Algebraic geometry

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• ConceptAffine space

  An affine space is a set of points on which the vectors of a vector space act by translation. It keeps the ideas of lines, planes and parallelism from Euclidean geometry, but it has no origin, no distance and no angle.

• ConceptAlgebraic set

  An algebraic set is the set of common zeros, in the affine space $k^n$ over a field $k$, of a family of polynomials. The polynomials are in $n$ variables. Algebraic sets are the basic geometric objects of classical algebraic geometry. They are the closed sets of the Zariski topology on $k^n$. Over an algebraically closed field they correspond to radical ideals of the polynomial ring.

• ConceptAffine variety

  An affine variety is an irreducible algebraic set in affine space $k^n$ over an algebraically closed field $k$. Affine varieties are the local building blocks of all algebraic varieties. An affine variety corresponds to a prime ideal of the polynomial ring, and it is determined up to isomorphism by its coordinate ring. Some authors drop the irreducibility condition and call every algebraic set an affine variety.

• ConceptIdeal of an algebraic set

  The ideal of a subset $X$ of affine space $k^n$ is the set of all polynomials that vanish at every point of $X$. It is the algebraic shadow of $X$ and is always a radical ideal. Over an algebraically closed field, Hilbert's Nullstellensatz says that this gives a bijection from algebraic sets to radical ideals. This turns geometry into algebra.

• ConceptZariski topology

  The Zariski topology is a topology on the spectrum of a commutative ring. Its closed sets are the sets of prime ideals containing a given set of elements. On an affine variety it restricts to the topology whose closed sets are the zero sets of polynomials. It has few open sets: it is quasi-compact and usually not Hausdorff. It is the topology used throughout algebraic geometry.

• TheoremHilbert's Nullstellensatz

  Let $k$ be an algebraically closed field, let $n \ge 1$, and let $I$ be an ideal of the polynomial ring $k[x_1, \ldots, x_n]$. Let $V(I) = \set{a \in k^n : f(a) = 0 \text{ for all } f \in I}$ be the zero set of $I$. Let $I(V(I)) = \set{f \in k[x_1, \ldots, x_n] : f(a) = 0 \text{ for all } a \in V(I)}$ be the ideal of polynomials vanishing on it. Then $$I(V(I)) = \sqrt{I},$$ where $\sqrt{I} = \set{f : f^m \in I \text{ for some } m \ge 1}$ is the radical of $I$.