Simple-minded systems and reduction for negative Calabi-Yau triangulated categories

Abstract

We develop the basic properties of w w -simple-minded systems in ( − w ) (-w) -Calabi-Yau triangulated categories for w ⩾ 1 w \geqslant 1 . We show that the theory of simple-minded systems exhibits striking parallels with that of cluster-tilting objects. The main result is a reduction technique for negative Calabi-Yau triangulated categories. Our construction provides an inductive technique for constructing simple-minded systems.

Reductions of triangulated categories and simple‐minded collections

“…The third aim of this paper is to connect our SMC reductions and the SMS reductions defined by Coelho Simões and Pauksztello [16]. We first show that the SMC reduction of a CY triple gives rise to a new CY triple.…”

“…The following proposition holds for any pre‐SMC, where Part (1) is due to [1, Corollary 3 and Proposition 4] or [31, Proposition 5.4] and Parts (2) and (3) are due to [16, Lemmas 2.7 and 2.8]. Proposition Let $$\mathcal {R}$$ be a pre‐SMC of $$\mathcal {T}$$.…”

“…By [16, Lemma 2.8], the condition (3) above is equivalent to saying that $$\mathcal {T}=\mathcal {H}[d-1]\ast \mathcal {H}[d-2]\ast \cdots \ast \mathcal {H}$$.…”

“…Simple‐minded systems (SMSs) in stable module categories were introduced in [30] and studied for negative CY triangulated categories in [15, 17]. Recently, there is increasing interest in negative CY triangulated categories (see, for example, [7, 12–16, 19]), including the stable categories of Cohen–Macaulay (CM) dg modules [21].…”

“…For a $$(1-d)$$‐CY triple $$(\mathcal {T}, \mathcal {T}^{\mathrm{p}}, \mathcal {S})$$, we know $$\mathcal {T}_{\operatorname{sg}\nolimits }$$ is a $$(-d)$$‐CY triangulated category by Theorem 1.2 (2), and we can consider the SMS reduction $$(\mathcal {T}_{\operatorname{sg}\nolimits })_{\mathcal {R}}$$ in $$\mathcal {T}_{\operatorname{sg}\nolimits }$$ with respect to $$\mathcal {R}$$ introduced in [16]. Our main theorem of this paper shows that SMS reduction is the shadow of SMC reduction in the following sense.…”