Abstract
In this paper we introduce new categorical notions and give many examples. In an earlier paper we proved that the Bridgeland stability space on the derived category of representations of K(l), the l-Kronecker quiver, is biholomorphic to {{mathbb {C}}} times {mathcal {H}} for lge 3. In the present paper we define a new notion of norm, which distinguishes {D^b(K(l)) }_{lge 2}. More precisely, to a triangulated category {mathcal {T}} which has property of a phase gap we attach a non-negative real number left| {mathcal {T}}right| ^{varepsilon }. Natural assumptions on a SOD {mathcal {T}} =langle {mathcal {T}}_1,{mathcal {T}}_2rangle imply left| langle {mathcal {T}}_1,{mathcal {T}}_2rangle right| ^{varepsilon }le {mathrm{min}}{left| {mathcal {T}}_1 right| ^{varepsilon },left| {mathcal {T}}_2right| ^{varepsilon } }. Using the norm we define a topology on the set of equivalence classes of proper triangulated categories with a phase gap, such that the set of discrete derived categories is a discrete subset, whereas the rationality of a smooth surface S ensures that [D^b(point)] in mathrm{Cl}([D^b(S)]). Categories in a neighborhood of D^b(K(l)) have the property that D^b(K(l)) is embedded in each of them. We view such embeddings as non-commutative curves in the ambient category and introduce categorical invariants based on counting them. Examples show that the idea of non-commutative curve-counting opens directions to new categorical structures and connections to number theory and classical geometry. We give a definition, which specializes to the non-commutative curve-counting invariants. In an example arising on the A side we specialize our definition to non-commutative Calabi–Yau curve-counting, where the entities we count are a Calabi–Yau modification of D^b(K(l)). In the end we speculate that one might consider a holomorphic family of categories, introduced by Kontsevich, as a non-commutative extension with the norm, introduced here, playing a role similar to the classical notion of degree of an extension in Galois theory.
Highlights
Page 3 of 60 45 and prove (Theorem 6.1) that if T = T1, T2 is a semi-orthogonal decomposition in which T is proper,1 rank(K0(T )) < ∞, T1 and T2 have phase gaps, T has phase gap as well and
For a proper T with rank(K0(T )) ≥ 3 we show in Proposition 11.9 that the following condition:2 for each l ∈ N there exists a full exceptional collection (E0, E1, . . . , En) and integers 0 ≤ i < j ≤ n for which hommin(Ei, E j ) ≥ l implies [Db( point)] ∈ Cl ([T ])
In [20] we extend our studies of CA,P (T ) beyond counting: besides numbers we extract from CA,P (T ) categorical versions of classical geometric structures which open new perspectives in non-commutative geometry
Summary
The function (3) depends on ε ∈ (0, 1), the three subsets of its domain determined by the three conditions on the first row in the following table do not depend on ε (Lemma 4.16): Categories with: examples:. 11 using (3) we introduce a topology on the class of proper triangulated categories with a phase gap up to equivalence. The class of discrete derived categories modulo equivalence is a discrete subset w.r. to it For a proper T with rank(K0(T )) ≥ 3 we show in Proposition 11.9 that the following condition: for each l ∈ N there exists a full exceptional collection If this conjecture holds, the presented results would imply that for any smooth projective surface S with a full exceptional collection holds (5) and
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