Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Serre-Swan Theorem in supergeometry

Archana S. Morye, Abhay Soman, V. Devichandrika

TL;DR

This work extends the classical Serre-Swan correspondence to the realm of supergeometry. By formulating and exploiting the adjoint pair (Γ, S) between global sections and the tensor-based functor S, it establishes an equivalence between finitely generated superprojective modules over the global function ring and locally free supersheaves of bounded rank on a locally ringed superspace, under the assumptions that such supersheaves are generated by global sections and acyclic. The key steps involve developing the sheaf-theoretic machinery for supermodules, proving full faithfulness of S, and deriving essential surjectivity through global generation and Hom-sheaf techniques. The results are exemplified in the affine setting and for split supermanifolds, where acyclicity and fineness conditions are satisfied, confirming the broad applicability of the super-Serre-Swan theorem in supergeometry. Overall, the paper provides a foundational dictionary linking algebraic and geometric objects in supercontexts, enabling concrete translations between finitely generated superprojective modules and locally free supersheaves.

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.

The Serre-Swan Theorem in supergeometry

TL;DR

This work extends the classical Serre-Swan correspondence to the realm of supergeometry. By formulating and exploiting the adjoint pair (Γ, S) between global sections and the tensor-based functor S, it establishes an equivalence between finitely generated superprojective modules over the global function ring and locally free supersheaves of bounded rank on a locally ringed superspace, under the assumptions that such supersheaves are generated by global sections and acyclic. The key steps involve developing the sheaf-theoretic machinery for supermodules, proving full faithfulness of S, and deriving essential surjectivity through global generation and Hom-sheaf techniques. The results are exemplified in the affine setting and for split supermanifolds, where acyclicity and fineness conditions are satisfied, confirming the broad applicability of the super-Serre-Swan theorem in supergeometry. Overall, the paper provides a foundational dictionary linking algebraic and geometric objects in supercontexts, enabling concrete translations between finitely generated superprojective modules and locally free supersheaves.

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.

Paper Structure

This paper contains 11 sections, 34 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and Notations
  4. Super-analogues of a few results from commutative algebra
  5. Basics of supersheaves
  6. Category of sheaves of supermodules
  7. Properties of an adjoint pair $\Gamma$ and $\mathcal{S}$
  8. Super Serre-Swan theorem
  9. Some examples
  10. Affine superschemes
  11. Split supermanifolds

Key Result

Theorem 1.1

Let $(X,\mathcal{O}_X)$ be a locally ringed superspace and $A=\Gamma(X,\mathcal{O}_X)$. Assume that every locally free supersheaf of bounded rank is acyclic and generated by finitely many global sections. Then the functor $\mathcal{S}$ defines an equivalence of categories from the category ${\rm\bf

Theorems & Definitions (91)

  • Theorem 1.1
  • Example 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4: Free supermodules and tensor product
  • proof
  • Lemma 2.5
  • Theorem 2.6: Cayley-Hamilton
  • ...and 81 more