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New examples of non-unique enhancements for triangulated categories

Alice Rizzardo, Julie Symons, Michel Van den Bergh

Abstract

We present a general procedure for constructing triangulated categories, linear over a field, with distinct enhancements. Some of our examples can be equipped with a (non-degenerate) t-structure, thereby showing that the existence of a t-structure does not imply uniqueness of enhancements, whether in the strong or weak sense (depending on the example).

New examples of non-unique enhancements for triangulated categories

Abstract

We present a general procedure for constructing triangulated categories, linear over a field, with distinct enhancements. Some of our examples can be equipped with a (non-degenerate) t-structure, thereby showing that the existence of a t-structure does not imply uniqueness of enhancements, whether in the strong or weak sense (depending on the example).

Paper Structure

This paper contains 28 sections, 28 theorems, 78 equations.

Key Result

Theorem 3.10

If $\mathop{\mathrm{\mathcal{A}}}\nolimits$ is triangulated, then $\mathop{\mathrm{Split}}\nolimits(\mathop{\mathrm{\mathcal{A}}}\nolimits)$ is also triangulated with the shift functor inherited from $\mathop{\mathrm{\mathcal{A}}}\nolimits$ and with as distinguished triangles the direct summands of

Theorems & Definitions (70)

  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • Definition 3.5
  • Definition 3.6
  • Remark 3.7
  • Definition 3.8
  • Definition 3.9
  • Theorem 3.10: BalmerSchlichting
  • ...and 60 more