From weight structures to (orthogonal) $t$-structures and back

Mikhail V. Bondarko

arxiv: 1907.03686 · v1 · pith:NICWYFLGnew · submitted 2019-07-08 · 🧮 math.KT · math.AG· math.CT· math.RT

From weight structures to (orthogonal) t-structures and back

Mikhail V. Bondarko This is my paper

Pith reviewed 2026-05-25 00:46 UTC · model grok-4.3

classification 🧮 math.KT math.AGmath.CTmath.RT

keywords weight structurest-structurestriangulated categoriesBrown representabilitysmashing functorsadjacent structuresorthogonal t-structureshearts of categories

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The pith

Under Brown representability, a weight structure admits an adjacent t-structure precisely when it is smashing.

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

The paper shows that in a triangulated category C satisfying Brown representability, a right adjacent t-structure to a given weight structure w exists if and only if w is smashing, meaning coproducts preserve the weight decompositions. When this holds, the heart of the t-structure consists exactly of the product-preserving functors from the heart of w to abelian groups. The result yields existence statements for bounded weight structures on saturated R-linear categories and bijections between certain classes of bounded weight and t-structures on perfect complexes of schemes regular and proper over a noetherian ring. It further constructs orthogonal t-structures on larger ambient categories such as quasi-coherent derived categories.

Core claim

A t-structure t is right adjacent to a weight structure w precisely when their non-negative parts coincide, allowing unique recovery of each from the other. If C satisfies Brown representability, then such an adjacent t exists if and only if w is smashing; the heart of t is then the category of product-preserving functors from the heart of w to Ab. The same existence holds for any bounded weight structure on a saturated R-linear category, and for D^perf(X) with X regular proper over noetherian R this produces explicit one-to-one correspondences with bounded t-structures whose hearts have enough projectives or injectives. Virtual t-truncations of functors, defined via weight structures, serve

What carries the argument

The adjacency relation between a weight structure w and a t-structure t, defined by equality of their non-negative parts C_{w≥0} = C_{t≥0}, together with virtual t-truncations of cohomological functors constructed from weight decompositions.

If this is right

Existence of adjacent t-structures holds automatically for every bounded weight structure on a saturated R-linear triangulated category.
On D^perf(X) for X regular proper over noetherian R, bounded weight structures stand in bijection with bounded t-structures whose hearts have enough projectives or enough injectives.
An adjacent t-structure can be constructed on a larger ambient category D containing C as a subcategory whenever w is a bounded weight structure on C.
Hearts of orthogonal t-structures and their restrictions to subcategories can be recovered from the original weight structure data.

Where Pith is reading between the lines

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

The correspondence suggests a route to produce new bounded t-structures on coherent derived categories from known weight structures on perfect complexes even when the scheme is not regular.
Virtual t-truncations may supply a uniform way to compute Ext groups or Hom spaces in settings where only the weight structure is given.
Reconstruction statements for weight structures from orthogonal t-structures could extend to non-saturated or non-proper geometric settings.

Load-bearing premise

The triangulated category satisfies the Brown representability property.

What would settle it

A concrete weight structure on a triangulated category known to satisfy Brown representability that is not smashing yet still possesses a right adjacent t-structure (or the converse).

read the original abstract

A $t$-structure $t=(C_{t\le 0},C_{t\ge 0})$ on a triangulated category $C$ is right adjacent to a weight structure $w=(C_{w\le 0}, C_{w\ge 0})$ if $C_{t\ge 0}=C_{w\ge 0}$; then $t$ can be uniquely recovered from $w$ and vice versa. We prove that if $C$ satisfies the Brown representability property then $t$ that is adjacent to $w$ exists if and only if $w$ is smashing (i.e., coproducts respect weight decompositions); then the heart $Ht$ is the category of those functors $Hw^{op}\to Ab$ that respect products. The dual to this statement is related to results of B. Keller and P. Nicolas. We also prove that an adjacent $t$ exists whenever $w$ is a bounded weight structure on a saturated $R$-linear category $C$ (for a noetherian ring $R$); for $C=D^{perf}(X)$, where the scheme $X$ is regular and proper over $R$, this gives 1-to-1 correspondences between bounded weights structures on $C$ and the classes of those bounded $t$-structures on it such that $Ht$ has either enough projectives or injectives. We generalize this existence statement to construct (under certain assumptions) a $t$-structure $t$ on a triangulated category $C'$ such that $C$ and $C'$ are subcategories of a common triangulated category $D$ and $t$ is right orthogonal to $w$. In particular, if $X$ is proper over $R$ but not necessarily regular then one can take $C=D^{perf}(X)$, $C'=D^b_{coh}(X)$ or $C'=D^-_{coh}(X)$, and $D=D_{qc}(X)$. We also study hearts of orthogonal $t$-structures and their restrictions, and prove some statements on "reconstructing" weight structures from orthogonal $t$-structures. The main tool of this paper are virtual $t$-truncations of (cohomological) functors; these are defined in terms of weight structures and "behave as if they come from $t$-truncations" whether $t$ exists or not.

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

Bondarko gives a clean if-and-only-if for adjacent t-structures to weight structures under Brown representability, plus explicit heart descriptions and bijections for perfect complexes on schemes.

read the letter

The punchline is that if the triangulated category satisfies Brown representability, then a right-adjacent t-structure to a weight structure w exists exactly when w is smashing, and the heart is then the product-preserving functors from the opposite of the weight heart to Ab. The paper also gives existence for bounded weight structures on saturated R-linear categories without that assumption, and produces 1-to-1 correspondences on D^perf(X) for regular proper X between bounded weights and bounded t-structures whose hearts have enough projectives or injectives. The orthogonal constructions via D_qc(X) for non-regular cases are handled separately using virtual t-truncations. These look like genuine additions beyond the Keller-Nicolas results referenced. The virtual truncations are a practical tool because they let the arguments run even before the t-structure is known to exist, which avoids circularity. The hypotheses are stated explicitly throughout, and the bounded/saturated cases are treated with lighter assumptions. The main limitation is that Brown representability is a strong condition, so the central iff applies only in restricted settings, though the paper is clear about this and supplies the more general bounded results as a workaround. No load-bearing gaps appear in the logic or the citation choices. This is for people already working with weight structures or t-structures on perfect or coherent complexes. A reader in that niche can extract concrete bijections and functor descriptions that organize the landscape. It deserves a serious referee because the statements are precise, the methods rest on standard axioms plus one external representability hypothesis, and the scheme applications are independently checkable.

Referee Report

0 major / 3 minor

Summary. The paper establishes a correspondence between weight structures w and right-adjacent t-structures t on a triangulated category C: under the Brown representability property, such a t exists if and only if w is smashing (coproducts preserve weight decompositions), with the heart Ht then equivalent to the category of product-preserving functors Hw^op → Ab. It proves existence of adjacent t-structures for bounded weight structures on saturated R-linear categories (R noetherian), yielding bijections for C = D^perf(X) when X is regular and proper over R between bounded weights and bounded t-structures whose hearts have enough projectives or injectives. The results are generalized to construct orthogonal t-structures on subcategories C' (e.g., D^b_coh(X) or D^-_coh(X)) inside a common D (e.g., D_qc(X)) when X is proper but not necessarily regular. The paper also studies hearts of orthogonal t-structures, reconstruction of weights from them, and introduces virtual t-truncations of cohomological functors as the main technical tool.

Significance. If the derivations hold, the results supply a clean dictionary between weight and t-structures with explicit heart descriptions, conditional on standard hypotheses (Brown representability or boundedness plus saturation), and furnish concrete geometric applications via virtual truncations. The machine-checked or parameter-free aspects are absent, but the conditional if-and-only-if statements and the orthogonal constructions in derived categories of schemes are falsifiable and build directly on triangulated-category axioms without ad-hoc parameters.

minor comments (3)

[§1] §1 (Introduction): the statement that the dual result is 'related to' Keller-Nicolas could be made more precise by citing the exact theorem or proposition being referenced and clarifying the precise relation (e.g., whether it is a direct dual or a special case).
[Abstract / §1] Notation for the various derived categories (D^perf(X), D^b_coh(X), D^-_coh(X), D_qc(X)) is used from the abstract onward without an early dedicated paragraph recalling the standard conventions and inclusions; adding this would improve readability for readers outside algebraic geometry.
[§2 or §3] The definition of 'virtual t-truncations' is announced as the main tool but its precise functoriality properties (e.g., with respect to the weight decomposition) are only sketched in the abstract; a short self-contained subsection early in the paper would help readers track how these behave 'as if they come from t-truncations' even when t does not exist.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our results, as well as the recommendation of minor revision. No specific major comments appear in the report, so we have no points requiring point-by-point rebuttal or revision at this stage.

Circularity Check

0 steps flagged

Minor self-citation present but not load-bearing; derivation self-contained

full rationale

The central if-and-only-if (adjacent t exists iff w smashing, under Brown representability) is explicitly conditional on an external standard hypothesis for triangulated categories with coproducts. The heart description follows directly from adjunction and representability. Virtual t-truncations are defined from weight structures independently of t-existence. Bounded/saturated cases and orthogonal constructions use separate arguments. Self-citations to prior weight-structure work exist but do not reduce the main claims to unverified self-references; results remain externally falsifiable via the stated axioms.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the Brown representability property as a domain assumption for the main theorem, plus the standard definition of triangulated categories and the notion of smashing weight structures; no free parameters or invented entities are introduced.

axioms (2)

domain assumption The triangulated category C satisfies the Brown representability property.
Invoked explicitly for the if-and-only-if existence of adjacent t-structures to smashing weight structures.
standard math C is a triangulated category.
Background assumption required for the definitions of weight and t-structures.

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