For instance, for every extension of the integers, but if one also allows quotients such as or instead, then and are no longer necessarily equal. Thus, as long as one thinks of a variety or scheme as cutting out points not just in the original base ring or field, but in all extensions of that base ring or field, one recovers an exact correspondence between the algebraic geometry perspective and the commutative algebra perspective. This is similar to the classical algebraic geometry position of viewing an algebraic variety as being defined simultaneously over all fields that contain the coefficients of the defining polynomials, but the crucial difference between scheme theory and classical algebraic geometry is that one also allows definition over commutative rings, and not just fields.
In particular, one needs to allow extensions to rings that may contain nilpotent elements, otherwise one cannot distinguish an ideal from its radical. There are of course many ways to extend a field into a ring, but as an analyst, one way to do so that appeals particularly to me is to introduce an epsilon parameter and work modulo errors of.
Consider the ring of real-valued quantities that depend on a parameter i. One can, if one wishes, impose some further continuity or smoothness hypotheses on how depends on , but this turns out not to be relevant for the following discussion. Algebraists often prefer to use the ring of Puiseux series here in place of , and a nonstandard analyst might instead use the hyperreals , but again this will not make too much difference for our purposes. Inside this commutative ring, we can form the ideal of quantities that are of size as , i. This can easily be seen to indeed be an ideal in.
We then form the quotient ring. Clearly, is a commutative ring extending. Hence, any algebraic variety. For instance, the scheme associated to the ideal , when interpreted over , becomes an neighbourhood of the origin. For instance, consider the scheme cut out by the polynomials in two dimensions. Over , this becomes. Note that the polynomial vanishes to order on this locus, but fails to lie in the ideal. Equivalently, we have , despite and being distinct ideals.
One can also view ideals and hence, schemes , from a model-theoretic perspective. Let be an ideal of a polynomial ring generated by some polynomials.
Then, clearly, if is another polynomial in the ideal , then we can use the axioms of commutative algebra which are basically the axioms of high school algebra to obtain the syntactic deduction. In particular, we have the semantic deduction. If we restrict to lie in only, then even if is an algebraically closed field , the converse of the above statement is false; there can exist polynomials outside of for which 1 holds for all assignments in.
For instance, we have. Of course, the nullstellensatz again explains what is going on here; 1 holds whenever lies in the radical of , which can be larger than itself. For instance, since. On the other hand, we still have. LO Tags: algebraic sets , algebraic varieties , Bezout's theorem , nonstandard analysis , ultrafilters , ultralimit analysis by Terence Tao 41 comments. One way to connect the two types of analysis is via compactness arguments and more specifically, contradiction and compactness arguments ; such arguments can convert qualitative properties such as continuity to quantitative properties such as bounded , basically because of the fundamental fact that continuous functions on a compact space are bounded or the closely related fact that sequentially continuous functions on a sequentially compact space are bounded.
One common way to achieve this is to embed everything inside some universal space and then use some weak compactness property of that space, such as the Banach-Alaoglu theorem or its sequential counterpart. This is for instance the idea behind the Furstenberg correspondence principle relating ergodic theory to combinatorics; see for instance this post of mine on this topic. However, there is a slightly different approach, which I will call ultralimit analysis , which proceeds via the machinery of ultrafilters and ultraproducts ; typically, the limit objects one constructs are now the ultraproducts or ultralimits of the original objects.
There are two main facts that make ultralimit analysis powerful. The first is that one can take ultralimits of arbitrary sequences of objects, as opposed to more traditional tools such as metric completions, which only allow one to take limits of Cauchy sequences of objects. This existence of elementary limits is a manifestation of the compactness theorem in logic; see this earlier blog post for more discussion.
So we see that compactness methods and ultrafilter methods are closely intertwined. See also my earlier class notes for a related connection between ultrafilters and compactness. Ultralimit analysis is very closely related to nonstandard analysis. I already discussed some aspects of this relationship in an earlier post , and will expand upon it at the bottom of this post.
Roughly speaking, the relationship between ultralimit analysis and nonstandard analysis is analogous to the relationship between measure theory and probability theory. To illustrate how ultralimit analysis is actually used in practice, I will show later in this post how to take a qualitative infinitary theory — in this case, basic algebraic geometry — and apply ultralimit analysis to then deduce a quantitative version of this theory, in which the complexity of the various algebraic sets and varieties that appear as outputs are controlled uniformly by the complexity of the inputs.
The point of this exercise is to show how ultralimit analysis allows for a relatively painless conversion back and forth between the quantitative and qualitative worlds, though in some cases the quantitative translation of a qualitative result or vice versa may be somewhat unexpected. In an upcoming paper of myself, Ben Green, and Emmanuel Breuillard announced in the previous blog post , we will rely on ultralimit analysis to reduce the messiness of various quantitative arguments by replacing them with a qualitative setting in which the theory becomes significantly cleaner.
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Ben Eastaugh and Chris Sternal-Johnson. Subscribe to feed. What's new Updates on my research and expository papers, discussion of open problems, and other maths-related topics. Tag Archive. If is any point in , then the distances from to are rationals that obey the equations for , and thus determine a rational point in the affine complex variety defined as By inspecting the projection from to , we see that is a branched cover of , with the generic cover having points coming from the different ways to form the square roots ; in particular, is a complex affine algebraic surface, defined over the rationals.
Hence the number of integer points of height comparable to should be about this is a convergent sum if ranges over say powers of two, and so from standard probabilistic heuristics see this previous post we in fact expect only finitely many solutions, in the absence of any special algebraic structure e. Such varieties can be viewed in at least four different ways: Algebraic geometry One can view a variety through the set of points over in that variety.
Commutative algebra One can view a variety through the field of rational functions on that variety, or the subring of polynomial functions in that field.
Dual algebraic geometry One can view a variety through a collection of polynomials that cut out that variety. Dual commutative algebra One can view a variety through the ideal of polynomials that vanish on that variety. For instance, the unit circle over the reals can be thought of in each of these four different ways: Algebraic geometry The set of points.
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Commutative algebra The quotient of the polynomial ring by the ideal generated by or equivalently, the algebra generated by subject to the constraint , or the fraction field of that quotient. Dual algebraic geometry The polynomial. Dual commutative algebra The ideal generated by. Any set of polynomials in indeterminate variables with coefficients in determines, on the one hand, an ideal in , and also cuts out a zero locus since each of the polynomials clearly make sense as maps from to.
Of course, one can also write in terms of : Thus the ideal uniquely determines the zero locus , and we will emphasise this by writing as. Indeed, suppose we have two ideals of that cut out the same non-empty zero locus for all extensions of , thus for all extensions of. Over , this becomes Note that the polynomial vanishes to order on this locus, but fails to lie in the ideal. Then, clearly, if is another polynomial in the ideal , then we can use the axioms of commutative algebra which are basically the axioms of high school algebra to obtain the syntactic deduction since is just a sum of multiples of.
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In particular, we have the semantic deduction for any assignment of indeterminates in or in any extension of. For instance, we have for all in an algebraically closed field, despite not lying in the ideal. For instance, since we no longer have a counterexample to the converse coming from and once we work in instead of.
On the other hand, we still have so the extension is not powerful enough to detect that does not actually lie in ; a larger ring which is less easy to assign an analytic interpretation to is needed to achieve this. What is good mathematics? Why global regularity for Navier-Stokes is hard. Top Posts Almost all Collatz orbits attain almost bounded values Career advice The Collatz conjecture, Littlewood-Offord theory, and powers of 2 and 3 Does one have to be a genius to do maths?
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