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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.
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 $D(QCoh(O_X)) \subset D(Mod(O_X))$.
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 $D(Coh(O_X)) \subset D(QCoh(O_X))$.
Let $X$ be a noetherian scheme and let $D^b(Coh(O_X))$ denote the bounded derived category of $Coh(O_X)$.
The canonical fully faithful functor
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).
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.
If $X$ is a smooth scheme, there is a canonical equivalence
This follows from the previous proposition combined with (SGA 6, Exp. I, Exemples 5.11).
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.
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
where $p : X \times Y \to X$ and $q : X \times Y \to Y$ are the projections.
See Fourier-Mukai functor for details.
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).
For the model structure on $Mod(O_X)$, see model structure on chain complexes and in particular
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.
For a summary of the results of Bondal-Orlov, see
For a detailed survey, see
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 $Pic^0 \rtimes Aut^0$, see
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
Last revised on March 27, 2020 at 10:23:30. See the history of this page for a list of all contributions to it.