Homological methods [Lecture notes]

79 Pages·2012·0.747 MB·English

by  Jenia Tevelev

HOMOLOGICAL METHODS
JENIATEVELEV
CONTENTS
§0. Syllabus 2
§1. Complexes. LongExactSequence. Jan23. 3
§2. Categoriesandfunctors. Jan25. 4
§3. Simplicialhomology. 5
§4. Singularhomology 7
§5. Functorialityofsingularhomology. Jan27 7
§6. DeRhamcohomology. 8
§7. Free,projective,andinjectiveresolutions. Jan30. 10
§8. Exactfunctors. Feb1. 12
§9. Abhasenoughinjectives. 12
§10. Adjointfunctors. 13
§11. R-modhasenoughinjectives. Feb3. 14
§12. Quasi-isomorphismandhomotopy. Feb6. 14
§13. Homotopyinvarianceofsingularhomology. Feb8 16
§14. HomotopyinvarianceofdeRhamcohomology. 17
§15. Dolbeautcomplexand∂¯-Poincarelemma. Feb10. 18
§16. Koszulcomplex. Feb13. 20
§17. Associatedprimes 21
§18. Regularsequences. Feb15. 22
§19. Mappingcone. Feb17. 25
§20. InductivedescriptionoftheKoszulcomplex. Feb22. 26
§21. Derivedfunctors. Feb24. 27
§22. δ-functors. Feb27. 29
§23. Tor. Feb29. 34
§24. Ext. 35
§25. Ext1 andextensions. Mar2. 36
§26. Spectralsequence. Mar5 37
§27. Spectralsequenceofafilteredcomplex. Mar7. 38
§28. Spectralsequenceofadoublecomplex. Mar9. 39
§29. ExtandTordefinedusingthesecondargument 39
§30. FiberBundles. Mar12 41
§31. Homotopygroups. 41
§32. Example: aHopffibration 43
§33. LerayspectralsequencefordeRhamcohomology. Mar14. 43
§34. Monodromy. Mar16 44
§35. Leray–HirshTheorem. Künnethformula. Mar26 45
§36. CalculationfortheHopfbundle. 46
§37. Sheaves. Ringedspaces. March28. 47
1
2 MATH797–HOMOLOGICALMETHODS
§38. Stalks. Sheafification. March30 50
§39. Abeliancategories. Exactsequencesofsheaves. April2 51
§40. Cohomologyofsheaves. April4. 52
§41. DeRhamresolution. Dolbeautresolution. April6. 53
§42. Cˇechcomplex. April9and11. 55
§43. InterpretationsofH1. Picardgroup. April13. 55
§44. Derived Categories – I. April 18. Guest Lecturer Rina Anno.
NotestypedbyTom. 55
§45. DerivedCategories–II.April20. GuestLecturerRinaAnno.
NotestypedbyJulie. 58
§46. Homestretch: Capproduct,Poincareduality,Serreduality. 62
Homework1. Deadline: February10. 63
Homework2. Deadline: February24. 66
Homework3. Deadline: March9. 68
Homework4. Deadline: March30. 71
Homework5. Deadline: April13. 74
Homework6. Deadline: May4. 76
§0. SYLLABUS
Calculations of various homology and cohomology groups are ubiqui-
tous in mathematics: they provide a systematic way of reducing difficult
geometric (=non-linear) problems to linear algebra. I will attempt to de-
velop homological machinery starting with the basic formalism of com-
plexes, exact sequences, spectral sequences, derived functors, etc. in the
direction of understanding cohomology of coherent sheaves on complex
algebraic varieties. There will be detours to algebraic topology and com-
mutative algebra. Prerequisites are graduate algebra, manifolds (for a de-
tourintoalgebraictopology),andcomplexanalysis. Somefamiliaritywith
algebraic varieties (a one-semester course or a reading course should be
enough). Algebraic varieties will not appear in the first half of the course,
and,inanycase,wewillstudysheavesfromscratch.
Iwon’tfollowanyparticularbook,butIwillborrowheavilyfrom
S.Weibel,AnIntroductiontoHomologicalAlgebra
S.Gelfand,Yu. Manin,MethodsofHomologicalAlgebra
R.Bott,L.Tu,DifferentialFormsinAlgebraicTopology
D.Eisenbud,CommutativeAlgebrawithaviewtowardsAlgebraicGeometry
V.I.Danilov,CohomologyofAlgebraicVarieties
S.Lang,Algebra
D.Huybrechts,Fourier–MukaitransformsinAlgebraicGeometry
C.Voisin,HodgeTheoryandComplexAlgebraicGeometry
Grading will be based on 6 biweekly homework sets. Each homework
will have a two-week deadlineand individual problems in the homework
will be worth some points (with a total of about 30 points for each home-
work). The passing standard (for an A) will be 90 points by the end of the
MATH797–HOMOLOGICALMETHODS 3
semester. Theideaistomakehomeworkassignmentsveryflexible. Begin-
ningstudentsareadvisedtosolvemany“cheap”problemstogainexperi-
ence. If you feel that you already have some familiarity with the subject,
pickafewhard“expensive”problems.
Homeworkproblemscanbepresentedintwoways. Anidealmethodis
to come to my office and explain your solution. If your solution is correct,
youwon’thavetowriteitdown. Ifyoursolutiondoesnotwork,Iwillgive
you a hint. Problems not discussed orally will have to be written down
andturnedinattheendofthetwo-weekperiod. Allofficehourswillbeby
appointment. Iwillaskyoutosignupforanindividualweekly30minutes
slot. You can also team up and convert two adjacent short slots into one
hour-longjointmeeting. IamavailableonM,Wfrom2:00-6:00andTu,Th
3:30-6:00. Pleasee-mailmeyourpreferredmeetingtimeassoonaspossible.
During the semester, please e-mail me in advance if by whatever reason
youwon’tmakeittoourweeklymeeting.
§1. COMPLEXES. LONG EXACT SEQUENCE. JAN 23.
FixaringR,notnecessarilycommutative.
ThecategoryR-modofleftR-modules: morphismsareR-linearmaps.
Examples: Z-mod = Ab (Abelian groups), k-mod = Vect (k-vector
k
spaces). For algebraic geometry: R = k[x ,...,n] (or its quotient algebra,
1
localization, completion, etc.). For complex geometry: R = O (U) for an
hol
open subset U ⊂ Cn. Representation theory: rings like R = k[G] (R-mod
=k-representationsofagroupG),etc.
R-mod is an example of an Abelian category, which we are going to
discuss later, after we have more examples. Constructions of homologi-
cal algebra apply to any Abelian category, for example to the category of
sheaves on a topological space, O -modules on a ringed space, etc (to be
X
discussed later as well). Things you can compute in an Abelian category
butnotinageneralcategory:
• 0object;
• f +g fortwomorphismsf,g : X → Y;
• X ⊕Y;
• Ker(f),Coker(f),Im(f)foramorphismf : A → B;
• thefirstisomorphismtheoremA/Ker(f) (cid:39) Im(f).
1.1. DEFINITION. • (Cochain)complexes
...−di→−1Ci−d→i Ci+1−di→+1..., d ◦d = 0 foranyi.
i+1 i
• Cohomologygroups: Hi = Kerd /Imd .
i+1 i
• Exactsequences: Kerd = Imd .
i+1 i
Onecanalsoconsiderchaincomplexesandhomologygroups.
1.2. LEMMA(SnakeLemma).GivenacommutativediagraminR-mod
X −−−−→ X −−−−→ X −−−−→ 0
1 2 3
  
  
(cid:121)f1 (cid:121)f2 (cid:121)f3
0 −−−−→ Y −−−−→ Y −−−−→ Y
1 2 3
4 MATH797–HOMOLOGICALMETHODS
withexactrows,onehasanexactsequence
δ
Ker(f ) → Ker(f ) → Ker(f )−→Coker(f ) → Coker(f ) → Coker(f )
1 2 3 1 2 3
where all maps are induced by maps in exact sequences except for δ (called con-
necting homomorphism), which is defined as follows: lift α ∈ Ker(f ) to
3
β ∈ X ,takeitsimageinY ,liftittoY ,takeacosetinCoker(f ).
2 2 1 1
Proof. Diagramchasing. Don’tforgettoshowthatδ isR-linear. (cid:3)
Mapsofcomplexes
f• : X• → Y•
aremapsfi : Xi → Yi thatcommutewithdifferentials. Theyinducemaps
ofcohomology
Hi(X) → Hi(Y) for any i.
1.3. LEMMA(LongExactSequence).Shortexactsequenceofcomplexes
0 → X• → Y• → Z• → 0
inducesalongexactsequenceofcohomology
...−→δ Hi(X) → Hi(Y) → Hi(Z)−→δ Hi+1(X) → Hi+1(Y) → ....
Proof. Followsfrom(andisequivalentto)theSnakeLemma. (cid:3)
§2. CATEGORIES AND FUNCTORS. JAN 25.
A reminder about categories and functors: a category C has a set of ob-
(cid:96)
jectsOb(C)andasetofmorphismsMor = Mor(X,Y),i.e.each
X,Y∈Ob(C)
morphismhasasourceandatarget. Italsohasacompositionlaw
Mor(X,Y)×Mor(Y,Z) → Mor(X,Z), (f,g) (cid:55)→ g◦f
and an identity morphism Id ∈ Mor(X,X) for each object X. Two ax-
X
iomshavetobesatisfied:
• f = Id ◦f = f ◦IdX foreachf ∈ Mor(X,Y).
Y
• Compositionisassociative: (f ◦g)◦h = f ◦(g◦h).
2.1. EXAMPLE.R-mod, Sets, Top (topological spaces as objects and con-
tinuousmapsbetweenthemasmorphisms).
A (covariant) functor F : C → C from one category to another is
1 2
a function F : Ob(C ) → Ob(C ) and functions F : Mor(X,Y) →
1 2
Mor(F(X),F(Y)). Twoaxiomshavetobesatisfied:
• F(Id ) = Id ;
X FX
• F(f ◦g) = F(f)◦F(g).
A contravariant functor F : C → C is a function F : Ob(C ) → Ob(C )
1 2 1 2
andfunctionsF : Mor(X,Y) → Mor(F(Y),F(X)). Thefirstaxiomisthe
same,thesecondchangestoF(f ◦g) = F(g)◦F(f).
2.2. EXAMPLE. . Foreachintegerithereisani-thcohomologyfunctor
Hi : Kom(R-mod) → R-mod.
Notice that this means that not only we can compute Hi(C•) of any com-
plexbutalsothatforeachmapofcomplexesC• → D•,thereisaninduced
MATH797–HOMOLOGICALMETHODS 5
map Hi(C•) → Hi(D•), which satisfies two axioms above. As a general
rule, morphisms are more important than objects (and functors are more
importantthancategories).
§3. SIMPLICIAL HOMOLOGY.
A simplicial complex X (not to be confused with its chain complex) is a
picture glued from points, segments, triangles, tetrahedra, etc. Formally,
X isasetoffinitesubsetsofthevertexsetV suchthat:
• {v} ∈ X foranyv ∈ V;
• ifS ∈ X thenT ∈ X foranyT ⊂ S. WecallT afaceofS.
We are also going to assume that V is well-ordered (for example V =
{0,1,2,...,n}). Atopologicalspace|X|(calledageometricrealizationofX)
is defined as follows. Let X ⊂ X be the set of subsets with n+1 points.
n
Let
(cid:88)
∆ = {(x ,...,x ) ⊂ Rn+1|x ≥ 0, x = 1}
n 0 n+1 i i
be the standard n-dimensional simplex (with n+1 vertices). For each i =
0,...,n,let
∂i : ∆ → ∆ (3.1)
n n−1 n
be the standard unique affine linear map which sends vertices of ∆ to
n−1
verticesof∆ (exceptforthei-thvertex). Themappingofverticesisgiven
n
byauniqueincreasingfunction{0,1,...,n−1} → {0,1,...,n}\{i}. Asa
(cid:96)
set,|X|isthedisjointunionofallsimplices (X ×∆ )moduloanequiv-
n n
alencerelationgeneratedbytherelation
(T,p) ∼ (S,δi(p))
n
eachtimeT isasubsetofS obtainedbydroppingitsi-thindex(recallthat
all vertices are well-ordered). Topology on |X| is defined as follows: a
subset U is open if and only if its preimage in each copy of ∆ is open.
n
Inotherwords,itistheweakesttopologysuchthatthemap
(cid:97)
(X ×∆ ) → |X|
n n
n≥0
iscontinuous.
Now let’s fix an Abelian group A and define the chain group C (X,A)
n
tobethegroupofformalfinitelinearcombinations
(cid:88)
a(x)x, a(x) ∈ A.
x∈Xn
Thechaindifferentialisdefinedasfollows:
(cid:88) (cid:88)
∂ ( a(x)x) = a(x)∂ (x),
n n
x∈Xn x∈Xn
and
n
(cid:88)
∂ ({i ,...,i }) = (−1)j{i ,...,ˆi ,...,i }
n 0 n 0 j n
j=0
(Recallthatasimplexisjustasubset{i ,...,i } ⊂ V,V iswell-ordered,so
0 n
Icanalwaysassumethati < ... < i ).
0 n
6 MATH797–HOMOLOGICALMETHODS
3.2. LEMMA.∂ isadifferential,i.e.∂n−1◦∂n = 0.
Proof. By linearity, it suffices to check that ∂ ◦∂ ({i ,...,i }) = 0. Ob-
n−1 n 0 n
serve that this is a linear combination of subsets of {i ,...,i } with n−1
0 n
(cid:3)
elements,whereeachsubsetappearstwice,withoppositesigns.
WedefinesimplicialhomologygroupsH (X,A)ashomologygroupsofthe
i
simplicialchaincomplexC (X,A)definedabove.
•
Noticethatthesegroupscanbeeffectivelycomputed. Forexample,con-
sideratriangulationofthesphereS2 givenbythefacesofthetetrahedron.
ThisgivesachaincomplexC (X,Z). Itlooksasfollows:
•
Z4 → Z6 → Z4
withmapsgivenbyveryexplicitmatrices. Thehomologygroupslooksas
follows:
Z, 0, Z.
The price to pay: it is not clear that these groups are invariants of S2,
i.e. that they do not depend on a choice of triangulation In fact this is true
andgives,amongotherthings,
3.3. THEOREM(Euler).Foranytriangulationofthesphere,
#vertices−#edges+#faces = 2. (3.4)
This follows from a more general theorem, for which we need a defini-
tion.
3.5. DEFINITION. LetΓbeanAbeliangroupandletRbeanR-module. An
Euler–Poincare mapping φ with values in Γ is a partially defined function
thatassignstoanR-moduleanelementofΓsuchthatthefollowingistrue.
Foranyexactsequence
0 → A → B → C → 0,
φ(B) is defined if and only if both φ(A) and φ(C) are defined. In this case
weshouldhave
φ(B) = φ(A)+φ(C).
The most basic example is dimension, which is an Euler–Poincare map-
ping for vector spaces. The definition above makes sense in any category
withaconceptofanexactsequence,i.e.inanyAbeliancategory.
3.6. THEOREM (Euler–Poincare).Consider a bounded chain complex C• of R-
modules (i.e. C = 0 for k (cid:28) 0 or k (cid:29) 0). Suppose φ(C ) is defined for any k.
k k
Thenφ(H )isdefinedforanyk andwehave
k
(cid:88) (cid:88)
(−1)kφ(C ) = (−1)kφ(H ).
k k
k k
This common quantity is called Euler characteristic. Analogous result holds for
cochaincomplexes.
Proof. Followsfromadditivityofφonexactsequence
0 → Kerd → C → Imd
k k k
MATH797–HOMOLOGICALMETHODS 7
and
0 → Imd → Kerd → H .
k+1 k k
(cid:3)
To prove Euler’s theorem on triangulations, simply notice that (3.4) is
an Euler characteristic of a simplicial chain complex of the triangulation,
whichbyEuler–Poincaretheoremisequalto1−0+1 = 2.
§4. SINGULAR HOMOLOGY
Nowwewouldliketodefinethehomologytheorycalledsingularhomol-
ogywithcoefficientsinanarbitraryAbeliangroupA,whichdependsonly
on the topological space X, more precisely on its homotopy type (we will
clarify this statement later). The construction is very similar to simplicial
homology, except that we will allow chains of arbitrary simplices rather
thanonlysimplicesappearinginthetriangulation.
Let X be the set of all continuous maps g : ∆ → X. Let C (X,A) be
n n n
thegroupofformalfinitelinearcombinations
(cid:88)
a(g)g
g∈Xn
withcoefficientsinA. Thedifferentialisdefinedasfollows:
 
(cid:88) (cid:88)
∂n a(g)g = a(g)∂n(g),
g∈Xn g∈Xn
and
n
(cid:88)
∂ (g) = (−1)ig◦∂i,
n n
i=0
where ∂i is given by (3.1). The same argument as in simplicial homology
n
gives ∂ ◦∂ = 0, and so we have a singular chain complex C (X,A) and
n−1 n •
singularhomologygroupsH (X,A).
•
§5. FUNCTORIALITY OF SINGULAR HOMOLOGY. JAN 27
ThesingularchaincomplexC (X,A)anditshomologygroupsH (X,A)
• •
dependonbothX andAinafunctorialway.
For example, for any continuous map f : X → Y, we have an induced
homomorphism
(cid:88) (cid:88)
f : C (X,A) → C (Y,A), a(x)x (cid:55)→ a(x)f ◦x
∗ n n
x∈Xn x∈Xn
x
for each n. Recall that x ∈ X is a continuous map ∆ −→X, and f ◦ x
n n
is just its composition with f. It is clear that f commutes with differen-
∗
tials and therefore induces a map of complexes f : C (X,A) → C (Y,A),
∗ • •
andinducedhomomorphismsofAbeliangroupsH (X,A) → H (Y,A). In
• •
short,wehaveafunctor
Top → Kom(Ab), X (cid:55)→ C (X,A).
•
8 MATH797–HOMOLOGICALMETHODS
For example, suppose we have an embedding i : Y (cid:44)→ X of topological
spaces. Thenwehaveamap,infactclearlyamonomorphism
i : C (Y,A) (cid:44)→ C (X,A).
∗ • •
We define a complex of relative singular chains C (X,Y;A) as a cokernel of
•
i ,i.e.C (X,Y;A) = C (X,A)/C (Y,A)foreachk.
∗ k k k
Similarly, for each homomorphism f : A → B, we have an induced
homomorphism
(cid:88) (cid:88)
C (X,A) → C (X,B), a(x)x (cid:55)→ f(a(x))x,
n n
x∈Xn x∈Xn
for each n. These homomorphisms obviously commute with differentials,
whichmeansthattheygiveamapofcomplexesC (X,A) → C (X,B)and
• •
ahomomorphismofhomologygroupsH (X,A) → H (X,B). Inshort,we
• •
haveafunctor
Ab → Kom(Ab), A (cid:55)→ C (X,A).
•
Forexample,anyshortexactsequencesofAbeliangroups
0 → A → B → C → 0
givesashortexactsequenceofsingularchaincomplexes
0 → C (X,A) → C (X,B) → C (X,C) → 0,
• • •
andaninducedlongexactsequenceofhomology
... → H (X,A) → H (X,B) → H (X,C) → H (X,A) → ...
k k k k−1
§6. DE RHAM COHOMOLOGY.
As an example of cohomology theory, let’s review basics of de Rham
cohomology. With each smooth manifold M, we are going to associate its
R
deRhamcomplexof -vectorspaces
Ω0(M,R)−d→Ω1(M,R)−d→Ω2(M,R)−d→...
For each smooth map f : M → N, and each k, we have a pull-back linear
map f∗ : Ωk(N,R) → Ωk(M,R). Altogether, this gives a contravariant
functor
Mflds → Kom(Vect )
R
fromthecategoryofsmoothmanifoldstothecategoryofcomplexesofvec-
tor spaces. Taking cohomology of de Rham complexes, we get de Rham
cohomologygroupsHk (M,R)foreachk ≥ 0.
dR
Nowthedetails. Sincesmoothmanifoldsofdimensionnaregluedfrom
opensubsetsofRn,itisusefultoconsiderthissituationfirst,wherewecan
andwillworkincoordinates.
ForanopensubsetU ⊂ Rn,wedefineΩk(U,R) = C∞(U)⊗RΛk(Rn)∨ as
avectorspaceofk-lineardifferentiableforms
(cid:88)
ω = f dx ∧...∧dx
i1...ik i1 ik
1≤i1<...<ik≤n
MATH797–HOMOLOGICALMETHODS 9
Here f ∈ C∞ is a smooth function for each subset i ,...,i . Symbols
i1...ik 1 k
dx satisfystandardrelationsoftheexterioralgebra:
i
dx ∧dx = −dx ∧dx ,
i j j i
which makes Ω•(U,R) into a graded R-algebra1 with respect to the exterior
product∧. Andofcoursewehavean“exterior”differential
d : Ωk(U,R) → Ωk+1(U,R)
 
n
(cid:88) fi1...ikdxi1∧...∧dxik (cid:55)→ (cid:88) (cid:88) ∂f∂i1x...ikdxidxi1∧...∧dxik
i
1≤i1<...<ik≤n 1≤i1<...<ik≤n j=1
Clairaut’s Theorem (equality of mixed partials) implies that d2 = 0, so
Ω•(U,R) is also a complex, called de Rham complex. The differential and
exteriormultiplicationarerelatedbythefollowingequation:
d(ω ∧ω ) = dω ∧ω +(−1)kω ∧ω for ω ∈ Ωk(U,R), ω ∈ Ωl(U,R).
1 2 1 2 1 2 1 2
Thisstructurecanbeaformalizedasfollows:
6.1.DEFINITION. Adifferentiablegradedalgebra(DGA)isagradedk-algebra
∞
A = (cid:76) An equippedwithadifferentialdsuchthat
n=0
• (A•,d)isacochaincomplex;
• d(w ·w ) = dw ·w +(−1)kw ·w ifw ∈ Ak,w ∈ Al.
1 2 1 2 1 2 1 2
SoΩ•(U,R)isaDGA.
Now let’s discuss functoriality: given open sets U ⊂ Rn (with coordi-
natesx ,...,x ),V ⊂ Rm (withcoordinatesy ,...,y ),andasmoothmap
1 n 1 m
α : U → V,wedefinethepullbackmap
α∗ : Ωk(V,R) → Ωk(U,R),
(cid:88) (cid:88)
f dy ∧...∧dy (cid:55)→ (f ◦g)d(y ◦g)∧...∧d(y ◦g),
i1...ik i1 ik i1...ik i1 ik
1≤i1<...<ik≤n 1≤i1<...<ik≤n
whereforanyfunctionhonU (forexampley ◦g),dhdenotesitsdifferential
i
fromcalculus
(cid:88) ∂h
dh = dx
i
∂x
i
i
Itisnothardtoshowthatthisgivesacontravariantfunctorforeachk
(cid:26) (cid:27)
open subsets of a vector space,
→ DGAR,
smooth maps
U (cid:55)→ Ω•(U,R), [U −α→V] (cid:55)→ [Ω•(V,R)−α→∗ Ω•(U,R)].
Indeed, it immediately follows from definitions that the exterior multipli-
cation and the exterior differential are preserved by pull-back. The most
nontrivialpartisthat
(f ◦g)∗ = g∗◦f∗
∞
1A ring S decomposed into a direct sum S = LSi of Abelian subgroups is called
i=0
gradedifSi·Sj ⊂Si+j.AringSwitharinghomomorphismk→Siscalledak-algebra.
10 MATH797–HOMOLOGICALMETHODS
g f
for a composition U −→V −→W of smooth maps. This is a Chain Rule
fromcalculus.
Now we extend our de Rham functor to manifolds. A manifold M has
anatlasM = ∪ U , whereeachU isisomorphictoanopensubsetofRn.
α α α
Fix this isomorphism. Each intersection U ∩U is isomorphic to an open
α β
subsetofRn intwodifferentways,viainclusions
iα : U ∩U (cid:44)→ U , iβ : U ∩U (cid:44)→ U
αβ α β α αβ α β β
Fixoneoftheseisomorphisms. Thenwedefine
(cid:40) (cid:41)
Ωk(M,R) = (ω ) ∈ (cid:89)Ωk(U ,R)|(iα )∗ω = (iβ )∗ω for anyα,β
α α αβ α αβ β
α
Functoriality of Ω•(U ,R) implies that the definition of Ω•(M,R) is inde-
α
pendent of any choices, is a DGA, and is functorial with respect to pull-
back. To show that Ω•(M,R) is independent of the choice of the atlas, one
chooses a common refinement and shows that Ω•(M,R) does not change
under refinement. More intrinsically, one can define differential forms as
sections of a sheaf or as sections of a vector bundle. We will return to this
whenwediscusssheaves.
6.2. EXAMPLE.
H0 (M,R) = {f ∈ C∞(M)|df = 0} = {locally constant fucntions on M}
dR
Inparticular,dimH0 (M,R)isthenumberofconnectedcomponentsofM.
dR
§7. FREE, PROJECTIVE, AND INJECTIVE RESOLUTIONS. JAN 30.
Nowwewilluse(co)homologytostudyaringRratherthanaspaceX.
7.1. LEMMA.AnyR-moduleM admitsafreeresolution,i.e.anexactsequence
... → F −f→2 F −f→1 F −f→0 M → 0,
2 1 0
whereF isafreeR-moduleforanyi. IfRisNoetherianandM isfinitelygener-
i
ated,thereexistsafreeresolutionsuchthatanyF isfinitelygenerated.
i
Proof. Choosegenerators{g } ofM anddefineasurjectivemap
i i∈S
(cid:77)
f : F = R → M, e (cid:55)→ g ,
0 0 i i
i∈S
iterate to construct a surjection f : F → Kerf , etc. Notice that Kerf
1 1 0 0
(and,byinduction,Kerf foranyi)isfinitelygeneratedifRisNoetherian
i
andM isfinitelygeneratedbyg ’s. (cid:3)
i
In more general Abelian categories (such as the category of complexes),
it’snotclearhowtodefinea“freeobject”. However,onecanalwaysdefine
a “projective object”, because the following definition makes sense in any
Abeliancategory:
7.2.DEFINITION. AnR-moduleP iscalledprojective,ifforanysurjectionof
p f g
R-modulesA−→B → 0,andanymapP −→B,thereexistsamapP −→A
suchthatp◦g = f.