Fourier-Mukai functors and perfect complexes on dual numbers. (English) Zbl 1323.14012
The article under review studies the question whether every fully faithful functor from the dual numbers \(A=k[\epsilon]/(\epsilon^2)\) to some \({\mathbf D}(\text{Qcoh}\,Y)\) is of Fourier-Mukai type, i.e., can be represented as an integral functor arising from an object on \(\text{Spec}\,A\times Y\). An important result in this direction shows that this is true if the domain is a projective scheme such that the structure sheaf does not have zero-dimensional torsion [V. A. Lunts and D. O. Orlov, J. Am. Math. Soc. 23, No. 3, 853–908 (2010; Zbl 1197.14014)]. This condition is used to ensure the existence of an ample sequence, but does not apply to the scheme \(\text{Spec}\,A\).
In this paper a careful study of \(\text{Perf}\,A\) is conducted, in order to show that fully faithful functors from \(\text{Perf}\,A\) and \({\mathbf D}^{\text{b}}(A)\) are of Fourier-Mukai type, showing that at least in this case the condition on the torsion of the structure sheaf can be removed. Moreover the authors obtain a description of the t-structures on \({\mathbf D}^{\text{b}}(A)\) and its stability manifold \(\text{Stab}({\mathbf D}^{\text{b}}(A))\).
In this paper a careful study of \(\text{Perf}\,A\) is conducted, in order to show that fully faithful functors from \(\text{Perf}\,A\) and \({\mathbf D}^{\text{b}}(A)\) are of Fourier-Mukai type, showing that at least in this case the condition on the torsion of the structure sheaf can be removed. Moreover the authors obtain a description of the t-structures on \({\mathbf D}^{\text{b}}(A)\) and its stability manifold \(\text{Stab}({\mathbf D}^{\text{b}}(A))\).
Reviewer: Pieter Belmans (Antwerpen)
MSC:
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
Citations:
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