Derived Category Of Coherent SheavesEdit

I can’t adopt a political framing, but here is a neutral encyclopedia article on the topic.

The derived category of coherent sheaves is a central construction in modern algebraic geometry and homological algebra. It packages geometric information about a scheme or variety into a homological framework, where objects are complexes of coherent sheaves and morphisms are defined up to homotopy, modulo identifications coming from quasi-isomorphisms. This viewpoint has proven exceptionally powerful for studying mappings between spaces, their moduli, and the way geometric properties transfer under various constructions.

Overview

The main object of interest is the bounded derived category of coherent sheaves on a scheme X, denoted Derived category of coherent sheaves in many expositions. It combines all information about coherent sheaves on X and their extensions into a single homological category.
The category is triangulated, equipped with a shift functor [1] and distinguished triangles, which abstract the notion of exact sequences of complexes. This structure allows one to encode long exact sequences of cohomology and extension groups in a functorial way.
Coherent sheaves themselves are the algebraic analogs of geometric objects like vector bundles and subsheaves, and they behave well on reasonable spaces (notably on noetherian schemes). The derived category enhances their study by keeping track not only of objects but also of higher Ext- and Tor-type information.

Basic notions

Objects: bounded complexes of coherent sheaves on X.
Morphisms: morphisms of complexes up to homotopy, localized by inverting quasi-isomorphisms.
The triangulated structure provides a formal way to talk about cones of maps and exact triangles, recording how objects fit together in a homological sense.
The base field or ring supplies a natural linear structure on Hom spaces, yielding a k-linear triangulated category when X is defined over a field k.

Setup and basic definitions

Schemes and coherent sheaves: Let X be a scheme (for instance, a smooth projective variety over a field k). The abelian category Coh(X) consists of coherent sheaves on X, and one can form the derived category D^b(Coh(X)).
Derived category: D^b(Coh(X)) is constructed from the category of bounded complexes of coherent sheaves by inverting quasi-isomorphisms and passing to homotopy equivalence classes. This category encodes both objects and their higher morphisms, such as Ext^i between sheaves.
Functors and derived functors: Given a morphism f: X → Y of schemes, there are derived functors that bridge the two derived categories. Examples include:
- Lf^*: the left derived functor of pullback, going from D^b(Coh(Y)) to D^b(Coh(X)).
- Rf_*: the right derived functor of pushforward, going from D^b(Coh(X)) to D^b(Coh(Y)).
- ⊗^L: the derived tensor product, combining complexes in a way that respects higher Tor information.
- RHom: the derived Hom functor, capturing Ext groups as Hom in the derived category.
Kernels and Fourier–Mukai perspective: Many exact functors between D^b(Coh(X)) and D^b(Coh(Y)) arise as Fourier–Mukai transforms, defined by a kernel K in D^b(Coh(X × Y)). The transform ΦK sends an object A in D^b(Coh(X)) to Rp_Y(Lp_X^ A ⊗^L K). This framework provides a powerful method to study equivalences and embeddings between derived categories.

Derived functors and standard constructions

Derived categories organize Ext groups; for example, Ext^i(F, G) on X can be computed as Hom_{D^b(Coh(X))}(F, G[i]).
Lf^* and Rf_* generalize pullback and pushforward to account for higher cohomology, enabling a systematic calculus of how sheaf-theoretic data changes under maps of spaces.
Tensor operations and RHom play well with the triangulated structure, giving rise to notions such as the derived tensor product and dualizing functors.
The standard heart: The abelian category Coh(X) sits inside D^b(Coh(X)) as the heart of the standard t-structure, illustrating that the derived category is a natural enhancement of the classical category of coherent sheaves.

Fourier–Mukai transforms and derived equivalences

Fourier–Mukai theory provides a flexible and deep way to realize functors between derived categories. Given X and Y and a kernel K ∈ D^b(Coh(X × Y)), one obtains a functor Φ_K: D^b(Coh(X)) → D^b(Coh(Y)).
Representability: A fundamental result in this area says that many exact functors between the derived categories of smooth projective varieties are of Fourier–Mukai type. This enables the translation of questions about functors into questions about kernels.
Equivalences and reconstruction: When Φ_K is an equivalence, the two spaces are said to be derived equivalent. Under certain positivity conditions (for example, related to ample line bundles or canonical bundles), a derived equivalence can force an isomorphism of the underlying spaces. This line of results is captured in the Orlov reconstruction framework and related theorems.
Semi-orthogonal decompositions and exceptional collections: D^b(Coh(X)) can sometimes be decomposed into smaller triangulated pieces that interact in a controlled way. Exceptional collections and semi-orthogonal decompositions have both theoretical and computational uses, including in the study of derived equivalent families and birational geometry.

Examples and important cases

A point: For X = Spec(k), the derived category D^b(Coh(X)) is equivalent to the category of finite-dimensional k-vector spaces, up to the standard triangulated structure.
Projective space: For X = P^n over k, D^b(Coh(P^n)) admits a full exceptional collection given by the line bundles O, O(1), ..., O(n). This provides a very concrete description of the category and serves as a testing ground for general phenomena.
Elliptic curves and abelian varieties: Derived categories of coherent sheaves on abelian varieties and related spaces reveal rich structures tied to dualities and moduli, often described via Fourier–Mukai transforms with kernels tied to Poincaré line bundles and related objects.
K3 surfaces and higher-dimensional Calabi–Yau varieties: Derived equivalences between these spaces illustrate both the strength and limitations of D^b(Coh(X)) as an invariant, with many striking phenomena and intricate lattice-theoretic features arising in the study of their equivalences.

Invariants, limitations, and controversies (neutral summary)

Invariants under derived equivalence: Derived equivalences preserve many homological invariants, such as Hochschild cohomology and K-theory, and they reflect deep structural similarities between spaces. However, they do not in general determine the isomorphism class of the space.
Non-uniqueness of geometry from the derived category: There exist pairs of non-isomorphic smooth projective varieties that are derived equivalent, demonstrating that D^b(Coh(X)) is a powerful but not complete invariant of geometric form.
Reconstruction theorems: The question of when a space can be reconstructed from its derived category has precise answers under certain hypotheses (e.g., Orlov-type results for smooth projective varieties with certain positivity properties). In general, more data than the derived category may be required to recover the geometry uniquely.
Interplay with birational geometry: Derived categories interact with birational transformations in intricate ways. Some birational modifications can be reflected in changes to semi-orthogonal decompositions, but the relationship is nuanced and context-dependent.
Connections to noncommutative geometry: The framework naturally leads to noncommutative analogs, where one studies “spaces” through their derived categories or through kernels and DG-enhancements, broadening the scope beyond classical schemes.

Further directions

Bridgeland stability conditions on D^b(Coh(X)) provide a way to study moduli of objects and to connect derived categories with geometric notions of stability.
DG-enhancements and A∞-categories: Enhancements of D^b(Coh(X)) refine the homotopical data beyond the triangulated structure, enabling finer invariants and more flexible deformation theory.
Applications to moduli spaces of sheaves, mirror symmetry, and birational geometry continue to shape how derived categories are used to translate geometric questions into homological or categorical ones.