algebraic sets and polynomial ideals
Suppose is a field. Let denote affine -space over .
For , define , the zero set of , by
We say that is an (affine) algebraic set if there exists such that . Taking these subsets of as a definition of the closed sets of a topology induces the Zariski topology over .
For , define the deal of in by
It is easily shown that is an ideal of .
Thus we have defined a function mapping from subsets of to algebraic sets in , and a function mapping from subsets of to ideals of .
We remark that the theory of algebraic sets presented herein is most cleanly stated over an algebraically closed field. For example, over such a field, the above have the following properties:
For any ideal , .
For any , , the closure of in the Zariski topology.
From the above, we see that there is a 1-1 correspondence between algebraic sets in and radical ideals of . Furthermore, an algebraic set is an affine variety if and only if is a prime ideal. As an example of how things can go wrong, the radical ideals and in define the same zero locus (the empty set) inside of , but are not the same ideal, and hence there is no such 1-1 correspondence.
|Title||algebraic sets and polynomial ideals|
|Date of creation||2013-03-22 13:05:40|
|Last modified on||2013-03-22 13:05:40|
|Last modified by||mathcam (2727)|
|Defines||ideal of an algebraic set|
|Defines||affine algebraic set|