Let F ⊆ P 3 be a smooth determinantal quartic surface which is general in the Nöther–Lefschetz sense. In the present paper we give a complete classification of locally free sheaves E of rank 2 on F ...such that h 1 ( F , E ( t h ) ) = 0 for t ∈ Z .
It has long been known that a key ingredient for a sheaf representation of a universal algebra A consists in a distributive lattice of commuting congruences on A. The sheaf representations of ...universal algebras (over stably compact spaces) that arise in this manner have been recently characterised by Gehrke and van Gool (J. Pure Appl. Algebra, 2018), who identified the central role of the notion of softness.
In this paper, we extend the scope of this theory by replacing varieties of algebras with Barr-exact categories, thus encompassing a number of “non-algebraic” examples. Our approach is based on the notion of K-sheaf: intuitively, whereas sheaves are defined on open subsets, K-sheaves are defined on compact ones. Throughout, we consider sheaves on complete lattices rather than spaces; this allows us to obtain point-free versions of sheaf representations whereby spaces are replaced with frames.
These results are used to construct sheaf representations for the dual of the category of compact ordered spaces, and to recover Banaschewski and Vermeulen's point-free sheaf representation of commutative Gelfand rings (Quaest. Math., 2011).
This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded ...derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a foundation for completing a triangulated category.
We exploit the theory of ∞-stacks to provide some basic definitions and calculational tools regarding stratified homotopy theory of stratified topological stacks.
We show that, for any fixed weight, there is a natural system of Hodge sheaves, whose Higgs field has no poles, arising from a flat projective family of varieties parametrized by a regular complex ...base scheme, extending the analogous classical result for smooth projective families due to Griffiths. As an application, based on positivity of direct image sheaves, we establish a criterion for base spaces of rational Gorenstein families to be of general type. A key component of our arguments is centered around the construction of derived categorical objects generalizing relative logarithmic forms for smooth maps and their functorial properties.
Let K be a field of characteristic 0. Fix integers r, d coprime with r ⩾ 2. Let XK be a smooth, projective, geometrically connected curve of genus g ⩾ 2 defined over K. Assume there exists a line ...bundle ${\cal L}_K$ on XK of degree d. In this paper we prove the existence of a stable locally free sheaf on XK with rank r and determinant ${\cal L}_K$. This trivially proves the C1 conjecture in mixed characteristic for the moduli space of stable locally free sheaves of fixed rank and determinant over a smooth, projective curve.
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