Course Digest for MATH 831 Spring 2021 Lecture 25 (Thursday, April 29) Our goal for the day was to prove the Theorem: If M is a nonzero finitely-generated D-module, then n ≤ dimD(M) ≤ 2n. To establish the upper bound, we related the D-module dimension and multiplicity of finitely-generated D-modules appearing in a short exact sequence of D-modules. As a corollary of this, we proved that D-module dimension cannot increase upon taking quotients, and that the behavior of D-module dimension and multiplicity is well-behaved with respect to the formation of direct sums. We then used this to establish the upper bound of 2n on the dimension of a nonzero finitely generated D-module. We also proved that the D-module dimension of a quotient M=N must be less than M when M = D, and N is a left ideal of D. This result used the fact that multiplicity is additive across short exact sequences when the dimensions of the outer terms agree. After this, we presented a sketch of a proof of Bernstein's Inequality, which is the lower bound on dimensions given above. This was only a sketch, because we did not prove one key step, which is described in the exercises below, and may appear in our last worksheet. Exercises Suppose that M is a finitely-generated D-module, and consider a good filtration G• such that G0, and hence Gi for all i, is nonzero (we recalled why such a filtration exists in lecture). Then, prove that the map i i 2i B ! Hom|(G ;G ) that maps an operator P to the |-linear map given by left multiplication by P . Hint: Induce on i. More hints may appear in the worksheet, but just give it a try on your own for now. Lecture 24 (Tuesday, April 27) We started off by discussing some more general facts of Hilbert polynomials. After that, we re-presented our proof that good filtrations are small (since it was rushed last time), and then used this fact to prove the following fact: If M is a finitely-generated • • D-module, with good filtrations G and H , then there exist positive integers `1; `2 such that i−` i i+` H 2 ⊆ G ⊆ H 1 for each i ≥ 0. Taking |-dimensions, and then interpreting this in terms of the associated Hilbert polynomials hG• (t) and hH• (t), we concluded that the degrees and leading terms of these polynomials must agree. Using this fact, we were able to define the notions of dimension and multiplicity for D-modules. To illustrate this definition, by considering some canonical good filtrations on D and R, we proved that the D-module dimension of D is 2n, that the D-module dimension of R is n, and that the D-module multiplicity of each of these modules is 1. We concluded by stating (but not proving) a result that shows that these two dimensions are extremal. Exercises Consider a function H : Z ! Z. Prove that there exists a numeric polynomial h(t) 2 Q[t] such that H(s) = h(s) for all s 0 if and only if there exists a numeric polynomial g(t) 2 Q[t] such that the function ΣH : Z ! Z defined by s 7! H(0) + ··· + H(s) agrees with g(s) for all s 0. Note: This is meant to show that the existence of a Hilbert polynomial for the function s 7! dim| M0 +··· dim| Ms, which is how we defined Hilbert functions associated to graded modules 1 M = ⊕i=0Mi over polynomial rings over |, is equivalent to the existence of a Hilbert polynomial for the function s 7! dim| Ms, which is how Hilbert functions associated to graded modules are also commonly defined. Of course, our choice of Hilbert function comes from the fact that the ΣH version is most convenient when dealing when graded modules coming from a filtration. 1 2 Lecture 23 (Thursday, April 22) Having developed the basic theory of Hilbert polynomials, we returned today to looking at D-module theory. We call a filtered (D; B•)-module (M; G•) good • 0 • if each component of G finite dimensional over | = B , and gr(M; G ) is finitely-generated over the polynomial ring S = gr(D; B•). In this case, the Hilbert function associated to gr(M; G•) s is simply s 7! dim| G , and we call the associated Hilbert polynomial hG• (t) 2 Q[t] the Hilbert polynomial associated to the filtration G•. Most of the lecture was dedicated to studying properties of good filtrations. For example, we proved that if M is a finitely-generated D-module, then good filtrations on M exist; when combined with a problem from our most recent worksheet, we see that good filtrations exist on a D-module M if and only if M is finitely-generated over D. We also characterized good filtrations (M; G•) as exactly those such that each Gi is finite-dimensional over 0 j∗ i j∗ i+j∗ | = B , and such that there exists a definitive component G , in the sense that B G = G for each i ≥ 0. Using this, we concluded the lecture by demonstrating that good filtrations must be small. Lecture 22 (Tuesday, April 20) Today, we focused on proving the existence of Hilbert polynomials. t t t We started off by describing how to use the Q-basis 0 ; 1 ;:::; d for Q[t]≤d to describe numeric polynomials of degree d, and using this, and some corollaries, we gave a proof of the existence of Hilbert polynomials. We concluded by returning to the setting of rings of differential operators, and explaining how we can use this to define the D-module dimension. There are still some issues concerning the existence of so-called good filtrations that we will need to consider. Exercises Take another close look at the Worksheet, which has updates that address some feed- back/questions from the class. In particular, the problems about exact sequences induced from filtrations has changed. The new problem is less ambitious, but it is correct, and should cover a number of applications, both in the worksheet, as well as in some upcoming lectures. Also, a sequence of ideals that more efficiently illustrates non-Noetherianity in prime characteristic has been included. If you used the original sequence, it still works, and you shouldn't need to modify your write-up, if you have already completed it. Lecture 21 (Thursday, April 15) We continued our discussion of filtered rings. After recalling the construction of (A; F •) 7! gr(A; F •). Then, we defined what it means for (M; G•) to be a left (A; F •)-module, and went over the construction (M; G•) 7! gr(M; G•), the associated graded module. Motivated by this, we discussed graded modules over graded rings, in general, and mo- tivated by our desire to have a good dimension theory in the graded case, we defined the notion of Hilbert functions associated to graded modules, and we computed them for polynomial rings in a small number of variables. Motivated by the existence of Hilbert polynomials (which we stated, but will only prove next time), we temporarily shifted our focus to studying numerical, or numeric, polynomials f(t) 2 Q[t]. We discussed a characterization using a nice basis of binomial polynomials in Q[t], and will pick up here next time. Exercises (1) Start the new worksheet. 1 (2) If R = |[x1; : : : ; xn] is a polynomial ring, and M = ⊕i=0Mi is a finitely-generated and graded R-module, then dim|(Mi) < 1 for every i. Hint: Use the worksheet to see that M 3 can be generated by finitely many homogeneous elements, and also to see how to describe elements in Mi in terms of such a generating set. Finally, use that dim| Ri < 1 for each i. Lecture 20 (Tuesday, April 13) Our focus today was on graded and filtered rings. We started by getting some practice with graded ideals, by proving that a graded ideal of a graded ring that is prime with respect to homogeneous elements is a prime ideal. After this, we defined what it means for (A; F •) to be an (ascending, exhaustive, multiplicative) filtered ring, and we connected this notion to graded rings. We pointed out some canonical examples from differential • • operators, defining the order filtration D = DR=k on D = DR=k for an arbitrary inclusion k ⊆ R of commutative rings. In the special case that R = |[x1; : : : ; xn], and | is a field of characteristic zero, then we also defined the Bernstein filtration B• = B• by the rule R=| i u v B = |fx @ : kuk + kvk ≤ ig We observed how this is a filtration of D by finite dimensional |-vector spaces, and after defining associated graded rings, we proved that gr(D; B•) is a polynomial ring in 2n-variables. We briefly discussed the consequences of this. p Exercises Prove that if R is a graded ring, and I is a graded ideal, the the radical I = ff 2 I : f n 2 Ig is a graded ideal of R. Lecture 19 (Thursday, April 8) Announcement: There is an upcoming worksheet on Noether- ian rings and modules, in the context of differential operators. We continued our discussion of D-modules. After revisiting the D-module C[x1; : : : ; xn]f , we shifted our attention to homo- geneous systems of linear PDEs. More precisely, we proved that if P1(f) = ··· Pm(f) = 0 is such a system, then the space of polynomial solutions of this system can be identified with P HomD(D= DPi; C[x1; : : : ; xn]).