Intro to alg geom 4
IV. AFFINE SCHEMES
If R is any commutative ring with 1, let X (= "specR") be the set of all prime ideals of R, with a topological closure operator where the closure of a set of prime ideals is the set of all prime ideals containing the intersection of the given set of primes. (Intuitively, each prime ideal contains the functions vanishing at the corresponding point, so their intersection is all functions vanishing at all the points of the set, and the prime ideals containing this intersection hence are all points on which that same set of functions vanishes. So the closure of a set is the smallest algebraically defined locus containing the set.) This closure operator defines the "Zariski topology" on X.
Now any ring map defines a morphism of their spectra by puling back prime ideals, and in particular a morphism is continuous, although this alone says little since the Zariski topology is so coarse. Notice now maximal ideals may pull back to non maximal ones, e.g. under the inclusion map Z-->Q of integers to the rationals, taking the unique point of specQ to a dense point of specZ. Maximal ideals now correspond to closed points, and in particular there are usually plenty of non closed points. Intuitively, every irreducible subvariety has a dense point, and together these "points", one for each irreducible subvariety, give all the points of specX.
If K is a ring, a "K valued point" of X is given by a ring homomorphism R-->K, not necessarily surjective. E.g. if K is a field, the pullback of the unique maximal ideal of K is a not necessarily maximal, prime ideal P of R, the K valued point. Even if the point is closed, i.e. if P is maximal, we get information on which maximal ideals correspond to points with coefficients in different fields. If say k = the real field, and f is a polynomial over k, then a k algebra map g:k[X,Y]/(f)-->k has as kernel a maximal ideal corresponding to a point of {f=0} in k^2, i.e. a point of {f=0} in the usual sense, with real coefficients. The coordinates of this point are given by the pair of images (g(X),g(Y)) in k^2 of the variables X,Y, under the algebra map g, which after all is evaluation of functions at our point. But if say f = Y-X^2, the map from k[X,Y]/(f) -->C taking X to i, and Y to -1, corresponds to the C (complex) - valued point (i,-1), in C^2 rather than k^2.
More generally, if I is any ideal in Z[X1,...,Xn] generated by integral polynomials f1,...fr, and A is a ring, a ring homomorphism Z[X1,...,Xn]/I -->A takes the variables Xj to elements aj of A such that all the polynomials fi vanish at the point of A^n with cordinates (a1,...an). I.e. the map defines an "A valued point " of the locus defined by I. E.g. if M is a maximal ideal of R,we can always view the coordinates of the corresponding point in the residue field R/M, i.e. the point M of specR is "R/M valued".
This approach let's us recover tangent vectors too, in case say of a variety V with ring R = k[X1,...,Xn]/I, where radI = I, and k is an algebraically closed field. Consider the ring S = k[T]/(T^2), with unique maximal ideal (t) generated by the nilpotent element t. Then we claim tangent vectors to V correspond to S valued points (over spec(k)), i.e. to k algebra maps R-->S. E.g. if R = k[X], and we map R-->S by sending X to a+bt, then the inverse image of the maximal ideal (t) is the maximal ideal (X-a), and two elements of (X-a) have the same image in S if and only if they have the same derivative at X=a. Thus S valued points of V are points of the "tangent bundle" of V.
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