We have defined exact sequences, but we have not defined exact (resp. left exact, resp. right exact) functors. These functors are important because one can view derived functors as a kind of machinery to measure how much a left (resp. right) exact functor fails to be exact. Let us define these.
Definition 6.1.1 (Exact functor). LetF : AÑB be an additive functor between abelian categories. If for any short exact sequence
0 X f Y g Z 0 (6.1)
inA, the sequence
0 FpXq Fpfq FpYq Fpgq FpZq 0
is exact, thenF is said to be exact. Similarly, if for any short exact sequence (6.1) the following sequence is exact 0 FpXq Fpfq FpYq Fpgq FpZq
thenF isleft exact. Dually, if for any short exact sequence (6.1) the sequence FpXq Fpfq FpYq Fpgq FpZq 0 is exact, thenF isright exact.
Consider an additive functor F : A Ñ B between abelian categories. This induces a functor C˚pFq, ˚ “ H,`,´, b, between the abelian categoriesC˚pAqand C˚pBq, by pointwise application of F. The functor C˚pFq maps homotopies to homotopies. Indeed, iff´g“dY‚χ`χdX‚, then
C˚pFqpf´gq “ pFpdi´1Y‚ qFpχiq `Fpχi`1qFpdiX‚qqiPZ. Thus it induces an additive functorK˚pFq:K˚pAq ÑK˚pBq.
We say that a complex in C˚pAq(orK˚pAq) isexact, if the cohomology complex is the zero complex.
Definition 6.1.2 (Adapted class). LetF :AÑBbe a left (resp. right) exact functor between abelian categories.
A class of objects R ĂObA containing the zero object and stable under biproducts is said to beadapted to F if any object Aof Ais a subobject (resp. quotient) of an object of RandC`pFq (resp. C´pFq) preserves exact complexes fromC`pRq(resp. C´pRq).
More precisely, a class of objects R is adapted to a left exact functor when it contains the zero object and satisfies the following properties
AC 1 For anyR1, R2PR, we haveR1‘R2PR.
AC 2 For any objectX PA, there exists a monomorphismX ÑR withRPR. AC 3 For any exact complexR‚PC`pRq,C`pFqpR‚qis an exact complex.
By abuse of notation, we will write Ralso for the full subcategory of Aconsisting of the objects of R. Also, the categoryK˚pRqis the full subcategory ofK˚pAqconsisting of all complexesX‚ such thatXiPRfor alli.
Lemma 6.1.3. Let F : A Ñ B be a left exact functor of abelian categories, R a class of objects adapted to F, and SR the class of quasi-isomorphisms in K`pRq. ThenSR is a localizing class of morphisms and the category K`pRqis a triangulated category.
Proof. By corollary 5.1.7 the class of quasi-isomorphisms is a localizing class inK`pAq. The same proof works for K`pRq because R is stable under direct sums, so mapping cones of any morphism is contained in K`pRq. The inclusion K`pRq Ñ K`pAq induces a triangulated category structure on K`pRq and one can easily verify the axioms of triangulated category forK`pRqusing the axioms forK`pAqand corollary 4.1.6.
We also have the dual version.
Lemma 6.1.4. Let F:AÑBbe a right exact functor of abelian categories,Ra class of objects adapted toF, and SR the class of quasi-isomorphisms inK´pRq. ThenSRis a localizing class andK´pRqis a triangulated category.
Lemma 6.1.5. LetF :AÑBbe an additive exact functor between abelian categories. ThenC˚pFq(resp. K˚pFq),
˚ “ H,`,´, b, preserves exact complexes.
By lemma 2.2.15 it suffices to show thatpFpmi´1q, Fpeiqqis an exact sequence. But this follows from the fact that F is exact.
Lemma 6.1.6. Let F :AÑB be a left exact functor of abelian categories andRa class of objects adapted to F.
For any complexX‚PK`pAqthere is a quasi-isomorphismt:X‚ÑR‚ withR‚PK`pRq.
Proof. Construction of R‚ and t: By application of the translation functor forX‚we may assume thatX‚is of the form
where i0is a monomorphism by corollary 2.3.5. Choose a monomorphism k0:Y0ÑR1, by AC2, and define d0R‚ “k0i0 andt1“k0j0. We have obtained the following commutative diagram with exact rows
Suppose that we have already chosen the objectsR0, . . . , Rn´1and differentials between them. Then we can obtain the following commutative diagram
Xn´1 Xn
Rn´1 cokerdn´2R‚ Yn´1 Rn
dn´1X‚
tn´1 jn´1
a b kn´1
(6.2)
by taking the pushout Yn´1 of the morphismsatn´1 anddn´1X‚ and then the monomorphismkn´1:Yn´1Ñ Rn, by AC2, for some objectRnofR. Letdn´1R‚ “kn´1baandtn“kn´1jn´1. By induction we have obtained a complexR‚ and a morphism of complexest:X‚ÑR‚.
Hiptqis an isomorphism: SinceAis an abelian category, to show thatHiptqare isomorphisms for alli, it suffices to check that Hiptqare monomorphisms and epimorphisms. Recall from proposition 2.5.5 that the pseudo-elements of HipX‚q are in one-to-one correspondence with the equivalence classes of pseudo-elements ofXi which are sent to the zero-morphism by diX‚ modulo the relation x1 : Y1 Ñ Xi „ x2 : Y2 Ñ Xi if and only if there exists epimorphisms h1 : V Ñ Y1 and h2 :V ÑY2 and a morphism k : V Ñ Xi´1 such that x1h1´x2h2“diX‚k.
Hiptqis an epimorphism: Let us use proposition 2.5.5 to show thatHiptqis an epimorphism. Let rP˚ Ri such that diR‚h“ 0. Sinceki is a monomorphism, by proposition 2.4.3 (ii) we get pci1aqprq “ 0, where c is the morphism cokerdi´1R‚ ‘XiÑYi like in corollary 2.3.4. By corollary 2.3.4 the following sequence is exact
Xi i1at cokerdi´1R‚ ‘Xi`1 Yi 0
i´i2diX‚ c
(6.3) By proposition 2.4.3 (iv) there exists a pseudo-element x P˚ Xi such that pi1ati ´i2diX‚qpxq “˚ pi1aqprq.
Hence atixv“arufor some epimorphismsu:AÑXi andv:AÑRi. We have the following diagram A
Ri´1 Ri cokerdi´1R‚
tixv ru di´1R‚ a
The row is exact by proposition 2.2.14, so by proposition 2.4.3 (iv) there exists a pseudo-elementr1 P˚Ri´1 such thatdi´1R‚ r1“˚tixv´ru. Therefore for some epimorphismsw1andw2we havedi´1R‚ r‚w1“tixvw2´ruw2. By the equivalence relation of proposition 2.5.5 this means thattixandrrepresent the same pseudo-element in HipR‚q. This shows by proposition 2.4.3 (iii) that HipR‚qis an epimorphism.
Hiptqis a monomorphism: We use proposition 2.5.5. By proposition 2.4.3 (ii) it suffices to show that a pseudo-element ofHipX‚qwhich is mapped by Hiptqto 0P˚HipR‚qis pseudo-equal to 0. LetxP˚ Xi correspond to a pseudo-elementx1P˚HipX‚qsuch that Hiptqpx1q “˚0. By commutativity of the following diagram
Xi kerdi´1X‚ HipX‚q
Ri kerdi´1R‚ HipR‚q
ti kerti Hiptq
and the correspondence of the pseudo-elements of Ri with HipR‚q given by proposition 2.5.5, we get that tix P˚ Ri corresponds to the pseudo-element 0 P˚ HipR‚q. Therefore there exists an epimorphism v and morphismusuch thatdi´1R‚ u“tixv. Since the morphismki´1 in (6.2) is a monomorphism, we haveci1au“ ci2xv. By corollary 2.3.5 the pushout and pullback diagrams coincide, so there exists a unique morphismφ such that di´1X‚φ “xv. Therefore xcorresponds to 0 P˚ HipX‚q by proposition 2.5.5. This completes the proof.
Similarly one proves similar lemma for right exact functors.
Lemma 6.1.7. LetF :AÑBbe a right exact functor of abelian categories andRa class of objects adapted toF. For any complexX‚PK`pAqthere is a quasi-isomorphismt:R‚ÑX‚ withR‚PK`pRq.
The following proposition is important in the construction of the right derived functor.
Proposition 6.1.8. Let F :AÑBbe a left exact functor of abelian categories, Ra class of objects adapted to F, andSR the class of quasi-isomorphisms inK`pRq. The canonical functor
D`pISRq:K`pRqrSR´1s ÑD`pAq is an equivalence of categories and commutes with the translation functor.
Proof. By theorem 5.1.4 we have an isomorphism of categoriesG´1:K`pAqrS´1s ÑD`pAq, where S is the class of quasi-isomorphisms inK`pAq. We letD`pISRq “G´1˝D`pIq, whereD`pIqis the unique morphism, obtained by theorem 3.1.3, such that the following diagram is commutative
K`pRq K`pAq
K`pRqrSR´1s K`pAqrS´1s
K`pIq
QR QA
D`pIq
Thus it suffices to show thatD`pIqis an equivalence of categories. By commutativityD`pIqis identity on objects and morphisms. By theorem 1.1.10 we have to show that D`pIq is fully faithful and essentially surjective. By lemma 6.1.6, any object ofK`pAqis quasi-isomorphic to an object ofK`pRq. ThusD`pIqis essentially surjective.
The lemma 6.1.6 show that the condition (ii) of proposition 3.2.7 holds, so K`pRqrSR´1s is a full subcategory of K`pAqrS´1s. Hence the functorD`pIqis fully faithful.
The inclusion functorI commutes with the translation functor. We know that the localization functorQA, and thus QSR, commute with the traslation functor. Therefore, the functor D`pISRq commutes with the translation functor.
The following is a version of the above proposition for right exact functors.
Proposition 6.1.9. Let F :AÑB be a right exact functor of abelian categories,Ra class of objects adapted to F, andSR the class of quasi-isomorphisms in K´pRq. The canonical functor
D´pISRq:K´pRqrSR´1s ÑD´pAq is an equivalence of categories and commutes with the translation functor.
The following shows that an inverse for the equivalence in proposition 6.1.8 also commutes with translations.
Corollary 6.1.10. Let D`pISRq : K`pRqrSR´1s Ñ D`pAq be the functor in proposition 6.1.8. There exists a functorΦ :D`pAq ÑK`pRqrSR´1ssuch that
IdK`pRqrSR´1s“Φ˝D`pISRq, D`pISRq ˝Φ–IdD`pAq
and the functorΦcommutes with the translation functor.
Proof. First we construct Φ on objects. Let Φp0‚q “ 0‚. For any object A‚ ‰ 0‚ of D`pAq such that A‚ R ObK`pRqrSR´1s, A0 ‰0 and Ai “0 for i ă0, fix by lemma 6.1.6 a quasi-isomorphism qA‚ : A‚ ÑR‚ to some objectR‚PK`pRqrSR´1s. For anyR‚PK`pRqrSR´1s, letqR‚ “IdR‚. In general, for an objectA‚‰0‚ ofD`pAq, letnPZsuch thatAn‰0 andAi“0 foriăn. Then, we defineqA‚ “qA‚r´nsrns.
For any object A‚ PD`pAqdefine ΦpA‚q “ CodpqA‚q. We see that this definition implies that Φ commutes with the translation functors on objects. Indeed, letA‚PD`pAqandnPZsuch thatAn‰0 andAi“0 foriăn.
Then
ΦpA‚r1sq “CodpqA‚r1sq “CodpqA‚r1sr´n´1sqrn`1s “CodpqA‚r´nsqrnsr1s “ΦpA‚qr1s.
To define the map on morphisms of the functor Φ, let f : A‚1 Ñ A‚2 be a morphism in D`pAq. Let Φpfq “ qA‚2fpqA‚1q´1. Since the subcategory K`pRqrSR´1s is a full subcategory of D`pAq this is well defined. A simple computation shows that Φ is a functor such that IdK`pRqrSR´1s “Φ˝D`pISRq. To see that IdD`pAq is isomorphic to D`pISRq ˝Φ, letτ : IdD`pAqÑD`pISRq ˝Φ be the natural transformation which sends an objectA‚ to qA‚. Then for any morphismps, fq:A‚1ÑA‚2 the following square commutes
A‚1 CodpqA‚1q
A‚2 CodpqA‚2q
qA‚ 1
ps,fq Ψps,fq“qA‚
2˝ps,fq˝pqA‚ 1q´1 qA‚
2
This shows thatτ is an isomorphism of functors and finishes the proof.
A similar corollary holds for right exact functors.
Corollary 6.1.11. Let D`pISRq : K´pRqrSR´1s Ñ D´pAq be the functor in proposition 6.1.9. There exists a functorΦ :D´pAq ÑK´pRqrSR´1ssuch that
IdK´pRqrSR´1s“Φ˝D´pISRq, D´pISRq ˝Φ–IdD´pAq
and the functorΦcommutes with the translation functor.
Lemma 6.1.12. Let F :AÑB be an additive functor between abelian categories and Ra full subcategory of A stable under biproducts such that the functor K˚pF ˝Iq “K˚pFq ˝K˚pIq:K˚pRq ÑK˚pAq ÑK˚pBq preserves exact complexes, whereI:RÑAis the inclusion functor. ThenK˚pFq ˝K˚pIqpreserves mapping cones, mapping cylinders, and quasi-isomorphisms. Moreover,F˝I induces an exact functorD˚pF˝Iq:K˚pRqrSR´1s ÑD˚pBq.
Proof. Letf :X‚ÑY‚ be a representative of a morphism ofK˚pRq. SinceF˝I is additive, by proposition 2.1.5 it preserves biproducts. A direct computation shows that
K˚pF˝IqpdiCpfqq “i1pF˝Iqpdi`1X‚qp1`i2pF˝Iqpfi`1qp1`i2pF˝IqpdiY‚qp2
and
K˚pF˝IqpCpfqq “K˚pF˝IqpX‚r1s ‘Y‚q
“ pK˚pF˝IqpX‚r1sqq ‘ pK˚pF˝IqY‚q
“ pK˚pFqX‚qr1s ‘ pK˚pF˝IqY‚q
“CpK˚pF˝Iqpfqq, soF˝I preserves the mapping cones. Similarly, from
K˚pF˝IqpdiCylpfqq “ i1pF˝IqpdiX‚qp1´i1p1p2
´i2i1pF˝Iqpdi`1X‚qp1p2`i2i2pF˝Iqpfi`1qp1p2`i2i2pF˝IqpdiY‚qp2p2. and
K˚pF˝IqpCylpfqq “K˚pF˝IqpX‚‘Cpfqq
“ pK˚pF˝IqpX‚qq ‘ pK˚pF˝IqCpfqq
“ pK˚pF˝IqpX‚qq ‘CpK˚pF˝Iqpfqq
“CylpK˚pF˝Iqpfqq
we see that K˚pF ˝Iqpreserves cylinders. To see that K˚pF ˝Iqpreserves quasi-isomorhisms, let f be a quasi-isomorphism in R. Then by lemma 6.1.6 applied in K˚pAq, Cpfq is an exact complex, so K˚pF˝IqpCpfqq “ CpK˚pF˝Iqpfqqis exact by assumption. Therefore lemma 6.1.6 shows thatK˚pF˝Iqpfqis a quasi-isomorphism.
SinceK˚pF˝Iqpreserves quasi-isomorphisms, by theorem 3.1.3 we have the following commutative diagram K˚pRq K˚pBq
K˚pRqrSR´1s D˚pBq
K˚pF˝Iq
QSR QB
D˚pF˝Iq
(6.4)
where SR is the class of quasi-isomorphisms in K˚pRq. The functor K˚pF ˝Iq preserves biproducts, cones and cylinders, so it maps any triangle of the form
X‚ i1 Cylpfq p2 Cpfq p1 X‚r1s
to a triangle of the same form in K˚pBq. This shows thatK˚pF˝Iqpreserves distinguished triangles. Since the localizing functorsQSR and QB preserve distinguished triangles, by commutativity of (6.4) shows thatD˚pF˝Iq preserves distinguished triangles. Hence it is an exact functor.
By lemma 6.1.5 C˚pFqof an exact functorF preserves exact complexes. Hence, by aboveD˚pFq: D˚pAq Ñ D˚pBqis an exact functor.
We come now to the definition of derived functors.
Definition 6.1.13. Theright derived functor of a left exact functor F : A Ñ B of abelian categories is a pair consisting of an exact functor RF :D`pAq ÑD`pBqand a natural transformationF :QB˝K`pFq ÑRF˝QB
such that for any exact functorG:D`pAq ÑD`pBqand any natural transformationG :QB˝K`pFq ÑG˝QA there exists a unique natural transformationη:RF ÑGsuch that the following diagram
QB˝K`pFq RF˝QA
Theleft derived functor of a right exact functorF :AÑBof abelian categories is a pair consisting of an exact functorLF :D´pAq ÑD´pBqand a morphism of functors F :QB˝K´pFq ÑLF ˝QAsuch that for any exact functor G:D´pAq ÑD´pBq and any natural transformationG :QB˝K´pFq Ñ G˝QA there exists a unique natural transformationη:GÑLF such that the following diagram
G˝QA QB˝K´pFq
LF˝QA
G
F η
is commutative.
We are ready to prove the main result of this section, the existence of derived functors. To prove this theorem we need the formalism of coroofs, see proposition 3.2.9.
Theorem 6.1.14. Let F :AÑB be a left (resp. right) exact functor of abelian categories, R a class of objects adapted toF, andSR the class of quasi-isomorphisms inK`pRq(resp. K´pRq). Then the right (resp. left) derived functorRF (resp. LF) exists.
Proof. We will prove only the existence of the right derived functorRF. Existence of the left derived functorLF is proved similarly.
Construction of RF: By corollary 6.1.10 we have a functor Φ :D`pAq ÑK`pRqrSR´1s which commutes with the translation functors and we have the following equality and isomorphism of functors
IdK`pRqrSR´1s“Φ˝D`pISRq, β : D`pISRq ˝Φ – IdD`pAq
. By proposition 6.1.8 and (6.4) the following diagram is commutative
K`pAq K`pRq K`pBq
Exactness ofRF: Since D`pF ˝Iq preserves distinguished triangles by lemma 6.1.12, it remains to show that Φ preserves distinguished triangles. Let pX1‚, Y1‚, Z1‚, f1, g1, h1qbe a distinguished triangle in D`pAq. By lemma 4.2.2 it is isomorphic to a triangle of the form pX2‚, Y2‚, Cpf2q, f2, i2,p1q. By lemma 6.1.6, we find isomorphisms X2‚ Ñ R0‚ and Y2‚ Ñ R‚1 in D`pAq such that R0‚, R‚1 P K`pRq. Let us denote by φ1 the composite X1‚ÑX2‚ÑR‚0 and byφ2 the compositeY1‚ÑY2‚ÑR‚1. Sinceφ1 andφ2 are isomorphisms, by TR5 and corollary 4.1.6, we obtain the following isomorphism of distinguished triangles inD`pAq.
X1‚ Y1‚ Z1‚ X1‚r1s
we find that the following diagram is an isomorphism of distinguished triangles
ΦpX1‚q ΦpY1‚q ΦpZ1‚q ΦpX1‚qr1s
Since the bottom triangle of this diagram is a distinguished triangle in K`pRq, by lemma 4.2.2, we have shown that Φ preserves distinguished triangles.
be a coroof which represents the isomorphismβpXqinD`pAq. By lemma 6.1.6 we have a quasi-isomorphism r : Z‚ Ñ K`pIqpR‚q for some R‚ P K`pRq. Since K`pRq is a full subcategory ofK`pAq the composite
K`pIqpY‚q Ñ Z‚ Ñ K`pIqpR‚q equals K`pIqpgq for some quasi-isomorphism g : Y‚ Ñ R‚ of K`pRq.
Application ofK`pFqyields the following diagram
K`pFqpX‚q K`pF˝IqpY‚q
K`pF˝IqpR‚q
K`pFqprfq
K`pF˝Iqpgq
whereK`pF˝Iqpgqis a quasi-isomorphism by lemma 6.1.12. We defineFpX‚qto be the morphism inD`pBq represented by this coroof.
FpXq well-defined: To show thatF is well-defined, we have to show that the morphismFpX‚qdoes not depend on the choice of the coroof inD`pAq. Let
X‚ K`pIqpY‚q
Z1‚ Z2‚
Z3‚
f g
t s
t1
s1
be an equivalence of two coroofs which represent the isomorphism βpX‚qofQApX‚qandQApK`pIqpY‚qqin D`pAq. By lemma 6.1.6 we have quasi-isomorphisms a1:Z1‚ÑK`pIqpR‚1qand a2 :Z2‚ ÑK`pIqpR‚2qand by transitivity of equivalence of coroofs, the coroofs
X‚ K`pIqpY‚q
K`pIqpR‚1q
a1f
K`pIqpa1tq
X‚ K`pIqpY‚q
K`pIqpR‚2q
a2g
K`pIqpa2sq
are equivalent. Therefore we can find morphismsb1:K`pIqpR‚1q ÑZ4‚ andb2 :K`pIqpR‚2q ÑZ4‚ such that the following diagram is an equivalence of coroofs
X‚ K`pIqpY‚q
K`pIqpR‚1q K`pIqpR‚2q
Z4‚
a1f a2g
a1K`pIqptq a2K`pIqpsq
b1
b2
Again, by lemma 6.1.6, we can find a quasi-isomorphisma3:Z4‚ ÑK`pIqpR‚3q. Thus we have the following
commutative diagram
X‚ K`pIqpY‚q
K`pIqpR‚1q K`pIqpR2‚q
K`pIqpR‚3q
a1f a2g
a1K`pIqptq a2K`pIqpsq
a3b1
a3K`pIqpb2q
Application ofK`pFqgives the following diagram
K`pFqpX‚q K`pF˝IqpY‚q
K`pF˝IqpR‚1q K`pF˝IqpR‚2q
K`pF˝IqpR‚3q
K`pFqpa1fq K`pFqpa2gq
k`pFqpa1qK`pF˝Iqptq K`pFqpa2qK`pF˝Iqpsq K`pFqpa3b1q
K`pFqpa3qK`pF˝Iqpb2q
which is an equivalence of coroofs inD`pBq. This shows thatFpX‚qis well-defined.
FpXq is a natural transformation: To show that FpX‚q is a natural transformation, let φ : X1‚ ÑX2‚ be a morphism in K`pAq. We have to show that the following diagram is commutative inD`pBq
pQB˝K`pFqqpX1‚q pRF ˝QAqpX1‚q
pQB˝K`pFqqpX2‚q pRF ˝QAqpX2‚q
FpX‚1q
pQB˝K`pFqqpφq pRF˝QAqpφq FpX‚2q
LetY1‚, Y2‚PObK`pRqbe such thatpQSRqpY1‚q “ pΦ˝QAqpX1‚qandpQSRqpY2‚q “ pΦ˝QAqpX2‚q. Letψbe a morphism of K`pRqsuch thatQSRpψq “ pΦ˝QAqpφq. By definition ofRF we have
pRF ˝QAqpφq “ pD`pF˝Iq ˝Φ˝QAqpφq “ pD`pF˝Iq ˝QSRqpψq “ pQB˝K`pF˝Iqqpψq, where the last equation follows from commutativity of proposition 6.1.8. Therefore we have to show that
FpX2q ˝ pQB˝K`pFqqpφq “ pQB˝K`pF˝Iqqpψq ˝FpX1q. (6.6) Sinceβ is a natural transformation, the following diagram
QApX1‚q pD`pISRq ˝QSRqpY1‚q
QApX2‚q pD`pISRq ˝QSRqpY2‚q
βpX1‚q
QApφq pD`pISRq˝QSRqpψq βpX2‚q
(6.7)
is commutative inD`pAq. We fix the following coroofs to represent the morphisms
βpX1‚q:“
X1‚ K`pIqpY1‚q
R1‚
βpX2‚q:“
X2‚ K`pIqpY2‚q
R‚2
pD`pISRq ˝QSRqpψq:“
K`pIqpY1‚q K`pIqpY2‚q
K`pIqpY2‚q
K`pIqpψq
IdK` pIqpY‚ 2q
where we have assumed by using lemma 6.1.6 that the bottom objects are objects coming from K`pRq.
By commutativity of (6.7), the following compositions of coroofs represent the same morphisms inD`pAq
X1‚ K`pIqpY1‚q K`pIqpY2‚q
K`pIqpR‚1q K`pIqpY2‚q
K`pIqpR‚3q
X1‚ X2‚ K`pIqpY2‚q
X2‚ K`pIqpR‚2q
K`pIqpR‚4q
φ
IdX‚ 2
where the objectsK`pIqpR‚3qandK`pIqpR‚4qand morphisms to them are constructed by using lemma 6.1.6.
An application ofK`pFqto these two coroofs yields two coroofs ofD`pBq, by lemma 6.1.12, which represent the same morphism. Since K`pFqpβpX1‚qq “ FpX1‚q K`pFqpβpX2‚qq “ FpX2‚q, and pK`pFq ˝D`pISRq ˝ QSRqpψq “K`pF˝Iqpψq, we have obtained the equation (6.6). This shows thatFis a natural transformation.
Universal property of RF: Let G:D`pAq Ñ D`pBq be an exact functor and G : QB˝K`pFq Ñ G˝QA a natural transformation. For anyX‚PK`pAq, let
X‚ K`pIqpY‚q
K`pIqpR‚q
f
K`pIqpsq
be a coroof, obtained by using lemma 6.1.6, which represents the isomorphism βpX‚q in D`pAq, where
Y‚PK`pR‚qsuch thatQSRpY‚q “ pΦ˝QAqpX‚q. We have the following commutative diagram s is a quasi-isomorphism,QAturns it into isomorphism, and any functor preserves isomorphisms. Therefore these two morphisms have inverses inD`pBqand we obtain the following commutative diagram
pQB˝K`pFqqpX‚q pG˝QAqpX‚q transfor-mation, we have to show that the following diagram is commutative
pRF ˝QAqpX1‚q pG˝QAqpK`pIqpY1‚qq pG˝QAqpX1‚q natural transformation. Sinceβ is a natural transformation, the following diagram
pG˝QAqpX1‚q pG˝QAqpK`pIqpY1‚qq This shows that the right square of 6.1 is commutative. Therefore η is a natural transformation.
Uniqueness ofη: To show that η is unique, let X‚ P K`pAq and let Y‚ P K`pRq such that pQAqpX‚q – pD`pISRq ˝QRqpY‚q. Then by definition ofη, (6.8), we have the following commutative diagram
pQB˝K`pFqqpX‚q pG˝QAqpX‚q
pRF˝QAqpX‚q pG˝QAqpY‚q
pRF ˝QAqpK`pIqpY‚qq
GpX‚q
FpX‚q GpβpX‚qq
FpK`pIqpY‚qq ηpQAqpX‚q
GpK`pIqpY‚qq
ηpQAqpY‚q
where FpK`pIqpY‚qqis the identity morphism by the definition ofF. Hence we obtain that ηpQAqpY‚q “GpK`pIqpY‚qq,
so the morphism ηpQAqpY‚qis uniquely determined, and by the fact that η is a natural transformation, we get
GpβpX‚qq ˝ηpQAqpX‚q “ηpQAqpY‚q.
Since GpβpX‚qqis an isomorphism, the morphismηpQAqpX‚qis uniquely determined. This shows that η is unique.
The following theorem shows that taking the derived functor behaves well under composition of functors.
Theorem 6.1.15. Let A1,A2, andA3 be abelian categories,F1:A1ÑA2 andF2:A2ÑA3 left exact functors, and R1 and R2 classes of objects adapted to the functors F1 andF2, respectively, such thatF1pR1q ĂR2. Then we have an isomorphism of functors
RpG˝Fq –RG˝RF. (6.9)
Proof. The composite
pQA3˝K`pGqq ˝K`pFqÑF2 RG˝ pQA2˝K`pFqq
F1
Ñ RG˝RF ˝QA1
is a natural transformation. Denote this by E. By the universal property of right derived functor there exists a natural transformationδ:RpG˝Fq ÑRG˝RF such that for any object X‚ PK`pA1qthe following diagram is commutative
pQB˝K`pG˝FqqpX‚q pRG˝RF˝QA1qpX‚q
pRpG˝Fq ˝QA1qpX‚q
E G˝FpX‚q
pδ˝QA1qpX‚q
By definition of F, G, and G˝F, and by assumption, for any object K`pIqpY‚q P ObK`pA1q, with Y‚ P ObK`pR1q, the morphisms FpK`pIqpY‚qq, GpK`pF1˝IqpY‚qq, and G˝FpK`pIqpY‚qq are identity morhisms.
This means thatEpQA∞˝K`pIqqpY‚qis an isomorphism, and because all the objects ofD`pA∞qare isomorphic to an object ofK`pR1q, given by β, we find that E is an isomorphism for all objects. By commutativity δ is an isomorphism. This proves the isomorphism (6.9).
The following is a version of the above for left derived functors
Theorem 6.1.16. LetA1,A2, andA3 be abelian categories,F1:A1ÑA2andF2:A2ÑA3right exact functors, andR1 andR2 adapted classes of object to the functors F1 andF2, respectively, such thatF1pR1q ĂR2. Then we
Theorem 6.1.16. LetA1,A2, andA3 be abelian categories,F1:A1ÑA2andF2:A2ÑA3right exact functors, andR1 andR2 adapted classes of object to the functors F1 andF2, respectively, such thatF1pR1q ĂR2. Then we