Whenever I decide to learn something – and especially when it’s learning for learning’s sake – I make sure to be meticulous with things. In particular, whenever I see propositions stated without proof, I break out the old pen and paper and start verifying.
The purpose of this post is to examine a few of the properties on page 661 of Dummit and Foote. Some of the background notation needed is discussed in this previous entry.
Claim. The following properties of the map are very easy exercises. Let and be subsets of .
(9) If is any subset of , then , and if is any ideal, then .
(10) If is an affine algebraic set then , and if then , i.e. and .
Proof. (7) Note that . In particular, a polynomial vanishes on if and only if it vanishes on each disjoint component of separately, which happens if and only if it vanishes on the entirety of and on the entirety of .
(9) Suppose . Clearly, any polynomial which vanishes on is in , and because if and only if on , is certainly contained in the zero set of . Therefore, . Note that the inclusion doesn’t necessarily reverse since may vanish for some element .
Next, suppose that is an ideal of the ring and that . By definition, the locus is the collection of all points for which for all . Then, certainly, for all , , i.e., is an element in the ideal that vanishes on . Hence, implies that .
(10) First, suppose that is an ideal and that is an affine algebraic set. It suffices to show that by way of two-sided inclusion. To that end, let . Then is in the zero-set of the ideal , whereby it follows that for all . But for any satisfying for arbitrary , and by definition. Hence, .
Conversely, if , then for all . But all such functions disappear for all values by definition of . This means that the polynomials for which are precisely the functions which satisfy for all . Hence, itself must be an element of , whereby the equality is proved.
The other expression is proved similarly and is omitted for brevity. Therefore, as claimed,
from which it follows that the maps and are inverses of one another under the construction given here.