[Paper Review] Monadicity theorem and weighted projective lines of tubular type

Jianmin Chen, Xiao‐Wu Chen+1 more|arXiv (Cornell University)|Jul 31, 2014

Algebraic structures and combinatorial models3 citations

TL;DR

This paper establishes explicit equivalences between the category of coherent sheaves on a weighted projective line of tubular type and that on an elliptic curve via equivariantization, using a novel formulation of Beck’s monadicity theorem in abelian categories. The key result shows that two different cyclic group actions—degree-shift and twisting—induce adjoint equivalences between these categories, unifying and generalizing known results in tubular algebra and elliptic curve sheaf theory.

ABSTRACT

We formulate a version of Beck's monadicity theorem for abelian categories, which is applied to the equivariantization of abelian categories with respect to a finite group action. We prove that the equivariantization is compatible with the construction of quotient abelian categories by Serre subcategories. We prove that the equivariantization of the graded module category over a graded ring is equivalent to the graded module category over the same ring but with a different grading. We deduce from these results two equivalences between the category of (equivariant) coherent sheaves on a weighted projective line of tubular type and that on an elliptic curve, where the acting groups are cyclic and the two equivalences are somehow adjoint to each other.

Motivation & Objective

• To provide a uniform, explicit treatment of the equivalence between coherent sheaves on weighted projective lines of tubular type and those on elliptic curves.
• To reformulate Beck’s monadicity theorem for abelian categories to facilitate applications in equivariantization.
• To prove that equivariantization commutes with quotient constructions via Serre subcategories.
• To establish equivalences between graded module categories under different gradings via group actions.
• To demonstrate that the homogeneous coordinate algebra of an elliptic plane curve arises as a restriction of that of a tubular weighted projective line.

Proposed method

• Formulate a version of Beck’s monadicity theorem tailored for abelian categories, with a self-contained proof.
• Use the isomorphism between equivariant objects and modules over a monad to link group actions to monadic structures.
• Prove compatibility of equivariantization with quotient abelian categories by Serre subcategories via exact monads.
• Establish equivalence between graded module categories under degree-shift and refined gradings using group actions.
• Apply these results to the homogeneous coordinate algebras of weighted projective lines and elliptic curves.
• Use the isomorphism between the coordinate algebra of an elliptic plane curve and a restriction subalgebra of a tubular weighted projective line to derive the main equivalences.

Experimental results

Research questions

• RQ1 How can Beck’s monadicity theorem be adapted for abelian categories to study equivariantization?
• RQ2 Under what conditions does equivariantization commute with quotient constructions by Serre subcategories?
• RQ3 How does equivariantization of a graded module category relate to changes in grading structure?
• RQ4 What is the precise relationship between the category of coherent sheaves on a weighted projective line of tubular type and that on an elliptic curve?
• RQ5 Can the two known equivalences between tubular weighted projective lines and elliptic curves be derived uniformly from general categorical principles?

Key findings

• A new formulation of Beck’s monadicity theorem is proven for abelian categories, enabling direct applications to equivariantization.
• Equivariantization of an abelian category is compatible with quotient constructions by Serre subcategories, as shown via exact monads.
• The equivariantization of the graded module category over a graded ring under a degree-shift action is equivalent to the graded module category over the same ring with a coarser grading.
• For finite abelian group actions, the equivariantization of the graded module category under a twisting action is equivalent to the graded module category with a refined grading.
• The homogeneous coordinate algebra of an elliptic plane curve is isomorphic to a restriction subalgebra of the coordinate algebra of a weighted projective line of tubular type.
• The main result establishes two adjoint equivalences: one between coherent sheaves on a tubular weighted projective line and those on an elliptic curve via degree-shift action, and the dual via twisting action, both with cyclic groups.