Commutative Algebra, Homological Algebra and Sheaves

124 Pages·2014·0.75 MB·English

by  Chambert-Loir A

COMMUTATIVE ALGEBRA,
HOMOLOGICAL ALGEBRA
AND SHEAVES
Antoine Chambert-Loir
Antoine Chambert-Loir
Laboratoire de mathématiques, Université Paris-Sud,
Bât. 425, Faculté des sciences d’Orsay, F-91405 Orsay Cedex.
E-mail :
[email protected]
Version of September 16, 2014, 9h45
The most up-do-date version of this text should be accessible online at address
http://www.math.u-psud.fr/~chambert/2014-15/ga/acahtf.pdf
©1998–2014, Antoine Chambert-Loir
CONTENTS
1. Commutative algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Recollections (Uptempo). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Localization (Medium up). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3. Nakayama’s lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4. Integral and algebraic dependence relations. . . . . . . . . . . . . . . . . . 9
1.5. The spectrum of a ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6. Finitely generated algebras over a field. . . . . . . . . . . . . . . . . . . . . . . 17
1.7. Hilbert’s Nullstellensatz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.8. Tensor products (Medium up). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.9. Noetherian rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.10. Irreducible components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.11. Dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.12. Artinian rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.13. Codimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2. Categories and homological algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.1. The language of categories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2. Functors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3. Limits and colimits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.4. Representable functors. Adjunction. . . . . . . . . . . . . . . . . . . . . . . . . 64
2.5. Exact sequences and complexes of modules. . . . . . . . . . . . . . . . . . 69
2.6. Differential modules and their homology. . . . . . . . . . . . . . . . . . . . 73
2.7. Projective modules and projective resolutions. . . . . . . . . . . . . . . . 79
2.8. Injective modules and injective resolutions. . . . . . . . . . . . . . . . . . . 83
iv CONTENTS
2.9. Abelian categories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2.10. Exact sequences in abelian categories. . . . . . . . . . . . . . . . . . . . . . . 92
2.11. Projective and injective objects in abelian categories. . . . . . . . . . 99
2.12. Derived functors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3. Sheaves and their cohomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.1. Presheaves and sheaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.2. Direct and inverse images of sheaves. . . . . . . . . . . . . . . . . . . . . . . . 108
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
CHAPTER 1
COMMUTATIVE ALGEBRA
1.1. Recollections (Uptempo)
1.1.1. Basic algebraic structures. — The concepts of groups, rings, fields,
modulesare assumed to be known, as well as the notion of morphisms of
groups, rings, fields, modules, etc.
In this course, rings are always commutative and possess a unit element,
generally denoted by 1. The multiplicative group of invertible elements of a
ring Awill be denoted by A∗ or A×.
1.1.2. Algebras. — Let k be a ring. A k-algebra is a ring Aendowed with a
morphism of rings f k A. When this morphism is injective, we will often
∶ →
understate the morphism f and consider that Ais an overring of k, or that
k is a subring of A... Let A, f k A and B, g k B we two k-algebras;
( ∶ → ) ( ∶ → )
a morphism of k-algebras is a ring morphism φ A B such that g φ f.
∶ → = ○
1.1.3. Polynomial algebras. — Let I be a set. One defines a k-algebra
k X of polynomials with coefficients in k in a family X of in-
[( i)i∈I] ( i)i∈I
determinates indexed by I. This algebra satisfies the following universal
property: foreveryfamily a ofelementsof A,thereexistsauniquemor-
( i)i∈I
phism φ k X Aof k-algebras such that φ X a for every i I.
∶ [( i)i∈I] → ( i) = i ∈
In other words, for every k-algebra A, the canonical map
Hom k X ,A Hom I,A , φ i φ X
k−Algebras( [ i] ) → Ens( ) ↦ ( ↦ ( i))
is a bijection.
2 CHAPTER1.COMMUTATIVEALGEBRA
When I has one, two, three,... elements, the indeterminates are often
denoted by individual letters, say X, Y, Z,...
Let J be a subset of I, and let K be its complementary subset. The
polynomial algebra k X is isomorphic to the polynomial algebra
[( i)i∈I]
k X X in the indeterminates X (for i K) with coeffi-
[( i)i∈J][( i)i∈K] i ∈
cients in the polynomial algebra k X with coefficients in k in the
[( i)i∈J]
indeterminates X (for i J).
i ∈
We do not detail the notion of degree in one of the indeterminates (of
degree, if I is a singleton).
Thereisanotionofeuclideandivisioninpolynomialrings. Let Abearing,
let f, g A X be polynomials in one indeterminate X with coefficients
∈ [ ]
in A. Iftheleadingcoefficientof g isinvertiblein A, thereexistauniquepair
q,r of polynomials in A X such that f gq r and deg r deg g .
( ) [ ] = + ( ) < ( )
1.1.4. Ideals. — Anidealofaring Aisanon-emptysubset I whichisstable
underaddition,andsuchthat ab I forevery a Aandeveryb I. Inother
∈ ∈ ∈
words, this is a A-submodule of A.
Thesubsets 0 and Aareideals. Theintersectionofafamilyofidealsof A
{ }
is an ideal. If S is a subset of A, the ideal generated by S is the smallest ideal
of Acontaining S (it is the intersection of all ideals of Awhich contain S).
Let I and J be ideals of A; the ideal I J (resp. the ideal I J, also denoted by
+ ⋅
IJ) is the ideal generated by the set of sums a b (resp. the set of products
+
ab) for a I and b J. The ideal generated by a family of elements of Ais
∈ ∈
often denoted by a ; for example a , a,b , a ,a ,a ...
(( i)i∈I) ( ) ( ) ( 1 2 3)
The image φ I of an ideal I of A under a morphism of rings φ A B
( ) ∶ →
is generally not an ideal of B; the ideal it generates is often denoted by IB.
However, the inverse image of an ideal J of B by such a morphism of rings is
alwaysanidealof A. Inparticular,thekernelker φ φ−1 0 ofamorphism
( ) = ( )
of rings is an ideal of A.
Let I be an ideal of A. The relation x y defined by x y I is an
∼ − ∈
equivalence relation. The quotient set A , denoted by A I, admits a unique
/∼ /
ring structure such that the canonical surjection π A A is a morphism
∶ → /∼
of rings. The so-called quotient ring A I possesses the following universal
/
1.1.RECOLLECTIONS(UPTEMPO) 3
property: for every ring B and every morphism of rings f A B such that
∶ →
f I 0 , there exists a unique morphism of rings φ A I B such that
( ) = { } ∶ / →
f φ π.
= ○
Thekernelofthecanonicalmorphism π istheideal I itself. Moregenerally,
the map associating with an ideal J of A I the ideal π−1 J of Ais a bijection
/ ( )
betweenthe(partiallyordered)setofidealsofA I andthe(partiallyordered)
/
set of ideals of Awhich contain I.
1.1.5. Domains. — Let A be a ring. One says that an element a A is a
∈
zero-divisor if there exists b A, such that ab 0 and b 0. One says that A
∈ = ≠
is an integral domain, or a domain, if A 0 and if 0 is its only zero-divisor.
≠ { }
Fields are integral domains.
1.1.6. Primeandmaximalideals. — Onesaysthatanideal I of Aisprime
if the quotient ring A I is an integral domain. This means that I A and
/ ≠
that for every a,b Asuch that ab I, either a I, or b I.
∈ ∈ ∈ ∈
One says that an ideal I of Ais maximal if the quotient ring A I is a field.
/
This means that I is a maximal element of the partially ordered set of ideals
of Awhich are not equal to A. A maximal ideal is a prime ideal.
One deduces from Zorn’s theorem that every ideal of Awhich is distinct
from A is contained in some maximal ideal. (Indeed, if I is an ideal of A
such that I A, the set of ideals J of A such that I J A, ordered by
≠ ⊂ ⊊
inclusion,isinductive—everytotallyorderedsubsetadmitsanupper-bound)
In particular, every non-zero ring contains maximal ideals.
Hilbert’s Nullstellensatz (theorem 1.7.1 below) gives a description of the
maximal ideals of polynomials rings over algebraically closed fields.
1.1.7. — Ifaringadmitsexactlyonemaximalideal,onesaysthatitisalocal
ring. A ring is local if and only if its set of non-invertible elements is an ideal
(exercise!).
1.1.8. — The intersection J of all maximal ideals of a ring A is called its
Jacobson radical. It admits the following characterization: one has a J if
∈
and only if 1 ab is invertible in Afor every b A(exercise!).
+ ∈
4 CHAPTER1.COMMUTATIVEALGEBRA
1.1.9. — Let A be an integral domain. One says that an element a A is
∈
irreducible if it is not invertible and if the equality a bc for b,c Aimplies
= ∈
that b or c is invertible. An element a is said to be prime if the principal
ideal a isprime; thisimpliesthat a isirreduciblebuttheconversedoesnot
( )
hold(exercise!;show,forexample,thattheelement1 i√5oftheringZ i√5
+ [ ]
is irreducible but not prime).
One says that the ring Ais a unique factorization domain (ufd, in short) if
the following two properties hold:
(a) Every strictly increasing sequence of principal ideals of Ais finite;
(b) Every irreducible element of Agenerates a prime ideal.
Indeed, these two properties are equivalent to the fact that every non-zero
element of Acan be written as the product of an invertible element and of
finitely many prime elements of A, in a unique way up to the order of the
factors and to multiplication of the factors by units.
Condition (ii) is sometimes stated under the name of ‘‘Gauss’s lemma’’:
If A is a ufd, then every irreducible element a which divides a product bc
must divide one of the factors b or c. Condition (i) obviously holds when A
is noetherian. Consequently, a noetherian ring for which Gauss’s lemma
holds is a ufd.
Principal ideal rings are unique factorization domains, as well as poly-
nomial rings over fields. In fact, if A is a ufd, then so is A X (a theorem
[ ]
proved by Gauss for A Z).
=
1.2. Localization (Medium up)
Let Abe a ring.
1.2.1. Nilpotent elements. — One says that an element a Ais nilpotent
∈
if there exists an integer n 1 such that an 0. The set of nilpotent elements
⩾ =
ofAisanidealofA,calleditsnilradical. When0istheonlynilpotentelement
of A, one says that Ais reduced. More generally, when I is an ideal of A, one
√
defines the radical of I, denoted by I, as the set of all a I for which there
∈
1.2.LOCALIZATION(MEDIUMUP) 5
exists an integer n 1 such that an I; it is an ideal of Awhich contains I.
⩾ ∈
An ideal which is equal to its radical is called a radical ideal.
1.2.2. Multiplicative subsets. — A multiplicative subset of A is a subset
S Awhich contains 1 and such that ab S for every a,b S.
⊂ ∈ ∈
1.2.3. — Let M be an A-module. The fraction module S−1M (sometimes
also denoted by M ) is the quotient of the set M S by the equivalence
S ×
relation such that m,s m′,s′ if and only if there exists t S such
∼ ( ) ∼ ( ) ∈
that t sm′ s′m 0. Let us denote by m s the class in S−1M of the pair
( − ) = /
m,s M S. The addition of the abelian group S−1M is given by the
( ) ∈ ×
familiar formulas
m s m′ s′ s′m sm′ ss′,
( / )+( / ) = ( + )/
for m,m′ M, s,s′ S; its zero is the element 0 1. Its structure of an
∈ ∈ /
A-module is given by a m s am s, for a A, m M and s S.
⋅( / ) = ( )/ ∈ ∈ ∈
For every s S, the multiplication by s is an isomorphism on S−1M—one
∈
saysthat S actsbyautomorphismson S−1M. Themap θ M S−1M givenby
∶ →
θ m m 1 is a morphism of A-modules; it satisfies the following universal
( ) = /
property: For every morphism of A-modules f M N such that S acts
∶ →
by automorphisms on N, there exists a unique morphism of A-modules
φ S−1M N such that f φ θ (explicitly: f m φ m 1 ) for every
∶ → = ○ ( ) = ( / )
m M).
∈
1.2.4. — Let B be an A-algebra. Then the module of fractions S−1B has a
natural structure of an A-algebra for which the multiplication is given by
the familiar formulas
b s b′ s′ bb′ ss′ ,
( / )⋅( / ) = ( )/( )
for b,b′ B and s,s′ S; its zero and unit are the elements 0 1 and 1 1. The
∈ ∈ / /
canonical map θ B S−1B is a morphism of A-algebras, and the images of
∶ →
the elements of S are invertible in S−1B. In fact, this morphism satisfies the
following universal property: For every morphism of A-algebras f B B′
∶ →
6 CHAPTER1.COMMUTATIVEALGEBRA
such that the images of elements of S are units of B′, there exists a unique
morphism of A-Algebras φ S−1B B′ such that f φ θ.
∶ → = ○
Inparticular,S−1AitselfisanA-algebra.. Moreover,foreveryA-module M,
the A-module S−1M has a natural structure of a S−1A-module.
The ring S−1Ais the zero ring if and only if 0 S.
∈
1.2.5. Examples. — Let us give examples of multiplicative subsets and let
us describe the corresponding ring of fractions.
– Let a A; the set S 1,a,a2,... is a multiplicative subset which
∈ = { }
contains 0 if and only if a is nilpotent. The corresponding fraction ring is
often denoted by A . Let φ A T A be the morphism of rings given by
a 1∶ [ ] → a
φ P P 1 a ;itissurjectiveanditskernelcontainsthepolynomial1 aT.
1( ) = ( / ) −
Let φ A T 1 aT A be the morphism of rings which is deduced
∶ [ ]/( − ) → a
from φ by passing to the quotient; let us show that φ is an isomorphism by
1
constructing its inverse.
The obvious morphism ψ A A T 1 aT maps a to an invertible
1∶ → [ ]/( − )
element of A T 1 aT ; by the universel property of the localization,
[ ]/( − )
there exists a unique morphism of rings ψ A A T 1 aT such that
∶ a → [ ]/( − )
ψ b b for every b A; one has ψ b an bcl T n for every b A and
( ) = ∈ ( / ) = ( ) ∈
every integer n 0. Moreover, φ ψ b an b an, so that φ ψ id. In the
⩾ ○ ( / ) = / ○ =
otherdirection,ψ φ b b foreveryb Aandψ φ T ψ 1 a cl T ;
○ ( ) = ∈ ○ 1( ) = ( / ) = ( )
consequently, ψ φ P cl P for every polynomial P A T , hence
○ 1( ) = ( ) ∈ [ ]
ψ φ id. This shows that φ is an isomorphism, with inverse ψ, as claimed.
○ =
– Let I be an ideal of A. The set S 1 I a I; a 1 I is a
= + = { ∈ − ∈ }
multiplicative subset of A.
– Let f A B be a morphism of rings, let T be a multiplicative subset
∶ →
of B and let S f−1 T . Then S is a multiplicative subset of Aand there is
= ( )
a unique morphism of rings φ S−1A T−1B such that φ a 1 f a 1 for
∶ → ( / ) = ( )/
every a A.
∈
– If Ais an integral domain, then S A 0 is a multiplicative subset
= { }
of A; the fraction ring S−1Ais a field, called the field of fractions of A.
– Letpbeanidealof Aandlet S A p. Then S isamultiplicativesubset
=
of Aif and only if p is a prime ideal of A; the fraction ring is denoted A .
p