Irreducibility and decomposition are key concepts in understanding affine varieties. They help us break down complex geometric objects into simpler parts, making them easier to study and analyze.

By examining whether a variety can be split into smaller pieces, we gain insights into its structure and properties. This knowledge is crucial for solving equations, classifying varieties, and exploring their geometric features.

Irreducibility of Affine Varieties

Definition and Characterization

• An affine variety $V$ is irreducible if it cannot be expressed as the union of two proper closed subsets
• Equivalently, an affine variety $V$ is irreducible if its coordinate ring $A(V)$ has no zero divisors, meaning $A(V)$ is an integral domain
  • The empty set and a single point (origin in $\mathbb{A}^n$) are considered irreducible affine varieties
  • The Zariski closure of any irreducible subset of an affine space is an irreducible affine variety (irreducible algebraic set in $\mathbb{A}^n$)
• The irreducibility of an affine variety is equivalent to the primality of its ideal in the corresponding polynomial ring ($I(V)$ is prime in $k[x_1, \ldots, x_n]$)

Properties and Examples

• An affine variety $V$ is irreducible if and only if its coordinate ring $A(V)$ is an integral domain
  • Example: The affine line $\mathbb{A}^1$ is irreducible because its coordinate ring $k[x]$ is an integral domain
• The graph of a non-constant polynomial function $f(x_1, \ldots, x_n)$ is an irreducible affine variety in $\mathbb{A}^{n+1}$
  • Example: The parabola $V(y - x^2)$ in $\mathbb{A}^2$ is irreducible
• The union of two distinct irreducible affine varieties is reducible (not irreducible)
  • Example: The union of the coordinate axes $V(xy)$ in $\mathbb{A}^2$ is reducible

Determining Irreducibility

Algebraic Techniques

• To determine the irreducibility of an affine variety $V$, examine its defining ideal $I(V)$ in the polynomial ring
  • If $I(V)$ is a prime ideal, then $V$ is irreducible. Conversely, if $V$ is irreducible, then $I(V)$ is a prime ideal
  • Use the Nullstellensatz to relate the irreducibility of $V$ to the primality of $I(V)$
• Apply algebraic techniques to check the primality of $I(V)$
  • Gröbner bases can be used to compute a reduced Gröbner basis of $I(V)$ and check if it generates a prime ideal
  • Primary decomposition can be used to decompose $I(V)$ into primary components and check if there is only one minimal prime component

Geometric Intuition

• Geometric intuition can sometimes be used to argue for the irreducibility or reducibility of an affine variety
  • Example: The union of two distinct parallel lines in $\mathbb{A}^2$ is reducible because it can be decomposed into two irreducible components
  • Example: The cone $V(x^2 + y^2 - z^2)$ in $\mathbb{A}^3$ is irreducible because it cannot be decomposed into proper closed subsets
• Visualizing the affine variety and its potential decompositions can provide insights into its irreducibility
  • Example: The circle $V(x^2 + y^2 - 1)$ in $\mathbb{A}^2$ is irreducible because it forms a single connected component

Decomposition into Irreducible Components

Existence and Uniqueness

• Every affine variety $V$ can be uniquely decomposed into a finite union of irreducible closed subsets, called its irreducible components
  • The irreducible components of $V$ are the maximal irreducible closed subsets of $V$ with respect to inclusion
  • The decomposition of $V$ into irreducible components is unique up to reordering
• The dimension of $V$ is equal to the maximum of the dimensions of its irreducible components
  • Example: The dimension of the union of a plane and a line in $\mathbb{A}^3$ is 2, the dimension of the plane component

Correspondence with Prime Ideals

• The irreducible components of $V$ correspond to the minimal prime ideals containing $I(V)$ in the polynomial ring
  • Each minimal prime ideal $P_i \supset I(V)$ defines an irreducible component $V(P_i)$ of $V$
  • The decomposition of $V$ into irreducible components corresponds to the primary decomposition of $I(V)$ into minimal prime components
• The prime ideals associated with the irreducible components can be used to study their properties and intersections
  • Example: The ideal $I(V) = \langle xy, xz \rangle$ in $k[x, y, z]$ has two minimal prime components $\langle x \rangle$ and $\langle y, z \rangle$, corresponding to the irreducible components $V(x)$ and $V(y, z)$

Finding Irreducible Components

Algebraic Methods

• To find the irreducible components of an affine variety $V$, decompose its defining ideal $I(V)$ into primary components
  • Use primary decomposition algorithms to find the minimal prime ideals containing $I(V)$
  • Each minimal prime ideal corresponds to an irreducible component of $V$
• The irreducible components can be described by their defining prime ideals or by their generating sets of polynomials
  • Example: The irreducible components of $V(xy, xz)$ in $\mathbb{A}^3$ are $V(x)$ and $V(y, z)$, defined by the prime ideals $\langle x \rangle$ and $\langle y, z \rangle$

Geometric Visualization

• Geometrically, sketch the irreducible components to visualize the decomposition of $V$
  • Example: The affine variety $V(xy, xz)$ in $\mathbb{A}^3$ can be visualized as the union of the $yz$-plane and the $x$-axis
• Apply the concept of dimension to determine the highest-dimensional irreducible components of $V$
  • Example: In the decomposition of $V(xy, x(z-1))$ in $\mathbb{A}^3$, the component $V(x)$ (the $yz$-plane) has dimension 2, while the component $V(y, z-1)$ (a line parallel to the $x$-axis) has dimension 1