Irreducible component

In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal (for set inclusion) for this property. For example, the set of solutions of the equation $xy = 0$ is not irreducible, and its irreducible components are the two lines of equations $x = 0$ and $y =0$ .

It is a fundamental theorem of classical algebraic geometry that every algebraic set may be written in a unique way as a finite union of irreducible componenents.

Theses concepts can be reformulated in purely topological terms, using the Zariski topology, for which the closed sets are the algebraic subsets: A topological space is irreducible if it is not the union of two proper closed subsets, and an irreducible component is a maximal subspace (necessarily closed) that is irreducible for the induced topology. Although these concepts may be considered for every topological space, this is rarely done outside algebraic geometry, since most common topological spaces are Hausdorff spaces, and, in a Hausdorff space, the irreducible components are the singletons.

In topology

A topological space X is reducible if it can be written as a union $X=X_{1}\cup X_{2}$ of two closed proper subsets $X_{1}$ , $X_{2}$ of $X.$ A topological space is irreducible (or hyperconnected) if it is not reducible. Equivalently, all non empty open subsets of X are dense or any two nonempty open sets have nonempty intersection.

A subset F of a topological space X is called irreducible or reducible, if F considered as a topological space via the subspace topology has the corresponding property in the above sense. That is, $F$ is reducible if it can be written as a union $F=(G_{1}\cap F)\cup (G_{2}\cap F),$ where $G_{1},G_{2}$ are closed subsets of $X$ , neither of which contains $F.$

An irreducible component of a topological space is a maximal irreducible subset. If a subset is irreducible, its closure is, so irreducible components are closed.

In algebraic geometry

Every affine or projective algebraic set is defined as the set of the zeros of an ideal in a polynomial ring. In this case, the irreducible components are the varieties associated to the minimal primes over the ideal. This is the identification that allows to prove the uniqueness and the finiteness of the decomposition. This decomposition is strongly related with the primary decomposition of the ideal.

In general scheme theory, every scheme is the union of its irreducible components, but the number of components is not necessarily finite. However, in most cases occurring in "practice", namely for all noetherian schemes, there are finitely many irreducible components.

Examples

In a Hausdorff space, the irreducible subsets and the irreducible components are the singletons. This is the case, in particular, for the real numbers. In fact, if $X$ is a set of real numbers that is not a singleton, there are three real numbers such that $x \in X$ , $y \in X$ , and $x < a < y$ . The set $X$ cannot be irreducible since $X=(X\cap \,]\infty ,a])\cup (X\cap [a,\infty [).$

The notion of irreducible component is fundamental in algebraic geometry and rarely considered outside this area of mathematics: consider the algebraic subset of the plane

X = {(x, y) | xy = 0}

.

For the Zariski topology, its closed subsets are itself, the empty set, the singletons, and the two lines defined by $x = 0$ and $y = 0$ . The set $X$ is thus reducible with these two lines as irreducible components.

The spectrum of a commutative ring is the set of the prime ideals of the ring, endowed which the Zariski topology, for which a set of prime ideals is closed if and only if it is the set of all prime ideals that contain a fixed ideal. In this case an irreducible subset is the set of all prime ideals that contain a prime ideal.

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