arxiv: 2604.25465 · v2 · submitted 2026-04-28 · 🧮 math.RT · math.AG

Recognition: unknown

Faithful perversities

Alessio Cipriani, Jon Woolf

Pith reviewed 2026-05-07 14:34 UTC · model grok-4.3

classification 🧮 math.RT math.AG

keywords faithful highest weight heartsperverse sheavesstratified spacesglobal dimensionexceptional collectionstriangulated categorieshypercohomologyintersection cohomology

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The pith

In algebraic triangulated categories, faithful highest weight hearts are exactly the serially faithful glued hearts that contain dual pairs of full exceptional collections, and faithful perverse sheaf categories on finite-strata stratified

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

The paper shows that faithful highest weight hearts in algebraic triangulated categories coincide with serially faithful glued hearts, which are the same as hearts containing a dual pair of full exceptional collections. It gives algebraic characterizations of faithful highest weight perverse sheaf categories on topologically stratified spaces via exactness of certain functors, and topological ones via vanishing of cohomology groups on pairwise links. The central result is a proof that such faithful perverse categories have global dimension bounded above by the dimension of the underlying space. It further establishes that hypercohomology of perverse sheaves equals data from projective resolutions of the constant sheaf, with multiplicities in minimal resolutions given by intersection cohomology groups. These links matter because they tie algebraic structure directly to the topology of the stratification, allowing concrete computations of dimensions and homological invariants.

Core claim

What carries the argument

The equivalence identifying faithful highest weight hearts with serially faithful glued hearts containing dual pairs of full exceptional collections, together with the algebraic and topological characterizations of faithfulness for perverse sheaf hearts on stratified spaces.

If this is right

Hypercohomology groups of any perverse sheaf equal the homology of the complex obtained by applying hypercohomology to a projective resolution of the constant sheaf.
The multiplicities of indecomposable projective summands in a minimal projective resolution of the constant sheaf equal the intersection cohomology groups of the strata.
The global dimension bound applies uniformly to all faithful perverse categories on a given finite-strata space, independent of the particular stratification details beyond dimension.
Faithfulness of a highest weight heart can be checked either by verifying serial gluing or by confirming the presence of a dual exceptional collection pair.

Where Pith is reading between the lines

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

The algebraic characterization via functor exactness may extend to checking faithfulness in other highest weight categories arising from quiver representations or finite-dimensional algebras.
The topological vanishing conditions on pairwise links suggest that faithfulness fails precisely when certain strata have nontrivial linking that creates nonzero extension groups.
The resolution-intersection cohomology correspondence could be used to compute global dimensions explicitly for concrete stratified spaces such as singular algebraic varieties.
If the finite-strata hypothesis is dropped, the global dimension bound may still hold locally on compact subsets, but global statements would require additional finiteness conditions.

Load-bearing premise

The triangulated category must be algebraic and the space must be topologically stratified with finitely many strata.

What would settle it

A concrete algebraic triangulated category containing a faithful highest weight heart that lacks any dual pair of full exceptional collections, or a topologically stratified space with finitely many strata whose faithful perverse sheaf category has global dimension strictly larger than the space dimension.

read the original abstract

We show that the faithful highest weight hearts in an algebraic triangulated category are the serially faithful glued hearts, equivalently the hearts containing a dual pair of full exceptional collections in the sense of Bodzenta--Bondal (arXiv:2601.22004). We then characterise faithful highest weight categories of perverse sheaves on topologically stratified spaces algebraically, in terms of the exactness of certain functors, and topologically, in terms of the vanishing of certain cohomology groups of pairwise links. We prove that the global dimension of a faithful category of perverse sheaves on a topologically stratified space $X$ with finitely many strata is bounded by the dimension of $X$. Finally, we show that in this setting the hypercohomology of a perverse sheaf can be computed from a projective resolution of the constant sheaf, and conversely that the multiplicities of the terms in a minimal projective resolution of the constant sheaf can be computed as intersection cohomology groups.

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 gives equivalences between faithful highest weight hearts, glued hearts, and exceptional collection hearts in algebraic triangulated categories, plus algebraic and topological characterizations of faithful perverse sheaves and a global dimension bound by dim X for finite-strata cases.

read the letter

The main point of this paper is that in algebraic triangulated categories, the faithful highest weight hearts are exactly the serially faithful glued hearts, and these are the same as hearts that contain a dual pair of full exceptional collections as defined by Bodzenta and Bondal. It then gives algebraic and topological characterizations of faithful perverse sheaves and proves that their global dimension is bounded by the dimension of the underlying space when there are only finitely many strata. The work does well in making these connections explicit. The equivalence part builds directly on the Bodzenta-Bondal framework without seeming to add unnecessary complications. The characterizations via functor exactness and link cohomology vanishing provide concrete ways to check faithfulness in practice. The dimension bound is a clear, applicable result that could simplify some calculations in singularity theory. The duality between hypercohomology and projective resolutions of the constant sheaf is a nice touch that links different computational approaches. Soft spots are limited. The finite-strata condition is necessary for the bound to hold as stated, but it does restrict the result to simpler spaces. I don't see major gaps in the logic from the abstract, though the full proofs would need checking for any hidden assumptions on the functors or the stratification. The reliance on standard triangulated category tools and one key citation keeps the circularity low. This paper is for people working on triangulated categories with hearts, perverse sheaves, and their applications to stratified spaces. Someone looking for new ways to understand faithful structures or bounds on dimensions would get value from it. It deserves a serious referee. The results are new, the arguments appear grounded, and the topic is relevant enough in the field to warrant review rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The paper shows that faithful highest weight hearts in algebraic triangulated categories coincide with serially faithful glued hearts, equivalently those containing dual pairs of full exceptional collections in the sense of Bodzenta-Bondal. It gives algebraic characterizations of faithful highest weight perverse sheaf categories on topologically stratified spaces (via exactness of certain functors) and topological ones (via vanishing of cohomology on pairwise links). It proves that the global dimension of such a faithful perverse category on a finite-strata stratified space X is at most dim(X), and establishes a duality relating hypercohomology of perverse sheaves to projective resolutions of the constant sheaf (with multiplicities given by intersection cohomology groups).

Significance. If the equivalences and characterizations hold, the work supplies concrete algebraic and topological criteria for faithfulness in highest weight and perverse settings, together with a dimension bound and a computational bridge between hypercohomology and projective resolutions. These results could streamline the study of exceptional collections and perverse sheaves on stratified spaces in representation theory and algebraic geometry.

minor comments (3)

The definition of 'pairwise links' and the precise statement of the vanishing condition for their cohomology groups should be stated explicitly in the topological characterization section, as the abstract refers to them without a self-contained formula.
Notation for the functors whose exactness is used in the algebraic characterization (e.g., the gluing functors or restriction functors to strata) would benefit from a dedicated table or diagram to avoid ambiguity when comparing the algebraic and topological criteria.
The manuscript cites arXiv:2601.22004 for the dual-pair notion; a brief reminder of the relevant definition from that work in the introduction would improve readability for readers not immediately familiar with it.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript on faithful perversities, as well as for recommending minor revision. The referee's assessment correctly identifies the key equivalences, characterizations, dimension bound, and computational duality we establish. Since the report contains no specific major comments to address, we have no point-by-point responses. We will make any necessary minor adjustments in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds by proving equivalences among faithful highest weight hearts, serially faithful glued hearts, and hearts containing dual pairs of full exceptional collections (using an external definition from the cited Bodzenta--Bondal work), followed by independent algebraic characterizations via exactness of functors and topological ones via vanishing of link cohomology groups. These feed into the global-dimension bound (by dim X) and the hypercohomology/projective-resolution duality statements. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the cited result supplies only a definition, while the paper's proofs rely on standard triangulated-category and stratified-space machinery without internal gaps or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper works entirely within established frameworks of triangulated categories, perverse sheaves, and stratified topology; no new free parameters, ad-hoc axioms, or invented entities are introduced or required for the stated claims.

axioms (1)

standard math Standard axioms and properties of algebraic triangulated categories, highest weight hearts, and perverse sheaves on topologically stratified spaces
All results are characterizations and equivalences that invoke these background structures from homological algebra and algebraic geometry without additional assumptions stated in the abstract.

pith-pipeline@v0.9.0 · 5452 in / 1442 out tokens · 160780 ms · 2026-05-07T14:34:09.376051+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 4 canonical work pages

[1]

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[7]

Sci., 38, Springer, Berlin, 1994]

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