Users Guide to Hartshorne Algebraic Geometry/Chapter 0

References

1. Algebraic Geometry - Hartshorne
2. Arithmetic - Serre
3. https://people.ucsc.edu/~weissman/Math222A/SerreAnn.pdf
4. http://web.mit.edu/18.705/www/12Nts-2up.pdf (for basic commutative algebra)
5. https://arxiv.org/pdf/1605.04832.pdf (for more advanced commutative algebra)
6. Algebraic Number Theory, A Computational Approach - Stein

Basic Commutative Algebra

Categories of Commutative Rings and Algebras

The starting point for this section is the definition of a commutative ring: a unital ring with commutative multiplication. In this book you can assume that all rings are commutative, so we will omit the 'commutative' adjective. The most basic rings include

• $\mathbb{Z}$
• $\mathbb{Z}/n$
• Fields $𝔽$
• Polynomial rings $R[x_1,\dots ,x_n]$

We can relate rings to one another using a morphism of rings. A function $\varphi :R\to S$ between rings is a morphism of rings if the following two axioms are satisfied

1. $\varphi (a+b)=\varphi (a)+\varphi (b)$ (Additivity)
2. $\varphi (ab)=\varphi (a)\varphi (b)$ (Multiplicativity)

we could have stated this succinctly as a function which respects the ring structure. It turns out that rings with ring morphisms form a category $\mathbf{\text{CRing}}$. As an important technical note, there is no zero-ring given by a single element in our category. This category has an initial object given by the ring of integers because given a ring morphisms

$\varphi :\mathbb{Z}\to R$

the ring morphism axioms forces

$\varphi (1)=1_R$, $\varphi (-1)=-1_R$, and $\varphi (n)=\varphi \left(\sum _{n}1\right)=\sum _n\varphi (1)$

Recall that the category of $R$-algebras has objects given by ring morphisms $R\to S$ and morphisms given by commutative diagrams

$\begin{array}{ccc}& & S\\ & \nearrow \\ R& & \downarrow \varphi \\ & \searrow \\ & & S^{\prime }\end{array}$

If we consider only algebras, the category

$\mathbb{Z}-\mathbf{\text{CAlg}}$

is equivalent to the category

$\mathbf{\text{CRing}}$

. Note that it is common to consider the categories

${𝔽}_q-\mathbf{\text{CAlg}}$

,

$ℂ-\mathbf{\text{CAlg}}$

,

${\mathbb{Q}}_p-\mathbf{\text{CAlg}}$

. The motivation for why will be readily apparent when considering categories of schemes.

Ideals

One of the ways to construct new rings is by taking quotient rings. An ideal of a ring is a subset $I\subset R$ which is

1. An abelian group under addition
2. $R\cdot I\subset R$

Then, we can take the quotient of abelian groups $R/I$ and use the multiplicative structure on $R$ to construct one on $R/I$. The second axiom of ideals guarantees that this is well-defined. This is called a quotient ring. Some typical examples of quotient rings are given by

• $\mathbb{Z}/(p)$
• $\mathbb{Z}[x]/(x^2-5)\cong \mathbb{Z}[\sqrt{5}]$
• $\mathbb{Q}[x]/(x^p-1)$
• $\frac{\mathbb{Z}[x_1,\dots ,x_n]}{(f_1,\dots ,f_k)}$

Playing with Presentations

As we have seen, there are many ways to construct polynomial ring; but, another interesting technique for creating new polynomial rings is to attach variables which have relations between them. For example, consider $\mathbb{Z}[x,x^2,x^3]$. We can relabel the elements we've attached, so we consider the ring $\mathbb{Z}[X,Y,Z]$, but there are a couple relations between these variables:

$X^2-Y,XY-Z$

note that these two relations can be used to show others such as $XZ-Y^2$ and $X^3-Z$. Hence

$\mathbb{Z}[x,x^2,x^3]\cong \frac{\mathbb{Z}[X,Y,Z]}{(X^2-Y,XY-Z)}$

Some other examples include

• $\mathbb{Z}[x,x^{3/2}]\cong \frac{\mathbb{Z}[X,Y]}{(Y^2-X^3)}$
• $\mathbb{Z}[x^2,xy,y^2]\cong \frac{\mathbb{Z}[X,Y,Z]}{(XZ-Y^2)}$
• $\mathbb{Z}[x^3,x^{2}y,x{y}^2,y^3]\cong \frac{\mathbb{Z}[X,Y,Z,W]}{(XW-YZ,XZ-Y^2,YW-Z^2)}$

Prime Ideals

There are a special class of ideals called prime ideals: an ideal $𝔭$ in a UFD $R$ is prime if

$xy\in 𝔭\iff x\in 𝔭\text{ or }y\in 𝔭$

For example, $(p)\subset \mathbb{Z}$ is the first known example of a prime ideal. It should be apparent that $(6)$ is not a prime ideal since $2\cdot 3\in (6)$ but $2,3\in ̸(6)$. Now, given an irreducible polynomial $f\in S[x_1,\dots ,x_n]$ the ideal $(f)$ will be prime. A simple non-example of a prime ideal is given by $(xy)\subset ℂ[x,y]$. This can be generalized to $(fg)\subset S[x_1,\dots ,x_n]$. Some other examples of prime ideals include

$(x^2+1)\subset \mathbb{Q}[x]$

$(x-\alpha )\subset ℂ[x]$

$(y^2-x^3+1)\subset ℂ[x,y]$

If you take the quotient ring of a prime ideal in a UFD $R$ you get an integral domain. This means your ring has the following multiplicative property:

$xy=0$ if $x=0$ or $y=0$

For example, in

$\frac{ℂ[x,y]}{(y^2-x^3)}$

you will never be able to multiply two non-zero elements together to get zero. The two key non-examples of a ring being an integral domain are

$ℂ[x,y]/(xy)$ since $xy=0$

$ℂ[x]/(x^2)$ since $x\cdot x=0$

In general, an ideal

$𝔭$

of a ring

$R$

is called prime if

$R/𝔭$

is an integral domain. If

$R/𝔪$

is also a field, then we call

$𝔪$

a maximal ideal. One useful exercise is to check that for a morphism

$f:R\to S$

and a prime ideal

$𝔭\subset S$

the inverse image

$f^{-1}(𝔭)$

is a prime ideal. The second example motivates the operation of taking radicals of an ideal. Given an ideal

$I\subseteq R$

we define its radical as

$\sqrt{I}=\{f\in R:f^k\in I\text{ for some }r\in \mathbb{N}\}$

For example, the radical of the ideal

$((x^2-y+z{)}^4(xyz-z^2{)}^5)\subset \mathbb{Z}[x,y,z]$

is

$((x^2-y+z)(xyz-z^2))$

. Given a quotient ring

$R/I$

we call the ring

$R/\sqrt{I}$

its reduction; sometimes this is denoted

$(R/I{)}_{\text{red}}$

. We define the nilradical of a ring

$R$

as

$\sqrt{(0)}$

. The nonzero elements in the nilradical are called nilpotents.

Eisenstein's Criterion and Constructing Prime Ideals

There is a generalization of Eisenstein's criterion for integral domains: given a ring $R$ and a polynomial $f(x)\in R[x]$ which can be written as

$f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$

then it cannot be written as a product of polynomials if the following conditions are satisfied: Suppose there exists a prime ideal $𝔭\subset R$ such that

$\begin{array}{c}{a}_i\in 𝔭\text{ for }i\ne n\\ a_n\in ̸𝔭\\ a_0\in ̸{𝔭}^2\end{array}$

then $f(x)$ cannot be written as a product of polynomials $g_1(x)\cdots g_k(x)$.

For example, consider the integral domain $ℂ[x]$ and the polynomial $f\in ℂ[x][y]$ given by

$f(x,y)=1\cdot y^3+0\cdot y^2+0\cdot y+x^3\cdot y^0=y^3+x^3$

Using the prime ideal $(x-1)\subset ℂ[x]$ we have that $a_1,a_2=0\in (x-1)$, $a_3=1\in ̸(x-1)$, and $a_0=x^3\in ̸(x-1{)}^2$. Hence

$(x^3+y^3)\subset ℂ[x,y]$

is a prime ideal. This example can be extended to show that

$x_1^{k_1}+\cdots +x_n^{k_n}\in ℂ[x_1,\dots ,x_n]$

generates a prime ideal.

Nullstellensatz

Now we are in the right place to discuss the foundational theorem of algebraic geometry: Hilbert's nullstellensatz. Here we fix $k$ as an algebraically closed field.

Theorem: The maximal ideals of $𝔪\subset k[x_1,\dots ,x_n]$ are in bijection with the set $k^n$.

For example, the kernel of

${\text{ev}}_{(1,2)}:ℂ[x,y]\to ℂ$

is the ideal

$(x-1,y-2)$

. This allows one to interpret quotient rings give by ideals

$I\subset k[x_1,\dots ,x_n]$

as algebraic subsets of

$k^n$

because an evaluation morphism

${\text{ev}}_{({\alpha }_1,\dots ,{\alpha }_n)}:ℂ[x_1,\dots ,x_n]/I\to ℂ$

is well-defined only if

$(x_1-{\alpha }_1,\dots ,x_n-{\alpha }_n)\subset k[x_1,\dots ,x_n]/I$

is a maximal ideal. For example, consider the following example and non-example:

• ${\text{ev}}_{(1,1)}:ℂ[x,y]/(y-x^2)\to ℂ$ is a well defined morphism since$(x-1,y-1,y^2-x)=(x-1,y-1,1^2-1)=(x-1,y-1)$This implies that $(1,1)\in \{(a,b)\in {ℂ}^2:b^2-a\}$
• ${\text{ev}}_{(1,2)}:ℂ[x,y]/(y-x^2)\to ℂ$ is not a well-defined morphism because $(x-1,y-2,y^2-x)=(x-1,y-2,4-1)=(1)$; there is no quotient ring $R/(1)$. Hence $(1,2)\in ̸\{(a,b)\in {ℂ}^2:b^2-a\}$

Now we can interpret rings which are not integral. For example, we saw that $ℂ[x,y]/(xy)$ is not an integral domain. Geometrically, this is the union of the $x$ and $y$ axes. The other main case of a non-integral ring is a non-reduced ring. For example, $ℂ[x,y]/(y^3)$ is the $x$-axis but there is extra algebraic information from the $y,y^2$ left over. The way you should interpret this ring as is a fat line.

Basic Scheme Theory

We can now confidently define an affine scheme: it is a functor

${\text{Hom}}_{𝐂𝐑𝐢𝐧𝐠}(R,-):𝐂𝐑𝐢𝐧𝐠\to \mathbf{\text{Sets}}$

for some fixed commutative ring

$R$

.

Localization

The next basic construction in commutative ring theory is localization. This defines a generalization of inverting the non-zero integers and getting the rational numbers. Let

$S\subset R$

be a multiplicatively closed subset with unity, meaning

$1\in S$

and

$s,s^{\prime }\in S\Rightarrow s{s}^{\prime }\in S$

. For example, for a fixed element

$f\in R$

consider the subset

$\{1,s,s^2,s^3,\dots \}$

. We define a commutative ring

$R[S^{-1}]$

as follows. First, consider the set

$R\times S/\sim$

where

$(r,s)\sim (r^{\prime },s^{\prime })\text{ if there exists }u\in S\text{ such that }u(r{s}^{\prime }-r^{\prime }s)=0$

(don't worry, we will given a motivating example for this seemingly random

$u$

). It is an exercise to verify that this indeed defines an equivalence relation — it is standard to write these equivalence classes as

$r/s$

. These equivalence classes have a well-define commutative ring structure given by

$\frac{r}{s}\cdot \frac{r^{\prime }}{s^{\prime }}=\frac{r{r}^{\prime }}{s{s}^{\prime }}\text{ and }\frac{r}{s}+\frac{r^{\prime }}{s^{\prime }}=\frac{r{s}^{\prime }+r^{\prime }s}{s{s}^{\prime }}$

Some basic examples of localization include

• The subset $S=\{1,p,p^2,p^3,\dots \}\subset \mathbb{Z}$ gives the ring $\mathbb{Z}[S^{-1}]=\mathbb{Z}[1/p]$. Notice that if we localized by the set $T=\{1,p^3,p^6,p^9,\dots \}\subset \mathbb{Z}$ then this gives the ring $\mathbb{Z}[1/p^3]$. But, because we could write $1/p$ as $p^2/p^3$, these two rings are isomorphic. For brevity, we could just say that we localized $\mathbb{Z}$ by $p$. Try localizing by some other non-zero integers and see why you find.
• An important geometric example is given by localizing by some non-zero polynomial $f=f_1^{i_1}\cdots f_k^{i_k}\in ℂ[x_1,\dots ,x_n]$.
• Given an integral domain $R$, we can take the set $S=R-\{0\}$. Then, $R[S^{-1}]$ is called the field of fractions of the integral domain. (It is an exercise to check that this is a field)
• Given a ring $R$ and a prime ideal $𝔭$, we can consider the set $S=R-𝔭$. This is multiplicatively closed because of the properties of primality of an ideal. The localization of $R$ by $S$ is typically denoted $R_{𝔭}$. For example, consider $(x,y)\subset ℂ[x,y]$. The localization can be described as

$ℂ[x,y][S^{-1}]=ℂ[x,y{]}_{𝔭}=\{\frac{f(x,y)}{g(x,y)}:f,g\in ℂ[x,y]\text{ and }g(0,0)\ne 0\}$

The last example is special because it motivates a definition: a ring is local if it has a unique maximal ideal. The pair

$(R_{𝔭},𝔭)$

is a local ring.

Basic Module Theory

A $R$-module is defined as an abelian group $M$ with a fixed ring morphism $\varphi :R\to {\text{End}}_{\mathbf{\text{Ab}}}(M)$. We will use the notation

$r\cdot m:=\varphi (r)(m)$ where $r\in R,m\in M$

for the ring action on $M$. A morphism of $R$-modules $\psi :M\to M^{\prime }$ is defined by a commutative diagram

$\begin{array}{ccc}& & {\text{End}}_{\mathbf{\text{Ab}}}(M)\\ & \nearrow \\ R& & \downarrow \psi \\ & \searrow \\ & & {\text{End}}_{\mathbf{\text{Ab}}}(M^{\prime })\end{array}$

We can use this construction to build a category of $R$-modules which is abelian. This means that it has a zero object, kernels and cokernels, products and coproducts, and images/co-images agree. Please note that we've had to enlarge our category of commutative rings to all rings since the endomorphism ring of an abelian group is generally non-commutative; This is one of the only cases where we use non-commutative unital rings in this book. Typical examples of $R$-modules includes

• the zero object $0$
• ideals $I\subset R$
• direct sums, such as $M_1\oplus \cdots \oplus M_k$
• a morphism $\varphi :R\to S$ of rings gives the structure of an $R$-module on the underlying abelian group of $S$

Another useful technique for constructing new modules is taking the cokernel of a morphism

$\psi :R^n\to R^m$

. For example, the cokernel of

$ℂ[x,y,z{]}^{\oplus 2}\stackrel{\cdot (x^4+y^4+z^4-1)\oplus \cdot (x^4-y^2+z^2+1)}{\to }ℂ[x,y,z]$

is

$ℂ[x,y,z]/(x^4+y^4+z^4-1,x^4-y^2+z^2+1)$

. We can generalize this example using exact sequences. A sequence of objects in an abelian category

$M_1\stackrel{{\varphi }_1}{\to }{M}_2\stackrel{{\varphi }_2}{\to }\to \cdots \stackrel{{\varphi }_{n-1}}{\to }{M}_n$

is called exact if each

$\frac{\text{Ker}({\varphi }_i)}{\text{Ker}({\varphi }_{i-1})}\cong 0$

in the last example, we had the exact sequence

$R^{\oplus 2}\to R\to M\to 0$

In general, if there is an exact sequence

$R^n\to R^m\to M\to 0$

for finite integers $m,n$, then we say that the module is of finite-type. If there is just a sequence

$R^n\to M\to 0$

then we say that the module is finite. For example, the module

$k[x_1,x_2,\dots ]\to k\to 0$

is finite but not finite-type since the kernel of the non-trivial morphism is the ideal

$(x_1,x_2,\dots )\cong \underset{i=1}{\overset{\mathrm{\infty }}{⨁}}R\cdot x_i$

Tensor Products

• construct tensor products for modules
• construct tensor products of algebras
  • show that tensor products of integral domains are integral
    • show that $k[\underset{\_}{x}]/(f(\underset{\_}{x}){\otimes }_{k}k[\underset{\_}{y}]/(g(\underset{\_}{y}))\cong k[\underset{\_}{x},\underset{\_}{y}]/(f(\underset{\_}{x},g(\underset{\_}{y}))$
    • show $k[\underset{\_}{x}]/(f(\underset{\_}{x})){\otimes }_{k[\underset{\_}{x}]}k[\underset{\_}{x}]/(g(\underset{\_}{x}))\cong k[\underset{\_}{x}]/(f(\underset{\_}{x}),g(\underset{\_}{x}))$

Finiteness, Chain Conditions

If we have an $R$-algebra $R\to S$ we say that $S$ it is a finite if it is finite as a module. We say that it is of finite-type if there exists a surjective morphism $R[x_1,\dots ,x_n]\to S$, implying that

$S\cong \frac{R[x_1,\dots ,x_n]}{(f_1,\dots ,f_k)}$

There are a couple other notions of "finiteness" which appear in commutative algebra called chain conditions. We say call a sequence of $R$-modules

$M_1\subseteq M_2\subseteq \cdots$

an ascending chain and

$N_1\supseteq N_2\supseteq \cdots$

a descending chain. They satisfy the ascending chain condition or descending chain condition if there is some $k$ such that $M_k=M_{k+1}=\cdots$, $N_k=N_{k+1}=N_{k+2}=\cdots$. If there exist chains

$M_1\subseteq M_2\subseteq \cdots$ where $\underset{i=1}{\overset{\mathrm{\infty }}{\cup }}{M}_i=R$

or

$R\supseteq N_2\supseteq \cdots$

then we say $R$ is Noetherian or Artinian, respectively. One can show that every Artinian ring is Noetherian. The basic examples of Noetherian rings include

• Fields
• $\mathbb{Z}$
• Finite algebras over fields
• Quotients of Noetherian rings.

A simple non-example is given by the ring

$k[x_1,x_2,x_3,\dots ]$

where

$k$

is a field. There is a fundamental theorem in algebra called Hilbert's Basis Theorem stating:

Theorem: If

$R$

is Noetherian, then

$R[x_1,\dots ,x_n]$

is Noetherian

Hence all rings of the form

$\frac{R[x_1,\dots ,x_n]}{(f_1,\dots ,f_k)}$

are Noetherian. Artinian rings are much simpler than Noetherian rings:

Theorem: Every Artin ring is a finite product of Artin local rings.

All we have to analyze is the structure of an Artin local ring

$(R,𝔪)$

. Notice that we have a descending chain

$R\supseteq 𝔪\supseteq {𝔪}^2\supseteq {𝔪}^3\supseteq \cdots$

which eventually stabilizes at some

${𝔪}^k$

; this is the zero ideal

$(0)$

. We can use this to show the underlying

$R/𝔪$

-vector space of

$R$

is finite dimensional. Some examples of artin local rings are

• $(ℂ[x]/(x^5),(x))$
• $(\mathbb{Q}[x,y]/((x-1{)}^3,(x-1{)}^2(y-2),(y-2{)}^5)),(x-1,y-2))$

Integrality

Given a morphism of commutative rings

$R\to R^{\prime }$

we say an element

$x\in R^{\prime }$

is integral over

$R$

if there is a monic polynomial

$f(t)=t^n+a_{n-1}{t}^{n-1}+\cdot +a_{1}t+a_0\in R[t]$

and a morphism

$\frac{R[t]}{f(t)}\to R^{\prime }$

sending

$t\mapsto x$

. For example,

$\sqrt{-5}\in \mathbb{Z}[\sqrt{-5}]$

is integral over

$\mathbb{Z}$

since

$\frac{\mathbb{Z}[t]}{(t^2+5)}\cong \mathbb{Z}[\sqrt{-5}]$

Adjoining all of the integral elements

$\{x_i\}$ $R$

is called the integral closure of

$R$

in

$R^{\prime }$

. An integral domain

$R$

is called integrally closed if every element in its fraction field is integral over

$R$

. For example, we can compute the integral closure of

$R=\frac{ℂ[x,y]}{(x^2-y^3)}$

fairly easily. Since it is isomorphic to the ring

$ℂ[x,x^{3/2}]$

we should see immediately that

$x^{1/2}$

is not contained in

$R$

. Adjoining this element to

$R$

gives a ring isomorphic to

$ℂ[s]$

. As an exercise, try and unpack

$\frac{ℂ[x,y,z,w]}{(x^2-y^5-y^3,z^3-w^4)}$

TODO:

- hyperelliptic curves

- quotient fields of curves

- https://math.stackexchange.com/questions/2304521/why-is-this-coordinate-ring-integral-over-kx

- rings of integers

Chapter 0

Basic Commutative Algebra

Structures

• tensor products of modules/algebras... localization of modules
• categories of commutative algebras
• support of modules
• graded rings
• colimits and stalks
• limits, completions, p-adics
• valuations - https://en.wikipedia.org/wiki/Valuation_(algebra)#P-adic_valuation_on_a_Dedekind_domain
• dimension
• transcendence degree
• functor formalism
• categorical structures such as pullbacks and pushforwards
• kahler differentials/ basic differential algebra
• smooth algebras/smooth morphisms
• complete intersection/local complete intersection morphisms
• etale algebras/etale morphisms
• galois theory with useful terminology...
• same w/ algebraic number theory terminology...
• basic homological algebra, ext, tor
• koszul complexes
• derived categories
• grothendieck group
• hilbert polynomial
• grobner bases/elimination theory - https://mathoverflow.net/questions/60957/how-to-determine-whether-an-ideal-is-prime-or-not-by-an-algorithm

Theorems

• Eisenstein's Criterion
• Primary Decomposition
• Noether Normalization
• Going up and down

Basic Differential/Complex Geometry

Constructions

• Smooth manifolds
• Morphisms
• Vector Bundles
• Topological K-theory
• de-Rham Cohomology
• Complex manifolds and sheaves
• hodge decomposition of complex manifolds

Theorems

• Whitney embedding theorem
• Submersion theorem
• Sard's theorem