##  [Algebraic Variety](/algebraic-variety-0) 

 Definition

A geometric object defined as the solution set of polynomial equations over a field and equipped with the Zariski topology; in classical terms often taken to be a reduced, separated scheme of finite type over a field, possibly irreducible (an integral variety) or reducible.

 

 

 

 

 

 





## Principle

Principle

Capture geometry by polynomial equations: the coordinate ring and its ideals encode geometric properties, dimensions, and maps between varieties via algebraic morphisms.

 

 

 

 

 





## Demonstration

Demonstration

An affine variety V(I) ⊂ A^n_k defined by an ideal I ⊂ k[x_1,...,x_n] has coordinate ring k[x_1,...,x_n]/I; projective varieties arise by homogeneous equations in projective space and are glued from affine pieces.

 

 

 

 

## Misapplication

Misapplication

Assuming every finite type scheme over a field is a variety (ignoring nonreduced schemes or separatedness), or treating varieties only over algebraically closed fields when arithmetic behavior over general fields matters.

 

 

 

 

 





## Consequence

Consequence

Varieties allow use of algebraic tools (coordinate rings, dimension theory, morphisms, rational maps) and support geometric notions like smoothness, singularities, and intersection theory over a field.

 

 

 

 

## Reversal

Reversal

General scheme or analytic space: relaxing the requirement of reducedness, finiteness, or a field base leads to schemes with nilpotents, formal schemes, or analytic spaces where polynomial equations are replaced by convergent power series.

 

 

 

 

 





## Boundary

Boundary

Typically restricted to reduced schemes of finite type over a field and often assumed separated; excludes schemes over general rings, stacks, formal schemes, and most analytic or transcendental spaces.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Variety versus scheme: varieties are classical, often reduced and over a field; schemes are broader, admitting nilpotents and arithmetic base rings — choosing one or the other depends on which phenomena one wishes to capture.

 

 

 

 

 





## Synthesis

Synthesis

An algebraic variety is a geometric locus cut out by polynomials over a field, whose coordinate ring encodes its algebraic structure; as a reduced finite type scheme it provides a flexible but classical setting for studying algebraic geometry.