Differential operators on reduced monomial rings

U.U.D.M. Project Report 2016:23

Examensarbete i matematik, 15 hp

Handledare: Ketil Tveiten

Examinator: Jörgen Östensson

Juni 2016

Differential operators on reduced monomial rings

Dierential operators on reduced

monomial rings

Contents

1 Introduction 3

1.1 Some notation . . . 3 1.2 The Weyl algebra . . . 3

2 Monomial ideals 8

2.1 Decomposition of ideals . . . 11

3 Calculation of Dk(R) 13

3.1 The characterizing theorem . . . 13 3.2 Some examples . . . 14

4 Visualization through simplicial complexes 17

1 Introduction

Throughout, if nothing else is stated, let k be a eld of characteristic zero. By k[x1, . . . , xn], we mean the ring of polynomials in the variables x1, . . . , xn with coecients in the eld k. By R/I we mean the quotient ring of R by the ideal I. Sometimes we will also write

R

I to denote the same quotient.

By I = (p) we mean that the ideal I is generated by the polynomial (p).

For two operators A, B, [A, B] denotes the commutator A · B − B · A. Note that [A, B] is itself an operator.

For two sets C and D, we denote by CD the set {c · d| c ∈ C, d ∈ D}.

1.2 The Weyl algebra

Denition 1.1. [1] The n-th Weyl algebra Wn is the ring generated by x1, . . . , xn, ∂x1, . . . ∂xn. We write Wn= khx1, . . . , xn, ∂x1, . . . , ∂xni.

The generators of Wn satisfy the commutator relations [∂xi, xj] =  1 if i = j 0 otherwise [∂xi, ∂xj] = 0 [xi, xj] = 0

Note that, at this point, the generators xi, ∂xi, are just symbols. Here, the Weyl algebra

is to be viewed abstractly, simply consisting of linear combinations over k of monomials in the generators x1, . . . , xn, ∂x1, . . . , ∂xn, satisfying the above relations.

Now, we want to instead view the generators of the Weyl algebra as operators acting on f ∈ k[x1, . . . , xn]. We dene the operators x1, . . . , xn as acting by multiplication, that is xi(f ) = xi· f. We dene the operators ∂x1, . . . , ∂xn as acting by partial dierentiation

with respect to the indexed variable, that is, ∂xi(f ) =

∂f ∂xi.

Remark 1.2. It is important to note the distinction between multiplication in the ring Wn, and the action of elements in Wnupon elements of k[x1, . . . , xn]. Therefore, to avoid ambiguity, we introduce the symbol • to denote action of Wn upon k[x1, . . . , xn]. For example, let f ∈ k[x]. Then ∂x • f (x) will denote action of the operator ∂x on f, so we have ∂x• f = _dxdf, whereas ∂x· f or just ∂xf will denote the product of the two operators ∂x, f in the Weyl algebra.

Remark 1.3. Consider the operator ∂xi· xi applied to a polynomial f ∈ k[x1, . . . , xn]. By

the product rule of dierentiation we have (∂xixi)•f = xi∂xi•f +f ∂xi•xi = xi∂xi•f +f.

If we let 1 denote the identity operator, we have ∂xixi = xi∂xi+ 1

or, in terms of the commutator, [∂xi, xi] = 1.

This can be taken as motivation as to why we demand the previously ungrounded be-havior of the commutators of generators when dening the Weyl algebra.

Proposition 1.4. The commutator is linear in both arguments. It is also alternating. Proof.

[A, B] = AB − BA = −(BA − AB) = −[B, A]

[A, αB+βC] = A(αB+βC)−(αB+βC)A = αAB+βAC−αBA−βCA = α[A, B]+β[A, C] Linearity in the second argument together with the alternating property gives linearity in the rst argument as follows

[αA + βB, C] = −[C, αA + βB] = −α[C, A] − β[C, B] = α[A, C] + β[B, C].

This, in fact, reveals the commutator to be what with fancy language is called an alter-nating bilinear form.

Denition 1.5. [2] Let R = k[x1, . . . , xn]/I be the quotient of k[x1, . . . , xn] by some ideal I. Then we dene the ring of dierential operators over R as

Dk(R) :=

{δ ∈ W_n | δ • I ⊂ I} I · Wn

.

The action of the operators of Dk(R) on R is well dened, as is shown by the following proposition.

Proposition 1.7. Let R = k[x1, . . . , xn]/I. Then Dk(R) • R ⊂ R.

Proof. Consider an arbitrary element of Dk(R). It has the form (δ + I · Wn).

We wish to show that this, acting upon an arbitrary element of R produces an element in R. An arbitrary element in R has the form (p + I) for some p ∈ k[x1, . . . , xn].

We then have:

(δ + IWn) • (p + I) = δ • p + δ • I + IWn• p + IWn• I We then observe the following:

1 δ • p ∈ k[x1, . . . , xn]

2 δ • I ∈ I, by denition of Dk(R). 3 IWn• p ∈ I

4 IWn• I ∈ I So, in fact, we have

δ • p + δ • I + IWn• p + IWn• I = δ • p + I ∈ R

This denition of the ring of dierential operators, Dk(R), can be viewed as gener-alizing the action of the Weyl algebra upon k[x1, . . . , xn], in the sense that Wn = Dk(k[x1, . . . , xn]) is the special case when I = (0). This fact is shown by proposition 1.8.

Proposition 1.8. Let R = k[x1, . . . , xn]/I and I = (0), then Dk(R) = Wn. Proof. If I = (0), we have I · Wn= {0}.

This means that Dk(R) =

{δ ∈ W_n | δ • (0) ⊂ (0)} (0) · Wn

= {δ ∈ Wn | δ • (0) ⊂ (0)} = Wn

since we have δ • 0 = 0 for every δ ∈ Wn.

Proposition 1.9. Let f ∈ k[x1, . . . , xn]be a polynomial . Then [∂xi, f ] =

∂f

For this proposition to make sense, it is important to note remark 1.6, which says that elements of k[x1, . . . , xn]can be thought of as elements of Wn.

Since ∂xi commutes with every xj with i 6= j, every such term can be moved out of the

commutator. This reduces the proof to the single variable-case. Before we give the proof of proposition 1.9, we state and prove a lemma that will be useful in the proof.

Lemma 1.10. [∂x, xk] = kxk−1 and [x, ∂xk] = −k∂xk−1. Proof of Lemma. We prove this by induction. Recall that

[∂x, x] = 1 ⇔ ∂xx − x∂x = 1. We then have that

[∂x, xk+1] = ∂xxk+1− xk+1∂x = ∂xxk+1− xk(∂xx − 1) = ∂xxk+1− xk∂xx + xk = [∂x, xk]x + xk = kxk−1· x + xk = (k + 1)xk

The second statement is shown using the same technique. Proof of Proposition 1.9. We have

f =X

k∈N ckxk so, for the commutator we have

[∂x, f ] = [∂x, X

k∈N ckxk] now, by proposition 1.4, we get

Remark 1.11. We consider elements of Wn that are of the form δ = P cabxa∂b to be in normal form. Here, a, b are multi-indices, that is

a = (a1, . . . , an), b = (b1, . . . , bn) ∈ Nn.

From the identities above, it is clear that any element not in normal form can be com-mutated around until every x is on the left and every ∂x is on the right.

Denition 1.12. We dene the operator ∇x := x∂x.

The next proposition is simply a collection of a few identities involving the operator ∇_x.

Proposition 1.13. Let ∇x be the operator dened above. Then the following identities hold: i) [∇x, xk] = kxk ii) [∇x, ∂xk] = −k∂xk iii) [∇x, xa∂xb] = (a − b)xa∂xb iv) (∇x− 1)x = x∇x v) xk_∂k x = ∇x(∇x− 1) . . . (∇x− (k − 1)) Proof. i) Using lemma 1.10

[∇x, xk] = x∂xxk− xkx∂x = x(∂xxk− xk∂x) = x[∂x, xk] = xkxk−1 = kxk ii) Using lemma 1.10

[∇x, ∂xk] = x∂xk+1− ∂xkx∂x = (x∂_xk− ∂k xx)∂x = [x, ∂_xk]∂x = −k∂_xk−1∂x = −k∂_xk iii) Using i) and ii), we get

and x∂x∂xk− ∂xkx∂x= −k∂xk⇔ ∂xkx∂x = x∂xk+1+ k∂kx. Now we have [∇x, xa∂xb] = x∂xxa∂xb− xa∂xbx∂x = (x∂xxa)∂xb − xa(∂xbx∂x) = (axa+ xa+1∂x)∂xb− xa(x∂xb+1+ b∂xb) = axa∂_xb + xa+1∂_xb+1− xa+1_∂b+1 x − bxa∂xb = (a − b)xa∂_xb

iv) Using [∇x, x] = x, that is, ∇xx − x∇x= x, we have that (∇x− 1)x = ∇xx − x

= (x∇x+ x) − x = x∇x

v) We use induction over k. First, we note that

∇x(∇x− 1) . . . (∇x− (k − 1))(∇x− k) = xk∂xk(∇x− k) by hypothesis. By lemma 1.10 xk∂k_x(∇x− k) = xk∂xkx∂x− kxk∂xk = xk(x∂_xk+ k∂_xk−1)∂x− kxk∂kx = (xk+1∂k_x+ kxk∂_xk−1)∂x− kxk∂xk = xk+1∂_xk+1+ kxk∂_xk− kxk∂_xk = xk+1∂_xk+1

2 Monomial ideals

Next, we aim to explicitly calculate the ring Dk(R) = {δ∈Wn_I·W| δ•I⊂I}_n for some rings of the form R = k[x1, . . . , xn]/I. Before we are equipped to do this, we need some additional theory.

Example 2.2. In R = k[x, y], the ideal I1 = (x + y) is not a monomial ideal, while I2 = (xy)is. Furthermore, I2 is reduced.

Denition 2.3. An ideal I 6= k[x1, . . . , xn]in k[x1, . . . , xn]is said to be prime if, for all f, g ∈ k[x1, . . . , xn], fg ∈ I ⇒ f ∈ I or g ∈ I.

To further our understanding of prime monomial ideals in k[x1, . . . , xn], we have the following result.

Proposition 2.4. Let M = (mi)be a monomial ideal, generated by the monomials mi, in k[x1, . . . , xn]. Then M is prime if and only if M is generated by a subset of the variables x1, . . . , xn.

Before we give the proof, we state and prove the following lemma.

Lemma 2.5. Let M = (mi) be a monomial ideal. Then we may assume that for every generator mi of M, no proper divisor of mi is in M.

Proof of Lemma. First, we note that every mi has nite degree. Thus, every mi has nitely many proper divisors. Let {di,j} denote the set of proper divisors of mi which are contained in M. Then we have

(di,j) ⊂ M = (mi) ⊂ (di,j) which gives us M = (di,j)

Proof of Proposition 2.4. Let M = (mi) be a prime monomial ideal. Let S be the sub-set of the variables x1, . . . , xn consisting of the variables which divide some mi. By our lemma, we may assume that no mi has a proper divisor in M. Now we want to show that M is generated by S, that is M = (S). The inclusion ⊂ is obvious.

Now, consider an xk∈ S. Since xk divides some mk, we have mk= xkqk. Now, since the ideal M is prime, either xk ∈ M or qk ∈ M. But by assumption, M contains no proper divisors of mk.

Thus, xk can not be a proper divisor. So instead, it follows that qk is in fact a unit and we have xk∈ M. Now we have (S) ⊂ M, and we have M = (S).

Conversely, consider some subset of the variables x1, . . . , xn and the ideal generated by it. If necessary, relabel the variables so that we get a set of the form S = {x1, . . . , xk}. Then we consider the quotient k[x1, . . . , xn]/(S) ∼= k[xk+1, . . . , xn] which is a domain. Thus, (S) is a prime ideal, since the quotient k[x1, . . . , xn]/S is a domain if and only if (S)is prime.

Denition 2.6. The radical of an idea I in k[x1, . . . , xn], denoted by √

I, is the set √

Denition 2.7. A ring R is called reduced if it has no non-zero nilpotent elements. That is, for every r ∈ R, if there exists a positive integer n such that rn_{= 0, we have r = 0.} Remark 2.8. Note that for a quotient ring R/I, the denition means that for every r ∈ R/I, if there exists a positive integer such that rn_{∈ I, we have r ∈ I.}

Proposition 2.9. Let R/I be a quotient ring. Then R/I is reduced if and only if the ideal I is radical.

Proof. `⇒': Suppose R/I is reduced and consider x ∈√I, that is, x is such that xn∈ I for some n. Then we have

I = xn+ I = (x + I)n⇒ x + I = I ⇔ x ∈ I since R/I is reduced. So x ∈ I and I is radical.

`⇐': Suppose I is radical. Consider x + I ∈ I such that (x + I)n_{= I. Then we have} (x + I)n= I ⇒ xn+ I = I ⇒ xn∈ I ⇒ x ∈ I ⇒ x + I = I

since I is radical. So R/I is reduced. Proposition 2.10. Every prime ideal is radical.

Lemma 2.11. Let f ∈ k[x1, . . . , xn] and let P be a prime ideal. Then, for a positive integer n, fn_{∈ P ⇒ f ∈ P.}

Proof of Lemma 2.11. We use induction over n. Then we have fn+1= f · fn∈ P ⇒ f ∈ P or fn∈ P so in either case we get f ∈ P since fn_{∈ P ⇒ f ∈ P by assumption.}

Proof of Proposition 2.10. Suppose f ∈ p(P ). Then fn_{∈ P for some n. By the lemma,} we have f ∈ P . So we have p(P ) ⊂ P . The converse inclusion is obvious.

Proposition 2.12. Let M = (mi) be a monomial ideal. Then M is generated by square-free monomials if and only if M is radical.

Proof. 0 _⇒0_{: By lemma 2.5 we may assume that every generator m}

i of M has no proper divisors in M. Suppose that some mkis not squarefree, that is mk= xqk11·, . . . , ·x

qr

kr

with some qs ≥ 2. Then p = xk1·, . . . , ·xkr ∈

√

M since pmax{qi} _{∈ M}. It is clear

that p | mk, and since qs≥ 2, p is a proper divisor of mk. Since by assumption, no proper divisor of mk is in M, we have p 6∈ M ⇒

√

M 6= M, so M is not radical. 0 _⇐0_{: Suppose every m}

i is squarefree. We want to show that M is radical, that is √

Since any polynomial p with pn _{∈ M can be written as a sum of the monomial} generators of M, it suces to show that this is the case for monomials. So suppose that p is a monomial such that pn_{∈ M for some n. Then we have p}n_{= x}α_{Q m}ki

i , where α is some multiindex and each mi is a generator. Since every exponent in pn is divisible by n, every exponent in xα as well as in Q mki

i must be too. Let g be such that gn_{= x}α _{and put c}

i= k_ni. Then we have pn= gnYmnci i ⇒ p = g Y mci i

So we have p as a product of generators and can conclude p ∈ M.

2.1 Decomposition of ideals

Analogously to prime ideals, one can dene the notion of a primary ideal as follows. Denition 2.13. An ideal I is said to be primary if for every xy ∈ I we have x ∈ I or yn_{∈ I for some positive integer n.}

Remark 2.14. It is a famous result that in a Noetherian ring, every ideal has a primary decomposition. That is, every ideal can be written as a nite intersection of primary ideals.

It is clear that the notion of a primary ideal is slightly weaker than that of a prime ideal. In the case of radical monomial ideals, we have the stronger result that every radical monomial ideal can be written as a nite intersection of prime ideals. That is, for every radical monomial ideal I we have

I = n \

i=1 Pi. where every Pi is a prime ideal.

Example 2.15. It is clear that we can write the ideal I = (xy) as I = (x) ∩ (y). We also see that these ideals are prime, by proposition 2.4.

It is almost as clear that we can write the ideal J = (xy, xz) as J = (x) ∩ (y, z). We again see that these are prime by proposition 2.4.

Ordering the set {Pi | 1 ≤ i ≤ n} by inclusion forms a partial order. The minimal elements of this partial order are called the minimal primes of the ideal I. Since by proposition 2.9 I is radical if and only if the quotient R/I is reduced, it makes sense to talk about the minimal primes over a reduced monomial ring.

Proposition 2.16. The minimal primes over a monomial ideal I = ({xα_{}) are those of} the form P = (xi1, . . . , xir) satisfying

1) Every minimal generator xα _{of I is divisible by some x} ij.

2) For each xij, there exists a minimal generator x

α _{such that x}

ij divides x

α _{and no} other xik divides x

α_.

Remark 2.17. We easily make the observation that the decomposition of the ideals given in example 2.15 agrees with the above result.

By proposition 2.4, we know that every prime monomial ideal is generated by a subset of the variables, i.e. is of the form in the statement of proposition 2.16.

Proof of proposition 2.16. For some monomial ideal I = ({xα_{})we consider a prime ideal} Pi containing I, that is, a prime ideal such that I ⊂ Pi.

We know Pito be of the form Pi= (xi1, . . . , xir). We now wish to show that the condition

1)is satised if and only if I ⊂ Pi.

⇒ Contrapositively, suppose that there exists a generator xα _{= x}α1

1 ·, . . . , ·xαnn of I which is not divisible by any xij. Then we must have αij = 0 for every index ij

appearing in the generators of Pi. This means that we have a generator xα without factors xi1, . . . xir, implying x

α_{∈ P/}

i. But then we have I 6⊂ Pi. ⇐ Conversely, suppose every generator xα _{is divisible by some x}

ij. Then every

gen-erator xα _{is a multiple of a generator of P}

i, so we have xα ∈ Pi for every generator of Pi, implying I ⊂ Pi.

Thus far we have established that a prime ideal P contains I = ({xα_{})if and only if the} condition 1) is satised.

Next, we wish to show that such a prime ideal containing I is minimal (with respect to inclusion) if and only if the condition 2) is satised.

⇒ Contrapositively, suppose that there exists a generator xα_{with at least two divisors} among the generators of Pi. That is xir | x

α _{and x} ik | x

α _{for some r 6= k.}

Simply dropping the generator xik from Pi = (xi1, . . . , xir, xik) gives the prime

ideal Pj = (xi1, . . . , xir). We obviously have Pj ⊂ Pi. It is easily seen that the

ideal Pj must contain I since the condition 1) will be satised. So in fact, xik is

redundant with regard to guaranteeing xα _{∈ P}

i. So we have I ⊂ Pj ⊂ Pi, where it is important to note that the inclusion Pj ⊂ Piis strict, meaning Piis not minimal. ⇐ Conversely, suppose the condition 2) is satised and that I ⊂ Pi = (xi1, . . . , xir).

Let I be generated by the monomials xαj. We now have that every generator xαj

is divisible by a unique generator of Pi. We can relabel the variables so that xi1 is

the unique generator dividing xα1_{, x}

i2 is the unique generator dividng x

α2, and so

We want to conclude that Pi is minimal. Supose that it is not. Then, there must exists an ideal P0

i strictly contained in Pi which contains I. For P 0 i to be strictly contained in Pi, it must have (at least) one less generator by proposition 2.4. Suppose this is the generator xik.

By condition 2) no other xij divides x

αk. But this means that when we consider

condition 1) for the ideal P0

i, we nd that it is not satised, since we have a generator, namely xαk, which is not divisible by any generator of P0

i. This means, by the above, that we can not have I ⊂ P0

i. Hence, Pi must be minimal.

We have now established that a given prime ideal is a minimal prime of the monomial ideal I = ({xα_{})if and only if the conditions of proposition 2.16 are satised.}

3 Calculation of D

(R)

3.1 The characterizing theorem

The following theorem, also stated in [2], will be the cherry on top of the theory we have developed thus far, characterizing the elements of the Weyl algebra which are in Dk(R) when R is a reduced monomial ring. This will enable us to explicitly calculate the ring of dierential operators for a given reduced monomial ring.

Theorem 3.1 (Traves). Let R = k[x1, . . . , xn]/I be a reduced monomial ring. An ele-ment δ = xα_∂β _{∈ W}

n is in Dk(R) if and only if for each minimal prime P of R, we have either xα_{∈ P or x}β _{∈ P/ .}

In particular, Dk(R) is generated as a k-algebra by the xα∂β satisfying the above condi-tion.

Proof. ⇒ Suppose xα_∂β _{∈ D}

k(R) and let P be a minimal prime of the ideal I. We can relabel the variables so that we get P = (x1, . . . , xr). If xβ ∈ P/ , we are done. Suppose therefore, that xβ _{∈ P. Then we have x}β _{· x}

r+1·, . . . , ·xn∈ I. Since xα_∂β _{∈ D}

k(R), xα∂β preserves the ideal I, so we have

(xα∂β) • xβ· x_r+1·, . . . , ·x_n= nxα· x_r+1·, . . . , ·x_n∈ I for some n ∈ Z. It is clear that xr+1·, . . . , ·xn∈ P/ . Since P is a prime ideal, we must have xα∈ P. ⇐ Suppose that xα_∂β _{∈ W}

We claim that then, we must have ∂β_{• x}γ_{∈ P. To see this, we suppose ∂}β_{• x}γ _{∈ P/} and aim for a contradiction.

If ∂β _{• x}γ _{∈ P/ we have, in particular ∂}β_{• x}γ _{6= 0 ⇒ β}

i ≤ γi for the multi-indices β, γ. This gives us xβ | xγ_{⇔ x}γ_{= x}β_xγ 0_{. But then we have x}β_(∂β_{• x}γ_{) = x}β_(∂β_• (xβxγ 0)) = nxβxγ 0 = nxγ ∈ P for some n ∈ Z. This is our desired contradiction, since we have assumed both xβ _{∈ P/ and ∂}β_{• x}γ _{∈ P/ .}

So we must have ∂β_{• x}γ _{∈ P ⇒ x}α_∂β_{• x}γ_{∈ P.}

Since we have now established that xα_∂β _{preserves every minimal prime P of I, it} is clear that it must also preserve I itself.

3.2 Some examples

Next, we will use the theory we've developed thus far to explicitly calculate the ring of dierential operators, Dk(R), for some given rings. To do this, we will use the following helpful lemma.

Lemma 3.2. Let R = k[x1, . . . , xn]/Ibe a reduced monomial ring and let xa11·, . . . , ·xann∂α∈ Dk(R). Then we have

xa1

1 ∂α ∈ Dk(R) ∨ · · · ∨ xann∂α ∈ Dk(R). Proof. For xa1

1 ·, . . . , ·xann∂α ∈ Dk(R) we have x1a1·, . . . , ·xann ∈ P ∨ xα ∈ P/ for every minimal prime P of I by proposition 3.1. If xα _{∈ P/ , we clearly have x}

i∂α ∈ Dk(R)∀i, also by proposition 3.1. Suppose instead xa1

1 ·, . . . , ·xann ∈ P. Then we have (xa1 1 ) ∈ P ∨ (x a2 2 ·, . . . , ·x an n ) ∈ P since P is prime. If xa1

1 ∈ P we are done. If not, repeat the same argument for the product xa2

2 ·, . . . , ·xann. This shows xaii ∈ P for at least one i. For such i, we now get xai

i ∂α ∈ Dk(R) by the condition of proposition 3.1. Example 3.3. We consider the ring R = k[x, y]/(xy).

The ideal (xy) is generated by a squarefree monomial, so it is radical. Since it is radical, the ring R is reduced, so we can apply proposition 3.1. Using proposition 2.16, we nd the minimal primes of (xy) as (x), (y). Since (xy) is radical, it is the intersection of its minimal primes. Hence, we have (xy) = (x) ∩ (y). An arbitrary element of the Weyl algebra can be written as δ = xa_yb_∂c

x∂yd.

We have δ ∈ Dk(R) if and only if xayb ∈ P or xcyd∈ P/ for each minimal prime of (xy). We write this condition as

Now, the δ satisfying the conditions of proposition 3.1 are given by the dierent combi-nations of the above conditions.

a ≥ 1, b ≥ 1gives δ1 = xmyn∈ I since m, n ≥ 1. a ≥ 1, d = 0 gives δ2 = xm∂xn, m ≥ 1, n ≥ 0. c = 0, b ≥ 1 gives δ3 = ym∂yn, m ≥ 1, n ≥ 0. c = 0, d = 0gives δ4 = xmyn∈ R since m, n ≥ 0.

For δ2, δ3 we have δ2 = xm∂xn= xm−1x∂xn and δ3= ym∂yn= ym−1y∂yn. To get a minimal set of generators of Dk(R), we observe the following

xm−1 ∈ R ⊂ k[x, y] (xy)W2 ⊂ Dk(R) ym−1 ∈ R ⊂ k[x, y] (xy)W2 ⊂ D_k(R) and conclude that we can reduce the generators xm_∂n

x to x∂xn and ym∂yn to y∂yn. Using this, we get

Dk(R) =

khx, y, x∂_xk, y∂_yk (k ≥ 1)i (xy)W2

For a relatively uncomplicated or small ring, like the one in this example, this method of computing Dk(R) is perfectly viable. However, it is not hard to realize that if we were dealing with a larger ring, we would quickly drown in cases of cases of hideous calculations. To spare ourselves, we introduce a dierent method of nding the generators of Dk(R)via the use of a clever table and lemma 3.2.

The table we want to use is the following.

1 xk yk xmyn 1

∂_xk ∂_yk ∂_xm∂_yn

Here, each cell represents a monomial of the form xα_∂β_{in the Weyl algebra. For example,} the cell row 2, column 2 represents xm_∂n

y. It is clear that the table is exhaustive with respect to possible monomials in the Weyl algebra. Now, for each monomial, represented by a cell of the table, we check if xα _{∈ P ∨ x}β _{∈ P/ for each minimal prime P of I and} then use proposition 3.1 to determine whether it is in Dk(R) or not. If it is, we check the cell. If not, we enter into the cell the minimal prime(s) for which the condition of proposition 3.1 does not hold. In the end, the checked cells will represent the generators of Dk(R).

1 xk _yk 1 X X X ∂_xk (x) X (x) ∂_yk (y) (y) X ∂m x ∂yn (x), (y) (y) (x)

This gives us the generators of Dk(R) as xk, yk, xm∂xn, ym∂yn. To get a minimal set of generators, we use the same argument as above to get

Dk(R) =

khx, y, x∂_xk, y∂_yk (k ≥ 1)i (xy)W2

Example 3.4. We consider the ring R = k[x, y, z, w]/(xz, xw, yw).

The ideal I = (xz, xw, yw) is generated by squarefree monomials, so it is radical. Since it is radical, the ring R is reduced. We nd the minimal primes of I as (x, y), (x, w), (z, w) using proposition 2.16. Since I is radical, it is the intersection of its minimal primes. So we have

(xz, xw, yw) = (x, y) ∩ (x, w) ∩ (z, w).

We then have Dk(R) =

hx, y, z, w, x∂m

x , y∂ym, z∂zm, w∂wm, x∂my , w∂mz , x∂xm∂yn, w∂zm∂wn (m, n ≥ 1)i (xz, xw, yw) · W4

Example 3.5. We consider the ring R = k[x, y, z, u, v]/(xz, xv, uz). The ideal I = (xz, xv, uz)is generated by squarefree monomials, so it is radical. Since it is radical, the ring is reduced. We nd the minimal primes as (x, u), (x, z), (z, v). Since the ideal is radical, it is the intersection of its minimal primes, so we have

(xz, xv, uz) = (x, u) ∩ (x, z) ∩ (z, v).

We again use the table method to nd the generators of Dk(R), see Table 1.

Now, the checked cells represent the generators of Dk(R). Using the same argument as before, we can the reduce the generators to get a minimal generating set. Then we have

Dk(R) =

khx, y, z, u, v, ∂_ym, z∂m_z , x∂_um, u∂_um, z∂_vm, v∂m_v , z∂_ym∂_zn, x∂_ym∂n_u, u∂_ym∂_un, z∂_ym∂_vn, v∂m_y ∂_vn, z∂_ym∂_vn, x∂_xl∂_ym∂_zn, z∂_yl∂_zm∂n_v (l, m, n ≥ 1)i

(xz, xv, uz) · W5

4 Visualization through simplicial complexes

Even though the table method used in the examples above made calculation of Dk(R) easier for larger rings, it is still rather unpleasant and suboptimal. Luckily, there exists an even neater way to calculate Dk(R). This is done by using simplicial complexes to visualize reduced monomial rings.

4.1 Denitions and properties

Denition 4.1. Let ∆ be a family of subsets of the set {x1, . . . , xn}. Then ∆ is an abstract simplicial complex on {x1, . . . , xn} if, for every set A in ∆, every B ⊂ A also belongs to ∆.

Another way of saying this is that an abstract simplicial complex is a family of sets that is closed under taking subsets. The sets belonging to ∆ are called the faces of ∆. If σ is a face of ∆ and τ ⊂ σ, then the face τ is said to belong to the face σ, so every face of a face of a complex ∆ is itself a face of ∆.

1 xk _yk _zk _uk _vk 1 _{X X X X X X} ∂k x (x, u), (x, z) X (z, v) (x, u) (x, z) (x, u), (x, z) ∂k y X X X X X X ∂k z (x, z), (z, v) (z, v) (x, z), (z, v) X (x, z), (z, v) (x, z) ∂_uk (x, u) _X (x, u) (x, u) _X (x, u) ∂k v (x, u), (x, z) (z, v) (z, v) X (z, v) X ∂m x∂ny (z, v) (z, v) (z, v) (x, u) (x, z) (x, u), (x, z)

∂m_x∂n_z (x, u), (x, z), (z, v) (x, z) (x, u), (x, z), (z, v) (x, u) (x, z), (z, v) (x, u), (x, z) ∂m

x∂nu (x, u), (x, z) (z, v) (x, u), (x, z) (x, u) (x, z) (x, u), (x, z)

∂m

x∂nv (x, u), (x, z), (z, v) (z, v) (x, u), (x, z), (z, v) (x, u) (x, z), (z, v) (x, u), (x, z)

∂m y ∂nz (x, z), (z, v) (z, v) (x, z), (z, v) X (x, z), (z, v) (x, z) ∂m y ∂nu (x, u) X (x, u) (x, u) X (x, u) ∂m y ∂nv (x, u), (x, z) (z, v) (z, v) X (z, v) X ∂m

z ∂nu (x, u), (x, z), (z, v) (z, v) (x, u), (x, z), (z, v) (x, u) (x, z), (z, v) (x, u), (x, z)

∂mz ∂nv (x, z), (z, v) (z, v) (x, z), (z, v) X (x, z), (z, v) (x, z)

∂m

u∂nv (x, u), (z, v) (x, u), (z, v) (x, u), (z, v) (x, u) (z, v) (x, u)

∂_xl∂_ym∂_zn (x, u), (x, z), (z, v) (z, v) (x, u), (x, z), (z, v) (x, u) (x, z), (z, v) (x, u), (x, z) ∂_xl∂_ym∂_un (x, u), (x, z) _X (x, u), (x, z) (x, u) (x, z) (x, u), (x, z) ∂_xl∂_ym∂_vn (x, u), (x, z), (z, v) (z, v) (x, u), (x, z), (z, v) (x, u) (x, z), (z, v) (x, u), (x, z) ∂_xl∂_zm∂_un (x, u), (x, z), (z, v) (z, v) (x, u), (x, z), (z, v) (x, u) (x, z), (z, v) (x, u), (x, z) ∂l

x∂zm∂vn (x, u), (x, z), (z, v) (z, v) (x, u), (x, z), (z, v) (x, u) (x, z), (z, v) (x, u), (x, z)

∂l

x∂um∂vn (x, u), (x, z), (z, v) (z, v) (x, u), (x, z), (z, v) (x, u) (x, z), (z, v) (x, u), (x, z)

∂_yl∂_zm∂_un (x, u), (x, z), (z, v) (z, v) (x, u), (x, z), (z, v) (x, u) (x, z), (z, v) (x, u), (x, z) ∂_yl∂_zm∂_vn (x, z), (z, v) (z, v) (x, z), (z, v) _X (x, z), (z, v) (x, z) ∂l

y∂um∂vn (x, u), (z, v) (x, u), (z, v) (x, u), (z, v) (x, u) (z, v) (x, u)

∂l

z∂um∂vn (x, u), (x, z), (z, v) (z, v) (x, u), (x, z), (z, v) (x, u) (x, z), (z, v) (x, u), (x, z)

∂xk∂yl∂zm∂un (x, u), (x, z), (z, v) (z, v) (x, u), (x, z), (z, v) (x, u) (x, z), (z, v) (x, u), (x, z)

∂xk∂yl∂zm∂vn (x, u), (x, z), (z, v) (z, v) (x, u), (x, z), (z, v) (x, u) (x, z), (z, v) (x, u), (x, z)

∂xk∂yl∂um∂vn (x, u), (x, z), (z, v) (z, v) (x, u), (x, z), (z, v) (x, u) (x, z), (z, v) (x, u), (x, z)

∂xk∂zl∂um∂vn (x, u), (x, z), (z, v) (z, v) (x, u), (x, z), (z, v) (x, u) (x, z), (z, v) (x, u), (x, z)

∂k

y∂zl∂um∂vn (x, u), (x, z), (z, v) (z, v) (x, u), (x, z), (z, v) (x, u) (x, z), (z, v) (x, u), (x, z)

∂j

x∂yk∂zl∂um∂vn (x, u), (x, z), (z, v) (z, v) (x, u), (x, z), (z, v) (x, u) (x, z), (z, v) (x, u), (x, z)

x y

z w

Figure 1: The simplicial complex described in example 4.6.

x

y

z

u v

Figure 2: The associated simplical complex of the ring k[x, y, z, u, v]/(xz, xv, uz) from example 3.5.

Denition 4.2. The vertex set of an abstract simplicial complex ∆ is the set V (∆) = ∪∆, that is, the union of all faces of ∆. The elements of the vertex set are called vertices of the complex. Note that for every vertex x of ∆, the set {x} is a face of ∆.

Denition 4.3. Let xa _{= x}a1

1 ·, . . . , ·xann be a monomial and ∆ a simplical complex on {x1, . . . , xn}. The support of xa in ∆, denoted supp(xa) is the set

supp(xa) = {xi|ai 6= 0}.

Denition 4.4. A subcomplex of a simplicial complex ∆ is a simplicial complex K such that every face of K belongs to ∆. In other words, K ⊂ ∆ and K is a simplicial complex. The following denition, given in [3], connects the notion of abstract simplicial complexes with that of reduced monomial rings.

Denition 4.5. Let ∆ be an abstract simplicial complex on the vertices x1, . . . , xn. The Stanley-Reisner ring of ∆ with coecients in k is the ring R∆= k[x1, . . . , xn]/I∆where I∆ = (xi1. . . xir|{xi1, . . . , xir} /∈ ∆) is the ideal of squarefree monomials corresponding

to the non-faces of ∆, called the face ideal of ∆.

Face rings are exactly the reduced monomial rings, that is, quotient rings of polynomial rings by squarefree monomial ideals.

Example 4.6. Consider the ring R = k[x, y, z, w]/(xz, xw, yw) from example 3.4. To nd the corresponding simplicial complex, we check the face ideal I∆ = (xz, xw, yw) to nd which monomials it is not generated by. We nd these to be xy, yz, zw. So the cor-responding simplicial complex on the vertices x, y, z, w has the faces {x, y}, {y, z}, {z, w} in addition to the trivial faces.

The next denition is also given in [3].

Denition 4.7. Let σ ∈ ∆ be a face. The closed star of σ in ∆ is the subcomplex st(σ, ∆) := {τ ∈ ∆|σ ∪ τ ∈ ∆}

In what follows, star we will mean closed star. The open complement of st(σ, ∆) is the set

Uσ(∆) = ∆\ st(σ, ∆) = {τ ∈ ∆|τ ∪ σ /∈ ∆}.

4.2 The characterizing theorem in terms of simplicial complexes

Now that we have established the connection between the notions of simplicial complexes and reduced monomial rings, we are ready to state and prove the theorems that will allow us to calculate Dk(R)using somplicial complexes.

The next proposition, given in [3], contains some useful properties of face ideals.

Proposition 4.8. i) If K, L are subcomplexes of ∆, then IK+ IL= IK∩L and IK∪ IL= IK∪L.

ii) Ist(σ)= (xτ|τ ∈ Uσ), where xτ =Qxi∈τxi.

iii) The minimal primes of I∆ are the face ideals Ist(σ) for the maximal simplices σ. Proof. i) The rst part is clear if we remind ourselves of the denitions. We have

IK = (xi1, . . . , xir|{xi1, . . . , xir} /∈ K) and IL = (xj1, . . . , xjr|{xj1, . . . , xjr} /∈ L).

So IK∩L= (xk1, . . . , xkr|{xk1, . . . , xkr} /∈ K ∩ L)and we see that IK∩L= IK+ IL.

IK∩ IL= IK∪L follows in a similar way. ii)

Ist(σ)= (xi1, . . . , xir|{xi1, . . . , xir} /∈ st(σ))

= (xi1, . . . , xir|{xi1, . . . , xir} ∈ Uσ)

= (xτ|τ ∈ U σ)

iii) We know that monomial ideals are prime if and only if they are generated by a subset of the variables. By ii) we also know that Ist(σ) = (xi|xi ∈ Uσ). For a maximal simplex, Uσ is a subset of the variables, so Ist(σ) is prime. We also know that every Ist(σ) is radical, since it is generated by squarefree monomials. Combining these with the second part of i), we get the result as

I∆=

[ maximal σ⊂∆

The following theorem, stated and proven in [3], characterizes the elements of the Weyl algebra which are in Dk(R)via the stars of certain faces in the simplicial complex asso-ciated to R.

Theorem 4.9. Let ∆ be a simplicial complex and R = R∆its Stanley-Reisner ring. Let xa = Q xai

i , xb = Q x bj

j be such that supp(xa) = st(σ) and supp(xb) = st(τ ) for some σ, τ ∈ ∆. Then xa∂b =Q xai

i ∂ bi

i ∈ Dk(R) if and only if st(σ) ⊂ st(τ).

Proof. Let Pxa denote the set of minimal primes of R that contain xa, and let P_¬xa

denote the set of minimal primes of R that do not contain xa_{. If we denote the set of} minimal primes of R by P, we clearly have Pxa∪ P_¬xa = P.

By proposition 4.8 we know that the minimal primes of R are the face ideals Ist(α) for the maximal simplices α. Now it is clear that Pxa is the set of ideals of the form I_st(α)

such that α is a maximal simplex and xa_{∈ I}

st(α). This is the same as the set of ideals Ist(α) such that α is maximal and α ∈ Uσ. The same argument gives us P¬xa as the set

of ideals Ist(α) such that α is a maximal simplex and α ∈ st(σ). By proposition 4.8 we now get Ist(σ)= ∩P¬xa. We then get

st(σ) ⊂ st(τ ) ⇔ Ist(σ)⊃ Ist(τ ) ⇔ P¬xa ⊃ P_¬xb

⇔ P_xa ⊂ P_xb

and combining with proposition 3.1 we then have

xa∂b ∈ D_k(R) ⇔ (xa∈ p ∨ xb ∈ p)∀p ∈ P/ ⇔ (p ∈ Pxa ∨ p ∈ P_¬xb)∀p ∈ P

⇔ P = P_xa ∪ P_¬xb

⇔ P_xa ⊂ P_xb∨ P_¬xb ⊂ P¬xa

⇔ st(σ) ⊂ st(τ )

Finally, we demonstrate proposition 4.9 by calculating Dk(R) for the same ring as in example 3.4, but with our new method.

σ st(σ) {x} {x, y} {y} {x, y, z} {z} {y, z, w} {w} {z, w} {x, y} {x, y} {y, z} {y, z} {z, w} {z, w}

Since we always have st(xi) ⊂ st(xi)we get the generators x∂xk, y∂yk, z∂zk, w∂wk in addition to the trivial generators x, y, z, w. Now we note that st(x) ⊂ st(y), st(w) ⊂ st(z), st(x) ⊂ st(x, y), st(w) ⊂ st(z, w). This gives us the additional generators x∂_yk, w∂_zk, x∂_xm∂_yn, w∂_zm∂_wn. So just like in the calculation in example 3.4 we get

Dk(R) =

hx, y, z, w, x∂m

x , y∂ym, z∂zm, w∂wm, x∂ym, w∂zm, x∂xm∂yn, w∂mz ∂wn (m, n ≥ 1)i (xz, xw, yw) · W4

. Example 4.11. Consider the ring k[x, y, z, u, v]/(xz, xv, uz) from example 3.5. The associated simplicial complex is given by Figure 2. We see that the the face ideal, corresponding to the non-faces of ∆ is I∆ = (xz, xv, uz). We also have the maximal simplices as σ1 = {x, y, u}, σ2 = {y, u, v}, σ3 = {y, z, v}, so we get the face ideals of the stars of the maximal simplices as Iσ1 = (v, z), Iσ2 = (x, z), Iσ3 = (x, u). We recognize

these ideals as the minimal primes of I∆, by example 3.5. This illustrates the connection between the face ideals of stars of maximal simplices and minimal primes of the face ideal of the entire complex given in proposition 4.8.

We get the stars of the faces of the complex by inspection of the complex given in Figure 2, so using the same technique we get, just like in example 3.5,

Dk(R) = khx, y, z, u, v, ∂_ym, z∂_zm, x∂m_u, u∂_um, z∂_vm, v∂_vm, z∂_ym∂_zn, x∂_ym∂_un, u∂_ym∂_un, z∂_ym∂_vn, v∂_ym∂_vn, z∂m_y ∂_vn, x∂_xl∂_ym∂_zn, z∂l_y∂_zm∂_vn (l, m, n ≥ 1)i (xz, xv, uz) · W5 .

References

[1] S. C. Coutinho, A primer of algebraic D-modules, London Mathematical Society Student Texts, vol. 33, Cambridge University Press, Cambridge, 1995.

[2] William N. Traves, Dierential operators on monomial rings, J. Pure Appl. Algebra 136 (1999), no. 2, 183197.