As an example we study the derived category of finite dimensional vector spaces over a field.
Example 5.2.1(Dpkvectq). Letkvectbe the category of finite dimensional vector spaces over a fieldk. It is well known to be a semisimple abelian category. This implies that for any morphism f :V1ÑV2 of finite dimensional k-vector spaces we haveV1–kerpfq ‘ImpfqandV2–V2{Impfq ‘Impfq. Hence any complexX‚ in Cpkvectqis isomorphic to a complex of the form
Y‚:. . .i1p3Imdi´2X‚ ‘ pkerdi´1X‚{Imdi´1X‚q ‘Imdi´1X‚ i1p3Imdi´1X‚ ‘ pkerdiX‚{Imdi´1X‚q ‘ImdiX‚i1p3. . . Consider the complex
Z‚:. . . 0 pkerdi´1X‚{Imdi´1X‚q 0 pkerdiX‚{Imdi´1X‚q 0 . . .
We show that the morphism p2 :Y‚ ÑZ‚ is an isomorphism in KpQ-vectqwith inverse given by i2 :Z‚ ÑY‚. By definition of biproduct p2i2 “IdZ‚. We show that the morphism i2p2 :Y‚ ÑY‚ is homotopic to IdY‚. Let χi :YiÑYi´1 be the morphismχi“i3p1. Then IdYi´i2p2“i1p1`i3p3“i1p3i3p1`i3p1i1p3, which shows that IdY‚ „i2p2. ThusY‚, and henceX‚, is isomorphic to the complexZ‚inKpkvectq. SinceH‚pZ‚q –Z‚ it is easy to see that the all the quasi-isomorphisms are already invertible in Kpkvectq. Thus Kpkvectq –Dpkvectq, and Dpkvectqis equivalent to the full abelian subcategory ofCpkvectqconsisting of complexes with zero differentials.
In the above example the derived category turned out to be an abelian category. This is because kvect is semisimple. Furthermore, the derived categoryDpAqis an abelian category if an only ifA is semisimple, [GM03, Exercise IV.1.1]. We have already seen in example 2.2.17 thatAbis not semisimple. Thus the derived category of an abelian category is not in general an abelian category.
The idea following example is taken from [HTR10, p.191 4.15]. In particular, it gives an example where local-ization of a category is not a category.
Example 5.2.2(Derived category not a category). In this example we show that the derived category of an abelian category is not necessarily a category because the collection of morphisms may fail to be a set. This also shows that localization of categories is not well-defined in general, if one does not use the axiom of strongly inaccessible cardinals.
Let U denote the class of all small cardinals and let A be the category of all finite dimensional pZ{2qrU s-Z{2-bimodules, where the dimension is taken as Z{2 vector space. One can verify that this category is an abelian category. We show thatDpAqis not a category by deriving a contradiction when assuming its existence.
Suppose that DpAqis a category. For any small cardinal λ, let Vλ be the pZ{2qrUs-Z{2-bimoduleZ{2‘Z{2 with the action
α.pz1, z2q “
#pz2,0q α“λ p0,0q α‰λ
Consider the following two morphisms inCpAq
. . . 0 Z{2 Vλ 0 . . .
. . . 0 0 Z{2 0 . . .
i1
p2
and
. . . 0 Z{2 Vλ 0 . . .
. . . 0 Z{2 0 0 . . .
i1
IdZ{2
where the morphisms i1 are differentials at index 0, the first morphism is a quasi-isomorphism and the bimodules Z{2 have trivial action of pZ{2qrUs. Denote the first morphism by sλ and the second by fλ. Denote the domain complex ofsλbyZλ‚, the codomain complex byX‚and the codomain complex of the morphismfλbyY‚“X‚r1s.
Consider morphisms from X‚ toY‚. For any small cardinalλthe following roof represents such a morphism Zλ‚
X‚ Y‚
sλ
fλ
We show that for two different cardinals α ‰ λ the corresponding roofs represent different morphisms, so that MorDpAqpX‚, Y‚qcannot be a set, because the class of all small cardinals is not a set.
Letα‰λbe different small cardinals. If the corresponding roofs would be equal, we would have a commutative diagram of the form inKpAq
W‚
Zλ‚ Zα‚
X‚ Y‚
s1λ
s1α
sλ
fλ sα
fα
Now sλs1λ : W‚ ÑX‚ is a quasi-isomorphism, and H0pX‚q “ Z{2, so W0 must be nonzero. By definition of a morphism ofpZ{2qrUs-Z{2-bimodules, for a nonzero elementwPW0 we have
ps1λq0pλ.pwqq “λ.pps1λq0pwqq ‰0 ps1αq0pλ.pwqq “λ.pps1αq0pwqq “0.
This shows that the image of λ.w under H0psλs1λq : H0pW‚q Ñ H0pX‚q is zero, but the image of λ.w under H0psαs1αq:H0pW‚q ÑH0pX‚qis nonzero. Hence the diagram cannot be commutative inKpAq.
5.3 Notes
Derived categories were initially developed by Grothendieck and Verdier to generalize Serre duality to relative case.
In this it was needed that the direct image functor f˚ to have a right adjoint, which is impossible in the category of schemes over a field k, becausef˚ is not right exact. When one passes to the derived category, the functor f˚
induces a derived functor between derived categories, see the next chapter, which preserves distinguished triangles and this functor has a right adjoint.
We have already noted that triangulated categories arise in many branches of mathematics, see section 4.4. This allows one to study connections between different branches of mathematics, by using the formalism of triangulated categories. In some sense, homological mirror symmetry can be seen to be an example of such connection.
Chapter 6
Derived functors
In this chapter we introduce right and left derived functors. We prove the existence of right derived functors and state the corresponding results for left derived functors. A right (resp. left) derived functor is an exact functor between derived categories which satisfies a certain universal property and is obtained from a left (resp. right) exact functor between the underlying abelian categories. This construction agrees with the classical derived functors. At the end of this chapter we give examples of derived functors.
Classically, to form theith right derived functorRiF (resp. left derived functorLiF) of a left (resp. right) exact functor F :AÑB, the idea is to assign an injective (resp. projective) resolution to each object of A, apply the functorFpointwise to this resolution, and then take theith cohomology of the resulting complex. Here we generalize this idea and construct the right (resp. left) derived functor by using an adapted classRof objects ofA, with respect to the functor F, and then we takeRF (resp. LF) to be the composite of an inverse of K`pRqrSR´1s Ñ D`pAq (resp. K´pRqrS´1R s ÑD´pAq) given by the universal property of localization of categories, followed by the unique functor K`pRqrSR´1s Ñ D`pBq (resp. K´pRqrSR´1s ÑD´pBq) given by the universal property of localization of
Note that in this chapter, as in the diagrams above, we occasionally abuse notation and identify the categories K˚pAqrS´1s andD˚pAq,˚ “ H,`,´, b, whereAis an abelian category andS is the class of quasi-isomorphisms in K˚pAq. This identification is justified by the fact that it is easier to manipulate morphisms by using roofs and coroofs than strings of morphisms, and that these categories are isomorphic by theorem 5.1.4 and all the properties we are interested in are true in isomorphic categories. If one is not satisfied with this approach, one can add the isomorphismG, or its inverseG´1, of theorem 5.1.4 to appropriate places. In particular in most of the places where one might want addGorG´1, the resulting functors are uniquely determined by theorem 3.1.3.