Algebraic Geometry Unit 2 Review: Commutative Algebra Foundations

Key Concepts and Definitions

• Commutative rings consist of a set with two binary operations, addition and multiplication, satisfying the axioms of a ring and the commutative property for multiplication ($ab = ba$ for all $a, b$ in the ring)
• Ideals are subsets of a ring that are closed under addition and absorption (if $I$ is an ideal and $r$ is any element of the ring, then $rI \subseteq I$)
  • Left ideals and right ideals satisfy absorption on the left or right, respectively
  • Two-sided ideals (or simply ideals) satisfy absorption on both sides
• Prime ideals are proper ideals $P$ such that for any two elements $a, b$ in the ring, if $ab \in P$, then either $a \in P$ or $b \in P$
• Maximal ideals are proper ideals that are not strictly contained in any other proper ideal
• Quotient rings are formed by "dividing out" an ideal $I$ from a ring $R$, denoted as $R/I$, where elements are equivalence classes of the form $a + I$ for $a \in R$
• Modules generalize the notion of vector spaces, with scalars from a ring instead of a field
  • Left $R$-modules have a left action of a ring $R$, satisfying axioms similar to those of vector spaces
• Algebras are rings that also have a compatible module structure over another ring

Rings and Ideals

• Rings are algebraic structures with two binary operations, typically denoted as addition and multiplication, satisfying certain axioms (associativity, distributivity, existence of additive identity and inverses)
  • Commutative rings satisfy the additional axiom of commutativity under multiplication ($ab = ba$ for all $a, b$ in the ring)
• Subrings are subsets of a ring that are closed under the ring operations and form a ring under the inherited operations
• Ideals are special subsets of a ring that absorb elements under multiplication
  • Left ideals satisfy $ra \in I$ for all $r$ in the ring $R$ and $a$ in the ideal $I$
  • Right ideals satisfy $ar \in I$ for all $r \in R$ and $a \in I$
  • Two-sided ideals (or simply ideals) satisfy both left and right absorption
• Principal ideals are ideals generated by a single element $a$, denoted as $(a)$, and consist of all multiples of $a$
• The sum of two ideals $I$ and $J$ is the ideal generated by all elements of the form $a + b$, where $a \in I$ and $b \in J$
• The product of two ideals $I$ and $J$ is the ideal generated by all finite sums of products $ab$, where $a \in I$ and $b \in J$

Prime and Maximal Ideals

• Prime ideals are proper ideals $P$ with the property that for any $a, b$ in the ring, if $ab \in P$, then either $a \in P$ or $b \in P$
  • Equivalently, the quotient ring $R/P$ is an integral domain
• Maximal ideals are proper ideals that are not strictly contained in any other proper ideal
  • Equivalently, the quotient ring $R/M$ is a field
• The Krull dimension of a ring is the supremum of the lengths of chains of prime ideals
• The spectrum of a ring, denoted $\text{Spec}(R)$, is the set of all prime ideals of $R$
  • The Zariski topology on $\text{Spec}(R)$ is defined by taking closed sets to be sets of the form $V(I) = \{P \in \text{Spec}(R) : I \subseteq P\}$ for some ideal $I$
• The maximal spectrum, denoted $\text{mSpec}(R)$, is the set of all maximal ideals of $R$
• The nilradical of a ring is the intersection of all prime ideals
  • Equivalently, it is the set of nilpotent elements (elements $a$ such that $a^n = 0$ for some $n > 0$)

Quotient Rings

• Given a ring $R$ and an ideal $I$, the quotient ring (or factor ring) $R/I$ is the set of equivalence classes of $R$ under the equivalence relation $a \sim b$ if and only if $a - b \in I$
  • The equivalence class of $a$ is denoted $a + I$ or $[a]$
• The quotient ring $R/I$ inherits a ring structure from $R$ by defining addition and multiplication on equivalence classes:
  • $(a + I) + (b + I) = (a + b) + I$
  • $(a + I)(b + I) = ab + I$
• The natural projection map $\pi: R \to R/I$ sends each element $a$ to its equivalence class $a + I$ and is a surjective ring homomorphism
• The First Isomorphism Theorem for rings states that if $f: R \to S$ is a surjective ring homomorphism with kernel $I$, then $R/I \cong S$
• The Chinese Remainder Theorem states that if $I_1, \ldots, I_n$ are pairwise coprime ideals in a ring $R$, then $R/(I_1 \cdots I_n) \cong R/I_1 \times \cdots \times R/I_n$

Modules and Algebras

• Modules generalize the notion of vector spaces by allowing scalars from a ring instead of a field
  • A left $R$-module $M$ is an abelian group with a left action of $R$ satisfying axioms similar to those of vector spaces
  • Right $R$-modules have a right action of $R$ instead
• Submodules are subsets of a module closed under addition and scalar multiplication, forming a module under the inherited operations
• Quotient modules are formed by "dividing out" a submodule $N$ from a module $M$, denoted $M/N$, analogous to quotient rings
• Module homomorphisms are maps between modules that preserve the module structure (addition and scalar multiplication)
• Algebras are rings that also have a compatible module structure over another ring
  • A left $R$-algebra is a ring $A$ with a left $R$-module structure satisfying $(ra)b = r(ab) = a(rb)$ for all $r \in R$ and $a, b \in A$
• Tensor products allow the construction of new modules and algebras from existing ones
  • The tensor product of a right $R$-module $M$ and a left $R$-module $N$, denoted $M \otimes_R N$, is a new abelian group satisfying a universal property

Localization

• Localization is a process that creates a new ring by formally inverting a specified subset of elements in a given ring
• The localization of a ring $R$ at a multiplicative subset $S$ (a subset containing 1 and closed under multiplication) is denoted $S^{-1}R$ or $R_S$
  • Elements of $S^{-1}R$ are equivalence classes of fractions $\frac{r}{s}$ with $r \in R$ and $s \in S$, where $\frac{r_1}{s_1} \sim \frac{r_2}{s_2}$ if there exists $t \in S$ such that $t(s_2r_1 - s_1r_2) = 0$
• The localization $S^{-1}R$ has a natural ring structure inherited from the fraction field of $R$
• The localization of $R$ at a prime ideal $P$, denoted $R_P$, is the localization at the multiplicative subset $S = R \setminus P$
  • This is called the local ring at $P$ and has a unique maximal ideal $PR_P$
• Localization is a functor from the category of $R$-modules to the category of $S^{-1}R$-modules
  • For an $R$-module $M$, the localized module $S^{-1}M$ is defined as $M \otimes_R S^{-1}R$
• Localization can be used to study local properties of rings and modules, as well as to simplify computations by working in the localized ring

Noetherian Rings

• A ring $R$ is Noetherian if it satisfies the ascending chain condition (ACC) on ideals: every ascending chain of ideals $I_1 \subseteq I_2 \subseteq \cdots$ eventually stabilizes
  • Equivalently, every ideal in $R$ is finitely generated
• The Hilbert Basis Theorem states that if $R$ is a Noetherian ring, then the polynomial ring $R[x]$ is also Noetherian
  • More generally, if $R$ is Noetherian, then $R[x_1, \ldots, x_n]$ is Noetherian for any $n \geq 1$
• Noetherian rings have several desirable properties:
  • Every submodule of a finitely generated module over a Noetherian ring is finitely generated
  • In a Noetherian ring, every prime ideal is the radical of a finitely generated ideal
• Examples of Noetherian rings include:
  • Fields, principal ideal domains (PIDs), and $\mathbb{Z}$
  • Finitely generated algebras over a Noetherian ring
• The Krull Intersection Theorem states that in a Noetherian local ring $(R, \mathfrak{m})$, the intersection of all powers of the maximal ideal is zero: $\bigcap_{n=1}^\infty \mathfrak{m}^n = (0)$

Integral Extensions and Algebraic Elements

• An extension of rings $R \subseteq S$ is integral if every element of $S$ is a root of a monic polynomial with coefficients in $R$
  • A monic polynomial is a polynomial with leading coefficient equal to 1
• An element $s \in S$ is integral over $R$ if it satisfies a monic polynomial equation with coefficients in $R$: $s^n + r_{n-1}s^{n-1} + \cdots + r_1s + r_0 = 0$ with $r_i \in R$
• The set of elements in $S$ that are integral over $R$ form a subring called the integral closure of $R$ in $S$, denoted $\overline{R}$ or $R^S$
• A ring $R$ is integrally closed in its field of fractions $K$ if $\overline{R} = R$
• An algebraic extension of fields $K \subseteq L$ is an extension where every element of $L$ is algebraic over $K$ (satisfies a polynomial equation with coefficients in $K$)
  • Finite extensions (where $L$ is a finite-dimensional vector space over $K$) are always algebraic
• The Going-Up Theorem states that if $R \subseteq S$ is an integral extension and $P_0 \subseteq P_1 \subseteq \cdots \subseteq P_n$ is a chain of prime ideals in $R$, then there exists a chain of prime ideals $Q_0 \subseteq Q_1 \subseteq \cdots \subseteq Q_n$ in $S$ such that $Q_i \cap R = P_i$ for each $i$
• The Going-Down Theorem is a partial converse, stating that if $R \subseteq S$ is an integral extension and $R$ is integrally closed, then given a chain of prime ideals $Q_0 \subseteq Q_1 \subseteq \cdots \subseteq Q_n$ in $S$ and a prime ideal $P_0$ in $R$ with $P_0 \subseteq Q_0$, there exists a chain of prime ideals $P_0 \subseteq P_1 \subseteq \cdots \subseteq P_n$ in $R$ such that $P_i = Q_i \cap R$ for each $i$