exclusion from closed sets:
Since we can only evaluate a polynomial up to some finite precision, we
start with zero precsion and then gradually get more precise. If some
is not a root of , then at some point, we will have
that . If is indeed a root, then we will never halt
this process; We will keep getting
# return NOT-IN-SET of x is not a zero of the polynomial
def is_not_in_zeros(poly, x0):
precision = 0 # start with zero precision
if poly(x0[:precision]) != 0: return NOT-IN-SET
precision += 1 # up the precision
poly(x0[:precision]) = 0 for all
levels of precision.
- To setup a topology for the prime spectrum of a ring, we define the topological space as , the set of all prime ideals of .
- The closed sets of the topology are , where the function each ideal to the set of prime ideals that contain it. Formally, .
- We can think of this differently, by seeing that we can rewrite the condition as : On rewriting using the prime ideal , we send the ideal to .
- Thus, the closed sets of are precisely the 'zeroes of polynomials' / 'zeroes of ideals'.
- To make the analogy precise, note that in the polynomial case, we imposed a topology on by saying that the closed sets were for some polynomial .
- Here, we are saying that the closed sets are for some ideal . so we are looking at ideals as functions from the prime ideal to the reduction of the ideal. That is, .
§ from this perspective
Since is a PID, we can think of numbers instead of ideals. The above
picture asks us to think of a number as a function from a prime to a reduced
number. . We then have that the closed sets are those primes
which can zero out some number. That is:
So in our minds eye, we need to imagine a space of prime ideals (points), which
are testing with all ideals (polynomials). Given a set of prime ideals (a tentative locus, say a circle),
the set of prime ideals is closed if it occurs as the zero of some collection of ideas
(if the locus occurs as the zero of some collection of polynomials).
§ Nilpotents of a scheme
- Consider the functions and . as functions on . They are indistinguishable based on Zariski, since their zero sets are the same ( ).
- If we now move to the scheme setting, we get two different schemes: , and .
- So the scheme is stronger than Zariski, as it can tell the difference
between these two situations. So we have a kind of "infinitesimal thickening"
in the case of .
For more, check the
- In the ring , we have an element such that , which is a nilpotent. This "picks up" on the repeated root.
math.se question: 'Geometric meaning of the nilradical'
- Another example is to consider (i) the pair of polynomials (the x-axis) and . The intersection is the zero set of the ideal .
- Then consider (ii) the pair of (the x-axis) and . Here, we have "more intersection" along the x-axis than in the previous case, as the parabola is "aligned" to the x-axis. The intersection is the zero set of the ideal .
- So (i) is governed by , while (ii) is governed by which has a nilpotent . This tells us that in (ii), there is an "infinitesimal thickening" along the -axis of the intersection.