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
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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 An) are considered irreducible affine varieties
The Zariski closure of any irreducible subset of an is an irreducible affine variety (irreducible in An)
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[x1,…,xn])
Properties and Examples
An affine variety V is irreducible if and only if its coordinate ring A(V) is an integral domain
Example: The A1 is irreducible because its coordinate ring k[x] is an integral domain
The graph of a non-constant polynomial function f(x1,…,xn) is an irreducible affine variety in An+1
Example: The parabola V(y−x2) in A2 is irreducible
The union of two distinct irreducible affine varieties is reducible (not irreducible)
Example: The union of the coordinate axes V(xy) in A2 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
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 A2 is reducible because it can be decomposed into two irreducible components
Example: The cone V(x2+y2−z2) in A3 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(x2+y2−1) in A2 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 A3 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 Pi⊃I(V) defines an irreducible component V(Pi) 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)=⟨xy,xz⟩ in k[x,y,z] has two minimal prime components ⟨x⟩ and ⟨y,z⟩, 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 A3 are V(x) and V(y,z), defined by the prime ideals ⟨x⟩ and ⟨y,z⟩
Geometric Visualization
Geometrically, sketch the irreducible components to visualize the decomposition of V
Example: The affine variety V(xy,xz) in A3 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 A3, 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