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Hochschild cohomology of graded gentle algebras and intrinsic formality

Sebastian Opper

Abstract

We describe the (bigraded) Hochschild cohomology of graded gentle algebras along with the Gerstenhaber bracket and cup product. In particular, this yields a description of the Hochschild cohomology of partially wrapped Fukaya categories of surfaces in the sense of Haiden-Katzarkov-Kontsevich which have at least one stop. Our results are an important ingredient in the author's recent description of the derived Picard group of partially wrapped Fukaya categories and graded gentle algebras. As another application we provide a characterisation of intrinsically formal graded gentle algebras under mild assumptions.

Hochschild cohomology of graded gentle algebras and intrinsic formality

Abstract

We describe the (bigraded) Hochschild cohomology of graded gentle algebras along with the Gerstenhaber bracket and cup product. In particular, this yields a description of the Hochschild cohomology of partially wrapped Fukaya categories of surfaces in the sense of Haiden-Katzarkov-Kontsevich which have at least one stop. Our results are an important ingredient in the author's recent description of the derived Picard group of partially wrapped Fukaya categories and graded gentle algebras. As another application we provide a characterisation of intrinsically formal graded gentle algebras under mild assumptions.
Paper Structure (18 sections, 30 theorems, 45 equations)

This paper contains 18 sections, 30 theorems, 45 equations.

Table of Contents

  1. Gentle algebras and Fukaya categories
  2. Graded gentle algebras and partially wrapped Fukaya categories
  3. Paths and anti-paths
  4. Smoothness and properness
  5. The AAG invariant and boundary components
  6. Hochschild cohomology of graded gentle algebras
  7. Bigraded Hochschild cohomology of a graded algebra
  8. Gerstenhaber bracket and cup product
  9. The bigraded Bardzell resolution
  10. Comparison maps between Hochschild and Bardzell complex
  11. Induced operations on the Bardzell resolution
  12. Computations of the Hochschild cohomology
  13. The graph and canonical decompositions
  14. Classification of atomic cocycles
  15. The cup product and the Gerstenhaber bracket
  16. ...and 3 more sections

Key Result

Theorem A

Let $A$ be a graded gentle algebras, possibly neither homologically smooth nor proper and let $\Sigma_A$ denote its graded marked surface. Then $\operatorname{HH}^{\bullet}(A,A)$ has a $\Bbbk$-linear (Schauder) basis whose elements are in bijection with the set consisting of the following elements: Moreover, $f_i \in \operatorname{HH}^1(A,A)$, $\mathcal{N}^s(\gamma) \in \operatorname{HH}^{\omega+

Theorems & Definitions (65)

  • Theorem A
  • Theorem B
  • Theorem C: \ref{['thm: intrinsic formality graded gentle algebras']}
  • Definition 1.1
  • Definition 1.2: HaidenKatzarkovKontsevich
  • Theorem 1.3: HaidenKatzarkovKontsevich
  • Remark 1.4
  • Remark 1.5
  • Definition 1.6
  • Proposition 1.7: HaidenKatzarkovKontsevich
  • ...and 55 more