Abstract
Decomposing a space into its connected components is a familiar topological idea which is immediately applicable to closed sets in kn and which we will proceed to generalize to group schemes. But the algebraic nature of our closed sets makes it easier to approach connectedness via a stronger concept, irreducibility. Consider for example the zeros of (x2 + y2 − 1)x in k2. This set is connected, but everyone would usually say it is made up of two pieces, the circle and line which arc the zeros of the factors. Minimal pieces of this kind are easily singled out in the Zariski topology: we call a topological space irreducible if it is not the union of two proper closed subsets.
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© 1979 Springer-Verlag New York Inc.
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Waterhouse, W.C. (1979). Irreducible and Connected Components. In: Introduction to Affine Group Schemes. Graduate Texts in Mathematics, vol 66. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-6217-6_5
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DOI: https://doi.org/10.1007/978-1-4612-6217-6_5
Publisher Name: Springer, New York, NY
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