(Perrin) Algebraic Geometry. Chapter 1 Highlights: The Algebra-Geometry Dictionary

Some applications of Ideal-Variety Correspondence

The Nullstellensatz establishes an algebra-geometry dictionary. This allows us to prove geometric statement by passing to algebra. For example, can you show this: 

$$\text{If \(k\) is an infinite field, then \(k^n\) is irreducible.}$$

Proof. Via the algebra-geometry dictionary, it suffices to show \( \mathbb{I}(k^n) = 0\). 

Remark: Why did we need the "infinite" condition? Consider \( k = \mathbb{F}_q \) then \(\mathbb{I}(k) = (x^q - x) \neq 0 \).

Conversely, we could use geometry to prove algebraic identity. The following useful little lemma (Extension of Algebraic Identity) is a consequence of the above topological fact that \(k^n\) is irreducible. 

Assume \( |k| = \infty\). Let \( X \)  be a proper affine algebraic subvariety of \(k^n\). Let \( f \in k[x_1, \ldots, x_n]\)  such that \( f = 0\) outside \(X\). Then \( f \equiv 0\).

 Proof. Note that \(f = 0\) on a non-empty open set \(U := k^n - X\) so \(\mathbb{V}(f)\) must contains the closure of \(U\), which is \(k^n\) by irreducibility of \(k^n\). 

The most common application of this is in "Density Arguments". To prove that an identity holds for all matrices for example, it suffices to prove it for invertible ones, as they form a dense open subset of the variety of \(n \times n \) matrices.