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CoHAs of Torsion Sheaves on Weighted Projective Curves

Timm Peerenboom

TL;DR

This work computes the cohomological Hall algebra $CoHA$ of torsion sheaves on weighted projective lines with weights $(2,\dots,2)$ by presenting it as the algebra $\mathscr{P}_n$ generated by explicit elements and relations. It leverages the derived equivalence between weighted projective lines and canonical algebras to identify $CoHA(\mathrm{Tor}(\mathbb{P}^1(\boldsymbol{\lambda};2^n)))$ with the regular-representation CoHA of the canonical algebra and employs stratifications to prove ChowHA$=$CoHA on these stacks. The paper builds a functorial framework relating Hall algebras across weight data, constructs the PBW-type algebra $\mathscr{P}_n$, and verifies it for small cases $n=0,1,2$, culminating in an explicit isomorphism $CoHA(\mathbb{P}^1(2^n))\cong\mathscr{P}_n$ and a closed-form Poincaré series. These results provide a concrete, generator–relation description of CoHAs for these geometric settings, enabling explicit computations of DT-type invariants and their graded structures in this family. The approach highlights the role of tilting equivalences, affine stratifications, and stability in controlling the algebraic structure of CoHAs.

Abstract

We describe the cohomological Hall algebra of torsion sheaves on a weighted projective line with weights $(2, \dots, 2)$ in terms of generators and relations.

CoHAs of Torsion Sheaves on Weighted Projective Curves

TL;DR

This work computes the cohomological Hall algebra of torsion sheaves on weighted projective lines with weights by presenting it as the algebra generated by explicit elements and relations. It leverages the derived equivalence between weighted projective lines and canonical algebras to identify with the regular-representation CoHA of the canonical algebra and employs stratifications to prove ChowHACoHA on these stacks. The paper builds a functorial framework relating Hall algebras across weight data, constructs the PBW-type algebra , and verifies it for small cases , culminating in an explicit isomorphism and a closed-form Poincaré series. These results provide a concrete, generator–relation description of CoHAs for these geometric settings, enabling explicit computations of DT-type invariants and their graded structures in this family. The approach highlights the role of tilting equivalences, affine stratifications, and stability in controlling the algebraic structure of CoHAs.

Abstract

We describe the cohomological Hall algebra of torsion sheaves on a weighted projective line with weights in terms of generators and relations.

Paper Structure

This paper contains 24 sections, 46 theorems, 139 equations.

Table of Contents

  1. CoHAs and ChowHAs for Categories of Homological Dimension One
  2. Moduli of Objects in Abelian Categories and Donaldson--Thomas Invariants
  3. Convolution Product and Cohomological Hall Algebras
  4. Chow Hall Algebra
  5. The Simple ChowHA
  6. Weighted Projective Line and the Canonical Algebra
  7. Weighted Projective Line
  8. Canonical Algebra
  9. The Stack of Regular Representations
  10. The Relationship between Canonical Algebras and Weighted Projective Lines
  11. Stratification of the Stacks Md
  12. Functoriality of Hall Algebras
  13. Morphisms between Weighted Projective Lines
  14. Functoriality of Hall Algebras
  15. The Algebras Pn
  16. ...and 9 more sections

Key Result

Proposition 1.12

If the cohomology of the stack $\mathfrak{M}_{\mathbf{d}}$ is pure for all dimension vectors ${\mathbf{d}}$, then we can recover the Poincaré series of the algebra $\mathop{\mathrm{CoHA}}\nolimits(\mathcal{A})$ from its class in the ring $K_0(\mathop{\mathrm{MHS}}\nolimits)[\mathbb{L}^{-1/2} , (\mat

Theorems & Definitions (104)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 1.7: meinhardt2015donaldsonthomasinvariantsvsintersection
  • Remark 1.8
  • Definition 1.9
  • Remark 1.10
  • ...and 94 more