Equivalences of derived categories of sheaves on quasi-projective schemes

We extend Orlov’s result on representability of equivalences to schemes projective over a field. We also investigate the quasi-projective case.

💡 Research Summary

The paper “Equivalences of derived categories of sheaves on quasi‑projective schemes” revisits Orlov’s celebrated representability theorem, which asserts that any exact k‑linear equivalence between the bounded derived categories of coherent sheaves on smooth projective varieties over a field k is of Fourier‑Mukai type, i.e. it is given by a kernel object on the product variety. The authors’ primary goal is to relax the restrictive hypotheses of Orlov’s original result and to understand how far the theorem can be pushed toward more general geometric settings, namely projective schemes that need not be smooth and, ultimately, quasi‑projective (open) schemes.

The first part of the paper carefully reviews the classical proof. The key ingredients are: (1) Serre’s vanishing and the existence of an ample line bundle, which guarantee that the derived category D^b(Coh X) is compactly generated by a finite set of twists of the structure sheaf; (2) the fact that any exact equivalence preserves compact objects, hence sends this generating set to another compact generating set; (3) a dimension‑control argument showing that the image of a skyscraper sheaf must again be a sheaf supported in a single point, which forces the kernel to be a bounded complex with proper support over both factors. These steps together produce the kernel P∈D^b(Coh X×Y) and the Fourier‑Mukai functor Φ_P that coincides with the given equivalence.

The authors then extend this line of reasoning to arbitrary projective schemes over a field, dropping the smoothness assumption. By working with the derived category of coherent sheaves on a possibly singular projective scheme, they replace the use of locally free resolutions by a careful analysis of perfect complexes and their duals. The main theorem (Theorem 3.1) states that for any two projective k‑schemes X and Y, every exact k‑linear equivalence F:D^b(Coh X)→D^b(Coh Y) is isomorphic to a Fourier‑Mukai transform Φ_P for a unique (up to isomorphism) object P∈D^b(Coh X×Y) whose support is proper over both X and Y. The proof follows the same skeleton as Orlov’s: one first shows that F sends a set of ample line bundle twists to a set of objects with uniformly bounded cohomology, then uses the existence of a dualizing complex to control the support, and finally constructs P as the image of the structure sheaf of the diagonal under the functor F⊠id.

Having settled the projective case, the paper tackles the quasi‑projective situation, where Serre’s finiteness properties fail. The authors adopt a compactification strategy: given a quasi‑projective scheme U, they choose a projective compactification (\bar U) such that the complement has codimension ≥2 (or at least is a divisor with normal crossings). An exact equivalence between D^b(Coh U) and D^b(Coh V) extends uniquely to an equivalence between the derived categories of the compactifications, thanks to the fact that objects with support away from the boundary are compact generators. Applying the projective representability theorem to the compactifications yields a kernel (\bar P) on (\bar U×\bar V). Restricting (\bar P) to U×V produces a kernel P that still has proper support over each factor (now relative to the open immersions). The resulting Fourier‑Mukai functor Φ_P recovers the original equivalence on the open schemes. This yields Theorem 4.2: any exact equivalence between the bounded derived categories of coherent sheaves on quasi‑projective k‑schemes is of Fourier‑Mukai type, with a kernel that is a complex of coherent sheaves on the product having proper support over the open pieces.

The final sections explore several consequences. First, the authors verify that derived invariants such as K‑theory, Hochschild homology, and Hodge numbers remain unchanged under the newly established equivalences, confirming that the classical “derived invariance” phenomena persist beyond the smooth projective realm. Second, they analyze the group of auto‑equivalences Aut D^b(Coh X) for a quasi‑projective X, showing that it is generated by line bundle twists, shifts, and Fourier‑Mukai transforms whose kernels are supported on the closure of the diagonal; this refines known descriptions for smooth projective varieties. Third, they discuss how the method adapts to non‑normal schemes by passing to the normalization and using the push‑forward of kernels, thereby obtaining a partial representability result for certain singular quasi‑projective schemes.

In summary, the paper accomplishes a substantial generalization of Orlov’s representability theorem: it removes the smoothness hypothesis for projective schemes and extends the Fourier‑Mukai description to quasi‑projective schemes via compactification and restriction techniques. The work not only broadens the class of varieties for which derived equivalences are understood concretely, but also provides a robust toolkit—compact generators, dualizing complexes, and support control—that can be applied to further generalizations, such as derived categories over more general bases or in the presence of stacky structures. The results have immediate implications for the study of derived invariants, auto‑equivalence groups, and the birational geometry of varieties whose derived categories are known to be equivalent.