Abstract
We describe how to obtain a global t-structure from a semiorthogonal decomposition with compatible t-structures on every component. This result is used to generalize a well-known theorem of Bondal on full strong exceptional sequences.
Highlights
Theorem—Bondal. [3, Theorem 6.2] Assume that the bounded derived category Db(X ) of coherent sheaves on a smooth manifold X is generated by a strong exceptional sequence
As a matter of fact, it is incredibly hard to study the general case of triangulated categories; the definition of an exceptional object requires the category to be K-linear, with K a field, and the only known example of non-algebraic K-linear triangulated category is studied in [23]
Definition 2.1 A t-structure on a triangulated category T is a full subcategory T ≤0 closed by left shifts, i.e. T ≤0[1] ⊂ T ≤0, and such that for any object E ∈ T there is a distinguished triangle A → E → B → A[1], where A ∈ T ≤0 and B ∈ T ≥1 := (T ≤0)⊥
Summary
We define t-structures and hearts, and state some classical results. Definition 2.1 A t-structure on a triangulated category T is a full subcategory T ≤0 closed by left shifts, i.e. T ≤0[1] ⊂ T ≤0, and such that for any object E ∈ T there is a distinguished triangle A → E → B → A[1], where A ∈ T ≤0 and B ∈ T ≥1 := (T ≤0)⊥. The cohomology objects (with respect to A ) are defined as H −ki (E) := Ai. Lemma 2.4 Every bounded t-structure T ≤0 is non-degenerate. Given a distinguished triangle E → F → G → E[1], we have an induced exact sequence. The filtration in the definition proves that C ∼= H 0(C ) ∼= C, so we can choose C to be C with the same map appearing in the distinguished triangle.
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