Non-normal affine monoids

0.29 MB·Italian

by  Lukas Katthän

NON-NORMAL AFFINE MONOIDS
2 LUKASKATTHA¨N
1
0
Abstract. Wegiveageometricdescriptionofthesetofholesinanon-normal
2
affine monoid Q and show how this relates to various properties of Q likelo-

t cal normality and Serre’s conditions (R1) and (S2). A combinatorial upper
c
boundforthedepththemonoidalgebrak[Q]isobtained. Moreover,weshow
O
the vanishing and non-vanishing of certain graded components of the local
cohomology of k[Q]. We then apply our results to simplicialand to seminor-
9
mal affine monoids. Finally, we prove a special case of the Eisenbud-Goto
1
conjecture.

]
C
A
. 1. Introduction
h
t Let Q be an affine monoid, i.e. a finitely generated submonoid of ZN for some
a
m N ∈ N. Further, let Q denote the normalization of Q. In this paper, we give
a geometric description of the set of “holes” Q\Q in Q and relate it to several
[
properties of Q. Our main result in this direction is the following.

2
v Theorem(Theorem 3.8). LetQbeanaffinemonoid. Thereexistsa(non-disjoint)
8 decomposition
5
l
2
6 (1) Q\Q= (qi+ZFi)∩R+Q
. i=1
9 [
0 withqi ∈Qandfaces Fi ofQ. Iftheunionisirredundant(meaningthatnoqi+ZFi
2 can be omitted), then the decomposition is unique.
1
: We call a set qi+ZFi appearing in (1) anj-*dimensional family of holes, where
v
j isthe*dimensionofF (SeeSection 2forthedefinitionofthe*dimension). There
i
X isanalgebraicinterpretationofthe sets appearingin(1). Letk be a fieldandk[Q]
r be the monoid algebra of Q. Then the faces appearing in (1) correspond to the
a
associated primes of the quotient k[Q]/k[Q]. The same face may appear several
times in (1), in fact, the number of times a face appears equals the multiplicity of
the corresponding prime.
In[1,Prop. 2.35]adifferentdecompositionoftheholesisconsidered. Itisshown
there that one can always find a decomposition of Q\Q into a disjoint union of
translates of faces of Q:
l
(2) Q\Q= q +F
i i
i=1
[
Date:October 22,2012.
2010 Mathematics Subject Classification. Primary52B20;Secondary 14M25,13D45.
Key words and phrases. monoidalgebra;affinemonoid;affinesemigroup;localcohomology.
ThisworkwassupportedbytheDAAD.
1
2 LUKASKATTHA¨N
(a) Thedecompositionof[1] (b)Ourdecomposition
Figure 1. Differentdecompositionoftheholesofa2-dimensional
affine semigroup
Infact,thisstatementanditsproofhavebeenthemotivationforprovingTheorem 3.8.
Figure 1 shows an example of both kinds of decomposition. The decomposition
given in (2) is disjoint, but far from being unique. On the other hand, our decom-
position is (in general) not disjoint, but it is unique. Moreover, it behaves nicely
under localization, see Proposition 3.10. Several ring-theoretic properties of the
monoid algebra k[Q] can be described in terms of the families of holes:
Theorem (Theorem 4.3). Let Q be an affine monoid of *dimension d. The fol-
lowing holds:
• If d ≥ 2, then ∗depthk[Q] = 1 if and only if there is a 0-*dimensional
family of holes.
• Q is locally normal if and only if there is no family of holes of positive
*dimension.
• k[Q]satisfiesSerre’scondition (R )ifandonlyifthereis nofamilyofholes
1
of *dimension d−1.
• k[Q] satisfies Serre’s condition (S ) if and only if every family of holes has
2
*dimension d−1.
Note that this implies that if Q is locally normal (very ample), but not normal,
then ∗depthk[Q] = 1. We generalize the first item of the preceding theorem to
obtain an upper bound on the *depth of k[Q]:
Theorem (Theorem 4.4). If Q has an i-*dimensional family of holes, then the
*depth of k[Q] is at most i+1.
This theoremstatesthata non-normalaffinemonoidwith “few”holeshasalow
*depth. Thisissomewhatcounterintuitive,becauseHochster’sTheoremstatesthat
the absence of holes, i.e. normality, implies maximal *depth. In small examples, it
is often not difficult to determine the bound given by this theorem geometrically.
Thiscanbe easierthanto compute the actual*depthalgebraically. Ingeneral,the
*depth may be strictly smaller than the bound given by Theorem 4.4. However,
NON-NORMAL AFFINE MONOIDS 3
in Proposition 4.5, Proposition6.2 and Proposition6.11 we identify some special
cases where equality holds. See Example 4.7 for an application.
In the second part of this paper, we consider the local cohomologyHi (k[Q]) at
m
the maximal graded ideal m. Recall that for an element q ∈ −intQ, it holds that
Hd(k[Q]) =k and Hi (k[Q]) =0 for all other i. So this part of Hi (k[Q]) is well
m q m q m
understood. It turns outthatthe remaining gradedcomponents arecloselyrelated
to the families of holes of Q.
Theorem(Theorem 5.3). Considerq ∈ZQsuchthatq ∈/ −intQ. IfHmi+1(k[Q])q 6=
0 for some i, then q is contained in a family of holes of *dimension at least i.
There is also a partial converse of the preceding result.
Corollary (Corollary 5.5). Every i-*dimensional family of holes contains an ele-
ment q ∈ZQ, such that Hmi+1(k[Q])q =k. If i>0, then there are in fact infinitely
many such elements (even up to units).
Note that this result extends Theorem 4.4. We will also giverestrictions oncer-
tain infinite parts of the local cohomology of k[Q] in Lemma 5.6. See Example 5.7
for an example of the geometric meaning of these results.
In Section 6, we apply our results to simplicial and seminormal affine monoids.
Forsimplicialaffinemonoids,thecharacterizationoftheCohen-Macaulayproperty
of[6]isextendedtothenon-positivecase. Forseminormalaffinemonoids,wegivea
new proofof the cohomologicalcharacterizationofseminormality of [3]. While our
proof is not actually simpler than the original one, we believe that it offers a new,
more geometric perspective. Moreover, we extend this and some other results of
[3] to the non-positive case. Further, a simplicial seminormal affine monoid whose
*depth depends on the characteristic of the field is constructed.
Inthelastsection,someadditionalresultsarelisted. Wegiveadirectproofthat
outcriterionforSerre’scondition(S )isequivalenttotheonegivenin[11,20]. Fur-
2
ther,aninterpretationofgreatestsize ofa family ofholesis given. Finally, wegive
aboundontheCastelnuovo-Mumfordregularityofseminormalhomogeneousaffine
monoids and prove a special case of the Eisenbud-Goto conjecture (Theorem 7.5).
2. Preliminaries and notation
AnaffinemonoidQisafinitelygeneratedsubmonoidoftheadditivemonoidZN
for an N ∈ N. Equivalently, Q is a commutative, finitely generated, cancellative
and torsionfree monoid. For general information about affine monoids see [1] or
[13]. We denote the groupgeneratedby Q by ZQ, the convex cone generatedby Q
by R Q ⊆ RN and the normalization by Q = ZQ∩R Q. Recall that an element
+ +
q ∈ZQ is contained in Q if and only if a multiple of q lies in Q. A face F ⊆Q of
Q is a subset such that for a,b∈Q the following holds:
a+b∈F ⇐⇒ a,b∈F
A unit is an element in u∈Q, such that −u∈Q. The set of units forms a face F
0
thatiscontainedineveryotherfaceofQ. We callQpositive if0isthe onlyunitin
Q. Foreveryelementq ∈Q,thereexistsauniqueminimalfaceF containingq. We
say that q is an interior point of F and write intF for the set of interior points of
F. Note that by definition 0∈intF . The dimensionof a face F is the rank of the
0
free abelian group ZF generated by F. Since we are working with not-necessarily
positive affine monoids, it is more convenient to consider a normalized version of
4 LUKASKATTHA¨N
the dimension. So we define the *dimension as ∗dimQ := dimQ− ∗dimF , and
0
∗dimF := dimF −dimF for every face F of Q. For a field k, we write k[Q] for
0
themonoidalgebraofQ. Further,foranelementq ∈Q,wewritexq ∈k[Q]forthe
corresponding monomial. For a face F we define p ⊆k[Q] to be the vector space
F
generated by those monomials xq such that q ∈ Q\F. Then p is a monomial
F
prime ideal of k[Q] and all monomial prime ideals are of this type. Moreover,
k[Q]/pF ∼= k[F]. k[Q] carries a natural ZQ-grading. With respect to this grading,
the homogeneous ideals k[Q] are exactly the monomial ideals. Thus the ideal p
F0
associated to the minimal face is the unique maximal graded ideal of k[Q]. We
will sometimes write m for this ideal. Its height equals the maximal length on an
descending chain of faces of Q, so (k[Q],m) is a *local ring of *dimension ∗dimQ.
More general, the height of p equals ∗dimF for every face F. The *depth of k[Q]
F
is the maximal length of a regular sequence in m. Equivalently, it is the depth of
the (inhomogeneous) localization k[Q] . For a face F of Q, we denote by
m
Q :={q−f q ∈Q,f ∈F }
F
the localization of Q at F. It holds that k[Q ] = k[Q] , where the later is the
F (pF)
homogeneous localization of k[Q] at p . Further, it holds that (Q) = (Q ), i.e.
F F F
localizationand normalizationcommutes. Note that localizations arealmostnever
positive.
A set M is called a Q-module if there is an operation Q×M → M (additively
written) of Q on M, such that (q +p)+m = q +(p+m) and 0+m = m for
q,p ∈ Q,m ∈ M. If M is a Q-module, then the vector space k{M} with basis
givenbythe elements ofM is naturallyak[Q]-module. IfM ⊆ZQ, thenwedefine
the localization M := {m−f m∈m,f ∈F } of M at a face F. One may also
F
consider the localization for general modules, but we only need this special case.
Note that if U ⊆M ⊆ZQ are modules, then k{MF}/k{UF}=k{M}/k{U}(pF).
Foragradedk[Q]-moduleN (inthealgebraicsense),thesupport ofN,suppN,is
definedtobethesetofthoseq ∈ZQ,suchthatthereexistsanelementofdegreeqin
N. IfM isaQ-moduleandU ⊆M asubmodule,thensuppk{M}/k{U}=M\U.
Definition 2.1. Let Q be an affine monoid. We call Q locally normal if every
localization Q at a face F 6=F is normal.
F 0
Since localizations of normal affine monoids are again normal, it is enough to
consider faces of *dimension 1. For a polytope P ⊂ RN−1 with integer vertices,
one often considers the polytopal affine monoid QP ⊆ ZN generated by the set
(p,1) p∈P ∩ZN−1 . In this case, QP is locally normal if and only if the poly-
topeP isvery ample. Toseethis,notethatthelocalizationofQ atavertex(v,1)
P
(cid:8) (cid:9)
splitsintoadirectsumofthecorner cone onv andacopyofZ. Thus,QP islocally
normal if and only if all the corner cones of P are normal, which is the definition
of very ampleness. It is known that P is very ample if and only if there are only
finitely many holes in Q . The corresponding statement in the general case is the
P
following:
Proposition 2.2. An affine monoid is locally normal if and only if
rank k[Q]/k[Q]<∞
k[F0]
for any field k. This rank does not depend on the field.
NON-NORMAL AFFINE MONOIDS 5
Recall that the face F consists of the units of Q, so this lemma amounts to
0
saying that there are only finitely many holes up to units. Every graded module
over k[F ] is free (cf. [7, Theorem 1.1.4]), so the rank is well-defined. We will not
0
use this lemma in the sequel, so we omit the proof.
3. The module of holes
In this section, we describe the structure of the set of holes Q\Q. Following an
idea from [1, p. 139], we consider a more general situation. Let M be a finitely
generated Q-submodule of ZQ and let U ⊆ M be a submodule of M. We are
interestedin the structure ofthe difference M\U. Clearly,in the caseM =Q and
U =Qthis correspondstotheholesQ\Q. Whileforourpurposeitwouldactually
suffice to consider this case, we believe that the additional generality makes the
exposition more clear. Another case of potential interest is N = Q and U ⊆ Q a
submodule. This correspondsto a monomialidealink[Q]. As notedabove,the set
M \U can be encoded as the support of the quotient k{M}/k{U}. The following
simple observation is the key idea: Consider an m ∈ M \U and a q ∈ Q. Let xm
and xq denote the corresponding monomials in k{M}/k{U} resp. in k[Q]. Then
q+m∈U ⇐⇒ xqxm =0.
Now let F be a face of Q. It holds that m ∈ M , because M ⊆ M . However,
F F
m∈U ifandonlyifxmgoestozerointhelocalizationofthequotientk{M}/k{U}
F
at p . This is the case if and only if the annihilator of xm is not contained in p ,
F F
i.e. if there is a q ∈ F such that q +m ∈ U. Consider the case that m ∈/ U
F
and F is maximal with this property. By this we mean that m ∈ U for all faces
G
G)F. Because pG ⊆pF, this is equivalent to pF being a minimal prime over the
annihilator of xm. We summarize what we have proven:
Lemma 3.1. Let F be a face of Q, m ∈ M \ U and xm be the corresponding
monomial in k{M}/k{U}. Then F +m ⊆ M \U if and only if p contains the
F
annihilator of xm. Moreover, F is a maximal face with this property if and only if
p is a minimal prime of the annihilator of xm
F
InviewofourobjectivetofindanirredundantdecompositionofthesetM\U,it
seemsnaturalto takethe largestpossible pieces. Therefore,weconsiderthe family
of sets
F(M)={ZF +m⊆ZQ m∈M \U,pF is a minimal prime over Annxm} .
These sets will yield the desired decomposition. Note that for m,n ∈ M \U with
m−n ∈ ZF, it holds that pF is a minimal prime over Annxm if and only if it
is minimal over Annxn. So we are free in choosing representatives of the sets in
F(M).
We first show that their union comprises all of M \U:
Lemma 3.2. It holds that
(3) M \U = S∩M
S∈[F(M)
Proof. Every monomial xm ∈k{M}/k{U}has at least one minimal prime overits
annihilator, so the left-hand side of (3) is clearly contained in the right-hand side.
On the other hand, consider n ∈ ZF +m∩M for ZF +m ∈ F(M). There exist
6 LUKASKATTHA¨N
f ,f inF suchthatf +m=f +n. Itfollowsthatn∈/ U,becauseF+m⊆M\U
1 2 1 2
by Lemma 3.1, and hence n∈M \U. (cid:3)
Next, we consider the behaviour of F(M) under localization.
Lemma 3.3. Let F ⊆G be faces of Q and let m∈ZQ. Then ZG+m∈F(M) if
and only if ZG+m∈F(MF).
Proof. If m ∈ M \U , then there exists an f ∈ F such that f +m ∈ M \U
F F
and, since F ⊆ G, it holds that ZG+m = ZG+f +m. So we can assume that
m ∈ M \U. In this case, ZG+m ∈ F(M) if and only if pG is minimal over the
annihilatorofxm ∈k{M}/k{U}. Butthis propertyispreservedunder localization
with p ⊇p . Hence the claim follows. (cid:3)
F G
Using the preceding lemma, we prove the finiteness of our decomposition:
Lemma 3.4. For every face F of Q, the number of sets of the form ZF +m ∈
F(M) equals the multiplicity of p on k{M}/k{U}. In particular, F(M) is finite.
F
Moreover, a face F appears in F(M) if and only if p is an associated prime of
F
k{M}/k{U}.
Proof. The multiplicity of p in k{M}/k{U} can be defined as the length of the
F
module
N :=H0 (k{M }/k{U })=span {xm ∈k{M }/k{U } pnxm =0 for n≫0},
pF F F k F F F
see [4, p. 102]. Note that the multiplicity is invariant under localization at F. By
Lemma 3.3, the same holds for the number of sets of the form ZF +m ∈ F(M).
So we may assume that F is the minimal face of Q.
Since N is a module of finite length (in the graded sense), we may consider a
composition series of N, i.e. a filtration 0 ( N1 ( N2 ( ··· ( Nr = N of graded
modules, such that each quotient N /N is a simple graded module.
i i−1
We claim that suppN is the union of the sets ZF +m ∈ F(M) (for our fixed
F). Indeed, m ∈ M \U is contained in suppN if and only if the annihilator of
xm is p -primary. Since p is the maximal graded ideal of k[Q], this is equivalent
F F
to saying that p is a minimal prime over the annihilator of xm. Now, for every
F
ZF +m∈F(M), there exists an i with m∈suppNi\suppNi−1. But this implies
thatZF+m=suppNi\suppNi−1,because Ni/Ni−1 is simple. Thus,the number
ofsetsoftheformZF+minF(M)equalsthelengthrofthecompositionseries. (cid:3)
We now turn to proving the irredundancy and the uniqueness of (3). First, we
giveavariantofthewell-knownfactthatavectorspaceoveraninfinitefieldcannot
be written as a union of finitely many subspaces.
Lemma 3.5. Let V be a vector space over Q and C ⊆V be a convex cone (i.e. a
subset such that for v,w ∈C and λ,µ≥0 it follows λv+µw ∈C). If C contains a
generating set of V, then it is not contained in any finite union of proper subspaces
of V.
Proof. Assume to the contrarythatthe coneC is containedin the unionoffinitely
many proper subspaces V ,...,V of V. We may further assume that none of the
1 l
subspaces is contained in the union of the others and that every V has a non-
i
empty intersection with C. We have certainly at least two subspaces, because C
containsageneratingsetofV. Hencewecanchooseelementsx ,x ∈C,suchthat
1 2
NON-NORMAL AFFINE MONOIDS 7
xi ∈ Vi\ j6=iVj for i= 1,2. For every i≥ 2 there exists at most one λ ∈Q with
λx +x ∈V . Indeed, if we hadλx +x ∈V andλ′x +x ∈V for two different
1 2 i 1 2 i 1 2 i
S
λ,λ′ ∈Q, then (λ−λ′)x1 ∈Vi, a contradictionto our choice of x1. Since there are
infinitely many non-negative rationalnumbers and it holds λx +x ∈C for every
1 2
such λ ≥ 0, we conclude that there exists a λ ∈ Q such that λx +x ∈ V . But
1 2 1
now x =(λx +x )−λx ∈V , a contradiction. (cid:3)
2 1 2 1 1
Next we prepare a discrete analogue of the preceding lemma.
Lemma 3.6. Let q,p1,...,pl ∈ ZQ be lattice points and let F,G1,...,Gl be (not
necessarily distinct) faces of Q. If F +q is contained in the union iZGi +pi,
then it is already contained in one of the sets ZGi+pi.
S
Note that this Lemma does notholdfor arbitrarysubgroupsofZN, for example
Z=2Z∪(2Z+1).
Proof. WemayassumethatF+qhasanon-emptyintersectionwitheveryZGi+pi
for 1≤i≤l. If F ⊆Gi for any i, then F +q ⊆ZF +q′ ⊆ZGi+q′ =ZGi+pi for
q′ ∈F +q∩ZGi+pi. Thus in this case our claim holds. We will show that there
exists always an i such that F ⊆G .
i
So, assume to the contrary that F *Gi for every i. As a notation, for a subset
S ⊆QN,wewriteQSfortheQ-subspacegeneratedbyS. Then,Q(ZF∩ZGi)(QF
for every i. Indeed, it holds that Q(ZF ∩ZGi) ⊆ QF ∩QGi ⊆ QF. The second
inclusion is strict except in the case that QF ⊆ QGi. But this would imply that
F ⊆Gi,becauseF =QF∩QandGi =QGi∩Q. Notethatthisisthepointwhere
we use that we deal with faces of an affine monoid.
By Lemma 3.5, we canfind anelement pˆin the cone generatedby F that is not
contained in any Q(ZF ∩ZGi). By multiplication with a positive scalar, we can
assume pˆ∈ F. For every non-negative integer λ, it holds that λpˆ+q ∈ F +q ⊆
iZGi +pi. Since the union is finite, there exists an index i and two different
integers λ,λ′ ∈ Z such that λpˆ+q,λ′pˆ+q ∈ ZGi +pi. But now it follows that
S(λ−λ′)pˆ∈ZF ∩ZGi and thus pˆ∈Q(ZF ∩ZGi), a contradiction to our choice of
pˆ. (cid:3)
Nowwearereadytoprovethatourdecompositionisinfactirredundantandunique.
Lemma 3.7. Consider a finite decomposition
M \U = (ZGi+mi)∩M
i
[
of M \U with m ∈M \U and faces G of Q. Then every set in F(M) appears in
i i
this decomposition. Thus, (3) defines the unique irredundant finite decomposition
of M \U.
Proof. Letm∈M\U andF afaceofQsuchthatZF+m∈F(M). ByLemma 3.6,
there exists an index i such that F +m ⊆ ZGi +mi. In particular, ZGi +m =
ZGi + mi. Hence F ⊆ ZGi and therefore F ⊆ ZGi ∩ Q = Gi. On the other
hand, by Lemma 3.1, F is a maximal face of Q such that F +m ⊆ M \U. But
Gi+m⊆ZGi+m∩M ⊆M\U,so we conclude thatGi =F. Whence ZF +m=
ZGi+mi. (cid:3)
8 LUKASKATTHA¨N
We summarize what we have proven in this section in the following theorem.
Theorem 3.8. Let Q be an affine monoid, let M ⊆ ZQ be a module and let
U ⊆M beasubmodule. Thenthereexistsauniqueirredundantfinite(non-disjoint)
decomposition
(4) M \U = (ZFi+mi)∩M.
i
[
Thenumberof timesF appears in (4)equals themultiplicity ofp onk{M}/k{U}.
F
In particular, a face F appears in (4) if and only if p is an associated prime of
F
k{M}/k{U}. Geometrically, a set ZFi +mi appears in (4) if and only if Fi is a
maximal face such that F +m ⊆M \U.
i i
Fromnowon,wespecializeourdiscussiontothecaseM =QandU =Q. Letuscall
a set ZF +m appearing in (4) a j-dimensional family of holes, where j = ∗dimF.
For the ease of reference, call a face F associated to Q if p is an associated prime
F
of k[Q]/k[Q]. The following is immediate:
Corollary 3.9. Q has a j-*dimensional family of holes if and only if there is a
j-*dimensional associated face of Q.
We get a description of the holes of the localization Q from Lemma 3.3:
F
Proposition 3.10. Let F be a face of Q. The families of holes of Q are exactly
F
those families of holes ZG+q of Q which satisfy F ⊆ G. In particular, QF is
normal if and only if no associated face contains F.
4. Special configurations of holes
In this section, we show various ring-theoretical properties of k[Q] correspond
to specialconfigurationsof the holesin Q. We startby establishing the connection
between the *depth and the module of holes:
Proposition 4.1. Assume that Q is not normal. Then
∗depthk[Q]=1+ ∗depthk[Q]/k[Q]
Proof. The idea is to consider the short exact sequence
0−→k[Q]−→k[Q]−→k[Q]/k[Q]−→0
of k[Q]-modules. We will prove that k[Q] is Cohen-Macaulay as a k[Q]-module.
This implies our claim by the depth lemma, [2, Prop. 1.2.9].
Everysystemofparametersx=x ,...,x ofk[Q]isalsoasystemofparameters
1 d
of k[Q] (as a module over itself). This follows from the graded analogue of [2,
TheoremA8]. Sincek[Q]isCohen-MacaulaybyHochster’sTheorem[1,Thm. 6.12],
x is a regular sequence for k[Q]. But this property depends only on the action of
x on k[Q], and thus x is also a k[Q]-regular sequence for k[Q] as an k[Q]-module.
Hence k[Q] is Cohen-Macaulay as a k[Q]-module. (cid:3)
Note that if ∗depthk[Q]< ∗dimk[Q], then by the previous proof depthk[Q] >
m
dimk[Q] as a k[Q] -module. Thus, by the Auslander-Buchsbaum Formula, its
m m
projective dimension is infinite. We recall the definition of Serre’s conditions.
Definition 4.2. Let R be a Noetherian ring.
NON-NORMAL AFFINE MONOIDS 9
(1) R satisfies Serre’s condition (R ), if R is a regular local ring for all p ∈
1 p
SpecR of height at most 1.
(2) R satisfies Serre’s condition (S ), if depthR ≥ min(2,dimR ) for all p ∈
2 p p
SpecR.
If R is graded, then for both properties it is enough to consider homogeneous
localizationsathomogeneousprimeideals. For(S ), thisfollowsfrom[7,Corollary
2
1.2.4] and for (R ) it was shown in [21, Proposition 2.1].
1
Theorem 4.3. Let Q be an affine monoid of *dimension d. The following holds:
• If d ≥ 2, then ∗depthk[Q] = 1 if and only if there is a 0-*dimensional
family of holes.
• Q is locally normal if and only if there is no family of holes of positive
*dimension.
• k[Q]satisfiesSerre’scondition (R )ifandonlyifthereis nofamilyofholes
1
of *dimension d−1.
• k[Q] satisfies Serre’s condition (S ) if and only if every family of holes has
2
*dimension d−1.
Proof. We start by proving the criterion for ∗depthk[Q] = 1. By Proposition 4.1,
it holds that ∗depthk[Q] = 1 if and only if ∗depthk[Q]/k[Q] = 0, which in turn
holdsifandonlyifthemaximalmonomialidealisanassociatedprimeofk[Q]/k[Q].
Since the maximal monomial ideal corresponds to the minimal face F of Q, the
0
claim follows from Corollary 3.9.
Next,weprovethecriterionforlocalnormality. Recallthatk[Q]islocallynormal
ifits localizationsatall1-*dimensionalfaces arenormal. By Proposition 3.10, this
is clearly equivalent to the statement that there are no families of holes of positive
*dimension.
To prove the criterion for Serre’s condition (R ), note k[Q] satisfies R if and
1 1
only if k[Q ] is regular for every facet F. On the other hand, by Proposition 3.10,
F
Q is normalfor everyfacet if and only if there is no (d−1)-dimensionalfamily of
F
holes. Everyregularringisnormal,sooneimplicationisclear. Since ∗dimQ =1,
F
it remains to show that 1-*dimensional normal monoid algebras are regular. But
every 1-*dimensional normal monoid is isomorphic to Zm⊕N for some m ∈N, so
the corresponding algebra is indeed regular.
Finally, we prove the criterion for Serre’s condition (S ). By above discussion
2
and Proposition 3.10, it holds for any face F of Q that ∗depthk[Q ] = 1 if and
F
only if F is associated to Q. On the other hand, it holds that ∗dimk[Q ] = 1 if
F
and only if F is a facet of Q. Therefore, Serre’s condition (S ) is satisfied if and
2
only if every associated face of Q is a facet. (cid:3)
NotethattheprecedingtheoremimpliesthatQisnormalifandonlyifitsatisfies
(R ) and (S ). Serre’s Theorem ([2, Theorem 2.2.22] states that this holds for
1 2
general Noetherian rings. Further, if ∗dimQ ≥ 2, then Q is normal if and only if
it is locally normal and ∗depthk[Q]≥2. It follows from Serre’s Theorem that this
statement also holds for general Noetherian rings.
The first part of the preceding Theorem can be generalized to an upper bound
on the *depth:
Theorem 4.4. If Q has an i-*dimensional family of holes, then the *depth of k[Q]
is at most i+1.
10 LUKASKATTHA¨N
Proof. ByProposition 4.1wecanconsiderthe*depthofk[Q]/k[Q]. ByTheorem 3.8
the families of holes of Q correspond to the associated primes of k[Q]/k[Q]. Now
the claim follows from the general fact that ∗depthk[Q]/k[Q] ≤ ∗dimk[Q]/p for
every p∈Assk[Q]/k[Q], see [2, Prop 1.2.13]. (cid:3)
We identify a special case where equality holds:
Proposition 4.5. If Q\Q=q+F for an element q ∈Q\Q and a face F of Q,
then ∗depthk[Q]=1+ ∗dimF.
Note that it is not sufficient to require that there is only one family ofholes, see
Example 5.7. Before we can prove Proposition 4.5, we prepare another Lemma:
Lemma 4.6. Let q ∈ Q and F a face of Q, such that F + q ⊆ Q \Q. Then
F +q ⊆Q\Q.
Proof. Assume the contrary,so there exists anelement f ∈F such that f+q ∈Q.
We can write f = f −f with f ,f ∈ F. But f +f +q = f +q ∈ Q\Q by
1 2 1 2 2 1
assumption, a contradiction. (cid:3)
Proof of Proposition 4.5. By Lemma 3.1, our hypothesis is equivalentto the state-
ment that the module k[Q]/k[Q] is cyclic with annihilator pF. Hence k[Q]/k[Q]∼=
k[Q]/p =k[F](Theisomorphismshiftsthegrading). Next,recallthatthe*depth
F
of a module M over a ring R equals its *depth over R/AnnM. Together with
Proposition 4.1 this yields
∗depth k[Q]=1+ ∗depth k[F].
k[Q] k[F]
NowweuseLemma 4.6toconcludethatQ\Q=F+q ⊆F+q ⊆Q\Q,soF =F.
Hence k[F] is normal and the result follows from Hochster’s Theorem. (cid:3)
Wegiveanexamplehowonecaneffectivelycomputethe*depthusingProposition4.5.
Example4.7. LetGbeagraphwithvertexsetV andedgesetE. Weassociatean
affinemonoidtoG, the toric edge ring k[G], introducedin[15]. This isthe monoid
algebra to the monoid generated by the vectors ei+ej ∈Z#V, where {i,j} is an
edge of G and e ,e denotes unit vectors indexed by the vertices of G.
i j
For positive k ∈N consider the graphGk+6 in Figure 2. In [9] the *depth ofthe
edge ring of this family of graphs is computed. We will show that these edge rings
satisfytheassumptionofProposition 4.5,andthusgiveanalternativecomputation
of the *depth.
First, it is knownthat k[Q] is generatedas a k[Q]-module by x x x x x x , i.e.
1 2 3 4 5 6
the monomial corresponding to the vector q ∈ Q ⊆ Rk+6 which assigns 1 to the
vertices 1,...,6. If we add one of the “middle” edges, e.g. {3,8}, to q, then it is
easyto seethat the resultlies inQ. Onthe otherhand, ifwe addanycombination
ofedgesfromthetrianglestoq,thentheresultwillalwaysbe inQ\Q. Toseethis,
notethatthesumovertheverticesofeachtriangleisalwaysodd. Thisimpliesthat
Q\Q=F +q, where F is the face spanned by the six edges in the triangles. The
dimension of F is 6, so by Proposition 4.5 it follows that depthk[Q]=1+6=7.
Wegeneralizethiscomputationtoshowthateverytoricedgeringcanberealized
as a combinatorial pure subring of a toric edge ring of *depth at most 7. The
construction is as follows: To a given graph G, add two triangles on six (in total)
newvertices. ThenconnecteveryvertexofGwitheverynewvertex. Obviously,the
toric edge ring of G is a combinatorial pure subring of the edge ring of this bigger