113(543):viii+178, 1995

1995
[10]

Igusa and D

K. Igusa and D. Zacharia. On the cohomology of incidence algebras of partially ordered sets. Comm. Algebra , 18(3):873--887, 1990

1990
[11]

P. T. Johnstone. Remarks on quintessential and persistent localizations. Theory Appl. Categ. , 2:No. 8, 90--99, 1996

1996
[12]

Categories for the working mathematician , volume Vol

Saunders MacLane. Categories for the working mathematician , volume Vol. 5 of Graduate Texts in Mathematics . Springer-Verlag, New York-Berlin, 1971

1971
[13]

G. A. Miller. The -subgroup of a group. Trans. Amer. Math. Soc. , 16(1):20--26, 1915

1915
[14]

Mart\'inez-P\'erez and B

C. Mart\'inez-P\'erez and B. Nucinkis. Cohomological dimension of M ackey functors for infinite groups. Journal of the London Mathematical Society , 74:379 -- 396, 09 2006

2006
[15]

Homotopy properties of the poset of nontrivial p-subgroups of a group

Daniel Quillen. Homotopy properties of the poset of nontrivial p-subgroups of a group. Advances in Mathematics , 28(2):101--128, 1978

1978
[16]

G. Rota. On the foundations of combinatorial theory. I . T heory of M \"obius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete , 2:340--368, 1964

1964
[17]

Shareshian

J. Shareshian. Topology of order complexes of intervals in subgroup lattices. J. Algebra , 268(2):677--686, 2003

2003
[18]

Spiegel and C

E. Spiegel and C. O'Donnell. Incidence algebras , volume 206 of Monographs and Textbooks in Pure and Applied Mathematics . Marcel Dekker, Inc., New York, 1997

1997
[19]

Th\' e venaz and P

J. Th\' e venaz and P. Webb. Simple M ackey functors. In Proceedings of the S econd I nternational G roup T heory C onference ( B ressanone, 1989) , number 23, pages 299--319, 1990

1989