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-, 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 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 → Homk(G ,G ) that maps an operator P to the k-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 `1, `2 such that i−` i i+` H 2 ⊆ G ⊆ H 1 for each i ≥ 0. Taking k-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) ∈ Q[t] such that H(s) = h(s) for all s  0 if and only if there exists a numeric polynomial g(t) ∈ 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→ dimk M0 +··· dimk Ms, which is how we defined Hilbert functions associated to graded modules ∞ M = ⊕i=0Mi over polynomial rings over k, is equivalent to the existence of a Hilbert polynomial for the function s 7→ dimk 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.

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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 k = B , and gr(M,G ) is finitely-generated over the polynomial S = gr(D,B•). In this case, the Hilbert function associated to gr(M,G•) s is simply s 7→ dimk G , and we call the associated Hilbert polynomial hG• (t) ∈ 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∗ k = 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) ∈ 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. ∞ (2) If R = k[x1, . . . , xn] is a , and M = ⊕i=0Mi is a finitely-generated and graded R-module, then dimk(Mi) < ∞ 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 dimk Ri < ∞ 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 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 = k[x1, . . . , xn], and k is a field of characteristic zero, then we also defined the Bernstein filtration B• = B• by the rule R/k i u v B = k{x ∂ : kuk + kvk ≤ i} We observed how this is a filtration of D by finite dimensional k-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. √ Exercises Prove that if R is a graded ring, and I is a graded ideal, the the radical I = {f ∈ I : f n ∈ I} 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]). We stressed that the solution space, as well as this set of D- module homomorphisms, is usually not a D-module, nor even an R-module, thought it is C-, and the above identification is as C-vector spaces. After this, we discussed our need to develop a suitable theory of dimension for D-modules. Towards this, we require the basics of graded rings. After defining these rings, and giving example, we provided three equivalent characterizations of what it means to be a homogeneous ideal.

Exercises Prove that if R is graded, and I is a homogeneous ideal of R, then R/I inherits a natural grading from R. More precisely, show that the i-th graded component of this quotient may ∼ be described as {f + I : f ∈ Ri} = Ri/Ri ∩ I.

Lecture 18 (Tuesday, April 6) We concluded our discussion of rings of differential operators of polynomial rings R in prime characteristic by taking a concrete look at the equality DR/k = ∞ ∪e=0 HomRpe (R,R) in the special case that R = Fp[x], a polynomial ring in a single variable x. By concrete, I mean that we explicitly described when divided power operators are linear over Rq, which forced us to understand the behavior of binomial coefficients modulo p. We did this by proving Lucas’ Theorem, and then applying it. After wrapping this up, we defined what we mean by the term D-module, and gave examples of D-modules. We expressed the philosophy that R-modules that are large become small when considered as modules over the larger ring D. For a particular example of this, we saw that C[x]x 4 is not finitely-generated over R = C[x], but is principally generated as a D = DC[x]/C-module by 1/x. We stated a somewhat surprising result that describes what happens when x is replaced by an arbitrary polynomial.

Lecture 17 (Thursday, April 1) We continued discussing rings of prime characteristic. Given an ideal I of such a ring R, we defined I[p] to be the extension of I along F . A more concrete definition is that I[p] = hf p : f ∈ Ii. We discussed how to describe I[p] in terms of a generating set for I, and using this, we proved an important comparison result. Proposition: If I is generated by n elements, then Inq ⊆ I[q] ⊆ Iq for every q = pe. With this in hand, we returned to our discussion of differential operators. If R = Fp[x1, . . . , xn], the we used the fact that ∆ = ker(R ⊗Fp R → R) is generated by n elements, along with our above comparison theorem, to prove that DR/Fp = ∞ ∪e=0 HomRpe (R,R). We also pointed out some really nice properties of this filtration.

[q] q Exercises Verify that if E = HomFp (R,R), then the annihilator of ∆ in E is HomR (R,R).

Lecture 16 (Tuesday, March 30) We continued our discussion of Frobenius, and considered the Frobenius fractal · · · ⊆ Rpe ⊆ · · · ⊆ Rp2 ⊆ Rp ⊆ R. To illustrate the fractal (i.e., self-similar) nature of this chain, we defined R to be F -finite if R is finitely generated over Rp, and, at least when R is reduced, we proved that this is the same thing as every module structure in this chain being finitely-generated, which is equivalent to a single such structure being finitely-generated. We defined F -split rings, discussed how it fits into the Frobenius fractal, and gave a lot of examples of rings that are F -split, as well as ones that are not.

Exercises If R is reduced, prove that Rp ⊆ R splits as a map of Rp-modules if and only if Rpe ⊆ R splits over Rpe for some e if and only if Rqpe ⊆ Rq splits over some q and e, if and only if this last map splits for every e and q.

Lecture 15 (Thursday, March 25) We started an in-depth discussion n rings of prime characteristic, which I expect will occupy us for a couple of weeks. Today, we covered the very basics of the Frobenius map F : R → R, and explained how the module structure of R over Rp = F (R) = {f p : f ∈ R} reflects some important ring-theoretic properties of R. For example, we proved that F is injective if any only if R is reduced. We also examined this module structure when R is a 2 3 polynomial ring over Fp, and also when R = F2[x , x ].

2 3 p Exercises Verify that R = Fp[x , x ] is finitely generated over R for an arbitrary prime p.

Lecture 14 (Tuesday, March 23) We started by briefly going over the second worksheet, which will be due on March 31st. After that, we continued our discussion concerning Stanley-Reisner rings. After establishing a sequence of interesting lemmas, we arrived at a concrete description of the ring of k-differential operators on the Stanley-Reisner ring k[x1, . . . , xn]/I, where I is an intersection of monomial primes. We looked at specific example of R = C[x, y]/hxyi, and concluded that DR/C is not finitely-generated over C. For more on differential operators over Stanley-Reisner rings, check out this article by Will Traves. 5

Exercises In a useful lemma, we proved that if k ,→ S was arbitrary, and I = ℘1 ∩ · · · ∩ ℘t is an irredundant intersection of prime ideals of S, then an operator P ∈ DS/k stabilizes I if and only if it stabilizes each component ℘i. The interesting direction (i.e., that stabilizing I forces the stabilization of each ℘i) can be generalized as follows: If a differential operator stabilizes I, then it must stabilize each minimal primary component of I. Recall that if I = q1 ∩ · · · ∩ pt is a primary decomposition of I, then a primary ideal is called minimal if its radical is minimal among the set of radicals of each component (recall that this sets of radicals are uniquely determined by I). Consult the excellent Wikipedia article for more on primary decomposition. If you are familiar with primary decomposition, try to prove the above generalization. Hint: You may need to invoke a version of prime avoidance that deals with a prime ideal containing an intersection of ideals.

Lecture 13 (Thursday, March 18) We continued our discussion of differential operators on quotient rings. To get an idea of the possibilities, we gave a quick overview on some basic properties of the 3 3 3 ring of C-linear differential operators on the C[x, y, z]/hx +y +z i. For more details, check out this paper. For a concrete example, we then considered the situation for Stanley-Reisner rings. We recalled the definition of these rings, as well as some equivalent characterizations, and motivated by the characterization of a Stanely-Reisner ring as the quotient of a polynomial ring by the intersection of monomial prime ideals, we considered the differential operators on a polynomial ring S that stabilizes such an intersection in terms of the differential operators that stablize each prime monomial component in an irredundant representation.

Exercises Suppose that J is a prime monomial ideal of S = k[x1, . . . , xn]. Verify that a k-linear differential operator on S stabilizes J if and only if it lies in the k-span of the set of all monomials xu∂[v] with either xu ∈ J or xv ∈/ J.

Lecture 12 (Thursday, March 11) We continued towards our goal of unraveling all of this T - formalism in the special case that R = k[x1, . . . , xn]. As a first step, we noted that, after iden- tifying T with R[x1, . . . , xn, y1, . . . , yn] = R[y1, . . . , yn] and ∆ with the ideal generated by the i+1 polynomials zi := yi − xi for 1 ≤ i ≤ n, then the module of principle parts T/∆ is identified i+1 with R[z1, . . . , zn]/hz1, . . . , zni , which is free over R, with free basis given by the monomials b b1 bn i+1 z = z1 ··· zn with kbk ≤ i. Therefore, the R-dual of T/∆ , which we earlier identifed with Di , is free over R, with free basis consisting of all dual maps (zb)∗ with kbk ≤ i. To understand R/k this dual map, we explained how to write elements of T/∆i+1 in terms of the above free basis. The key point was to know how to expand a monomial in the yi’s in terms of this basis, for after tracing everything around, we saw that the isomorphism G from (T/∆i+1)∗ ⊆ T ∗ to Di ⊆ E R/k sends an arbitrary map ψ to G(ψ): R → R defined by G(ψ)(r(x)) = ψ(r(y)). So, in particular, the dual maps (zb)∗ maps under G to the map that sends xu to (zb)(yu) = u1 ··· unxu−b. In other b1 bn words, these images are exactly the divided power maps that you investigated in your worksheet! With this, we have now presented three different methods that lead to our concrete description of the ring of differential operators on a polynomial ring! We concluded lecture with a short discussion of how to extend this computation to quotients.

Exercises I urge you to trace through the above arguments, so that it is crystal clear to you why the dual maps (zb)∗ correspond to the maps ∂[b]. 6

Lecture 11 (Tuesday, March 9) We started off the lecture by recalling more basics from module theory. Namely, we recalled the definition of free modules, and explained their . That is, to define an R-linear map from a , it suffices to specify its values on a free basis. In other words, if M is free over R, then the R-dual HomR(M,R) is free, with free bases for M inducing dual bases on the R-dual of M, as in linear algebra. After going over some suggestive examples, we returned to the isomorphism from E to the R-dual T ∗ of T . Previously, we had defined a T -module structure on E, and we defined the T -module structure on T ∗ via precomposition. We briefly noted that the isomorphism E =∼ T ∗ is actually T -linear, and using this T -linearity, along with our description of Di as the annihilator of ∆i+1 in E, we concluded that this set of differential R/k operators may be identified with the maps in T ∗ that map ∆i+1 to zero, which we saw may be i+1 canonically identified with HomR(T/∆ ,R). We concluded by understanding what all of this means in the simple case that R is a polynomial ring over k.

Exercises Verify, as asserted in class, that the isomorphisms E → T ∗ is T -linear. Also, verify the arguments presented in class that allowed us to simplify the ∆ of the multiplication map T → R in the case that R = k[x1, . . . , xn]. To make sure everything is clear in your head, you may prefer to run the argument through in the polynomial ring k[x1, . . . , xn, y1, . . . , yn].

Lecture 10 (Thursday, March 4) Let k ,→ R be an inclusion of rings. Our focus today was on studying the tensor product T = R⊗k R, and especially the T -module structure of E = Homk(R,R). After reviewing some basics from module theory, we returned to differential operators, and proved that if ∆ is the kernel of the multiplication map T → R, then Ann (∆i+1) = Di . We then E R/k set out to describe the k-linear differential operators on R of order at most i in terms of certain R-linear maps (as opposed to k-linear maps). Towards this, we exhibited an isomorphism ∼ ∼ ∗ E = Homk(R,R) = HomR(T,R) = T Before getting into details, it is worthwhile to mention that the R-module structure on T comes from multiplication on the left component of the tensor product. Of course, this choice is somewhat arbitrary, but it will account for some of the choices in the above isomorphisms of Hom modules. Let us recall the isomorphism. The forward map F maps φ ∈ E to the map F (φ): T → R defined by F (φ)(a ⊗ b) = aφ(b) on simple tensors, while the backwards map G maps ψ ∈ HomR(T,R) to the map G(ψ): R → R defined by G(ψ)(r) = ψ(1 ⊗ r).

Exercises Above, we specified the map F (φ): T → R on simple tensors. Using the unverisal properties of tensor products, verify that this map is well-defined. Furthermore, verify that F and G are inverses (recall the R-module structure on T , and convince yourself that this is why we placed the r in the second component in the definition of G(ψ) above), and the fact that these maps are linear over R.

Lecture 9 (Tuesday, March 2) Most of today’s lecture was spent discussing the basic properties of tensor products. After reviewing the construction and universal property of tensor products of k-modules, we noted that if k → R and k → S are maps of commutative rings (i.e., R and S are k-algebras), then the tensor product R ⊗k S is a (and hence, a k-algebra). We considered some examples, and then proved that the map R ⊗k R → R given by a ⊗ b 7→ ab is a 7 surjective ring map with kernel ∆R/k = ∆ being the ideal of R ⊗k R generated by the differences δf = 1 ⊗ f − f ⊗ 1 for all f ∈ R. We concluded by previewing the connection differential operators. We also discussed the recent worksheet. The suggested groups are below. Please let me know ASAP if you would like me to modify these.

• TA and JR and EH • AP and MD • DB and SD • RN and SG

Exercises

(1) Make certain that you understand the universal property of tensor products: There is a

one-to-one correspondence between k-module maps φ : R ⊗k S → T and map θ : R ×S → T that satisfies θ(r1 +r2, s) = θ(r1, s)+θ(r2, s), θ(r, s1 +s2) = θ(r, s1)+θ(r, s2), and θ(λr, s) = θ(r, λs) = λθ(r, s) for all r, ri ∈ R, s, si ∈ S, and λ ∈ k. (2) If f, g ∈ k[x], come up with a simplified description of the ring (k[x]/hfi) ⊗k (k[x]/hgi). (3) Suppose that R is a finitely-generated k-algebra. This just means that there exists some distinguished elements f1, . . . , fn ∈ R such that every element of R can be expressed (per- haps, non-uniquely) as a polynomial in the fi with coefficients in k. For example, we can take fi = xi if R = k[x1, . . . , xn] is the polynomial ring over k in n-variables, and fi equal to the class of xi in the quotient of such a polynomial ring by any ideal. In this case, find

a finite generating set for the ideal ∆ = ∆R/k.

Lecture 8 (Thursday, February 25) We started off by formally defining the divided power operators 1 ∂[n] = ∂p i n! i We established the basic properties of these operators, and then shifted our attention to using them to give a description of DR/k when R is a polynomial ring over the commutative ring k. Towards this, we proved that an element of Dm , with R as above, is determined by its values on monomials R/k in R of degree at most m. We concluded by describing how to use this result, along with another one that we stated, but did not prove, to give a description of Dm as the set of all left R-linear R/k [u] [u1] [un] combinations of the products ∂ = ∂1 ··· ∂n with kuk ≤ m. Note: Due to some technical difficulties, we were unable to complete the proof of the second aforementioned lemma. However, students will get a chance to work on this as part of an upcoming worksheet.

Lecture 7 (Tuesday, February 23) We recalled the definition and basic properties of the ring of k- i linear differential operators DR/k = ∪DR/k for an arbitrary inclusion of commutative rings k ,→ R. The rest of the lecture was spent proving the following Theorem: If k is a field of characteristic i i P u zero and R = k[x1, . . . , xn], then DR/k agrees with C := { fu∂ : fu ∈ R andkuk ≤ i}. In particular, DR/k = An, the n-th over k. A key point in the proof was the fact that i i−1 the map Q 7→ [Q, xm] defines a k-linear surjection C 7→ C for all m and i. The kernel of this map also played a role in our proof. We concluded lecture by defining the map 1 ∂2 : [x] → [x] 2 x Z Z 8

n n n−2 which maps the monomial x to 2 x . Of course, the factor of 1/2 appearing above makes no sense as an element of EndZ(Z[x]), so this map should not be thought of as a composition of maps within this endomorphism ring.

Exercises

(1) Coutinho Chapter 3: 3.1, 3.3 (2) Review the basics of tensor products. If you are only familiar with tensor products over fields (i.e., then tensor products of vector spaces) then that is OK. 1 (3) Verify the direct sum portion of the theorem DR/k = R ⊕ DerR/k. That is, prove that if f + P = g + Q with f, g ∈ R and P,Q two k-linear derivations of R, then f = g and P = Q. Hint: The difference of two derivations is a derivation.

Lecture 6 (Thursday, February 18) Consider an inclusion of arbitrary commutative rings k ,→ R. This lecture was focused on defining the ring of k-linear differential operators on R. We defined i i the subsets D = DR/k of E = Endk(R) as follows: (1) D−1 = 0 (2) D0 = R (3) If i ≥ 1, then P ∈ Di ⇐⇒ [P, f] ∈ Di−1 for every f ∈ R.

We then established the basic properties of these subsets. More precisely, we proved that Dm + Dm ⊆ Dm and DmDn ⊆ Dm+n. We concluded that each Dm is a left R-submodule of E. We also m m+1 ∞ i saw that D ⊆ D , and we defined DR/k = ∪m=1DR/k ⊆ E to be the ring of k-linear differential 1 operators on R. We then proved that D = R ⊕ DerR/k, where the second component consists of all k-linear derivations of R (i.e., the set of all k-linear maps T : R → R such that T (fg) = fT (g)+ gT (f) for every f, g ∈ R. We also saw that when R = k[x1, . . . , xn] and k an arbitrary commutative P ring, then DerR/k = ⊕R∂i. The isomorphism is the map ∆ 7→ i ∆(xi)∂i. We concluded the lecture by previewing the connection between RR/k and the Weyl algebra khx1, . . . , xn, ∂1, . . . , ∂ni when k is a field of characteristic zero.

Exercises Review your notes.

Lecture 5 (Tuesday, February 16) KU has cancelled all classes due to the weather. Stay warm!

Lecture 4 (Thursday, February 11) As an application of the basic properties of the Bernstein degree function that we derived last time, we started off by proving that An is a domain, and simple (i.e., has no two nonzero, proper two-sided ideals). We then had a brief discussion concerning the left/right ideal theory of An; see the Follow-up at the end of this entry for more information. The rest of the lecture was dedicated to studying and defining the notion of order. After gathering like terms in its canonical representation, every element of An can be written uniquely in the form P v P = pv∂ with each pv ∈ R = k[x1, . . . , xn], and the order of such an element is the maximum of all kvk with pv 6= 0. Using this definition, we proved that if P ∈ An and n ≥ 1, then P has order at most n if and only if [P, f] has order at most n−1 for every f ∈ R. This suggests an inductive, and purely algebraic, definition of order. We elaborated on this, and presented Grothendieck’s definition of the ring of differential operators associated to an arbitrary inclusion A ⊆ R of commutative rings. 9

Follow-up This is meant to address some questions that came up after class. Recall that the left ideals (modules) of any correspond to the right ideals (modules) of its . Furthermore, it is not too hard to see that the Weyl algebra is isomorphic to its opposite ring (see the exercises). So, any statement concerning left ideals of An (e.g., Stafford’s result that every left ideal of An can be generated by at most two elements) has a corresponding statement for right ideals of An. Soon (in the next lecture), we will consider a general definition of differential operators that agrees with the Weyl algebra for polynomial rings over fields of characteristic zero. In the general case, there are examples of rings of differential operators D with D =6∼ Dopp. In these cases, it is possible that the set of all left ideals of D may quite different from the set of all right ideals of D. For an example of this kind of behavior, see this. For some other examples in which D =∼ Dopp, see this. We will elaborate more on this later.

Exercises ∼ opp • Prove that there is a ring isomorphism A1 = A1 . Hint: If khx, yi is the noncommutative polynomial ring (i.e., the free algebra) in the variables x, y over k, then to define a map opp khx, yi → A1 , it suffices to pick targets for x and y. Where must a monomial map to? • Coutinho Chapter 1 4.10: As pointed out by Mark, the matrices in this problem are not elements of the ring M∞(K) defined in the previous problem. The problem is that the definition of M∞(K) is not the appropriate one for this context (for instance, it doesn’t have an identity element). Instead, we want to define M∞(K) to be the set of all matrices that encode K-linear transformations V → V , where V is a K-vector space with a countably infinite basis. That is, let M∞(K) be the set of all “matrices” with countably many rows and columns such that each column has only finitely many nonzero entries. Convince yourself ∼ that there is an isomorphism M∞(K) = EndK (V ) of K-vector spaces, and that matrix multiplication in this context corresponds to composition of linear transformations. • Coutinho Chapter 2: 4.1, 4.2

Lecture 3 (Tuesday, February 9) We started by recalling the basics of the canonical basis for the Weyl algebra. We then proved the following theorem, which provided a purely algebraic description for this algebra. Theorem: Let S = khy1, . . . , yn, z1, . . . , zni be the non-commutative polynomial ring over a field k of characteristic zero on the displayed variables. If I is the two-sided ideal of S generated by all non-commutative polynomials [yi, yj], [zi, zj], [zi, yj] − δij, then the ring map ∼ S → An defined by α 7→ α for every α ∈ k, yi 7→ xi and zi 7→ ∂i induces an isomorphism S/I = An. Note: This shows that the fundamental relations are enough to describe all of the relations on An. After this, we returned to discussing the (Bernstein) degree function on An. Recall that the degree of an operator is the maximum over all ku + vk such that xu∂v appears (with non-zero coefficient) in its canonical form. We then proved the Theorem: If P,Q ∈ An then (1) deg(P + Q) ≤ max{deg(P ), deg(Q)}, with equality if deg(P ) 6= deg(Q); (2) deg(PQ) = deg(P ) deg(Q); (3) deg([P,Q]) ≤ deg(P ) + deg(Q) − 2.

Exercises Coutinho: Chapter 1: 4.8, 4.9, 4.10

Lecture 2 (Thursday, February 4) Let k be a field. Consider R = k[x1, . . . , xn], and let E = Endk(R) be the (non-commutative) ring of all k-linear transformations of R. We described the 10 natural injective ring map R,→ E that sends a polynomial f ∈ R to the function [g 7→ fg], i.e., the linear transformation given by multiplication by f; as is often done in the literature, we also denote this linear transformation by f. We then defined the formal derivative ∂ = d : R → R, i dxi and defined the Weyl algebra An to be the of E generated by k, the image of R in E, and the all of the partial derivatives. That is, An is the smallest subring of E containing k, the image u v n of R, and all partial derivatives. We defined C = {x ∂ : u, v ∈ N }, which is called the canonical basis of An. To justify this terminology, we proved the Theorem: The canonical basis C is a P u v k-basis for An. That is, every element of An can be uniquely expressed as αu,vx ∂ . We then used the canonical basis to define the degree of an element of the Weyl algebra.

Exercises • Prove the product rule for the formal partial derivates. • Coutinho: Chapter 1: 4.1, 4.2, 4.3, 4.5, 4.6

Lecture 1 (Tuesday, February 2) After student introductions, we went over the syllabus, and then recalled the definition of a linear ODE with polynomial coefficients. Motivated by this, we defined dn d a differential operator on R[x] to be any expression of the form an dxn + ··· + a1 dx + a0 where each ai ∈ R[x]. We discussed how to interpret such an expression as an R-linear transformation on the R-vector space R[x]. The set of all such linear transformations forms a non-commutative ring, with operations being addition and composition of functions. We discussed how to multiply two expressions of this form to obtain another such expression, and concluded that the set of all such operators forms a non-commutative subring of the R-linear endomorphism ring of R[x].