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Starfish lemma via birational quasi-isomorphisms

Dmitriy Voloshyn

TL;DR

The paper develops a framework of birational quasi-isomorphisms between cluster structures of geometric type to transfer cluster-theoretic and algebraic properties across varieties. It axiomatizes these maps and proves a Starfish-lemma analogue that yields inclusions for upper cluster algebras and, under suitable conditions, equality with coordinate rings. A key contribution is showing how a single birational quasi-isomorphism, and then a pair of complementary ones, allow one to deduce the coordinate-ring status of upper cluster algebras in complex settings (including Gekhtman–Shapiro–Vainshtein-type constructions and Poisson-compatible structures). The results provide a robust toolkit for establishing when upper cluster algebras realize coordinate rings of varieties and for transferring regularity, coprimality, and primeness properties across birationally related cluster structures.

Abstract

We study birational quasi-isomorphisms between normal Noetherian domains endowed with cluster structures of geometric type. We prove an analogue of the Starfish lemma that allows one to transfer various cluster and algebraic properties of one variety onto another. In particular, we develop tools for proving that an upper cluster algebra equals the given commutative ring.

Starfish lemma via birational quasi-isomorphisms

TL;DR

The paper develops a framework of birational quasi-isomorphisms between cluster structures of geometric type to transfer cluster-theoretic and algebraic properties across varieties. It axiomatizes these maps and proves a Starfish-lemma analogue that yields inclusions for upper cluster algebras and, under suitable conditions, equality with coordinate rings. A key contribution is showing how a single birational quasi-isomorphism, and then a pair of complementary ones, allow one to deduce the coordinate-ring status of upper cluster algebras in complex settings (including Gekhtman–Shapiro–Vainshtein-type constructions and Poisson-compatible structures). The results provide a robust toolkit for establishing when upper cluster algebras realize coordinate rings of varieties and for transferring regularity, coprimality, and primeness properties across birationally related cluster structures.

Abstract

We study birational quasi-isomorphisms between normal Noetherian domains endowed with cluster structures of geometric type. We prove an analogue of the Starfish lemma that allows one to transfer various cluster and algebraic properties of one variety onto another. In particular, we develop tools for proving that an upper cluster algebra equals the given commutative ring.
Paper Structure (23 sections, 25 theorems, 37 equations, 3 figures)

This paper contains 23 sections, 25 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Background on cluster algebras
  4. Toric actions.
  5. Definitions and basic properties
  6. Related cluster structures
  7. Quasi-isomorphisms
  8. Birational quasi-isomorphisms
  9. Main results
  10. Single birational quasi-isomorphism
  11. Preliminary lemmas
  12. Transferring cluster and algebraic properties
  13. Prime elements
  14. Proof of Proposition \ref{['p:birat_single']}
  15. Proof of Proposition \ref{['p:fbirat_single']}
  16. ...and 8 more sections

Key Result

Proposition 2.7

Assume that the initial exchange matrix in $\mathcal{C}$ has full rank. Then the upper bounds $\bar{\mathcal{A}}(\mathcal{N})$ do not depend on the choice of the nerve $\mathcal{N}$ and $\bar{\mathcal{A}}(\mathcal{N}) = \bar{\mathcal{A}}(\mathcal{C})$.

Figures (3)

  • Figure 1: Initial quivers for $\bar{\mathcal{C}}$, $\tilde{\mathcal{C}}$, $\hat{\mathcal{C}}$ and $\mathcal{C}$; here, $\mathop{\mathrm{\mathbf{\Gamma}}}\nolimits = (\{2\},\{1\},2 \mapsto 1)$ and $\mathop{\mathrm{\mathbf{\Gamma}}}\nolimits_{\text{std}} = (\emptyset,\emptyset,\emptyset\rightarrow \emptyset)$.
  • Figure 2: Initial seed for cluster structure $\tilde{\mathcal{C}}$ in $\mathbb{C}[\widehat{\mathop{\mathrm{Gr}}\nolimits}(3,7)]$.
  • Figure 3: Initial seed for cluster structure ${\mathcal{C}}$ in $\mathbb{C}[\widehat{\mathop{\mathrm{Gr}}\nolimits}(3,7)]$.

Theorems & Definitions (66)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Definition 2.8
  • Definition 2.9
  • Proposition 2.10: Starfish Lemma
  • ...and 56 more