nLab triangulated categories of sheaves

Idea

Let $(X, O_X)$ be a ringed space. In analogy with modules over a commutative ring, one may study $(X, O_X)$ by studying the category $Mod(O_X)$ of sheaves of modules over $O_X$. There is a symmetric monoidal cofibrantly generated model structure on $Mod(O_X)$. The homotopy category with respect to this model structure, i.e. the derived category of $Mod(O_X)$, is a triangulated category $D(Mod(O_X))$ associated to $X$. Here we discuss various interesting subcategories of $D(Mod(O_X))$. 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 $QCoh(O_X) \subset Mod(O_X)$ of quasi-coherent sheaves, which are the ones corresponding locally to modules over a ring. One gets a triangulated subcategory $D(QCoh(O_X)) \subset D(Mod(O_X))$.

Derived category of coherent sheaves

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

Bounded derived category of coherent sheaves

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

Proposition

The canonical fully faithful functor

$$D^b(Coh(O_X)) \hookrightarrow D(Mod(O_X))$$

identifies $D^b(Coh(O_X))$ with the full subcategory of $D(Mod(O_X))$ 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 $X$ be a ringed space and let $Pf(X) \subset D(Mod(O_X))$ denote the full subcategory of perfect complexes of $O_X$-modules. This is a triangulated subcategory that is contained in $D_{coh}(Mod(O_X)) \subset D(Mod(O_X))$, 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 $X$ is a smooth scheme, there is a canonical equivalence

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

of triangulated categories.

This follows from the previous proposition combined with (SGA 6, Exp. I, Exemples 5.11).

Derived equivalence

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

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

Fourier-Mukai functors

Theorem (Orlov)

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

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

where $p : X \times Y \to X$ and $q : X \times Y \to Y$ 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.

• The canonical ring? is a derived invariant (Orlov 2003).

• Orlov has proved that for abelian varieties, the functor $X \mapsto X \times \hat{X}$ is a derived invariant (Orlov 2003).

• Rouquier? has proved that the functor $X \mapsto Pic^0(X) \rtimes Aut^0(X)$ valued in algebraic groups is a derived invariant (Rouquier 2010). (Note that this a generalization of Orlov’s result on abelian varieties above.)

• Orlov has proved that the derived category $D(X)$ determines the rational Chow motive $M(X)$ up to Tate twists. Under a certain condition on the corresponding object $E \in D(X \times Y)$, this may be refined to derived invariance of the integral Chow motive. In particular the corresponding statements are true for any Weil cohomology theory (e.g. etale cohomology, Betti cohomology, de Rham cohomology, etc.). See (Orlov, 2005).

• The derived category $D(X)$ determines the noncommutative Chow motive $NM(X)$ up to isomorphism. In particular, it follows that all additive invariants?, like K-theory or Hochschild homology, are derived invariants. See (Yusufzai).

Related concepts

• abelian sheaf cohomology

• derived category of coherent sheaves

• perfect complex

• deformation theory of sheaves

References

• SGA 6

For the model structure on $Mod(O_X)$, 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

• Aleksei Bondal, Dmitri Orlov, Derived categories of coherent sheaves, arXiv.

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

• Dmitri Orlov, Derived categories of coherent sheaves and motives.

  math.AG/0512620

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

• Adeel Khan Yusufzai, Perfect correspondences and Chow motives, master’s thesis, 2013. arXiv.

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

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

Triangulated category of perfect complexes

• Luc Illusie. Le complexe cotangent I et II, Lecture Notes in Math. 239, 283, Springer-Verlag, Berlin (1971,1972).

• Andre Hirschowitz, Carlos Simpson. Descente pour les n-champs, arXiv:math/9807049

• Stacks Project, tag 08CL