Two-sided ideals in the ring of differential operators on a Stanley-Reisner ring
aa r X i v : . [ m a t h . A C ] J u l TWO-SIDED IDEALS IN THE RING OF DIFFERENTIALOPERATORS ON A STANLEY-REISNER RING
KETIL TVEITEN
Abstract.
Let R be a Stanley-Reisner ring (that is, a reduced monomialring) with coefficients in a domain k , and K its associated simplicial complex.Also let D k ( R ) be the ring of k -linear differential operators on R . We givetwo different descriptions of the two-sided ideal structure of D k ( R ) as beingin bijection with certain well-known subcomplexes of K ; one based on explicitcomputation in the Weyl algebra, valid in any characteristic, and one validin characteristic p based on the Frobenius splitting of R . A result of Traves[Tra99] on the D k ( R ) -module structure of R is also given a new proof anddifferent interpretation using these techniques. Introduction
Rings of k -linear differential operators D k ( R ) on a k -algebra R are generallydifficult to study, even when the base ring R is well-behaved. Some descriptions of D k ( R ) are given in e.g. [Mus94] for the case of toric varieties, [Bav10a] and [Bav10b]for general smooth affine varieties (in zero and prime characteristic respectively),and [Tra99], [Tri97] and [Eri98] for Stanley-Reisner rings. Some criteria for simplic-ity of D k ( R ) exist (see [SVdB97] and [Sai07] among others), and the study of theirleft and right ideals, through the theory of D -modules, is well developed.When D k ( R ) is not simple, however, it is an interesting problem to give a de-scription of its two-sided ideals; the purpose of this paper is to do this for the caseof Stanley-Reisner rings. Every Stanley-Reisner ring is the face ring R K of a sim-plicial complex K , and we will give two different descriptions of the two-sided idealstructure of R in terms of the combinatorial structure of K ; namely the lattice ofideals is in a certain sense determined by the poset of subcomplexes of K that are stars of some face of K . The first description is based on explicit computations withmonomials in the Weyl algebra, and the second (valid only in prime characteristic)takes advantage of the Frobenius splitting of R .2. Some preliminaries
Let us fix some notation. Throughout, k is a commutative domain. K will de-note an abstract simplicial complex on vertices x , . . . , x n ; we will not distinguishbetween K as an abstract simplicial complex and its topological realization. In thecorresponding face rings (see 2.1) the indeterminate corresponding to a vertex x i will also be named x i to avoid notational clutter. Elements of K will be referred toas simplices or faces . For a face σ ∈ K , we let x σ := Q x i ∈ σ x i . R will always mean Mathematics Subject Classification.
Key words and phrases.
Rings of differential operators, Stanley-Reisner rings. a face ring R K for a simplicial complex K . We use standard multiindex notation: x a denotes x a · · · x a n n , and | a | = a + · · · a n .We briefly recall for the benefit of the reader some basics of Stanley-Reisnerrings, omitting the proofs. Definition 2.1.
Let K be an abstract simplicial complex on vertices x , . . . , x n .The Stanley-Reisner ring , or face ring , of K with coefficients in k is the ring R K = k [ x , . . . , x n ] /I K , where I K = h x i · · · x i r |{ x i , . . . , x i r } 6∈ K i is the ideal of square-free monomials corresponding to the non-faces of K , called the face ideal of K .Geometrically, R K is the coordinate ring of the cone on K , so dim R K = dim K +1 . Accordingly, when we talk about support of elements, we will refer to faces of K when strictly speaking we mean the cones on these faces. If K = ∆ n is a simplex, I K is the zero ideal, and R K is the polynomial ring in n variables. If K = K ′ ∗ K ′′ is the simplicial join of complexes K ′ and K ′′ , then R K ≃ R K ′ ⊗ k R K ′′ . Facerings are exactly the reduced monomial rings, i.e. quotients of polynomial rings bysquare-free monomial ideals.Given a simplicial complex K , we will have use for a well-known class of subsetsof K : Definition 2.2.
Let σ ∈ K be a face. The closed star of σ in K is the subcomplex st ( σ, K ) := { τ ∈ K | τ ∪ σ ∈ K } . The open star of σ in K is the set st ( σ, K ) ◦ := { τ ∈ K | σ ∪ τ ∈ K ∧ σ ∩ τ = ∅ } ; st ( σ, K ) ◦ is the interior of st ( σ, K ) in K , and st ( σ, K ) is the closure of st ( σ, K ) ◦ in K . The open complement of st ( σ, K ) is the set (not usually a subcomplex) U σ ( K ) = K \ st ( σ, K ) = { τ ∈ K | τ ∪ σ K } . Stars are important because the support of a principal monomial ideal of R K ,considered as an R K -module, is exactly equal to the open star of some face, and theclosed star is the smallest subcomplex containing it. For the remainder, we will take star to mean closed star . We will not have much need of comparing stars associatedto different subcomplexes and so will often write simply st ( σ ) , U σ if no confusion islikely to result. For completeness, we repeat a few simple facts: Lemma 2.3. (i) If σ ⊂ τ are faces in K , st ( σ, K ) ⊃ st ( τ, K ) ; (ii) If L ⊂ K is a subcomplex containing σ , st ( σ, L ) ⊂ st ( σ, K ) ; (iii) For a face σ = τ ∪ { x } , st ( σ, K ) = st ( x, st ( τ, K )) . (iv) st ( τ ) ⊂ st ( σ ) if and only if { maximal simplices in K that contain τ }⊂ { maximal simplices in K that contain σ } . (v) σ ∈ st ( τ ) ⇔ τ ∈ st ( σ ) . (vi) If σ ∪ τ is a face of K , st ( σ ) ◦ ∩ st ( τ ) ◦ = st ( σ ∪ τ ) ◦ . WO-SIDED IDEALS IN THE RING OF DIFFERENTIAL OPERATORS ON A STANLEY-REISNER RING3
Proof. ( i ) , ( ii ) and ( v ) are obvious. ( iv ) follows from the fact that a complex isdetermined by its maximal cells. ( iii ) follows from unwrapping the definitions: st ( x, st ( τ, K )) = { α ∈ st ( τ, K ) | α ∪ { x } ∈ st ( τ, K ) } (2.1) = { α ∈ st ( τ, K ) | α ∪ { x } ∪ τ ∈ K } (2.2) = { α ∈ st ( τ, K ) | α ∪ σ ∈ K } (2.3) = st ( τ, K ) ∩ st ( σ, K ) (2.4) = st ( σ, K ) (2.5)where the last equality follows from (i). To show ( vi ) , note that for any σ ∈ K , st ( σ ) ◦ is the interior of the union of maximal simplices containing σ . It follows that st ( σ ∪ τ ) ◦ is the interior of the union of maximal simplices containing both σ and τ , in other words the maximal simplices in st ( σ ) ∩ st ( τ ) . (cid:3) We will need some properties of the face ideals I st ( σ ) and face rings R st ( σ ) of thesubcomplexes st ( σ, K ) . Lemma 2.4. (1) If K , K are subcomplexes of K , I K + I K = I K ∩ K and I K ∩ I K = I K ∪ K . (2) I st ( σ ) = h x τ | τ ∈ U σ i . (3) The minimal primes of I K are the face ideals I st ( τ ) for the maximal sim-plices τ .Proof. The first two items follow from the definition of I st ( σ ) . For the last item,observe that I st ( σ ) is clearly prime when σ is a maximal simplex, as I st ( σ ) = h x i | x i ∈ U σ i and monomial ideals are prime exactly when they are generated by a subset ofthe variables; observe also that all I st ( σ ) are radical. These observations togetherwith item 1 give the result, as I K = T σ ⊂ K maximal I st ( σ ) . (cid:3) We intend to study the ring of differential operators on R , so let us define whatthat is: Definition 2.5.
The ring D k ( R ) of k -linear differential operators on a k -algebra R is defined inductively by D k ( R ) = [ n ≥ D nk ( R ) where D k ( R ) = R and for n > , D nk ( R ) := { φ ∈ End k ( R ) |∀ r ∈ R : [ φ, r ] ∈ D n − k ( R ) } . Elements of D nk ( R ) \ D n − k ( R ) are said to have order n , and there is anatural filtration D k ( R ) ⊂ D k ( R ) ⊂ D k ( R ) ⊂ · · · on D k ( R ) called the order filtration . Definition 2.6.
The
Weyl algebra in n variables over k is the ring of differentialoperators on the polynomial ring k [ x , . . . , x n ] . It is generated as an R -algebra bythe divided power operators ∂ ( a ) i = a ! ∂ a ∂x ai , with the relations [ x i , x j ] = [ ∂ ( a ) i , ∂ ( b ) j ] =0 for i = j , ∂ ( a ) i ∂ ( b ) i = (cid:0) a + ba (cid:1) ∂ ( a + b ) i and [ ∂ ( b ) i , x i ] = ∂ ( b − i (in particular [ ∂ i , x i ] = 1 ). Remark . We use the divided power operators rather than the usual vectorfields ∂∂x i as the latter do not generate the whole ring of differential operators inthe case of characteristic p ; the divided power operators however always generate KETIL TVEITEN everything regardless of the characteristic, as they define differential operators on Z and so descend to any commutative ring. In characteristic zero, the derivations ∂ i suffice to generate everything; in characteristic p we need the full set of elements ∂ p r i for r ≥ , which suffice due to the relation ∂ ( a ) i ∂ ( b ) i = (cid:0) a + ba (cid:1) ∂ ( a + b ) i .In the following, k will always be fixed, so we will omit it from the notationand write simply D ( R ) . Elements of k will be referred to as constants . One easilyverifies that an element x a ∂ ( b ) in the Weyl algebra has order | b | .3. The two-sided ideals of D ( R ) When R = R K is a face ring, there exist several descriptions of D ( R ) in theliterature, see [Tri97], [Eri98] and [Tra99]. We wish to give a description of thetwo-sided ideals of D ( R ) in terms of the combinatorics of K ; for our purposes, thefollowing description due to Traves ([Tra99]) is the most convenient. Theorem 3.1.
Let k be a commutative domain, and R = k [ X ] /J a reduced mono-mial ring. An element x a ∂ ( b ) = Q i x a i i ∂ ( b i ) i of the Weyl algebra over k is in D ( R ) ifand only if for each minimal prime p of R , we have either x a ∈ p or x b p . D ( R ) is generated as a k-algebra by these elements, and they form a free basis of D ( R ) as a left k -module. Example 3.2.
Let R = k [ x , x , x ] / ( x x x ) . The associated simplicial complex K is the boundary of a 2-simplex. Then by 3.1, D ( R ) = R h x a i i ∂ ( b i ) i | a i , b i ∈ N i . Example 3.3.
Let R = k [ x , x , x , x ] /I where I = ( x x , x x , x x ) . The asso-ciated complex K is a chain of three 1-simplices, connected in order x , x , x , x .Theorem 3.1 gives D ( R ) = R h x a ∂ ( b )1 , x a ∂ ( b )2 , x a ∂ ( b )3 , x a ∂ ( b )4 , x a ∂ ( b )2 , x a ∂ ( b )3 i (for a, b > ).Note that in both examples, generators of the form x ai ∂ ( b ) i appear; it is not hardto see that such “toric” operators are always in D ( R ) . In 3.3, we also have generatorsof the form e.g. x ai ∂ ( b ) j (where i = j ). To understand when this happens, we maygive a somewhat more geometric formulation of 3.1: Proposition 3.4.
Let K be a simplicial complex and R = R K its face ring. Alsolet x a = Q x a i i , x b = Q x b j j be such that supp ( x a ) = st ( σ ) and supp ( x b ) = st ( τ ) , forsome σ, τ ∈ K . Then x a ∂ ( b ) = Q i x a i i ∂ ( b i ) i is in D ( R ) if and only if st ( σ ) ⊂ st ( τ ) .Proof. Let P x a denote the set of minimal primes in R that contain x a , and P ¬ x a the set of minimal primes that does not contain x a . Clearly, P x a ∪ P ¬ x a is equalto the set of minimal primes in R ; denote this by P . Recalling from 2.4 that theminimal primes of R are the face ideals I st ( α ) for maximal simplices α , we canreformulate these definitions: P x a is the set of ideals I st ( α ) such that α is maximaland x a ∈ I st ( α ) , in other words those ideals I st ( α ) such that α is maximal and α ∈ U σ ; and P x a is the set of ideals I st ( α ) with α maximal and contained in st ( σ ) .Again using 2.4, the ideal I st ( σ ) defining st ( σ ) is equal to the intersection of allideals in P ¬ x a . Unwrapping definitions, we get st ( σ ) ⊂ st ( τ ) ⇔ I st ( σ ) ⊃ I st ( τ ) ⇔ P ¬ x a ⊃ P ¬ x b ⇔ P x a ⊂ P x b . WO-SIDED IDEALS IN THE RING OF DIFFERENTIAL OPERATORS ON A STANLEY-REISNER RING5
Putting this together with 3.1, we have x a ∂ ( b ) ∈ D ( R ) ⇔ ∀ p ∈ P : x a ∈ p ∨ x b p ⇔ ∀ p ∈ P : p ∈ P x a ∨ p ∈ P ¬ x b ⇔ P = P x a ∪ P ¬ x b ⇔ P x a ⊂ P x b ∨ P ¬ x b ⊂ P ¬ x a (and these are equivalent) ⇔ st ( σ ) ⊂ st ( τ ) . (cid:3) Example 3.5.
Let R = k [ x , x , x , x , x ] / ( x x , x x , x x ) , the associated K isthree 2-simplices { x , x , x } , { x , x , x } , { x , x , x } glued along the edges { x , x } and { x , x } ; x is a common vertex to all faces. Note that this makes K a simplicialjoin of { x } with the complex from Example 3.3. Looking at the closed stars of thefaces, we see that st ( x ) ⊂ st ( x ) ⊂ st ( x ) ⊃ st ( x ) ⊃ st ( x ) . As st ( x ) = st ( { x , x } ) , st ( x ) = st ( { x , x } ) and for any face σ , st ( σ ) = st ( σ ∪ x ) this accounts for all the stars. From this we should by 3.4 have the “toric”generators x ai ∂ ( b ) i , and also x a ∂ ( b )2 , x a ∂ ( b )5 , x a ∂ ( b )2 ∂ ( c )5 , x a ∂ ( b )5 and the same with x and x replaced by x and x respectively (by symmetry). In fact, st ( x ) = st ( ∅ ) = K , so we should also have ∂ ( a )5 = 1 · ∂ ( a )5 and the description is somewhat redundant.From 3.4 we deduce the following very useful criterion. Corollary 3.6. h x τ i ⊂ h x σ i if and only if st ( τ ) ⊂ st ( σ ) .Proof. If st ( τ ) ⊂ st ( σ ) , it follows from 3.4 that x τ ∂ σ = x τ Q i : x i ∈ σ ∂ i is in D ( R ) .Now observe that [ · · · [ x τ ∂ σ , x i ] , · · · , x i r ] = x τ (where x σ = Q ≤ j ≤ r x i j ), so wehave x τ ∈ T i : x i ∈ σ h x i i = h x σ i .To show the reverse implication, note that by definition of I st ( σ ) , we have I st ( σ ) ∩h x σ i = h i . If now st ( τ ) st ( σ ) , it follows that τ ∈ U σ , so x τ ∈ I st ( σ ) , which finallyimplies x τ
6∈ h x σ i . (cid:3) The following very useful result is surprising.
Theorem 3.7.
Any proper two-sided ideal in D ( R ) is generated by reduced mono-mials in the “ordinary” variables x , . . . , x n .Proof. The proof is in three parts:(1) The ideal h P ( a,b ) ∈ S c ab x a ∂ ( b ) i (for some index set S ⊂ N n ) is equal to theideal h x a ∂ ( b ) | ( a, b ) ∈ S i ;(2) the ideal h x a i is equal to the ideal h Q a i =0 x i i ;(3) the ideal h x a ∂ ( b ) i is equal to the ideal h Q a i =0 x i i .We will make heavy use of the fact that for any two-sided ideal I and any element φ ∈ D ( R ) , the set of commutators [ φ, I ] is contained in I .For the first part, recall that we have two natural concepts of grading on theWeyl algebra, that descend to D ( R ) . First, the natural Z n -grading on the Weylalgebra given by the degree deg ( x a ∂ ( b ) ) = ( a − b , . . . , a n − b n ) , KETIL TVEITEN which induces a grading on D ( R ) ; second we have the N n -grading given by the order ord ( x a ∂ ( b ) ) = ( b , . . . , b n ) . Note that [ x i ∂ i , x a ∂ ( b ) ] = x i ∂ i x a ∂ ( b ) − x a ∂ ( b ) x i ∂ i = x i ( x a ∂ i + a i x a − i ) ∂ ( b ) − x a ( x i ∂ ( b ) + ∂ ( b − i ) ) ∂ i = x i x a ∂ i ∂ ( b ) + a i x i x a − i ∂ ( b ) − x a x i ∂ ( b ) ∂ i − x a ∂ ( b − i ) ∂ i = a i x a ∂ ( b ) − (cid:18) b i − (cid:19) x a ∂ ( b ) =( a i − b i ) x a ∂ ( b ) (in the remainder we omit the proof of such identities to avoid tedium), and in thecase of characteristic p , if a i − b i = cp r , we have [ x p r i ∂ ( p r ) i , x a ∂ ( b ) ] = cx a ∂ ( b ) . Inother words, the operators [ x i ∂ i , − ] (and [ x p r i ∂ ( p r ) i , − ] ) give different weight to eachdegree-graded component. Note also that [ x a ∂ ( b ) , x i ] · ∂ i = x a ∂ ( b − i ) ∂ i = b i x a ∂ ( b ) , and if b i = cp r , we have [ x a ∂ ( b ) , x p r i ] ∂ ( p r ) i = cx a ∂ ( b ) . In other words the operators [ − , x i ] ∂ i (and [ − , x p r i ] ∂ p r ) give different weight to each order-graded component.Putting these together, we can isolate any term x a ∂ ( b ) by applying a suitable poly-nomial in the operators [ x i ∂ i , − ] , [ x p r i ∂ ( p r ) i , − ] , [ − , x i ] ∂ i and [ − , x p r i ] ∂ p r .For the second part, we may reduce to a single variable. We separate the casesby characteristic. If char ( k ) = p , we have [ x i ∂ ( p r ) i , x p r i ] = x i , so x i is in the idealgenerated by x p r i ; choosing a power of p larger than a i we have x p r i = x a i i · x p r − a i i and so x i ∈ h x a i i i . If char ( k ) = 0 , on the other hand, we have [ x i ∂ (2) i , x a i i ] = a i x a i i ∂ i + (cid:18) a (cid:19) x a i − i and [ x i ∂ (3) i , x a i i ] = a i x a i +1 i ∂ (2) i + (cid:18) a i (cid:19) x a i i ∂ i + (cid:18) a i (cid:19) x a i − i . If a i = 0 , there is nothing to prove, and if a i > , we can invert a i − (cid:0) a i (cid:1) − (cid:0) a i (cid:1) = a i ( a i − and get x a i − i = 12 a i ( a i − (cid:18) a i −
12 [ x i ∂ (2) i , x a i i ] − [ x i ∂ (3) i , x a i i ] + a i x a i i · x i ∂ i (cid:19) . This gives h x a i − i i ⊂ h x a i i i and by iterating this procedure, h x i i = h x a i i i .For the third part, observe that [ x n ∂ ( m ) , x j ] = x n ∂ ( m − j ) (for j such that m j =0 ) is a valid identity for all n, m > . Iterating this beginning with n = a, m = b gives h x a i ⊂ h x a ∂ ( b ) i . By applying part 2 this becomes h Q a i =0 x i i ⊂ h x a ∂ ( b ) i .To show the reverse implication h x a ∂ ( b ) i ⊂ h Q a i =0 x i i we show h x a ∂ ( b ) i ⊂ h x i i for the two cases a i , b i = 0 and a i = 0 , b i = 0 . For the first case, x a i i ∂ ( b i ) i is a factorof x a ∂ ( b ) ; and applying the above argument we have that x a i i ∂ ( b i ) i ∈ h x i i ; it followsthat x a ∂ ( b ) ∈ h Q i : a i ,b i =0 x i i . WO-SIDED IDEALS IN THE RING OF DIFFERENTIAL OPERATORS ON A STANLEY-REISNER RING7
For the second case, a i = 0 , b i = 0 , we may assume a i = 1 , for if a i > , thenclearly x a ∂ ( b ) = x i x a − i ∂ ( b ) ∈ h x i i . By the previous case, x i ∂ (2) i is in h x i i , and sois x a +1 i ∂ ( b ) = x i x a ∂ ( b ) ; then of course their commutator [ x a +1 i ∂ ( b ) , x i ∂ (2) i ] = − ( a i + 1) x a +1 i ∂ i ∂ ( b ) − a i x a ∂ ( b ) is also in h x i i . Rewriting this (with a i = 1 as we have assumed) we get x a ∂ ( b ) = [ x a +1 i ∂ ( b ) , x i ∂ (2) i ] − x a +1 i ∂ i ∂ ( b ) and so x a ∂ ( b ) ∈ h x i i ; it follows that x a ∂ ( b ) ∈ h Q i : a i =0 ,b i =0 x i i . Taking both casestogether we have shown that x a ∂ ( b ) ∈ h Q i : a i =0 ,b i =0 ∨ b i =0 x i i = h Q i : a i =0 x i i . (cid:3) We have shown that all ideals in D ( R ) are generated by reduced monomials Q x i in the variables of R ; the next question is of course which ones? Recall that we willnot distinguish between the vertices of the simplicial complex K and the variablesof the associated face ring R , but refer to either by the same name, e.g. x i . We alsoremind of the notation x σ = Q x i ∈ σ x i . Theorem 3.8.
Any proper ideal in D ( R ) is generated by monomials x σ with σ ∈ K such that st ( σ ) = K .Proof. From 3.7 it follows that any ideal in D ( R ) is generated by reduced monomialsin the variables x i , and clearly the monomials corresponding to non-faces cannotoccur as they are in I K , so what remains are the monomials x σ for σ ∈ K . Only those x σ such that st ( σ ) = K generate proper ideals, as otherwise we have st ( σ ) = K and by 3.4 the elements · ∂ i where x i ∈ σ are in D ( R ) , as both and ∂ i aremonomials with support contained in st ( σ ) = K ; if we write σ = { x i , . . . , x i t } , wehave [ ∂ i , [ ∂ i , [ · · · , [ ∂ i r , x σ ] · · · ]]] = 1 and so h x σ i = h i = R . (cid:3) This now gives us all the ideals in D ( R ) , as by sums of principal ideals h x σ i wecan make everything. We may however also take a different approach: Any two-sided ideal in D ( R ) is the kernel of some ring homomorphism; the combinatorialstructure of the associated simplicial complex K gives rise to several such maps. Anobvious choice for candidate homomorphisms is the localization at an element x σ ;we will see that the kernels of such maps is another generating set for the lattice oftwo-sided ideals in D ( R ) . We introduce the notation J for the extension to D ( R ) of an ideal J ⊂ R . Theorem 3.9.
The kernel of the localization map D ( R ) → D ( R )[ x σ ] is the exten-sion I st ( σ ) of the ideal I st ( σ,K ) ⊂ R to D ( R ) .Proof. By 3.8 it is enough to examine what happens in the localization to monomials x α for α ∈ K . Assume first that x σ = x i (in other words, σ is a vertex). Inverting x i has the effect that for any non-face β = ∪ x j containing x i , the monomial x β x i = Q x j ∈ β,j = i x j is zero in the localization. It is clear that no other monomials arekilled, so what remains after localization are those monomials supported on a face τ such that τ ∪ x i is not a non-face, or clearing negations, that τ ∪ x i is a face in K ;in other words the remaining monomials are those supported on a face of st ( x i ) .For the general case, note that inverting x σ = Q i x i is the same as inverting each x i successively, and observing that we have from 2.3(iii) that st ( σ, K ) = st ( x , st ( σ \ x , K )) , we are done by recursion. (cid:3) KETIL TVEITEN
Theorem 3.10.
The lattice of two-sided ideals in D ( R ) is generated by the ideals I st ( σ ) ⊂ D ( R ) .Proof. After applying 3.8 the question is whether we can generate any proper ideal h x τ i by sums and intersections of the ideals I st ( σ ) . Considering that I st ( σ ) = h x α | α ∈ U σ i , we can look at the intersection of all such ideals that contain x τ : \ σ : τ ∈ U σ I st ( σ ) = h x α | α ∈ \ σ : τ ∈ U σ U σ i (3.1) = h x α |∀ σ ∈ K : τ ∈ U σ ⇒ α ∈ U σ i (3.2) = h x α |∀ σ ∈ K : α U σ ⇒ τ U σ i (3.3) = h x α |∀ σ ∈ K : α ∪ σ ∈ K ⇒ τ ∪ σ ∈ K i (3.4) = h x α |∀ σ ∈ K : σ ∈ st ( α ) ⇒ σ ∈ st ( τ ) i (3.5) = h x α | st ( α ) ⊂ st ( τ ) i (3.6) = h x τ i (3.7)where the last step is applying Corollary 3.6. (cid:3) Example 3.11.
Consider again the ring from 3.3, R = k [ x , x , x , x ] /I where I = ( x x , x x , x x ) ; the associated complex K is a chain of three 1-simplices. In-verting x gives us that x and x go to zero in the localization as x = x x x ∈ I ,etc; it follows that the generators x a ∂ ( b )3 are also killed; the kernel of the localization D ( R ) → D ( R )[ x ] is then (using 3.7 and 3.4) the ideal ( x , x ) , which is the faceideal of st ( x , K ) . Localizing at x gives x = x x x = 0 , and the kernel of thelocalization is indeed equal to the ideal ( x ) , the face ideal of st ( x , K ) . Proceedingin the same manner for the remaining faces x , x , { x , x } , { x , x } , and { x , x } ,we get as possible kernels the ideals ( x ) , ( x ) , ( x , x ) , ( x , x ) and ( x , x ) . By 3.4we have ( x , x ) = ( x ) and ( x , x ) = ( x ) ; in other words our possible kernels oflocalization are the ideals ( x ) , ( x ) , ( x ) and ( x ) ; in light of 3.7 these obviouslygenerate all the ideals by sums and intersections.Let us round off this section with some applications. In [Tra99], Traves examinesthe D ( R ) -module structure of R when k is a field, and determines what the (left) D ( R ) -submodules of R are. These are the ideals I ⊂ R such that D ( R ) • I = I ,so we follow Traves’ terminology and call such a submodule a D ( R ) -stable ideal .The reason for restricting k to be a field is that elements of D k ( R ) are k -linearendomorphisms of R , so any ideal of k extends to a D k ( R ) -submodule of R . Theorem 3.12 (Traves) . When k is a field, the D k ( R ) -submodules of the reducedmonomial ring R are exactly the ideals given by intersections of sums of minimalprimes of R . Based on our results about the ideal structure of D ( R ) , we can give a new proofof this result. We denote the module action of D ( R ) by • (e.g. D ( R ) • I ) and theproduct in D ( R ) by · (e.g. D ( R ) · I ). We prove the result by means of a generalfact which to our knowledge is previously unknown. Proposition 3.13.
Let k be a field and R be a k -algebra. An ideal J ⊂ R is D ( R ) -stable if and only if J = J ∩ R , where J denotes the extension of J to D ( R ) . WO-SIDED IDEALS IN THE RING OF DIFFERENTIAL OPERATORS ON A STANLEY-REISNER RING9
Proof.
Observe first that R is isomorphic as a D ( R ) -module to D ( R ) /D > ( R ) , thequotient by the left ideal of positive order elements; we can see this by writing D ( R ) = D ( R ) + D > ( R ) = R + D > ( R ) , as R = D ( R ) . In other words, if S ⊂ R is a subset, then under this isomorphism D ( R ) • S = D ( R ) · S + D > ( R ) . Further,if J ∈ D ( R ) is some subset, then J · D ( R ) + D > ( R ) = J · ( D ( R ) + D > ( R )) + D > ( R )= J · R + D > ( R ) . Now, if I ⊂ R is an ideal, the extension of I to D ( R ) is I = D ( R ) · I · D ( R ) , so wehave I + D > ( R ) = D ( R ) · I · D ( R ) + D > ( R )= D ( R ) · I + D > ( R )= D ( R ) • I. A D -stable ideal is an ideal I ⊂ R such that D ( R ) • I = I , so it follows that the D -stable ideals are exactly those such that I + D > ( R ) = I .It remains to show that for an ideal J ⊂ D ( R ) , J + D > ( R ) = J ∩ R . Let f ∈ J be some element, and write it as the sum f = f + f + · · · + f ord ( f ) where f i are theterms of order i ; it then follows from 3.7 that also each f i ∈ J . Reducing modulo D > ( R ) we get J + D > ( R ) = { f | f ∈ J } , and restricting to the homogenouselements of order zero we have J ∩ R = J ∩ D ( R ) = { f ∈ J | f = f } ; these setsclearly are equal. (cid:3) Theorem 3.14.
The D ( R ) -stable ideals of R are those generated by sums andintersections of the ideals I st ( σ ) for σ ∈ K .Proof. As we have shown (3.8, 3.10) that any ideal of D ( R ) is an extension of anideal of R , we only have to restrict these to R to recover the D ( R ) -stable ideals.Theorem 3.10 tells us that the lattice of ideals in D ( R ) is generated by sums andintersections of ideals I st ( σ ) , and it is easy to see that I st ( σ ) ∩ R = I st ( σ ) : Indeed,the only possible problem is that in D ( R ) , h x α i ⊂ h x β i if and only if st ( α ) ⊂ st ( β ) ,and this may cause additional monomials not in I to appear in I ∩ R . For I st ( σ ) however, this does not happen. Consider that I st ( σ ) = h x τ | τ ∈ U σ i and I st ( σ ) ∩ R = h x τ | τ ∈ U σ i = h x α |∃ τ ∈ U σ : st ( α ) ⊂ st ( τ ) i . In other words, we need to check ifthere are faces τ ∈ U σ and α ∈ st ( σ ) such that st ( α ) ⊂ st ( τ ) , as then x α wouldbe in I st ( σ ) ∩ R , but not in I st ( σ ) . This is impossible, however: by 2.3 ( v ) , α ∈ st ( σ ) if and only if σ ∈ st ( α ) , and if st ( α ) ⊂ st ( τ ) , we have σ ∈ st ( τ ) , which again by2.3 ( v ) gives τ ∈ st ( σ ) , which contradicts the assumption τ ∈ U σ . (cid:3) To recover 3.12, recall that by 2.4, the minimal primes are exactly the face idealsof the maximal faces of K , and any I st ( σ ) is the intersection of the face ideals ofthe maximal faces of st ( σ ) . Remark . Recall that the partially ordered set of two-sided ideals of D ( R ) (or bijectively, the D ( R ) -stable ideals of R ) is in order-reversing bijection withthe partially ordered set of closed stars of K . This partially ordered set can becompleted to a simplicial complex ( e K , say), homotopic to the nerve of the coverof K by open stars. The results about two-sided ideals of D ( R ) and D ( R ) -stableideals of R imply that subcomplexes L of K such that I L is D ( R ) -stable or I L is a two-sided ideal of D ( R ) are exactly those that are unions of intersections of closedstars; in other words the complex e K classifies such subcomplexes. This interestingconnection is perhaps worthy of further study.4. Characteristic p The constructions in the previous section are independent of the characteristicof k , and so solve the problem of finding the two-sided ideal structure of D ( R ) .In characteristic p however, there is a qualitatively different construction of D ( R ) ,which perhaps offers more interesting possibilities for generalization. From here on,we assume k is a field of characteristic p .The major tool when working in characteristic p is the Frobenius automorphismof k , given by x x p . This induces an endomorphism F : R → R given by F ( f ) = f p , and the image F ( R ) is the subring R p ⊂ R of p ’th powers; as R isreduced F is also an isomorphism onto its image. Any R -module M gets a new R -module structure through the pullback by the Frobenius map, namely F ∗ M isequal to M as an abelian group, but has R -module structure given by f · m = f p m .This is equivalent to considering M as an R p -module, as the maps F : R → R and R p ֒ → R both are injections with image R p . We will have need for considering alsoiterates of F , so if we let q = p r we write F r : R → R or R q = R p r ⊂ R . For ourpurposes in examining D ( R ) , it will be most convenient to use the description interms of the subrings R q , as we will see.Considering the behaviour R itself as an R p -module gives rise to several classify-ing properties of the ring R . We will simply recall the definitions of the particularproperties that are relevant for us, other such properties and further details maybe found in [SVdB97]. If R is finitely generated as an R p -module, we say that R is F -finite ; if R is F -finite and the map R p ֒ → R splits as a map of R p -moduleswe say R is F -split ; if F r ∗ R ≃ M r ⊕ · · · ⊕ M qn ( r ) as an R -module and the set ofisomorphism classes { [ M ri ] | r ∈ N , ≤ i ≤ n ( r ) } of modules appearing in such adecomposition for some r is finite, we say that R has finite F -representation type ,or FFRT .For our purposes, the key property of face rings R K in this respect is that theyare F -split and have FFRT. Even better, we can give a concrete decomposition of R as an R q -module: Lemma 4.1.
As an R q -module, R is isomorphic to L st ( σ ) ⊂ K ( R qst ( σ ) ) m st ( σ ) ( q ) ,where m st ( σ ) ( q ) = P α : st ( α )= st ( σ ) ( q − (dim( α )+1) . Note that the direct sum runs over those subcomplexes of K that is the star ofsome simplex. Proof.
As we have R ≃ R p ⊕ R p x ⊕ · · · ⊕ R p x p − · · · x p − n (where only the ap-propriate monomials appear), this expresses R as an R p -module. We can rewritethis using R p · x α ≃ R p /Ann R p ( x α ) , and observing that as the monomials x α thatappear in the decomposition are those supported on a face supp ( α ) =: σ , and thatthe annihilator of x α is the face ideal of the complex st ( σ, K ) , we get the decompo-sition R = L σ ∈ K ( R pst ( σ ) ) m st ( σ ) , where R pst ( σ ) is the ( p ’th power) face ring of st ( σ ) and by simply counting monomials we have m st ( σ ) = P α : st ( α )= st ( σ ) ( p − (dim( α )+1) (using the convention that dim( ∅ ) = − ). Iterating the same construction, we get R = L σ ∈ K ( R qst ( σ ) ) m st ( σ ) ( q ) , where m st ( σ ) ( q ) = ( q − (dim( σ )+1) . (cid:3) WO-SIDED IDEALS IN THE RING OF DIFFERENTIAL OPERATORS ON A STANLEY-REISNER RING11
Let us make use of this to compute some invariants of R that only make sense incharacteristic p , namely the Hilbert-Kunz function and the
Hilbert-Kunz multiplic-ity . This invariant was introduced by Kunz [Kun69] for local rings, and extendedto graded rings by Conca [Con96]; see also [Hun13] and [Mon83].
Definition 4.2.
Let R be a local ring with maximal ideal m , or a graded ring withhomogenous maximal ideal m , over a field k of characteristic p , and let q = p r . The Hilbert-Kunz function of a ring R is the function HK R ( q ) = l ( R/ m [ q ] ) where I [ q ] is the ideal generated by q ’th powers of elements in the ideal I . The Hilbert-Kunz multiplicity is the number e HK ( R ) = lim q →∞ HK R ( q ) q dim R , in other words the leading coefficient of HK R ( q ) .The Hilbert-Kunz function gives a measure of singularity of R , roughly speakinghigher multiplicities correspond to worse singularities. It is a theorem of Kunz that HK R ( q ) = q dim R if and only if R is regular (see [Kun69]), so if R is regular, e HK ( R ) = 1 . The converse holds for unmixed rings, but not in general, and inparticular not for face rings. The following is equivalent to Remark 2.2 in [Con96],though we prove it in a different way. Proposition 4.3.
Let R K be a face ring, then HK R ( q ) = P dim( R ) − i = − f i ( q − i +1 ,where f i is the number of i -simplices in K , so ( f − , . . . , f dim( R ) − ) is the f -vectorof K (we recall the usual convention dim( ∅ ) = − , so f − = 1 ). In particular, e HK ( R K ) = f dim K , the number of top-dimensional faces of K .Proof. The number of indecomposable summands of R as an R q -module is P σ ∈ K ( q − dim ( σ )+1 by 4.1. By simply rearranging the sum, this is equal to P dim( R ) − i = − f i ( q − i +1 . The claim now follows from the fact that none of the generators of thesesummands are in m [ q ] = h x q , . . . , x qn i , so the number of summands in the splittingof R is the same as the length of R/ m [ q ] . (cid:3) The promised different construction of D ( R ) is due to Yekutieli [Yek92]. Weomit the proof here, but mention that in addition to [Yek92], the reader can findan excellent exposition in [SVdB97]. Proposition 4.4. D k ( R ) ≃ S q End R q ( R ) , where q = p r , r ∈ N and R q is thesubring of q -th powers. Let us now give the summands appearing in 4.1 a more convenient notation, anddefine M qst ( σ ) := ( R qst ( σ ) ) m st ( σ ) ( q ) . It follows from 4.1 that End R q ( R ) ≃ M st ( σ ) ,st ( τ ) ⊂ K Hom R q ( M qst ( σ ) , M qst ( τ ) ) . As each M qst ( σ ) is generated as an R q -module by monomials of degree in each vari-able up to q − , we can see that as an R pq -module it is contained in L st ( α ) ⊂ st ( σ ) M pqst ( α ) ,because the elements of M qst ( σ ) contain monomials of degree larger than q − , whichhave support on smaller stars (recall that as q = p r , pq = p r +1 ). In particular thisimplies the following: Lemma 4.5.
Hom R q ( M qst ( σ ) , M qst ( τ ) ) ⊂ L st ( α ) ⊂ st ( σ ) ,st ( β ) ⊂ st ( τ ) Hom R pq ( M pqst ( α ) , M pqst ( β ) ) . This lets us think of elements φ ∈ End R q ( R ) as block matrices with each blockhaving entries in some R q /I st ( σ ) ; it is vital to remember that this means that theentries have degree equal to a multiple of q . Definition 4.6.
Let J q ( st ( α ) , st ( β )) denote the ideal in D ( R ) generated by theelements of Hom R q ( M qst ( α ) , M qst ( β ) ) , and let J ( st ( α ) , st ( β )) := P q J q ( st ( α ) , st ( β )) .For convenience we denote J ( st ( σ ) , st ( σ )) by simply J ( st ( σ )) .The following result is essentially the same as 3.7 in a different guise. Proposition 4.7.
Assume st ( σ ) ⊃ st ( τ ) , and let φ ∈ Hom R q ( M qst ( σ ) , M qst ( τ ) ) be anonzero element. Then h φ i , the ideal in D ( R ) generated by φ , is equal to the ideal J ( st ( τ )) . Furthermore, we have that J ( st ( τ ) ⊂ J ( st ( σ )) .Proof. Clearly, J ( st ( τ )) is generated by the identity maps id qst ( τ ) : M qst ( τ ) → M qst ( τ ) (for each q ), so it suffices to show that these are in h φ i .Recall that any element of End R q ( R ) has entries with degree a multiple of q .We claim that for s > q a sufficiently large power of p , φ considered as an element of End R s ( R ) will have at least some constant entries in each block Hom R s ( M sst ( σ ) , M sst ( τ ) ) .To see this, suppose φ (as an element of End R q ( R ) ) has an entry x qi in a block Hom R q ( R q · x a , R q · x b ) (with all ≤ a j , b j < q ), in other words φ ( x a + cq ) = x a +( c +1 i ) q + b . It follows from 4.5 that this block has image in End R pq ( R ) containedin L ≤ c,d
q a sufficiently large power of p .Now let s be such a sufficiently large power of p , and consider φ as an element of End R s ( R ) ; by 4.5, Hom R q ( M qst ( σ ) , M qst ( τ ) ) is contained in L st ( α ) ⊂ st ( σ ) ,st ( β ) ⊂ st ( τ ) Hom R s ( M sst ( σ ) , M sst ( τ ) ) .We can see that φ , considered as a matrix ( φ ij ) in End R s ( R ) , will have (amongothers) some constant entries in each block End R s ( M sst ( β ) ) such that st ( β ) ⊂ st ( τ ) .Each of these entries can be “picked out” in the following manner: Let ii be thematrix in End R s ( R ) with the appropriate identity map in position ( i, i ) and zeroesotherwise. It is clear that ii · φ · jj is the matrix with entry φ ij in position ( i, j ) andzeroes otherwise; we may assume φ ij = 1 as it is constant. Applying permutationsof End R s ( M sst ( β ) ) (on both sides), we can now place this entry wherever we wantwithin the matrix block corresponding to End R s ( M sst ( β ) ) ; taking sums of these wecan produce any matrix with constant entries. In particular, we can make id sst ( β ) .Thus, we have that each id sst ( β ) such that st ( β ) ⊂ st ( τ ) is in h φ i , and in thesame way any such id tst ( β ) for t > s any larger power of p . To recreate id tst ( τ ) for smaller powers t < s we observe that those maps, considered as elements of End R s ( R ) , are in L st ( β ) ⊂ st ( τ ) End R s ( M st ( β ) ) and as such are contained in theideal generated by the identity maps id sst ( β ) , in other words contained in h φ i . Wehave shown J ( st ( τ )) ⊂ h φ i ; the opposite inclusion follows from the observation that φ = id qst ( τ ) ◦ φ , and so φ ∈ J ( st ( τ )) .The final claim is similar: φ = φ ◦ id qst ( σ ) , and so φ ∈ J ( st ( σ )) . (cid:3) Proposition 4.8.
The ideal J ( st ( σ ) , st ( τ )) is equal to J ( st ( σ ∪ τ )) , if σ ∪ τ is aface of K , and the zero ideal otherwise. WO-SIDED IDEALS IN THE RING OF DIFFERENTIAL OPERATORS ON A STANLEY-REISNER RING13
Proof.
The module
Hom R q ( M qst ( σ ) , M qst ( τ ) ) has support st ( σ ) ◦ ∩ st ( τ ) ◦ . From 2.3 ( vi ) it follows that this is st ( σ ∪ τ ) ◦ , if σ ∪ τ ∈ K .If σ ∪ τ is a non-face, st ( σ ) ∩ st ( τ ) does not contain any maximal simplices, andso the cone on st ( σ ) ∩ st ( τ ) is not a union of irreducible components of Spec ( R ) ,and so is not the closure of the support of any element in Hom R q ( M qst ( σ ) , M qst ( τ ) ) ,so this must be the zero module. It follows that J ( st ( σ ) , st ( τ )) is the zero ideal.For the case when σ ∪ τ is a face of K , recall that by Lemma 4.5, Hom R q ( M qst ( σ ) , M qst ( τ ) ) ⊂ M st ( α ) ⊂ st ( σ ) ,st ( β ) ⊂ st ( τ ) Hom R pq ( M pqst ( α ) , M pqst ( β ) ) . In particular, there will be entries in the block
Hom R pq ( M pqst ( σ ∪ τ ) , M pqst ( σ ∪ τ ) ) , so by4.7 we have that J ( st ( σ ∪ τ )) ⊂ J ( st ( σ ) , st ( τ )) .For the converse, note that as an R q -module, Hom R q ( M qst ( σ ) , M qst ( τ ) ) ≃ (cid:0) ( I qst ( τ ) : I qst ( σ ) ) /I qst ( τ ) (cid:1) m st ( σ ) ( q ) × m st ( τ ) ( q ) (where I q is the restriction of I ⊂ R to R q ). Any element of Hom R q ( M qst ( α ) , M qst ( β ) ) has, as a matrix, entries with degree (in each variable) a multiple of q , with constant(nonzero) entries only when st ( β ) ⊂ st ( α ) , as then ( I qst ( β ) : I qst ( α ) ) is the unit ideal in R q (otherwise it is generated by elements of degree ≥ q ). It follows that elements ofthe image of Hom R q ( M qst ( σ ) , M qst ( τ ) ) in End R s ( R ) for s > q (considered as matrices)have entries with degree some multiple of s , with constant (nonzero) entries only inthose blocks Hom R s ( M sst ( α ) , M sst ( β ) ) with st ( β ) ⊂ st ( α ) . In the direct limit, theseelements become infinite matrices with entries in k , in other words there can only benonzero entries in those blocks corresponding to st ( β ) ⊂ st ( α ) (any nonzero entryin a different block must have infinite degree, which is impossible). This implies that J ( st ( σ ) , st ( τ )) is contained in P st ( σ ) ⊃ st ( α ) ⊃ st ( β ) ⊂ st ( τ ) J ( st ( α ) , st ( β )) , which by 4.7is equal to P st ( σ ) ⊃ st ( β ) ⊂ st ( τ ) J ( st ( β )) = J ( st ( σ ∪ τ )) and we are done. (cid:3) Theorem 4.9.
The ideals J ( st ( σ )) generate the lattice of ideals in D ( R ) by sumsand intersections.Proof. Let I be an ideal in D ( R ) ; it is of course true in general that I = P φ ∈ I h φ i .By 4.7 and 4.8 this is equal to P J ( st ( σ )) , where the sum goes over all σ ∈ K suchthat I contains elements from some Hom R q ( M qst ( α ) , M qst ( σ ) ) .Finally, the intersection J ( st ( σ )) ∩ J ( st ( τ )) contains elements in those End R q ( M qst ( α ) ) with st ( α ) ⊂ st ( σ ) ∩ st ( τ ) ; the maximal such star is st ( σ ∪ τ ) if σ ∪ τ is a face of K ,and if σ ∪ τ is not a face, there are no such α ; in other words J ( st ( σ )) ∩ J ( st ( τ )) = J ( st ( σ ∪ τ )) . (cid:3) We have now given two essentially different descriptions of the ideals of D ( R ) ,and we may wonder how to translate between the two languages. This is not toohard, as the obvious suggestion turns out to be true. Theorem 4.10.
The ideal J ( st ( σ )) is equal to the ideal h x σ i .Proof. It follows from 4.7 and 4.8 that J ( st ( σ )) = L q> ,st ( β ) ⊂ st ( σ ) Hom R q ( M qst ( α ) , M qst ( β ) ) ,in other words all the endomorphisms with support contained in st ( σ ) . We can thinkof x σ as an endomorphism of R , given by f f x σ , and considering that whateverelement f we choose, f x σ has support contained in st ( σ ) . This means that theendomorphism x σ is in J ( st ( σ )) and not in any larger ideal, and as x σ (1) = x σ has support equal to st ( σ ) ◦ , it is not in any smaller ideal J ( st ( τ )) with st ( τ ) ⊂ st ( σ ) .From 4.7 it follows that x σ generates all of J ( st ( σ )) and the two ideals are equal. (cid:3) Acknowledgements
I would like to thank my advisor Rikard Bøgvad for all the usual reasons, and Ialso thank Anders Björner for some helpful remarks.
References [Bav10a] V. V. Bavula,
Generators and defining relations for the ring of differential operators ona smooth affine algebraic variety , Algebr. Represent. Theory (2010), no. 2, 159–187.[Bav10b] , Generators and defining relations for the ring of differential operators on asmooth affine algebraic variety in prime characteristic , J. Algebra (2010), no. 4,1036–1051.[Con96] Aldo Conca,
Hilbert-Kunz function of monomial ideals and binomial hypersurfaces ,Manuscripta Math. (1996), no. 3, 287–300.[Eri98] Anders Eriksson, The ring of differential operators of a Stanley-Reisner ring , Comm.Algebra (1998), no. 12, 4007–4013.[Hun13] Craig Huneke, Hilbert-Kunz multiplicity and the F-signature , Commutative algebra,Springer, New York, 2013, pp. 485–525.[Kun69] Ernst Kunz,
Characterizations of regular local rings for characteristic p , Amer. J. Math. (1969), 772–784.[Mon83] P. Monsky, The Hilbert-Kunz function , Math. Ann. (1983), no. 1, 43–49.[Mus94] Ian M. Musson,
Differential operators on toric varieties , J. Pure Appl. Algebra (1994), no. 3, 303–315.[Sai07] Mutsumi Saito, Primitive ideals of the ring of differential operators on an affine toricvariety , Tohoku Math. J. (2) (2007), no. 1, 119–144.[SVdB97] Karen E. Smith and Michel Van den Bergh, Simplicity of rings of differential operatorsin prime characteristic , Proc. London Math. Soc. (3) (1997), no. 1, 32–62.[Tra99] William N. Traves, Differential operators on monomial rings , J. Pure Appl. Algebra (1999), no. 2, 183–197.[Tri97] J. R. Tripp,
Differential operators on Stanley-Reisner rings , Trans. Amer. Math. Soc. (1997), no. 6, 2507–2523.[Yek92] Amnon Yekutieli,
An explicit construction of the Grothendieck residue complex ,Astérisque (1992), no. 208, 127, With an appendix by Pramathanath Sastry.
Ketil Tveiten, Matematiska Institutionen, Stockholms Universitet, 106 91 Stock-holm.
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