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triangulated categories of sheaves

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Idea

Let (X,O X)(X, O_X) be a ringed space. In analogy with modules over a commutative ring, one may study (X,O X)(X, O_X) by studying the category Mod(O X)Mod(O_X) of sheaves of modules over O XO_X. There is a symmetric monoidal cofibrantly generated model structure on Mod(O X)Mod(O_X). The homotopy category with respect to this model structure, i.e. the derived category of Mod(O X)Mod(O_X), is a triangulated category D(Mod(O X))D(Mod(O_X)) associated to XX. Here we discuss various interesting subcategories of D(Mod(O X))D(Mod(O_X)). In particular, the bounded derived category of coherent sheaves is an important invariant studied by the school of Bondal, Orlov and others. The triangulated category of perfect complexes is a very significant construction in noncommutative algebraic geometry.

Definitions

Derived category of quasi-coherent sheaves

Recall that one has the full abelian subcategory QCoh(O X)Mod(O X)QCoh(O_X) \subset Mod(O_X) of quasi-coherent sheaves, which are the ones corresponding locally to modules over a ring. One gets a triangulated subcategory D(QCoh(O X))D(Mod(O X))D(QCoh(O_X)) \subset D(Mod(O_X)).

Derived category of coherent sheaves

When XX is a noetherian scheme, one has the full abelian subcategory Coh(O X)QCoh(O X)Coh(O_X) \subset QCoh(O_X) of coherent sheaves; these are the ones corresponding locally to finitely generated modules. Hence one gets the triangulated subcategory D(Coh(O X))D(QCoh(O X))D(Coh(O_X)) \subset D(QCoh(O_X)).

Bounded derived category of coherent sheaves

Let XX be a noetherian scheme and let D b(Coh(O X))D^b(Coh(O_X)) denote the bounded derived category of Coh(O X)Coh(O_X).

Proposition

The canonical fully faithful functor

D b(Coh(O X))D(Mod(O X)) D^b(Coh(O_X)) \hookrightarrow D(Mod(O_X))

identifies D b(Coh(O X))D^b(Coh(O_X)) with the full subcategory of D(Mod(O X))D(Mod(O_X)) of bounded complexes whose cohomology objects are coherent sheaves.

See (SGA 6, Exp. II, Corollaire 2.2.2.1).

Triangulated category of perfect complexes

Let XX be a ringed space and let Pf(X)D(Mod(O X))Pf(X) \subset D(Mod(O_X)) denote the full subcategory of perfect complexes of O XO_X-modules. This is a triangulated subcategory that is contained in D coh(Mod(O X))D(Mod(O X))D_{coh}(Mod(O_X)) \subset D(Mod(O_X)), the full subcategory of complexes with coherent cohomology. It is stable under derived tensor product, derived inverse image, and derived direct image of proper morphisms.

Proposition

If XX is a smooth scheme, there is a canonical equivalence

Pf(X)D b(Coh(O X)) Pf(X) \stackrel{\sim}{\to} D^b(Coh(O_X))

of triangulated categories.

See (SGA 6, Exp. I).

Derived equivalence

The bounded derived category of coherent sheaves D b(Coh(O X))D^b(Coh(O_X)) is usually called simply the derived category of XX and denoted D(X)D(X). It is an important invariant of XX, which has been studied extensively by Mukai?, Bondal-Orlov, and others.

There are several geometrically interesting examples of non-isomorphic varieties XX and YY with D(X)D(Y)D(X) \simeq D(Y); for example, the derived category of an abelian variety XX is equivalent to the derived category of its dual X^\hat{X}. In such cases one often says that XX and YY are derived equivalent. However, the derived category is not a weak invariant of XX, indeed in some cases it is as strong as isomorphism; see Bondal-Orlov reconstruction theorem. In fact, the construction XD(X)X \mapsto D(X) seems to lose just enough information so that derived equivalence becomes a geometrically interesting invariant.

Fourier-Mukai functors

Theorem (Orlov)

Let XX and YY be smooth projective varieties over a field KK. Let F:D(X)D(Y)F : D(X) \to D(Y) be a triangulated fully faithful functor. There exists an object ED(X×Y)E \in D(X \times Y) which is unique up to isomorphism and is such that FF is isomorphic to the Fourier-Mukai transform of EE, that is the functor

FRq *(Lp *(F) LE)) F \mapsto Rq_*(Lp^*(F) \otimes^L E))

where p:X×YXp : X \times Y \to X and q:X×YYq : X \times Y \to Y are the projections.

See Fourier-Mukai functor for details.

Derived invariants

Let us say that a functor on the category of smooth projective varieties is a derived invariant if it maps derived equivalent varieties to isomorphic objects.

References

For the model structure on Mod(O X)Mod(O_X), see model structure on chain complexes and in particular

  • Denis-Charles Cisinski, F. Déglise, Local and stable homologial algebra in Grothendieck abelian categories, Homology, Homotopy and Applications, vol. 11 (1) (2009) (url)

Derived categories of quasi-coherent sheaves

  • R. Thomason, T. Trobaugh. Higher algebraic K-theory of schemes and of derived categories. The Grothendieck Festschrift, Vol. III Birkh¨auser, Boston (1990), 247-436.

  • Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc., vol. 9, no. 1, 1996, pp. 205-236, doi.

Bounded derived category of coherent sheaves

For a summary of the results of Bondal-Orlov, see

For a detailed survey, see

  • Dmitri Orlov, Derived categories of coherent sheaves and equivalences between them, Russian Math. Surveys, 58 (2003), 3, 89-172, translation.

On the relationship between derived categories and Chow motives, see

For some discussion of the above result and the relationship between derived categories and additive invariants? see

For the derived invariance of Pic 0Aut 0Pic^0 \rtimes Aut^0, see

  • Rouquier?, Automorphismes, graduations et categories triangulees, arXiv.

Triangulated category of perfect complexes

Last revised on January 2, 2016 at 18:15:28. See the history of this page for a list of all contributions to it.