Subset (often algebraic set) that is not the union of subsets of the same nature
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.
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are in general position). If mi denotes the multiplicity of an irreduciblecomponent Zi in the intersection (i.e., intersection multiplicity), then the...
V_{1}\subsetneq \cdots \subsetneq V_{\ell }\subset X.} An irreducible subset of X is an irreduciblecomponent of X if and only if the codimension of it in X is...
follows: hypersurfaces are exactly the algebraic sets whose all irreduciblecomponents have dimension n – 1. A real hypersurface is a hypersurface that...
∣ Z an irreduciblecomponent of f − 1 ( f ( x ) ) containing x } . {\displaystyle e(x)=\max\{\dim Z\mid Z{\text{ an irreduciblecomponent of }}f^{-1}(f(x)){\text{...
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representation has a subrepresentation, but only has one non-trivial irreduciblecomponent. For example, the additive group ( R , + ) {\displaystyle (\mathbb...
algebraic set may be uniquely decomposed into a finite union of irreduciblecomponents. It has a straightforward extension to modules stating that every...
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is unique. Thus its elements are called the irreduciblecomponents of the algebraic set. An irreducible algebraic set is also called a variety. It turns...
dim A + dim B − dim X, then A · B is a linear combination of the irreduciblecomponents of A ∩ B, with coefficients the intersection multiplicities. At...
subscheme. If X {\displaystyle X} is of pure dimension r; i.e., every irreduciblecomponent has dimension r, then C W / X {\displaystyle C_{W/X}} is also of...
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