Category of complexes - An Introduction To Homological Algebra

In this section we construct a new category from an additive category, the category of complexes. The objects of this category are sequences of objects of the underlying category connected with differentials. It turns out that the category of complexes over an additive category is additive and the category of complexes over an abelian category is abelian, see lemma 2.5.6 and theorem 2.5.7. The language of complexes allows us to define cohomology, which is central in homological algebra.

Definition 2.5.1(Category of complexes). For any additive categoryAwe can construct the category of(cochain) complexes CpAq. The objects ofCpAqare sequencesA^‚“ pAⁱ, dⁱ_A‚ :AⁱÑA^i`1qiPZ, whereAⁱare objects ofAand

The construction of cochain complexes can be expressed also in terms of functors. Let I be a category where objects are the elements ofZY t˚u. Suppose˚ is the zero object inI, and for any iPZ, there exists a morphism iÑi`1 such that iÑi`1Ñi`2“iÑ ˚ Ñi`2. Then a complex over an additive category Ais a functor F : I Ñ A which preserves the zero object. A morphism of complexes F : I Ñ A and G : I Ñ A is a natural transformationF ÑG.

Definition 2.5.2(Cohomology complex). LetCpAqbe the category of chain complexes over an abelian category.

LetX^‚PObCpAqand consider the commutative diagram cokerd^i´1_X‚

where existence of morphismsaⁱandbⁱ follows from the universal property of kernel and cokernel and the equation dⁱ_X‚d^i´1_X‚ “0. Thei-th cohomology ofX^‚ is the object

HⁱpX^‚q “cokeraⁱ–kerbⁱ and thecohomology complex ofX^‚ is the complex

H^‚pX^‚q: . . . ⁰ H^i´1pX^‚q ⁰ HⁱpX^‚q ⁰ H^i`1pX^‚q ⁰ . . . .

We need to verify that the above definition is well-defined, that is cokeraⁱ – kerbⁱ. Let d^i´1_X‚ “ m1e1 and dⁱ_X‚ “m2e2 be the factorizations given by theorem 2.2.10. We get the following commutative diagram where the morphismsa¹ andb¹ are obtained from equationsm2e2m1“0 ande2m1e1“0. φⁱ₁ is a monomorphism, so a¹h“ψ. The morphismh is unique becausea¹ is a monomorphism. This shows that a¹ is the kernel of φⁱ₂φⁱ₁. Similarly one shows that b¹ is the cokernel of φⁱ₂φⁱ₁. Putting these together and using theorem 2.2.10 we get

cokeraⁱ“cokera¹“cokerpkerφⁱ₂φⁱ₁q –kerpcokerφⁱ₂φⁱ₁q “kerb¹“kerbⁱ. This shows that cohomology is well-defined.

We prove an alternative characterization for cohomology which will be used to obtain a long exact sequence from a short exact sequence of complexes.

Lemma 2.5.3. For any complexX^‚ there exists a unique morphismhsuch that the following diagram Xⁱ cokerd^i´1_X‚

β1φⁱ₂ “β1hφⁱ₁“0. Therefore there exist unique morphisms γ1 : cokerhÑH^i`1pX^‚qand γ2 :H^i`1pX^‚q Ñ cokerhsuch thatγ1γ2β2“β2 andγ2γ1β1“β1, soγ1γ2“IdH^i`1pX^‚qand γ2γ1“Idcokerh.

For any morphism f : X^‚ Ñ Y^‚ of complexes over an abelian category, we have an induced morphism of complexesHⁱpfq:HⁱpXq ÑHⁱpYqfor alli. Indeed,

kerdⁱ_X ÑXⁱ ÑYⁱÑY^i`1“kerdⁱ_X ÑXⁱÑX^i`1ÑY^i`1“0,

so there exists a unique morphism kerfⁱ: kerdⁱ_XÑkerdⁱ_Y such that the following diagram is commutative kerdⁱ_X kerdⁱ_Y

Xⁱ Yⁱ

kerfⁱ

fⁱ

(2.6)

By commutativity

X^i´1ÑkerdⁱX ÑkerdⁱY ÑYⁱ“X^i´1ÑkerdⁱXÑXⁱÑYⁱ

“X^i´1ÑXⁱÑYⁱ

“X^i´1ÑY^i´1ÑYⁱ

“X^i´1ÑY^i´1Ñkerdⁱ_Y ÑYⁱ. The morphism kerdⁱ_Y ÑYⁱ is a monomorphism, so we have

X^i´1ÑkerdⁱXÑkerdⁱY “X^i´1ÑY^i´1ÑkerdⁱY. (2.7) Therefore

X₁^i´1Ñkerdⁱ_X Ñkerdⁱ_Y ÑHⁱpYq “X^i´1ÑY^i´1ÑkerdⁱÑHⁱpYq “0,

and by the cokernel property for HⁱpXqthere exists a unique morphism Hⁱpfq : HⁱpXq Ñ HⁱpYqsuch that the following diagram is commutative

kerdⁱ_X kerdⁱ_Y

HⁱpX^‚q HⁱpY^‚q

kerfⁱ

Hⁱpfq

(2.8)

By fixing kernels and cokernels of each morphism, this allows us to define a functorCpAq ÑCpAqwhich maps a complex to its cohomology complex and morphisms to induced morphisms. This functor is called thecohomology functor. Indeed, by using the property that the morphisms kerfⁱ andHⁱpfqin the commutative diagrams (2.6) and (2.8) are unique it is easy to see that this map preserves identity morphisms and composition of morphisms.

We have proved the following proposition.

Proposition 2.5.4. Let A be an abelian category. There exists a functor H^‚ : CpAq Ñ CpAq which sends a complex X^‚ to the cohomology complex H^‚pX^‚q and a morphism f :X^‚Ñ Y^‚ to the induced morphism H^‚pfq: H^‚pX^‚q ÑH^‚pY^‚q.

The following proposition gives an alternative characterization of the pseudo-elements of i-th cohomology of a complex.

Proposition 2.5.5. Let X^‚ P ObCpAq where A is an abelian category. The pseudo-elements of HⁱpX^‚q are in natural one-to-one correspondence with pseudo-elements x P^˚ Xⁱ such that dⁱ_X‚pxq “ 0 under the following equivalence relation „: for x:Y1ÑXⁱ, y:Y2ÑXⁱP^˚ Xⁱ we have x„y if and only if there exist epimorphisms e1:V ÑY1 ande2:V ÑY2 such that xe1´ye2“d^i´1_X‚z for some pseudo-elementzP^˚X^i´1.

This one-to-one correspondence ψ is given for a pseudo-element x:U ÑXⁱ such thatdⁱ_X‚x“0 by first taking the unique morphism x¹:U Ñkerdⁱ_X‚ such that the following diagram commutes

U Xⁱ

kerdⁱ_X‚

x¹ x

φⁱ₁

and then we letψpxqto be the composite ofx¹ followed by the morphismγ: kerdⁱ_X‚ ÑHⁱpX^‚q, the cokernel of the morphism aⁱ:X^i´1Ñkerdⁱ_X‚ constructed in the diagram (2.4).

Proof. Fix notation by the following commutative diagram X^i´1 Xⁱ

kerdⁱ_X‚

HⁱpX^‚q

d^i´1_X_‚ aⁱ

φⁱ₁

First we need to verify that the mapψis well-defined. Letx, yP^˚Xⁱbe pseudo-elements such thatdⁱ_X‚x“dⁱ_X‚y“0 andxe1´ye2“d^i´1_X‚zfor some pseudo-elementzofX^i´1and epimorphismse1ande2. By definition of kernel there exist unique morphisms x¹ and y¹ such that φⁱ₁x¹ “x andφⁱ₁y¹ “y. Since φⁱ₁px¹e1´y¹e2q “d^i´1_X‚z “φⁱ₁aⁱz, and x¹e1´y¹e2is the unique morphism with this property, we havex¹e1´y¹e2“aⁱz, becauseφⁱ₁ is a monomorphism.

Nowγaⁱz“0, by the fact thatγis the cokernel ofaⁱ, and we find thatγx¹e1“γy¹e2. This shows thatψpxq “^˚ψpyq andψis well-defined.

To show that every pseudo-element ofHⁱpX^‚qis an image of a pseudo-element ofXⁱ, letwP^˚ HⁱpX^‚q. By 2.4.3 (iii) there existsyP^˚kerdⁱ_X‚such thatφⁱ₁pyq “^˚w. Thenφⁱ₁pyqis a pseudo-element ofXⁱsuch thatdⁱ_X‚pφⁱ₁pyqq “0.

By construction ofψ, φⁱ₁pyqis sent to a pseudo-element of kerd^i´1_X‚ pseudo-equal toy. Thusψis surjective.

It remains to show that ψ is injective. Let x, yP^˚ Xⁱ be pseudo-elements which are mapped to pseudo-equal pseudo-elements of HⁱpX^‚q. We have to show that xe1´ye2 “ d^i´1_X‚z for some pseudo-element z of X^i´1 and some epimorphisms e1 and e2. Let x¹ and y¹ be the unique morphisms such that φⁱ₁x¹ “ x and φⁱ₁y¹ “ y. By assumption γx¹p1 “ γy¹p2 for some epimorphisms p1 and p2. Now γpx¹p1´y¹p2q “ 0 and by 2.4.3 (iv) there exists a pseudo-element z¹ of X^i´1 such that aⁱzq1 “ px¹p1´y¹p2qq2 for some epimorphisms q1 and q2. Now d^i´1_X‚pz¹q1q “φⁱ₁px¹p1´y¹p2qq2“xp1q2´yp2q2. Hence we can takez“z¹q1. This shows thatψis injective.

Lemma 2.5.6. IfA is an additive category, thenCpAqis additive.

Proof. Abelian group: Letf, g:K^‚ÑL^‚ be two morphisms inCpAq. The following diagram commutes This shows that addition is bilinear with respect to composition.

Zero object: Clearly the complexp0_A,0ⁱ_AqiPZ is the zero complex inCpAq. easily check that these are morphisms in CpAqsatisfying the following formulas

p1i1“IdA^‚, p2i2“IdB^‚, p1i2“0, p2i1“0, p1i1`p2i2“IdA^‚‘B^‚.

Let C^‚ P CpAq be a complex and f : A^‚ Ñ C^‚ and g : B^‚ Ñ C^‚ be morphisms of complexes. Then h^‚“ fⁱp1`gⁱp2

(

iPZis the unique morphism fromA^‚‘B^‚toC^‚making the following diagram commutative C^‚

Indeed, the fact that h^‚ is a morphism follows from the equations

h^i`1dⁱA^‚‘B^‚ “ pf^i`1p1`g^i`1p2qpi1dⁱA^‚p1`i2dⁱB^‚p2q

“ pf^i`1dⁱ_A‚p1`g^i`1dⁱ_B‚p2q

“dⁱC^‚pf^i`1p1`g^i`1p2q “dⁱC^‚hⁱ.

Uniqueness of the morphismh^‚follows from uniqueness of each of thehⁱ. This shows thatCpAqhas biproducts.

Theorem 2.5.7. LetA be an abelian category. ThenCpAqis abelian.

Proof. Zero object: Clearly the complex

. . . ⁰ 0 ⁰ 0 ⁰ . . . is the zero object inCpAq.

Product: Since every abelian category is additive, by theorem 2.3.3, by lemma 2.5.6 the categoryCpAqis additive.

By proposition 2.1.3,CpAqhas the product and coproduct of each pair of objects because it has the biproduct of each pair of objects.

Kernel: By duality it suffices to show that every morphism has the kernel. Consider a morphismf :K^‚ÑL^‚ of complexes. Let kerf denote the complex

. . . ^d kerf^i´1 kerfⁱ kerf^i`1 . . .

i´2

kerf d^i´1_ker_f dⁱ_ker_f d^i`1_ker_f

where dⁱ_ker_f is the unique morphism making the following diagram commutative kerfⁱ kerf^i`1

Kⁱ K^i`1

Lⁱ L^i`1

dⁱ_ker_f

γⁱ γ^i`1

dⁱ_K‚

fⁱ f^i`1

dⁱ_L_‚

The equation d^i`1_ker_fdⁱ_ker_f “ 0 follows from commutativity,d^i`1_K‚dⁱ_K‚ “0, and the fact that kerfⁱ ÑKⁱ is a monomorphism.

Now, given any morphism g: M^‚ ÑK^‚ such that f g “0, there exist unique morphisms φⁱ :Mⁱ Ñ kerfⁱ such thatγⁱφⁱ“gⁱ. It remains to verify that these φⁱ define a morphisms of complexes. Now

γ^i`1φ^i`1dⁱ_M‚ “g^i`1dⁱ_M‚“dⁱ_K‚gⁱ“dⁱ_K‚γⁱφⁱ“γ^i`1dⁱ_ker_fφⁱ,

so by the fact thatγ^i`1is a monomorphismφcommutes with differentials. This shows that the complex kerf satisfies the universal property for kernel off in CpAq.

Monomorphism is kernel: By duality it suffices to show that every monomorphism is the kernel of some mor-phism. Let f : K^‚ Ñ L^‚ be a monomorphism. Then each fⁱ is a monomorphism. Let γ : L^‚ Ñ cokerf be the cokernel of f. By the construction of cokerf in CpAq, γⁱ is the cokernel of fⁱ. By corollary 2.2.11, fⁱ“kerγⁱ. By construction of kernels inCpAqthis shows thatf is the kernel of its cokernel.

In document An Introduction To Homological Algebra (sivua 41-46)