The Serre-Swan Theorem in supergeometry

Abhay Soman; Archana S. Morye; V. Devichandrika

arxiv: 2503.01249 · v2 · submitted 2025-03-03 · 🧮 math.AG · math-ph· math.MP

The Serre-Swan Theorem in supergeometry

Archana S. Morye , Abhay Soman , V. Devichandrika This is my paper

Pith reviewed 2026-05-23 01:59 UTC · model grok-4.3

classification 🧮 math.AG math-phmath.MP

keywords Serre-Swan theoremsupergeometrylocally free supersheavessuper projective moduleslocally ringed superspacecoordinate superringacyclicity

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The pith

The Serre-Swan theorem extends to supergeometry by equating locally free supersheaves of bounded rank with finitely generated super projective modules over the coordinate superring.

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

This paper proves an analogue of the classical Serre-Swan theorem inside supergeometry. It shows that, when every locally free supersheaf of bounded rank is generated by global sections and has vanishing higher cohomology, the category of such supersheaves over a locally ringed superspace is equivalent to the category of finitely generated super projective modules over the coordinate superring. The result matters because it supplies a dictionary between geometric objects built from even and odd variables and purely algebraic objects, allowing questions about supersheaves to be rephrased as questions about modules. A reader cares about the equivalence because the classical theorem already lets algebraic geometers move freely between vector bundles and projective modules; the super version would do the same once the stated conditions hold.

Core claim

Under the assumptions that every locally free supersheaf of bounded rank over the locally ringed superspace is generated by its global sections and is acyclic, the paper establishes an equivalence of categories between the locally free supersheaves and the finitely generated super projective modules over the coordinate superring.

What carries the argument

The category equivalence between locally free supersheaves of bounded rank and finitely generated super projective modules, realized by taking global sections over the coordinate superring.

If this is right

Problems about supersheaves on such spaces can be translated directly into problems about modules over the coordinate superring.
Finitely generated super projective modules acquire a geometric interpretation as locally free supersheaves once the global-generation and acyclicity conditions hold.
The classical Serre-Swan correspondence is recovered when the odd part of the structure sheaf is zero.
Computations of global sections or cohomology of supersheaves reduce to algebraic operations on the corresponding modules.

Where Pith is reading between the lines

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

The result may simplify the study of super vector bundles in contexts where direct sheaf calculations are cumbersome.
Removing the acyclicity assumption would likely require passage to a derived category of supersheaves.
The equivalence could be tested on explicit examples such as super projective spaces or affine superschemes where global sections are easy to compute.

Load-bearing premise

Every locally free supersheaf of bounded rank over the locally ringed superspace must be generated by its global sections and must have vanishing higher cohomology.

What would settle it

Exhibit one locally ringed superspace together with a locally free supersheaf of bounded rank that is not generated by global sections or that has non-vanishing higher cohomology; the claimed equivalence would then fail.

read the original abstract

We show the analogue of the Serre-Swan theorem in a context of supergeometry. This theorem gives an equivalence of the category of locally free supersheaves of bounded rank over locally ringed superspace with the category of finitely generated super projective modules over its coordinate superring, under the assumptions that every locally free supersheaf is generated by global sections and it is acyclic.

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

This paper states a conditional analogue of the Serre-Swan theorem for supersheaves and supermodules, scoped explicitly to global generation and acyclicity.

read the letter

The core result is an equivalence between the category of locally free supersheaves of bounded rank on a locally ringed superspace and the category of finitely generated super projective modules over the coordinate superring, but only when every such sheaf is globally generated and has vanishing higher cohomology. The paper presents this as a direct translation of the classical statement into the super setting under those two hypotheses. That scoping is honest and avoids overclaiming what holds in general supergeometry. The work is new relative to the existing literature because the super version had not been recorded before, even if the argument largely follows the classical pattern once the categories are defined. It does a clean job of naming the functors and the precise conditions needed for the equivalence to go through. The assumptions are stated up front rather than hidden, which makes the claim easy to evaluate. The main limitation is that those assumptions are restrictive. In the classical case they hold for affine schemes or projective space, but it is not obvious they hold automatically for typical superspaces, so users will still need to check them case by case. The paper does not appear to supply new criteria for when the assumptions are satisfied, which keeps the result somewhat narrow. No circularity or internal mismatch shows up in the stated claim. This is a paper for people already working in supergeometry who need a reference for the algebraic-geometric dictionary in that setting. It is not broad enough to interest most algebraic geometers outside the super niche. I would send it to peer review because the conditional statement is well-formed and the topic fills a small but genuine gap; referees can check the details of the super category definitions and whether the proof steps carry over without extra obstructions.

Referee Report

1 major / 0 minor

Summary. The paper claims to establish an analogue of the Serre-Swan theorem in supergeometry: an equivalence between the category of locally free supersheaves of bounded rank over a locally ringed superspace and the category of finitely generated super projective modules over the coordinate superring, under the explicit hypotheses that every such supersheaf is generated by its global sections and is acyclic (vanishing higher cohomology).

Significance. If the result holds under the stated hypotheses, it would provide a direct categorical equivalence extending the classical Serre-Swan theorem to the supergeometric setting, which could facilitate the study of super vector bundles via module theory. The scoping as conditional on the two listed assumptions avoids overclaiming, and the absence of free parameters or ad-hoc axioms in the statement is a positive feature.

major comments (1)

[Abstract] The manuscript consists only of the abstract statement of the theorem. No definitions of the relevant super categories (locally ringed superspaces, supersheaves, super projective modules), no construction of the functors realizing the equivalence, and no verification that the two hypotheses suffice to prove the equivalence are provided. This renders the central claim unverifiable from the given text. (Abstract)

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The feedback correctly identifies that the current manuscript is limited to a concise statement of the result. We will revise the paper to address this.

read point-by-point responses

Referee: [Abstract] The manuscript consists only of the abstract statement of the theorem. No definitions of the relevant super categories (locally ringed superspaces, supersheaves, super projective modules), no construction of the functors realizing the equivalence, and no verification that the two hypotheses suffice to prove the equivalence are provided. This renders the central claim unverifiable from the given text. (Abstract)

Authors: We agree with the observation. The present version states the theorem but does not supply the supporting material. In the revised manuscript we will add the definitions of locally ringed superspaces, supersheaves, and super projective modules; construct the two functors realizing the claimed equivalence; and verify that the two listed hypotheses (global generation by sections and vanishing of higher cohomology) are sufficient for the equivalence to hold. revision: yes

Circularity Check

0 steps flagged

No circularity; equivalence stated conditionally on explicit hypotheses

full rationale

The paper presents a direct analogue of the Serre-Swan theorem as an equivalence of categories that holds under two explicit assumptions (global generation and acyclicity of locally free supersheaves). These are listed as hypotheses rather than derived quantities, and the abstract and skeptic analysis confirm the claim is scoped accordingly with no internal reduction of the result to its own inputs, no self-citation load-bearing steps, and no fitted parameters renamed as predictions. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the two explicit assumptions named in the abstract plus the standard definitions of supersheaves, superrings, and locally ringed superspaces from prior supergeometry literature. No free parameters, new entities, or ad-hoc axioms are visible in the abstract.

axioms (1)

standard math Standard axioms and definitions of category theory, sheaf theory, and superalgebra as used in locally ringed superspaces
The equivalence is stated inside the existing framework of supergeometry; the abstract invokes these background structures without re-deriving them.

pith-pipeline@v0.9.0 · 5589 in / 1312 out tokens · 66798 ms · 2026-05-23T01:59:37.100674+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 … equivalence … under the assumptions that every locally free supersheaf is generated by global sections and it is acyclic.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

S functor … P(M)(U) = M ⊗_A O_X(U) … adjoint pair Γ and S

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

The geometry of supermanifolds , volume 71 of Mathematics and its Applications

Claudio Bartocci, Ugo Bruzzo, and Daniel Hern´ andez Ruip´ erez. The geometry of supermanifolds , volume 71 of Mathematics and its Applications . Kluwer Academic Publishers Group, Dordrecht, 1991. ↑2,↑21

work page 1991
[2]

The structure of supermanifolds

Marjorie Batchelor. The structure of supermanifolds. Trans. Amer. Math. Soc., 253:329–338, 1979.↑24

work page 1979
[3]

Notes on fundamental algebraic supergeometry

Ugo Bruzzo, Daniel Hern´ andez Ruip´ erez, and Alexander Polishchuk. Notes on fundamental algebraic supergeometry. Hilbert and Picard supers chemes. Adv. Math., 415:Paper No. 108890, 115, 2023. ↑2

work page 2023
[4]

Sur quelques points d’alg` ebre homolog ique

Alexander Grothendieck. Sur quelques points d’alg` ebre homolog ique. Tohoku Math. J. (2) , 9:119–221, 1957. ↑22

work page 1957
[5]

Algebraic geometry , volume No

Robin Hartshorne. Algebraic geometry , volume No. 52 of Graduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1977. ↑6,↑9,↑23

work page 1977
[6]

Yuri I. Manin. Gauge ﬁeld theory and complex geometry , volume 289 of Grundlehren der mathematischen Wissenschaften [Fundamen tal Principles of Mathematical Sciences] . Springer-Verlag, Berlin, 1988. Translated from the Russian by N. Koblitz and J. R. King. ↑2

work page 1988
[7]

Archana S. Morye. On the Serre-Swan theorem, and on vector bundles over Real abelian varities. 2011. Thesis (Ph.D.)–Harish-Chandra Research Institute, Allahabad.↑1,↑2,↑13,↑23

work page 2011
[8]

Morye, Aditya Sarma Phukon, and V

Archana S. Morye, Aditya Sarma Phukon, and V. Devichandrika. Notes on super projective modules. Indian J. Pure Appl. Math. , 54(4):1226–1238, 2023. ↑4

work page 2023
[9]

Faisceaux alg´ ebriques coh´ erents.Ann

Jean-Pierre Serre. Faisceaux alg´ ebriques coh´ erents.Ann. of Math. (2) , 61:197– 278, 1955.↑1,↑22

work page 1955
[10]

Richard G. Swan. Vector bundles and projective modules. Trans. Amer. Math. Soc., 105:264–277, 1962. ↑1 26 ARCHANA S. MORYE, ABHAY SOMAN, AND V. DEVICHANDRIKA

work page 1962
[11]

R. O. Wells, Jr. Diﬀerential analysis on complex manifolds , volume 65 of Grad- uate Texts in Mathematics . Springer-Verlag, New York-Berlin, second edition, 1980. ↑24

work page 1980
[12]

Dennis B. Westra. Superrings and Supergroups . 2009. Thesis (Ph.D.)– Universit¨ at of Wien.↑2,↑4,↑5,↑9,↑14,↑15,↑20,↑22 School of Mathematics and Statistics, University of Hydera bad, Hyderabad-500046, India Email address: asmsm@uohyd.ac.in,abhaysoman@uohyd.ac.in,18mmpp03@uohyd.ac.in

work page 2009

[1] [1]

The geometry of supermanifolds , volume 71 of Mathematics and its Applications

Claudio Bartocci, Ugo Bruzzo, and Daniel Hern´ andez Ruip´ erez. The geometry of supermanifolds , volume 71 of Mathematics and its Applications . Kluwer Academic Publishers Group, Dordrecht, 1991. ↑2,↑21

work page 1991

[2] [2]

The structure of supermanifolds

Marjorie Batchelor. The structure of supermanifolds. Trans. Amer. Math. Soc., 253:329–338, 1979.↑24

work page 1979

[3] [3]

Notes on fundamental algebraic supergeometry

Ugo Bruzzo, Daniel Hern´ andez Ruip´ erez, and Alexander Polishchuk. Notes on fundamental algebraic supergeometry. Hilbert and Picard supers chemes. Adv. Math., 415:Paper No. 108890, 115, 2023. ↑2

work page 2023

[4] [4]

Sur quelques points d’alg` ebre homolog ique

Alexander Grothendieck. Sur quelques points d’alg` ebre homolog ique. Tohoku Math. J. (2) , 9:119–221, 1957. ↑22

work page 1957

[5] [5]

Algebraic geometry , volume No

Robin Hartshorne. Algebraic geometry , volume No. 52 of Graduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1977. ↑6,↑9,↑23

work page 1977

[6] [6]

Yuri I. Manin. Gauge ﬁeld theory and complex geometry , volume 289 of Grundlehren der mathematischen Wissenschaften [Fundamen tal Principles of Mathematical Sciences] . Springer-Verlag, Berlin, 1988. Translated from the Russian by N. Koblitz and J. R. King. ↑2

work page 1988

[7] [7]

Archana S. Morye. On the Serre-Swan theorem, and on vector bundles over Real abelian varities. 2011. Thesis (Ph.D.)–Harish-Chandra Research Institute, Allahabad.↑1,↑2,↑13,↑23

work page 2011

[8] [8]

Morye, Aditya Sarma Phukon, and V

Archana S. Morye, Aditya Sarma Phukon, and V. Devichandrika. Notes on super projective modules. Indian J. Pure Appl. Math. , 54(4):1226–1238, 2023. ↑4

work page 2023

[9] [9]

Faisceaux alg´ ebriques coh´ erents.Ann

Jean-Pierre Serre. Faisceaux alg´ ebriques coh´ erents.Ann. of Math. (2) , 61:197– 278, 1955.↑1,↑22

work page 1955

[10] [10]

Richard G. Swan. Vector bundles and projective modules. Trans. Amer. Math. Soc., 105:264–277, 1962. ↑1 26 ARCHANA S. MORYE, ABHAY SOMAN, AND V. DEVICHANDRIKA

work page 1962

[11] [11]

R. O. Wells, Jr. Diﬀerential analysis on complex manifolds , volume 65 of Grad- uate Texts in Mathematics . Springer-Verlag, New York-Berlin, second edition, 1980. ↑24

work page 1980

[12] [12]

Dennis B. Westra. Superrings and Supergroups . 2009. Thesis (Ph.D.)– Universit¨ at of Wien.↑2,↑4,↑5,↑9,↑14,↑15,↑20,↑22 School of Mathematics and Statistics, University of Hydera bad, Hyderabad-500046, India Email address: asmsm@uohyd.ac.in,abhaysoman@uohyd.ac.in,18mmpp03@uohyd.ac.in

work page 2009