Hubbry Logo
Irreducible componentIrreducible componentMain
Open search
Irreducible component
Community hub
Irreducible component
logo
7 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
Irreducible component
Irreducible component
Irreducible component
View on Wikipedia
from Wikipedia

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 of an algebraic set 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 components.

These 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

[edit]

A topological space X is reducible if it can be written as a union of two closed proper subsets , of A topological space is irreducible (or hyperconnected) if it is not reducible. Equivalently, X is irreducible if all non empty open subsets of X are dense, or if 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, is reducible if it can be written as a union where are closed subsets of , neither of which contains

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

Every irreducible subset of a space X is contained in a (not necessarily unique) irreducible component of X.[1] Every point is contained in some irreducible component of X.

The empty topological space

[edit]

The empty topological space vacuously satisfies the definition above for irreducible (since it has no proper subsets). However some authors,[2] especially those interested in applications to algebraic topology, explicitly exclude the empty set from being irreducible. This article will not follow that convention.

In algebraic geometry

[edit]

Every affine or projective algebraic set is defined as the set of the zeros of an ideal in a polynomial ring. An irreducible algebraic set, more commonly known as an algebraic variety, is an algebraic set that cannot be decomposed as the union of two smaller algebraic sets. Lasker–Noether theorem implies that every algebraic set is the union of a finite number of uniquely defined algebraic sets, called its irreducible components. These notions of irreducibility and irreducible components are exactly the above defined ones, when the Zariski topology is considered, since the algebraic sets are exactly the closed sets of this topology.

The spectrum of a ring is a topological space whose points are the prime ideals and the closed sets are the sets of all prime ideals that contain a fixed ideal. For this topology, a closed set is irreducible if it is the set of all prime ideals that contain some prime ideal, and the irreducible components correspond to minimal prime ideals. The number of irreducible components is finite in the case of a Noetherian ring.

A scheme is obtained by gluing together spectra of rings in the same way that a manifold is obtained by gluing together charts. So the definition of irreducibility and irreducible components extends immediately to schemes.

Examples

[edit]

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 ∈ X, y ∈ X, and x < a < y. The set X cannot be irreducible since

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 with 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 fixed prime ideal.

Notes

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Irreducible component

View on Grokipedia
from Grokipedia
In , an irreducible component of an or scheme is a maximal irreducible closed that appears in the unique decomposition of the space into irreducible pieces, corresponding to the minimal prime ideals in the associated ideal or ring. This decomposition allows the study of complex geometric objects by breaking them down into simpler, indecomposable building blocks. A , such as an algebraic set defined by polynomial equations over an , is called irreducible if it is nonempty and cannot be expressed as the union of two proper closed subsets. Equivalently, in the affine case, an algebraic set is irreducible its vanishing ideal is a in the . For a general algebraic set VV, which may be reducible, the irreducible components are the maximal irreducible closed subsets whose union is VV, and this is unique up to . These components are themselves closed in the and play a central role in understanding the , singularities, and birational properties of the variety. In the more general setting of schemes, the irreducible components of Spec⁡(A)\operatorname{Spec}(A) for a AA are the closed subsets V(p)V(\mathfrak{p}) where p\mathfrak{p} ranges over the minimal prime ideals of AA, each of which is irreducible and maximal among irreducible closed subsets. This perspective unifies the classical variety case with more abstract algebraic structures, enabling applications in , , and moduli problems. The study of irreducible components also extends to rigid analytic spaces and other geometric contexts, where they facilitate decompositions for computational and theoretical purposes.

Preliminaries

Irreducibility

In , an irreducible is defined as a nonempty XX that cannot be expressed as the union of two proper closed subsets. Equivalently, XX is irreducible if every pair of nonempty open subsets of XX has nonempty . This condition ensures that the space lacks a nontrivial into closed parts, reflecting a high degree of "indivisibility" in its structure. Another standard characterization is that every nonempty open of an irreducible XX is dense in XX. To see this equivalence, note that if some nonempty open U⊂XU \subset X is not dense, then its closure is a proper closed subset, and the complement provides another proper closed subset whose union is XX, contradicting irreducibility. Conversely, if every nonempty open is dense, then any two such opens intersect, as each is dense in the other. Irreducibility is a stronger property than connectedness, particularly in non-Hausdorff spaces, since every irreducible space is connected (as a disconnection into disjoint nonempty opens would yield proper closed complements whose union is XX), but connected spaces need not be irreducible. For instance, in Hausdorff spaces, the only irreducible subsets are singletons, as distinct points admit disjoint open neighborhoods whose closures are proper closed sets.

Space decomposition

In a topological space XX, an irreducible component is a maximal irreducible closed subset, meaning it cannot be expressed as the union of two proper closed subsets and is not properly contained in any larger irreducible closed subset. Every topological space XX admits a decomposition into irreducible components: XX is the union of all its irreducible components, and every point of XX lies in at least one such component. The existence of these maximal irreducible closed subsets follows from , applied to the collection of irreducible closed subsets of XX ordered by inclusion; for any point x∈Xx \in X, the irreducible closed subsets containing xx have a maximal element under this ordering. This decomposition is unique in the sense that the irreducible components are precisely the maximal elements among the irreducible closed subsets, and any expression of XX as a union of irreducible closed subsets without redundancies (where no subset is contained in the union of the others) must consist exactly of these components. The components cover XX but are generally not disjoint; distinct irreducible components may intersect, though their intersection cannot be dense in either. In Hausdorff spaces, such intersections are typically controlled by separation properties, often reducing to lower-dimensional overlaps if dimensional structure is present. In Noetherian topological spaces—those satisfying the descending chain condition on closed subsets—the decomposition is finite, with XX expressible as a union of finitely many irreducible components. More generally, the number of irreducible components may be infinite. Every irreducible component is connected, as an irreducible space cannot admit a disconnection into two nonempty disjoint open sets (which would yield a cover by proper closed subsets). Conversely, connected components need not be irreducible and may decompose further into unions of irreducible components.

In topology

Definition

In topology, a topological space XX is said to be irreducible if it is nonempty and cannot be expressed as the union of two proper closed subsets, meaning that whenever X=Z1∪Z2X = Z_1 \cup Z_2 with Z1Z_1 and Z2Z_2 closed in XX, then either X=Z1X = Z_1 or X=Z2X = Z_2. Equivalently, XX is irreducible if every pair of nonempty open subsets of XX has nonempty . A subset Z⊆XZ \subseteq X of a XX is called irreducible if ZZ, equipped with the induced from XX, is an irreducible topological space. Note that the closure of an irreducible is itself irreducible. An irreducible component of a XX is defined as a maximal irreducible of XX, meaning an irreducible that is not properly contained in any larger irreducible of XX. Every irreducible component is closed in XX. Moreover, the irreducible components of XX cover XX, so XX is the union of its irreducible components, and every point of XX lies in at least one irreducible component.

Properties

Irreducible components of a XX are defined as the maximal irreducible closed subsets of XX. These subsets are closed by construction, as irreducibility is typically considered for closed sets in this context. Every point in XX belongs to at least one irreducible component, ensuring that the collection covers the entire space. Moreover, XX is the union of its irreducible components, though these components need not be disjoint; their intersections can be non-empty, as seen in examples like the union of two lines intersecting at a point in certain topologies. An irreducible space is necessarily connected, because if it were a union of two non-empty disjoint open sets, their closures would form a decomposition into proper closed subsets. Equivalently, in an irreducible space, any two non-empty open subsets have non-empty , a also known as hyperconnectedness. Furthermore, every non-empty open subset of an irreducible space is both dense and itself irreducible. In Noetherian topological spaces—those satisfying the ascending chain condition on open sets—the decomposition into irreducible components is finite and unique up to ordering, with no component contained in another. The closure of any irreducible is irreducible, and every irreducible is contained in a maximal one, namely an irreducible component. In Hausdorff spaces, irreducible components reduce to singletons, since distinct points can be separated by disjoint open neighborhoods, preventing larger irreducible sets.

In algebraic geometry

Definition

In , an algebraic set V⊆AknV \subseteq \mathbb{A}^n_k over an kk is irreducible if it is nonempty and cannot be written as the union of two proper algebraic subsets. Equivalently, for an affine algebraic set, VV is irreducible if and only if its vanishing ideal I(V)I(V) is a in the k[x1,…,xn]k[x_1, \dots, x_n]. An irreducible component of an algebraic set VV is a maximal irreducible closed of VV (in the ), meaning an irreducible algebraic that is not properly contained in any larger irreducible algebraic of VV. Every algebraic set decomposes uniquely (up to ordering) as a finite union of its irreducible components, by the Lasker–Noether , assuming the polynomial ring is Noetherian. In the more general setting of schemes, for Spec⁡(A)\operatorname{Spec}(A) where AA is a , the irreducible components are the closed subsets V(p)V(\mathfrak{p}) corresponding to the minimal prime ideals p\mathfrak{p} of AA. Each such component is irreducible and maximal among the irreducible closed subsets.

Properties

The irreducible components of an algebraic set or scheme are closed in the and cover the entire space, though they may intersect nontrivially. In Noetherian schemes, the decomposition into irreducible components is finite and unique. Irreducible varieties are connected in the , and their dimension is well-defined as the of the coordinate ring. The irreducible components determine the of the ideal defining the variety. Singularities and birational equivalences are often analyzed componentwise. In projective space, irreducible components behave similarly, with the added structure from homogeneous ideals. This framework extends to more abstract geometric objects like stacks and motives, where irreducible components aid in computing invariants.

Examples

Topological examples

In the real line R\mathbb{R} equipped with the standard , which is Hausdorff, every irreducible subset is a singleton, so the irreducible components of R\mathbb{R} are precisely the single-point sets {x}\{x\} for each x∈Rx \in \mathbb{R}. The , denoted S={a,b}S = \{a, b\} with open sets ∅\emptyset, {a}\{a\}, and {a,b}\{a, b\}, is irreducible because its only proper nonempty closed subset is {b}\{b\}, and no union of two proper closed subsets covers SS. Thus, the single irreducible component of SS is SS itself. Consider a XX equipped with the discrete topology, where every is both open and closed; this coincides with the on the set of points of an over a . In this case, any with more than one point can be expressed as a union of two proper nonempty closed subsets (e.g., singletons), so the irreducible subsets are the singletons, and hence the irreducible components of XX are its individual points. The empty ∅\emptyset has no irreducible components, as irreducibility is defined only for nonempty spaces that cannot be decomposed into unions of proper closed subsets.

Geometric examples

In , a classic example of a reducible affine is the zero set of the xy=0xy = 0 in the affine plane Ak2\mathbb{A}^2_k over a field kk. This variety decomposes into two irreducible components: the xx-axis, defined by the ideal (x)(x), and the yy-axis, defined by the ideal (y)(y). These components intersect at the origin, illustrating how the of the ideal (xy)(xy) corresponds to the geometric union of the lines V(x)V(x) and V(y)V(y). Another illustrative case is the nodal cubic curve defined by the equation y2=x3+x2y^2 = x^3 + x^2 in A2\mathbb{A}^2. This curve is irreducible as a variety, despite having a singularity at the origin, because its defining ideal (y2−x3−x2)(y^2 - x^3 - x^2) is prime in k[x,y]k[x, y]. The polynomial y2−x3−x2y^2 - x^3 - x^2 is irreducible over C\mathbb{C}, confirming that the entire curve forms a single irreducible component, which is rational and can be parametrized, but cannot be decomposed further into proper subvarieties. For a reducible quadric surface, consider the variety defined by xy=0xy = 0 in affine 3-space A3\mathbb{A}^3, which extends to the projective closure in P3\mathbb{P}^3. This hypersurface is reducible, consisting of the union of the two planes V(x)V(x) and V(y)V(y), each an irreducible component of dimension 2. In the projective setting, these planes intersect along the projective line V(x,y)V(x,y), which includes the point at infinity [0:0:1:0][0:0:1:0] on the line at infinity V(w)V(w), demonstrating how a degenerate quadric factors into linear components. Such reducible quadrics arise when the defining quadratic form has rank 2 and factors into distinct linear terms. The provides a scheme-theoretic perspective on these decompositions; for instance, Spec⁡(k[x,y]/(xy))\operatorname{Spec}(k[x, y]/(xy)) has exactly two irreducible components, corresponding to the prime ideals (x)(x) and (y)(y), which geometrically realize the axes as closed subschemes. This aligns with the affine example above, emphasizing that the irreducible components are the spectra of the quotient rings by these minimal primes.
Add your contribution
Related Hubs
User Avatar
No comments yet.