代数几何入门：Steven Dale Cutkosky著

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“Introduction to Algebraic Geometry 2018” 是一本关于代数几何的高清电子书，由Steven Dale Cutkosky撰写，是Graduate Studies in Mathematics系列的第188卷，最新英文版。代数几何是一门数学学科，它结合了抽象代数和几何学的元素，研究的是多变量的代数方程组所定义的空间结构。Steven Dale Cutkosky的这本著作是该领域的入门教材，旨在为研究生和对代数几何感兴趣的学者提供基础理论和概念。本书可能涵盖了以下几个关键知识点： 1. **基本概念**：书中可能介绍了代数簇、簇上的点、投影空间、射影坐标以及线性系统等基础概念，这些都是代数几何的基石。 2. **代数结构**：代数几何的核心是研究多项式方程的解集，书中可能会深入讨论多项式环、理想、格罗布纳基（Groebner basis）和同调代数等工具，这些工具用于理解和描述代数簇的代数结构。 3. **几何性质**：通过引入簇的拓扑和几何属性，如维数、奇点、连通性，Cutkosky可能会解释如何从代数数据中提取几何信息。 4. **簇的局部和全局性质**：局部环、局部坐标、邻域理论等局部概念与全局构造如同调群、上同调类、Chow环和K理论等相结合，构成代数几何的丰富内容。 5. **几何与代数的相互作用**：书中可能会讨论几何变换如仿射变换、射影变换，以及这些变换如何影响代数结构，同时也会涉及代数不变量在几何中的应用。 6. **解析方法**：虽然主要关注代数方法，但Cutkosky可能也会提及解析几何和复分析在解决代数几何问题时的作用，例如黎曼-Roch定理和典范丛理论。 7. **实例和应用**：为了帮助读者理解抽象概念，书中可能会包含许多具体例子，如平面曲线、阿贝尔簇、射影空间中的簇等，并探讨它们的实际应用，如编码理论、密码学和物理学。 8. **现代发展**：由于是最新版本，书里可能包含了代数几何的最新研究成果，如 Mori理论、极小模型纲领、Gromov-Witten不变量等，这些反映了代数几何的现代发展方向。 9. **学习资源**：除了正文内容，书中的参考文献和索引将为读者提供进一步研究的路径，帮助他们深入探索代数几何的各个分支。 10. **教学方法**：作为教材，本书可能采用了易于理解的解释和逐步引导的方式，适合初学者逐步建立对代数几何的理解。 “Introduction to Algebraic Geometry 2018” 是一个全面介绍代数几何的资源，适合想要进入这个领域或深化理解的学者。通过阅读此书，读者可以系统地学习到代数几何的基础理论和现代发展，并掌握相关计算和证明技巧。

1.1. Basic algebra 3

we have an induced isomorphism R[x

,...,x

]/I

∼

S. Letting R =Φ(R)

∼

R/I ∩ R, we have that S =

R[t

,...,t

]wheret

=Φ(x

). More abstractly,

if I is an ideal in the polynomial ring R[x

,...,x

], let S = R[x

,...,x

]/I.

Let

R = R/(I ∩ R) ⊂ S and x

= x

+ I in S.ThenS = R[x

,...,x

An element x ∈ R is a zero divisor if x = 0 and there exists 0 = y ∈ R

such that xy =0. Anelementx ∈ R is nilpotent if x = 0 and there exists

n ∈ N such that x

= 0. The radical of an ideal I in R is

√

I = {f ∈ R | f

∈ I for some n ∈ N}.

AringR is reduced if whenever f ∈ R is such that f

=0forsome

positive integer n, we have that f = 0. Suppose that I is an ideal in a ring

R. The ring R/I is reduced if and only if

√

I = I.

An R-algebra A is ﬁnitely generated if A is generated by a ﬁnite number

of elements as an R-algebra, so that A is a quotient of a polynomial ring

over R in ﬁnitely many variables.

If A is nonzero and is generated by u

,...,u

as a R-algebra, then

R = R1

is a subring of A and A = R[u

,...,u

]. In particular, A is a

quotient of a polynomial ring over R.

If K is a ﬁeld and A is a nonzero K-algebra, then we can view K as a

subring of A by identifying K with K1

The following lemma will be useful in some of the problems in Chapter 2.

Lemma 1.3. Suppose that K is a ﬁeld, K[x

,...,x

,z] is a polynomial ring

over K,andf

,...,f

,g ∈ K[x

,...,x

].Then

A = K[x

,...,x

,z]/(f

,...,x

),...,f

,...,x

),z− g(x

,...,x

))

∼

K[x

,...,x

]/(f

,...,x

),...,f

,...,x

)).

Proof. Let

,...,x

, z be the classes of x

,...,x

,z in A. Wehavethat

A is generated by

,...,x

and z as a K-algebra, and z = g(x

,...,x

so A = K[

,...,x

] is generated by x

,...,x

as a K-algebra.

By the universal property of polynomial rings, we have a K-algebra

homomorphism Φ : K[x

,...,x

] → A deﬁned by Φ(x

)=x

for 1 ≤ i ≤ n.

Since A = K[

,...,x

], Φ is surjective.

We now compute the kernel of Φ. The elements f

,...,x

)arein

Kernel(Φ) since Φ(f

)=f

,...,x

) = 0. Suppose

h(x

,...,x

) ∈ Kernel(Φ).

Then Φ(h(x

,...,x

)) = h(x

,...,x

)=0inA, and so h(x

,...,x

)isin

the ideal

,...,x

),...,f

,...,x

),z− g(x

,...,x

))

6 1. A Crash Course in Commutative Algebra

Exercise 1.12. Suppose that R is a local ring and f

,...,f

∈ R generate

an ideal I of R. Suppose that I is a principal ideal. Show that there exists

an index i such that I =(f

). Give an example to show that this is false in

a polynomial ring k[x].

Exercise 1.13. Let κ be a ﬁeld, and deﬁne Map(κ

,κ) to be the set of maps

(of sets) from κ

to κ.Sinceκ is a κ-algebra, Map(κ

,κ)isaκ-algebra,

with the operations (φ + ψ)(α)=φ(α)+ψ(α), (φψ)(α)=φ(α)ψ(α), and

(cφ)(α)=cφ(α)forφ, ψ ∈ Map(κ

,κ), α ∈ κ

, and c ∈ κ.

a) Let κ[x

,...,x

] be a polynomial ring over κ and deﬁne

Λ:κ[x

,...,x

] → Map(κ

,κ)

by Λ(f )(α)=f (α)forf ∈ k[x

,...,x

] and α ∈ κ

. Show that

Λisaκ-algebra homomorphism. The image, Λ(κ[x

,...,x

]), is a

subring of Map(κ

,κ) which is called the ring of polynomial func-

tions on κ

b) Show that Λ is an isomorphism onto the polynomial functions of

if and only if κ is an inﬁnite ﬁeld.

1.2. Field extensions

Suppose that K is a ﬁeld and A is a K-algebra. Suppose that Λ is a subset

of A. The set Λ is said to be algebraically independent over K if whenever

we have a relation

f(t

,...,t

)=0

for some distinct t

,...,t

∈ Λ and a polynomial f in the polynomial ring

K[x

,...,x

], we have that f = 0 (all the coeﬃcients of f are zero).

Suppose that K is a subﬁeld of a ﬁeld L and Λ is a subset of L.The

subﬁeld K(Λ) of L is the smallest subﬁeld of L which contains K and Λ.

A subset Λ of L which is algebraically independent over K and is max-

imal with respect to inclusions is called a transcendence basis of L over

K. Transcendence bases always exist. Any set of algebraically independent

elements in L over K can be extended to a transcendence basis of L over

K. Any two transcendence bases of L over K have the same cardinality

([95, Theorem 1.1, page 356] or [160, Theorem 25, page 99]). This cardi-

nality is called the transcendence degree of the ﬁeld L over K and is written

as trdeg

Suppose that L ⊂ M ⊂ N is a tower of ﬁelds. Then

(1.1) trdeg

N =trdeg

N +trdeg

by [160, Theorem 26, page 100].

1.2. Field extensions 7

An algebraic function ﬁeld over a ﬁeld K is a ﬁnitely generated ﬁeld

extension L = K(y

,...,y

)ofK. After possibly permuting y

,...,y

there exists an integer r with 0 ≤ r ≤ m such that y

,...,y

is a transcen-

dence basis of L over K. The ﬁeld L is then said to be an r-dimensional

algebraic function ﬁeld. We have that L is ﬁnite over K(y

,...,y

). The

ﬁeld K(y

,...,y

) is isomorphic as a K-algebra to the quotient ﬁeld of a

polynomial ring over K in r variables, so K(y

,...,y

) is called a rational

function ﬁeld over K.

The ﬁeld L is said to be separably generated over K if there exists a

transcendence basis z

,...,z

of L over K such that L is separably algebraic

over K(z

,...,z

). The set of elements z

,...,z

is then called a separating

transcendence basis of L over K.

Theorem 1.14. If K is a perfect ﬁeld (K has characteristic 0,orK has

characteristic p>0 and all elements of K have a p-th root in K), then all

ﬁnitely generated ﬁeld extensions over K are separably generated over K.

Proof. [160, Theorem 31, page 105]. 

In any algebraic extension of ﬁelds, there is a maximal separable exten-

sion.

Theorem 1 .15. Suppose that L is an algebraic extension of a ﬁeld K.Then

there exits a maximal subﬁeld M of L which is separable algebraic over K

and such that L is purely inseparable over M.

Proof. [95, Theorem 4.5, page 241]. 

The M of the conclusions of Theorem 1.15 is called the separable closure

of K in L. With the notation of the above theorem, we deﬁne

(1.2) [L : K]

=[M : K] and [L : K]

=[L : M].

The primitive element theorem gives a nice description of ﬁnite separable

extensions.

Theorem 1.16 (Primitive element theorem). Suppose that L is ﬁnite exten-

sion ﬁeld of a ﬁeld K. There exists an element α ∈ L such that L = K(α) if

and only if there exist only a ﬁnite number of ﬁelds F such that K ⊂ F ⊂ L.

If L is separable over K then there exists such an element α.

Proof. [95, Theorem 4.6, page 243]. 

Suppose that L is a ﬁnite extension ﬁeld of a ﬁeld K. We will write

Aut(L/K) for the group of K-automorphisms of L. In the case that L is

Galois over K, we will write G(L/K) for the Galois group Aut(L/K).

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