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MAS439/MAS6320 COMMUTATIVE ALGEBRA

创建日期：2024-05-08 16:54:35

所属分类：数学mathematics

标签： MAS439

COMMUTATIVE ALGEBRA

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MAS439/MAS6320 COMMUTATIVE ALGEBRA

AND ALGEBRAIC GEOMETRY
Contents
1. Introduction 3
2. Commutative rings– definitions and examples 5
3. Subrings 7
4. Homomorphisms of rings 9
5. Ideals 11
6. Prime ideals 15
7. Radicals and nilradicals 18
8. Sneak preview: Algebraic sets 19
8.1. The Zariski Topology 22
8.2. The Nullstellensatz 23
8.3. Affine varieties 26
8.4. Coordinate rings 28
8.5. Polynomial maps 30
9. Modules 32
10. Homomorphisms of R-modules 33
11. Quotients of modules 34
12. The first Isomorphism Theorem 35
13. Generators of modules 35
14. Direct sums and products 37
15. Free modules 37
16. Localization of rings 38
17. Localization of modules 41
18. Exact sequences 42
19. The exactness of localization 43
20. Local properties 44
21. Further properties of localization 45
22. Noetherian rings 46
23. Hilbert’s Basis Theorem 48
24. Primary decomposition 49
1
MAS439/MAS6320 COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY 2
25. Artinian rings 53
26. The height on an ideal 54
27. Matrix algebra over commutative rings 56
28. Algebras 57
29. Integral extensions 58
30. Noether’s Normalization Theorem 60
31. Hilbert’s Nullstellensatz 62
32. The Going Up Theorem 63
33. Dimensions of rings 64
34. Algebraic sets, again 65
35. Hilbert’s Nullstellensatz– strong form 66
36. Irreducible algebraic sets, again 68
37. The dimension of an algebraic set 69
MAS439/MAS6320 COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY 3
1. Introduction
Commutative algebra is the study of comuutative rings. These are sets R having
two composition laws (called addition and multiplication) which behave in the same
way as addition and multiplication of integers. Commutative rings come up all over
pure mathematics. Here are two examples:
• Number theory. The Gaussian integers
Z[i] = {a+ bi : a, b ∈ Z} ⊂ C
form a commutative ring under the usual operations of addition and mul-
tiplication of complex numbers. Similarly, if we consider the third root of
unity
ω = exp(2pii/3) = (−1 +
√
3 i)/2
then the subset
Z[ω] = {a+ bω : a, b ∈ Z} ⊂ C
form a commutative ring. To see this note the relations
ω3 = 1, ω2 + ω + 1 = 0.
Rings of algebraic integers like these are useful for studying Diohantine equa-
tions. For example, to prove the first case n = 3 of Fermat’s last theorem one
writes
z3 = x3 + y3 = (x+ y)(x+ ωy)(x+ ω2y)
and considers prime factorizations of both sides in the ring Z[ω].1
• Geometry. We can study a topological space X (for example a subset X ⊂ Rn)
via its ring of functions. This is the set of all continuous functions f : X → R.
These can be added and multiplied pointwise:
(f + g)(x) = f(x) + g(x), (f · g)(x) = f(x) · g(x).
Similarly we can consider rings of differentiable or analytic functions.
Algebraic geometry is the study of systems of polynomial equations. The set of
solutions to such a system is called an affine variety. We first choose a field K we wish
to work over (e.g. K = Q, R or C). Given a collection of polynomials f1, f2, · · · , fr
in n variables x1, · · · , xn with coefficients in K, the corresponding affine variety is the
subset
V (f1, · · · , fr) ⊆ Kn
consisting of those points (a1, · · · , an) ∈ Kn satisfying
f1(a1, · · · , an) = f2(a1, · · · , an) = · · · = fr(a1, · · · , an) = 0.
1See Hardy and Wright, An Introduction to the Theory of Numbers, Chapters 12–13.
MAS439/MAS6320 COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY 4
Perhaps surprisingly, algebraic geometry occupies a central place in modern pure
mathematics. The basic reason is that the study of polynomial equations has many
different aspects, which relate to lots of other areas of pure mathematics. We give a
few examples of this:
• Number theory. If we take the field k = Q then polynomial equations become
Diophantine equations. For example Fermat’s last theorem is the statement
that if n > 2 the variety
xn + yn = 1
has no points over the field k = Q with x, y nonzero.
• Geometry and topology. Working over C we can study the geometry and
topology of varieties. For example, consider the variety
y2 = x3 − x.
This is an example of an elliptic curve. Topologically it is a torus (dough-
nut) with 3 points removed. To get the full torus we must work with the
corresponding projective variety2.
• Physics. In string theory the world is supposed to have 10 dimensions. 4 of
these dimensions form spacetime, the other 6 are supposed to be curled up very
small, and account for the properties of the fundamental forces. For string
theory to work properly these curled up dimensions must have a special shape:
they should be Calabi-Yau threefolds3. Examples of Calabi-Yau threefolds are
most easily described using polynomial equations. For example, again working
over C, the variety
x51 + x
5
2 + x
5
3 + x
5
4 + x1x2x3x4 = 0
is a Calabi-Yau threefold known as the quintic threefold. Note that it has
4− 1 = 3 complex dimensions, and hence 6 real dimensions.
• Algebra. The basic calculational tool in algebraic geometry is commutative
algebra. Every affine variety X has an associated co-ordinate ring
K[V ] = K[x1, · · · , xn]/(f1, · · · , fr),
which is a commutative ring. It is obtained by quotienting the polynomial
ring K[x1, · · · , xn] by the ideal generated by the polynomials defining V .
This course will focus on the relationship between commutative algebra and alge-
braic geometry. We aim to introduce the basic commutative algebra needed to study
2See Frances Kirwan, ‘Complex algebraic curves’, Chapter 5.
3See Brian Greene, ‘The Elegant Universe’.
MAS439/MAS6320 COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY 5
affine varieties, and to build up an intuition for the relationship between the geomet-
rical properties of an affine variety V and the algebraic properties of the associated
co-ordinate ring K[V ].
2. Commutative rings– definitions and examples
Definition 2.1. A ring is a set R together with two binary operations + : R×R→ R
(“addition”) and · : R×R→ R (“multiplication”) such that
(a) (R,+) is an Abelian group (we call the additive identity 0R or just 0),
(b) multiplication is associative: (a · b) · c = a · (b · c) for all a, b, c ∈ R, and there
exists a multiplicative identity 1R: a1R = 1Ra = a for all a ∈ R, and
(c) for all a, b, c ∈ R, a · (b+ c) = a · b+ a · c and (a+ b) · c = a · c+ b · c.
If a ring R satisfies the additional condition that a · b = b · a for all a, b ∈ R then we
call R a commutative ring.
Remark. Notice that for any r ∈ R, −1R · r = −r and 0R · r = (1R + (−1R))r =
r + (−r) = 0R.
Remark. Notice that a field is just a commutative ring R in which every element other
that 0R has a multiplicative inverse.
Example. The integers Z with its usual addition and multiplication is a commutative
ring. Also, Z/nZ the integers mod n form a commutative ring.
Example. Let C[0, 1] be the set of continuous real functions defined on [0, 1]. For
f, g ∈ C[0, 1] define (f + g)(x) = f(x) + g(x) and (f · g)(x) = f(x)g(x). This turns
C[0, 1] into a commutative ring.
Example (Polynomial rings). Let R be a commutative ring. The set of polynomials
in one variable with coefficients in R forms a commutative ring R[x]. The elements
are finite sums of the form
f(x) =
d∑
i=0
aix
i = a0 + a1x+ a2x
2 + · · ·+ adxd
with a0, a1, · · · ad ∈ R. If ad 6= 0 we call the number d ≥ 0 the degree of the polynomial
f(x) (we consider the zero polynomial to have degree 0). Polynomials of degree 0 are
called constant polynomials. The addition and multiplication laws are( n∑
i=0
aix
i
)
+
( n∑
i=0
bix
i
)
=
n∑
i=0
(ai + bi)x
i.
( m∑
i=0
aix
i
) · ( n∑
i=0
bjx
j
)
=
mn∑
k=0
ckx
k where ck =
k∑
i=0
ai · bk−i.
MAS439/MAS6320 COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY 6
The zero element is the zero polynomial 0R. The identity is the constant polynomial
1R.
Now we can define R[x1, . . . , xn], the polynomial ring in x1, . . . , xn with coefficients
in R, recursively as R[x1, . . . , xn] = R[x1, . . . , xn−1][xn].
Example (Power series rings). Let R be a commutative ring and define R[[x]] to be
the set of all power series in x with coefficients in R, i.e., all expressions a0 + a1x +
· · ·+ anxn + . . . where a0, . . . , an, · · · ∈ R with the following operations( ∞∑
i=0
aix
i
)
+
( ∞∑
j=0
bjx
j
)
=
∞∑
i=0
(ai + bi)x
i,
( ∞∑
i=0
aix
i
)
·
( ∞∑
j=0
bjx
j
)
=
∞∑
d=0
d∑
k=1
(ak · bd−k)xd.
Its not hard to see that R[[x]] is a commutative ring with additive identity 0R + 0Rx+
. . . 0Rx
n + . . . and multiplicative identity 1R + 0Rx+ 0Rx
2 . . . 0Rx
n + . . . .
Now we can define R[[x1, . . . , xn]], the power series ring in x1, . . . , xn with coefficients
in R, recursively as R[[x1, . . . , xn]] = R[[x1, . . . , xn−1]][[xn]].
Example (Direct Products). Let R and S be commutative rings. Define the direct
product of R and S to be the Cartesian product R×S with addition (r1, s1)+(r2, s2) =
(r1 + r2, s1 + s2) and multiplication (r1, s1) · (r2, s2) = (r1 · r2, s1 · s2). This is a
new commutative ring, which we denote R × S, with additive identity (0R, 0S) and
multiplicative identity (1R, 1S).
Definition 2.2. Let R be a commutative ring. An element a ∈ R is a zero-divisor if
there exists a b ∈ R \ {0R} such that ab = 0R.
We say that R is an integral domain (or just a domain, for short) if 0R is its only
zero-divisor.
Example. Z is a domain and so is any field; Z/nZ is a domain if and only if n is
prime.
Example. Let R and S be commutative rings; R × S is never a domain because
(1R, 0S) · (0R, 1S) = (0R, 0S) = 0R×S.
Proposition 2.3. If R is a domain, so are R[x] and R[[x]].
MAS439/MAS6320 COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY 7
Proof. Suppose that for non-zero f, g ∈ R[[x]] we have fg = 0. Since f, g are not
zero we can write f = amx
m + am+1x
m+1 + . . . and g = bnx
n + bn+1x
n+1 + . . . with
am, bn 6= 0R. Now
0R[[x]] = f · g = ambnxm+n + (am+1bn + ambn+1)xm+n+1 + . . .
so ambn = 0R. But R is an integral domain and am, bn 6= 0R so these are not zero-
divisors and we obtain a contradiction.
The proof that R[x] is an integral domain is similar.
Remark. From now on we will refer to the additive and multiplicative identities of a
ring R as 0 and 1 rather that 0R and 1R if no confusion is likely.
3. Subrings
Definition 3.1. Let S be a commutative ring. We say that R ⊆ S is a subring of S
if
(a) the addition and multiplication of S restricted to R make R into a commuta-
tive ring, and
(b) 1S ∈ R.
Proposition 3.2 (The subring criterion). Let S be a commutative ring and let R ⊆ S
be a subset. Then R is a subring of S if and only if
(a) 1S ∈ R,
(b) R is closed under addition and multiplication,
(c) for all a ∈ R, −a ∈ R.
Example. The following is a chain of subrings
Z ⊆ Q ⊆ R ⊆ C
and so is
R ⊆ R[[x1]] ⊆ R[[x1, x2]] ⊆ R[[x1, x2, x3]] ⊆ . . . .
Example (Polynomial rings). Let R be a commutative ring; let R[x] be the subset
of R[[x]] consisting of finite sums, i.e., all elements
∑∞
i=0 aix
i ∈ R[[x]] for which there
exists an n ≥ 0 such that ai = 0R for all i ≥ n.
Lemma 3.3. Let R be a commutative ring, let Λ be a set, and for each element
λ ∈ Λ, let Sλ be a subring of R. Then the intersection S =
⋂
λ∈Λ Sλ ⊂ R is a subring.
MAS439/MAS6320 COMMUTATIVE ALGEBRA AND ALGEBRAIC GEOMETRY 8
Proof. This is immediate from the definitions. Let us check for example that the
intersection S is closed under addition. Suppose given elements a, b ∈ S. This means
precisely that a, b ∈ Sj for all j ∈ J . Then since the subsets Sj ⊂ R are subrings we
have a+ b ∈ Sj for all j ∈ J . This then implies that a+ b ∈ S.
Definition 3.4. Let R be a ring and T ⊂ R a subset. The subring of R generated
by T is the intersection of all subrings S ⊂ R containing T . It is denoted 〈T 〉 ⊂ R.
Note that 〈T 〉 ⊂ R is indeed a subring by Lemma 3.3. In fact it is the smallest
subring of R containing T : if S ⊂ R is any other subring containing T then by
definition we have 〈T 〉 ⊂ S. To describe 〈T 〉 ⊂ R explicitly we first define the subset
(1) Tˆ = {t1 · t2 · · · tn : n ≥ 0, ti ∈ T} ⊂ R,
consisting of all finite (possibly empty) products of elements of T . Note that 1R ∈ Tˆ
by definition, since we interpret the empty product as meaning 1R.
Lemma 3.5. The subring 〈T 〉 ⊂ R consists of those elements of R which can be
written as finite (possibly empty) sums of the form
(2) r = ±p1 ± p2 ± · · · ± pk with k ≥ 0 and pi ∈ Tˆ .
Here, when k = 0, we interpret the empty sum as meaning 0R.
Proof. Let X ⊂ R be the set of all elements of the form (2). It is easy to see that any
subring S ⊂ R containing T also contains X, since by definition S is closed under
addition, multiplication and additive inverses. In particular X ⊂ 〈T 〉. On the other
hand, we claim that the subset X ⊂ R is itself a subring. Since T ⊂ X it then follows
from Definition 3.4 that 〈T 〉 ⊂ X, and hence that 〈T 〉 = X.
To prove the claim note that X is clearly closed under addition and additive inverses
and contains 1R. To see that it is closed under multiplication we use
(±p1 ± · · · ± pk) · (±q1 ± · · · ± ql) =
∑
i,j
±(pi · qj).
Since pi · qj ∈ Tˆ for all i, j, the sum on the right also lies in X.
Definition 3.6. We say that a ring R is generated by a subset T ⊂ R if R = 〈T 〉.
We say that a ring R is finitely-generated if it is generated by a finite subset.
Example.

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