ON ALGEBRAS ASSOCIATED TO PARTIALLY ORDERED SETS
MORTEN BRUN and TIM RÖMER
Abstract
We continue the work [2] on sheaves of rings on finite posets. We present examples where the ring of global sections coincide with toric faces rings, quotients of a polynomial ring by a monomial ideal and algebras with straightening laws. We prove a rank-selection theorem which generalizes the well-known rank-selection theorem of Stanley-Reisner rings. Finally, we determine an explicit presentation of certain global rings of sections.
1. Introduction
In the present paper we continue the work [2] on sheaves of commutative rings on finite partially ordered sets (posets for short).
A finite posetP can be considered as a topological space with theAlexan- drov topology[1], that is, the topology where the open sets are the subsetsU ofP such thaty∈U andx≤yimpliesx∈U.
ApresheafTonPconsists of an abelian groupT(U )for every open subset U ofP andrestriction mapsTV U:T(U )→T(V )forV ⊆U open subsets ofP. The restriction maps are subject to the conditions thatTU Uis the identity onT(U )and thatTW V ◦TV U =TW U forW ⊆V ⊆ U open subsets ofP. ThestalkTxofT atx ∈P is the set
x∈UT(U )/∼of equivalence classes represented by elementst ∈ T(U )forU an open subset ofP containingx under the equivalence relation generated by requiringt ∼ TV U(t)whenever V ⊆ U are open subsets ofP containingx. The presheafT is asheaf if the sequence
0→T(U1∪U2)→T(U1)×T(U2)→T(U1∩U2)
is exact for every pair(U1, U2)of open subsets ofP. Here the mapT(U1∪ U2) → T(U1)×T(U2) is (TU1U1∪U2,TU2U1∪U2), and the map T(U1)× T(U2) → T(U1 ∩ U2) is given by TU1∩U2U1 on the first factor and by
−TU2∩U2U1on the second factor.
Received January 2, 2006; in revised form March 7, 2007.
A sheafT onP is uniquely determined by its stalks(Tx)x∈P and the ho- momorphismsTxy:Ty →Txforx ≤y inP induced by the restriction maps ofT. More precisely, the stalks ofT form aZP-algebraT, that is, a system (Tx)x∈P of abelian groups and homomorphismsTxy:Ty →Tx forx ≤yinP with the property thatTxx is the identity onTx and thatTxy ◦Tyz = Txz for everyx ≤y≤zinP. Conversely, there is a sheafT, withT(U )equal to the (inverse)limitlimx∈UTx, associated to everyZP-algebraT. Here limx∈UTx
is the subgroup of the product
x∈UTx consisting of families(fx)x∈U with Txy(fy) = fx for allx ≤ y inU. This defines a one to one correspondence between sheaves onP andZP-algebras, and thus the concepts of sheaves and ofZP-algebras are equivalent.
LetRbe aZm-graded commutative Noetherian ring. By abuse of termino- logy we say thatT is asheaf ofZm-gradedR-algebrasifT is a sheaf onP and the restriction mapsTU V are homomorphisms ofZm-gradedR-algebras.
Similarly we say thatT is anRP-algebraif the homomorphismsTxy are ho- momorphisms ofZm-gradedR-algebras. The above discussion also shows that the concept of sheaves ofZm-gradedR-algebras is equivalent to the concept ofRP-algebras. TheRP-algebraT is calledflasqueif the associated sheafT is flasque, that is, if the restriction mapTV U:T(U )→T(V )is surjective for every inclusionV ⊆U of open subsets ofP.
In this paper we study the ringH0(P ,T):=T(P )= limPT and call it thering of global sections ofT. Under this name it was for example studied by Yuzvinsky [22], [23] and Caijun [6]. We are particularly interested inRP- algebras that appear in the literature on algebraic combinatoric and commutat- ive algebra, including face rings for fans and simplicial complexes, monomial ideals and algebras with straightening laws. Applying results from [2] on local cohomology ofH0(P ,T)we obtain a rank selection theorem in the tradition of Duval [8], Hibi [10], Munkres [13] and Stanley [15]. In order to state it we need a little terminology. Firstly, in [10] Hibi calls a subsetXofP excellentif for every elementzofP there exists at most onex∈Xwithx≤z. Secondly, therank of P is the supremum of the numbers|C| −1 forC a chain in P, and the rank of x ∈ P is the rank of the poset(0P, x] = {z ∈ P:z ≤ x}. Thirdly, if there exists exactly oneZm-graded idealᒊinRsuch thatR/ᒊis a field, then we callRaZm-graded local ring. Note thatR/ᒊis concentrated in degree zero, and thusᒊcontains every element ofR of non-zero degree.
In particular, ifK=R0is a field, thenᒊ=
a∈Zm\{0}Ra. Finally, we denote byHᒊi(N)the local cohomology modules of a finitely generatedZm-graded R-module N with respect to ᒊ (see [5] for details). Sinceᒊ is a maximal ideal the definition of graded local cohomology modules and usual local co- homology modules coincide. With the notation recollected in Definition 4.1, the following generalizes Hibi’s result [10, Theorem 1.7].
Theorem1.1. Let(R,ᒊ)be aZm-graded local ring withK=R0a field and letXbe an excellent subset of a finite posetP. LetTbe a flasqueZm-graded RP-algebra with dim limPT = rank(P ), and such that the homomorphism R→limPT is surjective. Assume that:
(i) Tyis Cohen-Macaulay of dimensionrank(y)for everyy∈P.
(ii) For everyx ∈ Xwe have thatrank(star(x)) = rank(P )and we have thatrank(star(x)∩PX)=rank(P )−1.
(iii) For everyx ∈Xandy ∈star(x)we have thatHi((y,1star(x));K)=0 fori=rank(P )−rank(y)−1.
(iv) For everyx∈Xandy∈star(x)∩PXwe haveHi((y,1star(x)∩P X);K)=0 fori=rank(P )−rank(y)−2.
Then the natural homomorphism
Hᒊi(lim
P T )→Hᒊi(lim
PX T )
is surjective fori =rank(P ), it is injective fori =rank(P )−1, and otherwise it is an isomorphism.
Here the cohomology groupsHi(P;K)are reduced singular cohomology groups of the underlying topological space of the posetP, or equivalently the cohomology groups of the associated order complex.
As an application of Theorem 1.1 we consider the following situation. We define theithskeletonof a finite posetP to be the sub-posetP(i)= {y ∈P : rank(y)≤i}.If for exampleP =P ()is the face poset of a rational pointed faninRmandT =T ()is theKP-algebra as constructed in Example 2.1, then we obtain that
depth lim
P T =max
i∈N:i ≤rank(P )and lim
P(i)T is Cohen-Macaulay . IfP is the face poset of a simplicial complexandT is the Stanley-Reisner KP-algebra associated to, then this is a well-known result, because limT ∼= K[] is the Stanley-Reisner ring associated to and limP(i)T ∼= K[(i)] is the Stanley-Reisner ring associated to the ith skeleton (i) of in this situation.
We end the paper by discussing presentations of RP-algebras. In The- orem 5.6 we give a criterion, namely thatT issectionedand has apresentation , ensuring that the ring limP Tof global sections of anRP-algebraT is of the formK[F]/I for a finite setF, where the idealIis a sumI =IT +Iof the Stanley-Reisner ideal of a simplicial complexrelated to the order complex ofP and the sumIT of defining ideals for the stalksTx ofT. A presentation
of T roughly consists of a subsetFx of F and a surjective homomorphism K[Fx]→Txfor everyxinP. Using the main result of [2] again we obtain the following description of the depth and the dimension of certainKP-algebras.
(Kis a field.)
Corollary1.2. Let(, p)be a presentation of a sectioned flasque KP- algebraT consisting ofZm-graded Cohen-Macaulay rings such thatdx < dy
andfx < fyforx < yinP, wheredx = dim(Tx)andfx = |Fx|. We have that
depth lim
P T =min{i ∈N:Hᒊi+(fx−dx)(K[])=0for somex ∈P}, dim lim
P T =max{i∈N:Hᒊi+(fx−dx)(K[])=0for somex∈P}.
In particular, if there exists ann∈ Zsuch thatdim(K[Fx]) = dim(Tx)+n for everyx∈P and such thatdim(K[])=dim(limPT )+n, thenlimPT is Cohen-Macaulay if and only ifis Cohen-Macaulay.
The paper is organized as follows: In Section 2 we present classes ofKP- algebras associated to toric face rings (as introduced in [3]), monomial ideals and algebras with straightening laws (ASL’s for short). In Section 3 we give a criterion on aKP-algebra with stalks given by ASL’s to have an ASL as ring of global sections. As an application we generalize a construction of Stanley [17] who defined the face ring of a simplicial poset. In Section 4 we prove Theorem 1.1 and study some of its consequences. Finally, in Section 5 we study presentations ofKP-algebras.
2. Examples ofRP-algebras
In this section we presentRP-algebras appearing naturally in algebraic combin- atoric and commutative algebra. Of course everyK-algebraSequals limPT for the posetP = {x}consisting of one element and theKP-algebraT with Tx =S. However this gives no new information aboutS. Our goal is to choose a finite posetP and a suitableKP-algebraT such that the stalksTxare as nice as possible (e.g. Cohen-Macaulay rings) and to characterize ring properties of limPT in terms of combinatorial properties ofP and ring properties of the ringsTxforx ∈P. Our first example goes back to a construction of Stanley [16].
Example2.1 (Toric face rings). We consider a rational faninRm, that is, is a finite collection of rational cones inRmsuch that forC ⊆ C with C∈we have thatCis a face ofCif and only ifC∈, and ifC, C ∈, thenC∩Cis a common face ofCandC. For each coneC ∈we choose an affine monoidMC ⊆Zmsuch that:
(i) cone(MC)=CforC ∈;
(ii) IfC, C ∈,C⊆C, thenMC =MC∩C.
Observe that we do not require thatMC is normal. LetP () = (,⊆)be the partially ordered set of faces of ordered by inclusion and letK be a field. ForC ∈ P () we let TC denote the affine monoid ringK[MC]. For C ⊆ C in P () the homomorphismsTCC:TC → TC are induced by the natural face projectionsK[C∩Zm]→K[C∩Zm]. This is aZm-gradedKP- algebra, and withR=limT it is anRP-algebra ofZm-gradedR-algebras. We writeT = T ()ifMC = C∩Zm, for everyC ∈ . In this case the ring limT ()is called thetoric face ringof, and it was studied by Stanley [16], Bruns-Gubeladze [4] and Brun-Römer [3].
Stanley-Reisner rings of finite simplicial complexes are covered by the above example: To a simplicial complexwe associate a fanwith the same face poset as, and we can chooseMCisomorphic toNdim(C)for everyC ∈. In this situation we writeP ()= P ()and callT () =T ()theStanley- Reisner KP-algebra. The ring limT ()is isomorphic to the Stanley-Reisner ringK[]. (See [2, Example 5.2].)
Our next example shows that rings defined by monomial ideals give rise to KP-algebras.
Example 2.2 (Monomial ideals). Let S = K[x1, . . . , xn] be the poly- nomial ring over a fieldKwith the usualZn-grading. Recall that irreducible monomial ideals inSare of the formᒊb=(x1b1, . . . , xnbn)for 0=b∈Nn. A monomial idealI has a unique irredundant irreducible decomposition of the form
I =ᒊb1 ∩ · · · ∩ᒊbt for b1, . . . , bt ∈Nn.
(See [11, Section 5.2] for details.) For each subsetCof [t]= {1, . . . , t}there exists a unique maximal subsetC ⊆ [t] with the property that
i∈Cᒊbi =
i∈Cᒊbi. LetP (I )denote the poset of the subsetsCof [t] for∅ =C⊆[t] ordered by reverse inclusion. ForC∈P (I )letT (I )C =S/
i∈Cᒊbi and for C, D ∈ P (I )withC ⊆ Dwe defineT (I )DC : TC →TD to be the natural projection map. Now:
(i) T (I )is aZn-gradedSP (I )-algebra.
(ii) By using Example 3.3 of [2] we see thatT (I )is flasque with limT (I )∼= S/I.
(iii) T (I )C is a complete intersection and thus Cohen-Macaulay forC ∈ P (I ).
The result [2, Theorem 4.1] has the following corollary. (See [20, The- orem 1] for a more general description ofHᒊi(S/I ).)
Corollary2.3.LetI =ᒊb1∩· · ·∩ᒊbt ⊂K[x1, . . . , xn]be an irredundant irreducible decomposition of a monomial ideal and let dC = dimT (I )C. If dC > dD for C, D ∈ P (I ) with D ⊂ C then Hᒊi(K[x1, . . . , xn]/I ) is isomorphic to
F∈P (I )
Hi−dF−1
(F,1P (I ) );K
⊗KHᒊdF
K[x1, . . . , xn]/
i∈F
mbi
as aZn-gradedK-vector space.
For the next example we recall the notion of analgebra with straightening law(ASL for short). We call a functionm:P →N(i.e.m∈NP) amonomial on P, and consider it as an element of the polynomial ring K[P] on the elements ofP. Thesupportofmis the set supp(m)= {x ∈P :m(x) =0}. The monomial is called astandard monomialif supp(m)is a chain inP. Let Rbe an algebra over a fieldKwith an injectionφ:P →R. We associate to each monomialmthe elementφ(m) =
x∈Pφ(x)m(x) ∈ Rand callφ(m)a monomial inR. Following [7] we callR analgebra with straightening law (ASL for short) onPopoverKif
ASL1 The set of standard monomials is aK-basis ofR, and ASL2 ifxandyare incomparable inP and if
(1) φ(x)φ(y)=
m
rm,xyφ(m)
is the unique representation ofφ(x)φ(y)as a linear combination of standard monomials guaranteed by ASL1, thenrm,xy = 0 implies that the maximal elementxmof supp(m)satisfiesx < xmandy < xm.
Note that for technical reasons we work with Pop instead of P. By ASL1 every monomialφ(n)in R has a unique representation of the formφ(n) =
m∈NP rm,nφ(m), whererm,n=0 implies thatmis a standard monomial. We write
r(n)=
m∈NP
rm,nm
for the element in the polynomial ringK[P] associated to this representation ofφ(m).
Example2.4 (ASL). LetRbe an ASL onPop. ThenR=K[P]/IP where the idealIP ⊆K[P] is generated by the straightening relationsn−r(n). For
Q ⊆Qan inclusion of subsets ofP we consider the projection map pQQ:K[Q]→K[Q], x →
x ifx∈Q, 0 ifx∈Q.
We let IQ = pQP(IP) ⊆ K[Q], we define TQ = K[Q]/IQ and we let TQQ:TQ → TQ denote the map induced by pQQ. For Q open in P the ideal IQ is generated by the elements n− pQP(r(n)) for n ∈ NQ. By [7, Proposition 1.2(b)] (withI =P \Q) theK-algebraTQis an ASL onQop.
Now we letTx = T(0P,x]forx ∈ P andTxy = T(0P,y](0P,x]forx, y ∈ P withx ≤ y. It follows thatT is anRP-algebra. Using thatTQ is an ASL on Qop if Q ⊆ P is open, we see by working with standard monomials that limPT ∼=TP.
3. KP-algebras with an ASL-structure
We saw in Example 2.4 that an ASLRcan be seen as the ring of global sections of a suitableKP-algebra. Sometimes it is possible to reverse this construction in the sense that giving locally ASL’s one can construct aKP-algebra such that the global ring of sections is again an ASL with prescribed presentation.
Recall thatK[P] =K[x :x ∈P] denotes the polynomial algebra on the elementsx of P. At first we present a criterion which ensures that a given KP-algebra has an ASL-structure.
Theorem3.1. LetT be a KP-algebra on a finite posetP such thatTxis anASLon(0P, x]opfor everyx∈P. Letφx:K[(0P, x]]→Txbe the natural projection induced by theASL-structure onTx forx ∈P. If the diagram
K[(0P, y]]−−−−−−→φy Ty
↓||
p(0P,x](0P,y] Txy↓||
K[(0P, x]]−−−−−−→φx Tx
commutes for everyx, y∈P withx≤y, thenlimPT is anASLonPop. In particular, ifS ⊆ NP denotes the set of standard monomials inK[P] thenlimPT ∼=K[P]/IP, where the idealIP is generated by relations of the form
n−
m∈S
rm,nm for n∈NP.
Proof. Letφ:K[P]→limPT denote the homomorphism induced by the natural projectionsK[P]→K[(0P, x]] forx∈P.
We claim thatφ is surjective and moreover limPT is generated as a K- vector space by the elementsφ(m)withm∈S.
Lets =(sx)x∈P ∈limP T. Letx∈P and note thatsx ∈Tx =lim(0P,x]T. SinceTxis an ASL on(0P, x]opthere exist unique scalarsλxm,s ∈Ksuch that
sx=
m∈N(0P,x]∩S
λxm,sφx(m).
For every chainminP we setxm=max(supp(m)). Note that by the assump- tions we have thatλxm,r =λym,rfor allx, y∈P such thaty ≥xandxm≤x, y. We definerm,s=λxm,sm . Let
fs =
m∈S
rm,sm∈K[P]. Then by the choice of the coefficientsrm,s
φ(fs)=s∈lim
P T .
This shows the first claim. Observe that ifx, y ∈ P are incomparable, then xm> x, y for allmsuch thatrm,φ(xy) =0. Thus ASL2 follows ones we have proved that the standard monomials areK-linearly independent in limPT. Assume that
0=
m∈S
amφ(m)∈lim
P T .
Choosen∈S and consider the projectionπ(T )xn: limPT →Txn. Hence
0=
m∈S
π(T )xn(amφ(m))=
m∈N(0P,xn]∩S
amφxn(m)∈Txn.
Since Txn is an ASL (0P, xn]op, we get that an = 0. This shows that the standard monomials are indeedK-linearly independent in limPT and thus we have proved that ASL1 holds. As noted above also ASL2 holds and this shows that limPT is an ASL onPop.
Example3.2 (Locally distributive lattices). We refer to [19] for the notion of distributive lattice. A finite posetP is called alocally distributive lattice, if P has a terminal element 0P and for all elementsz∈P the interval [0P, z] is a distributive lattice. For example the simplicial posets as considered in [8], [12]
and [18], i.e.Phas 0Pand for allz∈Pthe interval [0P, z] is a boolean algebra, are locally distributive lattices. Letz∈P. Since [0P, z] is a distributive lattice it follows from a result of Hibi (see [9]) that
Tz=K[[0P, z]]/(xy−(x∧y)(x∨y))
is a graded ASL onPopwith straightening relationrz(x, y)=(x∧y)(x∨y) (which is an integral domain). Here ∧ and ∨ are the meet and join in the distributive lattice [0P, z]. Note thatx∧yis the meet ofxandyinP andx∨y is a minimal upper bound ofx, yinP depending onz.
LetR =K[P]. The restriction homomorphismsTzz:Tz→Tz forz, z∈ P withz ≤ z, define aZ-gradedKP-algebra. We call limT thegeneralized Hibi ringassociated toP. By Theorem 3.1 limPT is an ASL onPopand it is easy to see that for incomparablex, y∈P the straightening relations are
r(x, y)=(x∧y)
z
z
wherezranges over all minimal upper bounds forx, y. The sum is defined to be zero if there are no such elements.
Note that ifRis agradedASL, i.e.Ris a gradedK-algebra and all elements of P are homogeneous and have positive degree, then it is known that the straightening relation (1) gives a presentation ofR(see Proposition 1.1 in [7]).
This is not true in the general case (see [14] and [21] for counterexamples).
Together with this observation we proved in Example 3.2 in fact the fol- lowing corollary:
Corollary3.3. Let P be a locally distributive lattice and letT be the KP-algebra constructed in 3.2. ThenlimP T is aZ-gradedASLonPop and forlimPT =K[P]/IT we have that the idealIT is generated by
xy−(x∧y)
z
z
wherezranges over all minimal upper bounds forx, y. The sum is defined to be zero if there are no such elements.
Hence we obtain exactly the poset ringA˜P constructed by Stanley in [17]
for simplicial posets with the same ASL structure. One immediately gets the following result:
Corollary 3.4. Let P and T be as in Corollary 3.3. IfP is a Cohen- Macaulay poset, i.e.(P )is Cohen-Macaulay simplicial complex, thenlimPT is a Cohen-Macaulay ring.
Proof. Since limP T isZ-graded and the elements ofP have positive de- gree, it follows from [7, Cor. 7.2] that limP T is a Cohen-Macaulay ring.
4. Quotients of rings of global sections
LetP be a finite poset andT be a flasqueRP-algebra. Given an open subset QofP, the restriction ofT toQinduces a surjective homomorphism
limP T →lim
Q T .
Thus limQT is a quotient of limP T. We are interested in the relationship between ring properties of these two rings. In general there is no close con- nection, but in special situations there is more hope.
The goal of this section is to generalize so-calledrank-selection theorems in the tradition of Duval [8], Hibi [10], Munkres [13] and Stanley [15] to our situation. We follow the ideas of Hibi [10] and at first we define:
Definition4.1.
(i) LetXbe a subset ofP. We denote byPXthe sub-posetP\
x∈X[x,1P) . (ii) Forx ∈P thestarofxinP is the sub-poset starP(x)consisting of the
elementsy∈P such that there existsz∈P withx, y≤z.
(iii) A non-empty subsetX⊆P is calledexcellentif for every elementzof P there exists at most onex∈Xwithx≤z.
Given an elementx ∈P we letIxdenote the kernel of the homomorphism limP T → lim
P\[x,1)T .
Lemma4.2. LetRbe aZm-graded ring,Xbe an excellent subset of a finite posetP andT be a flasqueZm-graded RP-algebra. Then there exists a short exact sequence
0→
x∈X
Ix →lim
P T →lim
PX T →0 ofZm-gradedR-modules.
Proof. The homomorphism
x∈X
Ix →lim
P T
is the sum of the inclusionsIx → limP T. This homomorphism is injective becauseXis excellent. It follows from the definition ofPXthat the sequence in question is a zero-sequence. It is right-exact, sinceT is flasque. We leave it as an exercise to the reader to check middle-exactness.
We can now apply the main result in [2].
Lemma4.3. Let(R,ᒊ)be aZm-graded local ring withK=R0a field and letP be a finite poset. LetT be a flasqueZm-graded RP-algebra such that the homomorphismR →limP T is surjective. Assume that:
(i) Tyis Cohen-Macaulay of dimensionrank(y)for everyy∈P. (ii) For everyy∈P we have that
Hi((y,1P);K)=0 for i=rank(P )−rank(y)−1. ThenlimPT is Cohen-Macaulay withdim limPT = rank(P ). In particular, ifP is a Cohen-Macaulay poset then the assumption(ii)is satisfied.
Proof. By [2, Theorem 4.1] we have that Hᒊi(lim
P T )∼=
y∈P
Hi−rank(y)−1((y,1P);K)⊗KHᒊrank(y)(Ty)
asZm-gradedK-vector spaces. By assumption (ii) we have that Hi−rank(y)−1((y,1P);K)=0
for i−rank(y)−1=rank(P )−rank(y)−1. The latter condition is equivalent toi=rank(P ). Hence
Hᒊi(lim
P T )=0 for i=rank(P ) and thus limP T is Cohen-Macaulay of dimension rank(P ).
We are ready to prove Theorem 1.1.
Proof of Theorem1.1. For everyx∈Xthe sets star(x)and star(x)∩PX
are open subsets ofP and thus the ranks of elements in these poset coincide with the ones in P. Let Jx be the kernel of the surjective homomorphism limstar(x)T →limstar(x)∩PXT. Since star(x)∩PX=star(x)\[x,1P)we get a commutative diagram ofR-modules of the form
0−−−−→ Ix −−−−→ limPT −−−−→ limP\[x,1P)T −−−−→0
↓|| ↓|| ↓|| ↓|| ↓||
0−−−−→Jx−−−−→limstar(x)T −−−−→limstar(x)∩PXT −−−−→0 Note that limPT →limstar(x)T and limP\[x,1P)T →limstar(x)∩PXT are sur- jective and thatIx →Jxis an isomorphism.
By Lemma 4.3 the rings limstar(x)Tand limstar(x)∩PXT are Cohen-Macaulay of dimension rank(star(x))=rank(P )and rank(star(x)∩PX)=rank(P )−
1 respectively. The long exact local cohomology sequence induced by the second row above shows thatJxand thusIxare Cohen-Macaulay of dimension rank(P ).
Next we consider the short exact sequence
0→
x∈X
Ix →lim
P T →lim
PX T →0
and read the statement of the theorem off from the associated long exact local cohomology sequence.
IfXis an excellent subset of a finite posetP, thenPXis calledhereditaryif the assumptions (ii)–(iv) of Theorem 1.1 are satisfied. In view of the above res- ult and a theorem of Hochster implying that the normal monoid ringK[D∩Zd] is Cohen-Macaulay of dimensiond for everyd-dimensional rational pointed coneDinRd, we obtain the following
Corollary 4.4. Let P () be the face poset of a rational pointed d- dimensional fan in Rm, let R = limPT and letT ()be the associated KP()-algebra of Example 2.1. IfXis a hereditary subset ofP ()then the homomorphism
Hᒊi
P ()lim T
→Hᒊi
P ()limX
T
is surjective fori = d, it is injective fori = d −1, and otherwise it is an isomorphism.
A sub-posetQofP is calledn-hereditary(or hereditary for short) if there exists a sequenceX1, X2, . . . , Xnof subsets ofP and a sequenceQ=P1 ⊆ P2⊆ · · · ⊆ Pn+1=P of sub-posets ofP such thatXi is an excellent subset ofPi+1andPi =(Pi+1)Xi hereditary inPi+1fori =1, . . . , n.
Corollary4.5. LetQbe ann-hereditary sub-poset of a finite posetP. Let (R,ᒊ)be aZm-graded local ring withR0 = K a field and letT be a flasqueZm-graded RP-algebra such that the homomorphismR →limPT is surjective and such thatTyis Cohen-Macaulay of dimensionrank(y)for every y∈P. Then
Hᒊi
limP T∼=Hᒊi limQ T
for i =0, . . . ,rank(P )−1−n as Zm-graded limPT-modules. In particular, iflimPT is Cohen-Macaulay andrank(Q)=rank(P )−n, thenlimQT is Cohen-Macaulay.
Theithskeletonof a posetP is the sub-poset P(i) = {y∈P : rank(y)≤i}.
Example4.6. LetP = P ()be the face poset of a rational pointed fan inRmand letT =T ()be theKP()-algebra of Example 2.1. We claim thatP(i)is a hereditary sub-poset ofP(i+1)with respect to the excellent subset
X(i) = {C∈P(i+1): rank(C)=i+1}.
ForC ∈ X(i) we have that star(C)∩P(i+1)contains its supremum, and thus it is contractible. For everyC ∈P the poset star(C)\ {C}is the face poset of the boundary of a cone and thus Cohen-Macaulay. This shows the claim.
We obtain immediately from Corollary 4.5 that depth lim
P T =max
i∈N:i ≤rank(P )and lim
P(i)T is Cohen-Macaulay . IfP is the face poset of a simplicial complexandT is the Stanley-Reisner KP-algebra associated to, then this is a well-known result.
5. Presentations of sectionedKP-algebras
Recall that for a setFwe denote byK[F] the polynomial ring on the variables a ∈ F and that thesupport of a monomialm ∈ K[F] is the set supp(m) = {a ∈ F : m(a) > 0}. Given an inclusionFx ⊆ F of finite sets we denote by πx:K[F] → K[Fx] the natural projection with πx(a) = a if a ∈ Fx
and πx(a) = 0 otherwise. There is a KP-algebra T associated to every system = (Fx)x∈P of subsets of a finite setF with Fx∩Fy = ∪z≤x,yFz
for every x, y ∈ P defined by (T)x = K[Fx] and (T)xy(f ) = πx(f ) for x ≤ y in P and f ∈ K[Fy] ⊆ K[F]. The projections πx:K[F] → K[Fx] induce a surjective homomorphismπ:K[F] →limPTwith kernel Ker(π)=
x∈PKer(πx)equal to the ideal generated by the set of monomials m in K[F] such that supp(m) is not contained in any of the setsFx. Thus the ring limPT is isomorphic to the Stanley-Reisner ring of the simplicial complexconsisting of the subsetsGofF contained inFxfor somex∈P. Definition5.1. LetT be aKP-algebra. We call a pair(, p)of a system = (Fx)x∈P of subsets of a finite setF withFx∩Fy = ∪z≤x,yFzfor every x, y ∈ P and a surjectionp:T → T ofKP-algebras apresentationofT. The homomorphisms induced bypare denotedpQ: limQT →limQT for Qopen inP andpx:(T)x →Txforx∈P.
Given a KP-algebra T and x ∈ P, the structure-homomorphism from limPT toTxis denotedπ(T )x: limPT →Tx.
Definition5.2. Asectioned KP-algebra(T , ι(T ))consists of aKP-alge- bra T and K-linear homomorphisms ι(T )x:Tx → limP T with π(T )x ◦ ι(T )x=idTx forx∈P.
Example5.3 (Monomial ideals). LetI be a monomial ideal in the poly- nomial ringS = K[x1, . . . , xn] with irredundant irreducible decomposition I = ᒊb1 ∩ · · · ∩ᒊbt. We consider the KP-algebra T = T (I )on the poset P = P (I ) of Example 2.2. ForD ⊆ C ⊆ [t], there is a preferred section S/
i∈Dᒊbi → S/
i∈Cᒊbi of the projection TDC induced by aK-linear homomorphismS→Sacting either as the identity or as zero on a monomial of S. The KP-algebra T is sectioned with ι(T )C such that the composition π(T )D◦ι(T )C is given by the composition
TC=S/
i∈C
ᒊbi →S/
i∈C∩D
ᒊbi →S/
i∈D
ᒊbi,
where the first homomorphism is the section described above and the second homomorphism is the natural projection.
The KP-algebras of the form T considered above are sectioned KP- algebras with K-linear sections ι(T)x:Tx → limPT defined by ι(T)x(f )= (py(ιx(f )))y∈P, whereιx:K[Fx] →K[F] denotes the natural inclusion.
Definition5.4. We call(, p)apresentationof the sectionedKP-algebra T if it is a presentation of T considered as a KP-algebra and the identity pP ◦ι(T)x =ι(T )x◦pxholds for everyx∈P.
The notation of the above definition can be summarized in the following commutative diagram:
K[F] K[F] K[F]
↑||
ιx π↓|| πx↓||
K[Fx]−−−−−−→ι(T)x limPT−−−−−−→π(T)x K[Fx]
↓||
px pP↓|| px↓||
Tx −−−−−−→ι(T )x limPT −−−−−−→π(T )x Tx. Let us record some facts.
Lemma5.5.If(, p)is a presentation of a sectioned KP-algebraT, then (i) 0=
x∈PKer(π(T )x)⊂limT. (ii) Ker(pP)=
x∈PKer(π(T )x◦pP).
(iii) Ker(π(T )x◦pP ◦π)=Ker(px◦πx)=ιx(Ker(px))+(F \Fx). (iv) ιx(Ker(px)) = Ker(px ◦πx)∩ιx(K[Fx]) = Ker(π(T )x◦pP ◦π)∩
ιx(K[Fx]).
The following theorem is our main result in this section.
Theorem5.6. Let(, p)be a presentation of a sectioned KP-algebraT. We have that
(i) Ker(π) =
x∈P Ker(πx) = I is the Stanley-Reisner ideal of in K[F].
(ii) Ker(pP ◦π)=
x∈P(ιx(Ker(px)))+
x∈P Ker(πx).
In particular,
limP T ∼=K[F]/
x∈P
Ix+I,
whereIx=(Ker(px◦πx)∩ιx(K[Fx]))forx∈P.
Proof. Since we have explained (i) above we only need to prove (ii).
Clearly the right hand side is contained in Ker(pP◦π). Givenf ∈Ker(pP◦π) we consider the setP (f )consisting of the elementsx ∈ P such thatf con- tains a monomialm with supp(m) ⊆ Fx. We prove the opposite inclusion by induction on the cardinality of the setP (f ). First note that P (f ) = ∅ impliesf ∈
x∈PKer(πx). IfP (f )= ∅we choose an elementx ∈P (f ). Sinceιxπx(f ) ∈ ιx(Ker(px))the elementf = f −ιxπx(f )is an element of Ker(pP ◦π) withP (f)a proper subset ofP (f ). This takes care of the induction step.
In some cases there is a relation between the simplicial complex and theK-algebra limP T. In fact, we have:
Corollary5.7. Let(, p)be a presentation of a sectioned flasque KP- algebraT consisting ofZm-graded Cohen-Macaulay rings such thatdx < dy
andfx < fyforx < yinP, wheredx = dim(Tx)andfx = |Fx|. We have that
depth lim
P T =min
i∈N:Hᒊi+(fx−dx)(K[])=0 for somex∈P , dim lim
P T =max
i ∈N:Hᒊi+(fx−dx)(K[])=0 for somex ∈P . In particular, if there exists ann∈ Zsuch thatdim(K[Fx]) = dim(Tx)+n for everyx∈P and such thatdim(K[])=dim(limPT )+n, thenlimPT is Cohen-Macaulay if and only ifis Cohen-Macaulay.
Proof. TheK-algebra limPT is isomorphic to the Stanley-Reisner ring of. The statements are a direct consequence of [2, Corollary 4.2].
Example 5.8 (Toric face rings). Let be a rational pointed fan in Rm and choose affine monoids MC for C ∈ as in Example 2.1. The KP- algebraT of Example 2.1 withTC =K[MC] is a sectionedKP-algebra with
ι(T )C:TC→limPT determined by requiring the compositionπ(T )D◦ι(T )C
to be the composition of the inclusionK[MC] ⊆ K[ME] and the face pro- jectionK[ME] →K[MD] ifC andD are faces of a common coneEin, and requiringπ(T )D◦ι(T )Cto be zero otherwise. Denoting the Hilbert basis of MC by FC, the surjectionspC:K[FC] → K[MC] define a presentation (, p)of the sectionedKP-algebraT with = (FC)C∈. In the particular case whereMC =Zm∩Cfor everyC∈, theKP-algebraT is denotedT (), andT ()C is Cohen-Macaulay of dimension dim(C)since it is the monoid ring of a normal affine monoid. Note that Corollary 5.7 applies toT ().
Finally, we want to compute the initial ideals of the presentation ideal of limPTwith respect to weight orders inωonK[F] induced by a mapω:F →R.
Theorem5.9.Let(, p)be a presentation of a sectioned KP-algebraT. For a mapω:F →Rwe have that
K[F]/inω(Ker(pP ◦π))∼=lim
P K[Fx]/inω(Ker(px)).
Proof. TheKP-algebra inωT with inωTx =K[Fx]/inω(Ker(px))is pre- sented by(, q), whereqxis the projectionK[Fx]→K[Fx]/inω(Ker(px)). Hence by Theorem 5.6
Ker(qP ◦π)=
x∈P
ιx(Ker(qx))
+
x∈P
Ker(πx).
Clearly ιx(Ker(qx)) = ιx(inω(Ker(px))) is contained in inω(Ker(pP ◦π)), and thus the identity on K[F] induces a homomorphism K[F]/Ker(qP ◦ π)→K[F]/inω(Ker(pP ◦π)). On the other hand,πx(inω(Ker(pP ◦π)))⊆ inω(Ker(px)), and thus the projectionK[F]→K[Fx]/inω(Ker(px))induces a homomorphism
K[F]/inω(Ker(pP ◦π))→K[Fx]/inω(Ker(px)).
These homomorphisms in turn assemble to a homomorphism K[F]/inω(Ker(pP ◦π))→lim
P K[Fx]/inω(Ker(px))∼=K[F]/Ker(qP◦π).
We leave it as an exercise to the reader that this is an inverse isomorphisms to K[F]/Ker(qP ◦π)→K[F]/inω(Ker(pP ◦π)).
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DEPT. OF MATHEMATICS UNIVERSITY OF BERGEN JOHS. BRUNSGT. 12 N-5008 BERGEN NORWAY
E-mail:morten.brun@math.uib.no
FB MATHEMATIK/INFORMATIK UNIVERSITÄT OSNABRÜCK 49069 OSNABRÜCK GERMANY
E-mail:troemer@mathematik.uni-osnabrueck.de
