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Affine varieties are the basic geometric objects of classical algebraic geometry. They arise from a simple but powerful reversal: instead of studying a polynomial by its coefficients, study the set of points where many polynomials vanish at once. This turns algebra in a polynomial ring into geometry in affine space, and it is the entry point for ideas such as [Polynomial Ring](/page/Polynomial%20Ring), [Ideal](/page/Ideal), Zariski Topology, Coordinate Ring, and Scheme.

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The definition is flexible enough to describe familiar curves and surfaces, but rigid enough to remember algebraic structure. A line, a parabola, a plane conic, a finite set of points, and a hypersurface can all be described by polynomial equations. The reason affine varieties matter is that many geometric questions become algebraic questions about ideals, quotient rings, and homomorphisms.

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Over an [algebraically closed field](/page/Algebraically%20Closed%20Field), affine varieties are tightly controlled by the Hilbert Nullstellensatz. Over a non-algebraically closed field, there is a useful distinction between equations with coefficients in $k$ and points whose coordinates lie in $k$; that distinction is one of the first signs that algebraic geometry must eventually separate schemes, rational points, and base change.

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## Definition

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Affine varieties are the basic geometric objects studied through polynomial equations, so we begin by isolating the ambient algebraic setting in which those equations live.. The definition below packages the ambient affine space and the common-zero-locus condition together; the separate definitions of [Affine Space](/page/Affine%20Space) and vanishing set then isolate the notation used throughout the rest of the page.

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[definition: Affine Variety] Let $k$ be a field and let $n \in \mathbb{N}$. An affine variety over $k$ is a subset $X \subset \mathbb{A}^n_k := k^n$ for which there exists a set $S \subset k[x_1, \ldots, x_n]$ such that \begin{align*} X = \{p \in \mathbb{A}^n_k : f(p)=0 \text{ for every } f \in S\}. \end{align*} [/definition]

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Polynomial equations need a fixed universe of points before their zero sets can be discussed. The same symbols $x_1, \ldots, x_n$ can describe coordinates on many different ambient spaces, depending on the base field and the number of coordinates. Affine space supplies that ambient universe, so later notation can distinguish the space of all possible coordinate tuples from the particular solution sets cut out inside it.

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[definition: Affine Space] Let $k$ be a field and let $n \in \mathbb{N}$. The affine $n$-space over $k$ is the set \begin{align*} \mathbb{A}^n_k := k^n. \end{align*} A point $p \in \mathbb{A}^n_k$ is written $p=(a_1, \ldots, a_n)$ with $a_i \in k$. [/definition]

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Once the ambient space is fixed, the next recurring operation is to pass from equations to their simultaneous solutions. Isolating this operation lets us compare different equation lists, describe intersections and unions later, and speak about the zero locus of an ideal without restating the evaluation condition each time.

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[definition: Vanishing Set] Let $k$ be a field, let $n \in \mathbb{N}$, and let $S \subset k[x_1, \ldots, x_n]$. The vanishing set of $S$ in $\mathbb{A}^n_k$ is \begin{align*} V(S) := \{p \in \mathbb{A}^n_k : f(p)=0 \text{ for every } f \in S\}. \end{align*} [/definition]

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Many authors reserve the word variety for irreducible algebraic sets over an algebraically closed field. On this page, an affine variety means any common zero locus $V(S)$, including reducible and empty examples. When irreducibility matters, it is stated as an additional condition.

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For example, in $\mathbb{A}^2_k$ the equation $y=0$ defines

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\begin{align*} V(y)=\{(a,0):a\in k\}, \end{align*}

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the affine line sitting inside the affine plane. This small example is the template: a polynomial equation is algebra, while its vanishing set is the geometric object.

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A subset of affine space also determines polynomials: namely, the polynomials that vanish on every point of the subset. To compare different equation lists for the same variety, we need this reverse construction from geometry back to algebra.

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[definition: Vanishing Ideal] Let $k$ be a field, let $n \in \mathbb{N}$, and let $X \subset \mathbb{A}^n_k$. The vanishing ideal of $X$ is \begin{align*} I(X) := \{f \in k[x_1, \ldots, x_n] : f(p)=0 \text{ for every } p \in X\}. \end{align*} [/definition]

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The [vanishing ideal is an ideal](/theorems/9404) in the polynomial ring. It records all polynomial equations satisfied by $X$, not just one chosen list of equations.

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We need the coordinate ring because polynomial functions on $X$ should identify ambient polynomials that agree at every point of $X$. Quotienting by $I(X)$ performs exactly that identification and turns the geometry of $X$ into a commutative algebra object.

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[definition: Coordinate Ring of an Affine Variety] Let $k$ be a field and let $X \subset \mathbb{A}^n_k$ be an affine variety. The coordinate ring of $X$ is the [quotient ring](/page/Quotient%20Ring) \begin{align*} k[X] := k[x_1, \ldots, x_n]/I(X). \end{align*} [/definition]

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The coordinate ring is the algebra of polynomial functions on the chosen set of $k$-points $X$. Over algebraically closed fields this point-set construction matches the usual classical dictionary especially well; over finite or non-algebraically closed fields, $I(X)$ may remember only what happens on the displayed $k$-rational points, so the field of definition must remain part of the data. It is still the main reason affine varieties are algebraically tractable: maps between affine varieties are encoded by homomorphisms between coordinate rings in the opposite direction.

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