Use in Algebraic Geometry
In general algebraic variety or scheme X is the union of its irreducible components Xi. In most cases occurring in "practice", namely for all noetherian schemes, there are finitely many irreducible components. There is the following description of irreducible affine varieties or schemes X = Spec A: X is irreducible iff the coordinate ring A of X has one minimal prime ideal. This follows from the definition of the Zariski topology. In particular, if A has no zero divisors, Spec A is irreducible, because then the zero-ideal is the minimal prime ideal.
As a matter of commutative algebra, the primary decomposition of an ideal gives rise to the decomposition into irreducible components; and is somewhat finer in the information it gives, since it is not limited to radical ideals.
An affine variety or scheme X = Spec A is connected iff A has no nontrivial (i.e. ≠0 or 1) idempotents. Geometrically, a nontrivial idempotent e corresponds to the function on X which is equal to 1 on some connected component(s) and 0 on others.
Irreducible components serve to define the dimension of schemes.
Read more about this topic: Irreducible Component
Famous quotes containing the words algebraic and/or geometry:
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)
“... geometry became a symbol for human relations, except that it was better, because in geometry things never go bad. If certain things occur, if certain lines meet, an angle is born. You cannot fail. It’s not going to fail; it is eternal. I found in rules of mathematics a peace and a trust that I could not place in human beings. This sublimation was total and remained total. Thus, I’m able to avoid or manipulate or process pain.”
—Louise Bourgeois (b. 1911)