On simple A-multigraded minimal resolutions

0.19 MB·English

by  Hara Charalambous

9
0 On simple A-multigraded minimal resolutions
0
2
Hara Charalambous and Apostolos Thoma
n
a
J Abstract. Let Abeasemigroupwhoseonlyinvertibleelement is0. For an

9 A-homogeneous ideal we discuss the notions of simple i-syzygies and simple
minimalfreeresolutionsofR/I. WhenIisalatticeideal,thesimple0-syzygies

] ofR/I arethebinomialsinI. Weshowthatforanappropriatechoiceofbases
C everyA-homogeneous minimalfreeresolutionofR/I issimple. Weintroduce
A the gcd-complex∆gcd(b)foradegree b∈A. We showthat thehomology of
. ∆gcd(b) determines the i-Betti numbers of degree b. We discuss the notion
h of an indispensable complex of R/I. We show that the Koszul complex of a
t completeintersectionlatticeidealIistheindispensableresolutionofR/Iwhen
a
theA-degreesoftheelementsofthegeneratingR-sequenceareincomparable.
m
[

1
v
1. Notation
6
9 LetL⊂Zn be alattice suchthatL∩Nn ={0}andletAbethe subsemigroup
1 of Zn/L generated by {a = e +L : 1 ≤ i ≤ n} where {e : 1 ≤ i ≤ n} is the
1 i i i
canonical basis of Zn. Since the only element in A with an inverse is 0, it follows
.
1
that we can partially order A with the relation
0
9 c≥d⇐⇒ there is e∈A such that c=d+e.
0
: Let k be a field. We consider the polynomial ring R = k[x ,...,x ]. We set
v 1 n
i degA(xi)=ai. If xv =xv11···xvnn then we set
X
deg (xv):=v a +···+v a ∈A .
r A 1 1 n n
a
ItfollowsthatRispositivelymultigradedbythesemigroupA,see[13]. Thelattice
idealassociatedtoListheidealIL(orIA)generatedbyallthebinomialsxu+−xu−
where u+,u− ∈ Nn and u = u+ −u− ∈ L. We note that if xu+ −xu− ∈ IL
then deg xu+ = deg xu−. Prime lattice ideals are the defining ideals of toric
A A
varieties and are calledtoric ideals, [23]. In generallattice ideals arise in problems
fromdiverseareasof mathematics,including toric geometry,integer programming,
dynamical systems, graph theory, algebraic statistics, hypergeometric differential
equations, we refer to [12] for more details.
WesaythananidealIofRisA-homogeneousifitisgeneratedbyA-homogeneous
polynomials, i.e. polynomials whose monomial terms have the same A-degree.
1991 Mathematics Subject Classification. 13D02, 13D25.
Key words and phrases. Resolutions, lattice ideal, syzygies, indispensable syzygies, Scarf
complex.
1
2 H.CHARALAMBOUSANDA.THOMA
Lattice ideals are clearly A-homogeneous. For the rest of the paper I is an A-
homogeneous ideal. For b ∈ A we let R[−b] be the A-graded free R-module of
rank 1 whose generator has A-degree b. Let
(F ,φ): 0−→F −φ→p ······−→F −φ→1 F −→R/I−→0,
• p 1 0
be a minimal A-graded free resolution of R/I. The i-Betti number of R/I of A-
degree b, β (R/I), equals the rank of the R-summand of F of A-degree b:
i,b i
βi,b(R/I)=dimkTori(R/I,k)b
andis aninvariantof I, see [13]. The degreesb for whichβ (R/I)6=0 are called
i,b
i-Betti degrees. The minimal elements of the set {b : β (R/I) 6= 0} are called
i,b
minimali-Betti degrees. The elements ofImφ =kerφ arethe i-syzygiesofR/I
i+1 i
in F .
•
TheproblemofobtaininganexplicitminimalfreeresolutionofR/Iisextremely
difficult. One of the factors that make this problem hard to attack, is that given a
minimalfreeresolutiononecanobtainbyachangeofbasisadifferentdescriptionof
thisresolution. Toobtainsomecontroloverthis,in[10]wedefinedsimple minimal
freeresolutions. We alsodefined andstudiedthe gcd-complex∆ (b)fora degree
gcd
b∈A. We used this complex to generalize the results in [19] and to construct the
generalizedalgebraicScarfcomplexbasedontheconnectedcomponentsof∆ (b)
gcd
fordegreesb∈A. WhenI isalatticeidealweshowedthatthegeneralizedalgebraic
Scarf complex is present in every simple minimal free resolution of R/I. This
current paper analyzes in more detail the notions presented in [10]. We note that
the original motivation for this work came from a question in Algebraic Statistics
concerning conditions for the uniqueness of a minimal binomial generating set of
toric ideals.
The structure of this paper is as follows. In section 2 we discuss the notion of
simple i-syzygies of R/I. The simple 0-syzygies of R/I when I is a lattice ideal
are exactly the binomials of I. We also discuss the notion of a simple minimal
free resolution of R/I. This notion requires the presence of a system of bases for
the free modules of the resolution. We show that for an appropriate choice of
bases every A-homogeneous minimal free resolution of R/I is simple. In section
3 we discuss the gcd-complex ∆ (b) for a degree b ∈ A. We show that the
gcd
homology of ∆ (b) determines the i-Betti numbers of degree b. We count the
gcd
numbers of binomials that could be part of a minimal binomial generating set of
a lattice ideal up to a constant multiple. In section 4 we discuss the notion of
indispensable i-syzygies. Intrinsically indispensable i-syzygies are present in all A-
homogeneoussimple minimal free resolutions. For the 0-stepandfor alattice ideal
I this means that there are some binomials of the ideal I that are part (up to a
L L
constant multiple) of all A-homogeneous systems of minimal binomial generators
of I . A strongly indispensable i-syzygy needs to be present in every minimal free
L
resolution of R/I even if the resolution is not simple. For the 0-step and for a
lattice ideal I this means that there are some elements of I that are part (up to
L L
a constant multiple) of all A-homogeneous minimal sets of generators of I , where
L
the generators are not necessarily binomials. We consider conditions for strongly
indispensable i-syzygies to exist. We show that the Koszul complex of a complete
intersection lattice ideal I is indispensable when the A-degrees of the elements of
the generating R-sequence are incomparable.
SYZYGIES 3
2. Simple syzygies
We recall and generalize the definition of a simple i-syzygy,see [10, Definition
3.1], to arbitrary elements of an A-graded free module. Let F be a free A-graded
module of rank β and let B = {E : t = 1,...,β} be an A-homogeneous basis of
t
F. Let h be an A-homogeneous element of F:
h= X (X catxat)Ei .
1≤t≤β cat6=0
The S-support of h with respect to B is the set
S (h)={xatE : c 6=0} .
B i at
We introduce a partial order on the elements of F:
h′ ≤h if and only if S (h′)⊂S (h) .
B B
Definition 2.1. Let F and B be as above, let G be an A-graded subset of F
and let h be an A-homogeneous nonzero element of G. We say that h is simple in
GwithrespecttoB ifthereisnononzeroA-homogeneoush′ ∈Gsuchthath′ <h.
In [10, Theorem 3.4] we showed that if (F ,φ) is a minimal free resolution
•
of R/I then for any given basis B of F there exists a minimal A-homogeneous
i
generatingsetofkerφ consistingofsimplei-syzygieswithrespecttoB. Theproof
i
ofthe next propositionis animmediate generalizationofthe proofof thattheorem
and is omitted.
Proposition2.2. LetF beafreeA-gradedmodule,letB beanA-homogeneous
basis of F and let G be an A-graded submodule of F. There is a minimal system
of generators of G each being simple in G with respect to B.
GivenanA-homogeneouscomplexoffreemodules(G ,φ)wespecifyA-homoge-
•
neous bases B for the homological summands G . The collection of theses bases
i i
forms a system of bases B. We write B=(B ) and we say that B is in B.
i i
Definition 2.3. A based complex (G ,φ,B) is an A-homogeneous complex
•
(G ,φ) together with a system of bases B = (B ). Let (G ,φ,B) and (F ,φ,C)
• i • •
be two based complexes, B = (B ) and C = (C ). We say that the complex
i i
homomorphism ω : G −→F is a based homomorphism if for each E ∈ B , there
• • i
exists an H ∈C such that ω(E)=cH for some c∈k∗.
i
Let I be an A-homogeneous ideal and let (F ,φ) be a minimal A-graded free
•
resolutionof R/I. We let s be the projective dimension of R/I and β be the rank
i
of F . For each i we suppose that B is an A-homogeneous basis of F and we let
i i i
B=(B ,B ,...,B ).
0 1 s
Definition 2.4. ([10,Definition 3.5])Let(F ,φ,B)be asabove. We saythat
•
(F ,φ,B) is simple if and only if for each i and each E ∈ B , φ (E) is simple in
• i i
kerφ with respect to B .
i−1 i−1
We remark that when I is a lattice ideal then for any choice of basis B , the
0
simple 0-syzygies of R/I are the binomials of I. It is an immediate consequence
of Proposition 2.2 that one can construct a minimal simple resolution of R/I with
respecttoasystemB=(B ,...,B )startingwithB ={1},seealso[10,Corollary
0 s 0
3.6]. In the next proposition we show that any minimal free resolution (F ,φ) of
•
R/I becomes simple with the right choice of bases.
4 H.CHARALAMBOUSANDA.THOMA
Proposition 2.5. Let I be an A-homogeneous ideal and let (F ,φ) be a min-
•
imal free resolution of R/I. There exists a system of bases B so that (F ,φ,B) is
•
simple.
Proof. Let C = {1} and for each i > 0 choose a basis C = {H : t =
0 i ti
1,...,β }ofF . Let(G ,θ)beasimpleminimalfreeresolutionofR/I withrespect
i i •
toD=(D ,...,D )whereD ={1}andD ={E : t=1,...,β }. SinceG ,F
0 s 0 i ti i • •
arebothminimalprojectiveresolutionsofR/Ithereisanisomorphismofcomplexes
h : G −→F that extends the identity map on R/I. In particular h =id . For
• • • 0 R
each i we let H′ = h (E ) and consider the set B = {H′ : t = 1,...,β }. We
ti i ti i ti i
note that B = {1}. It is immediate that B is a basis for F . We claim that
0 i i
(F ,φ,B) is simple.
•
Indeed for t=1,...,β using the commutativity of the diagram we get that
1
φ (H′ )=φ (h (E ))=h (θ (E ))=θ (E ) .
1 t1 1 1 t1 0 1 t1 1 t1
Since θ (E ) is simple with respect to C it follows at once that φ (H′ ) is simple
1 t1 0 1 t1
with respect to B . For i>1 and t=1,...,β we have that
0 i
φ (H′ )=φ (h (E ))=h (θ (E )) .
i ti i i ti i−1 i ti
Suppose that φ (H′ ) were not simple with respect to B . Since h is bijective
i ti i−1 i−1
it follows that θ (E ) is not simple with respect to D , a contradiction. (cid:3)
i ti i−1
Let I be an A-homogeneous ideal and let (F ,φ) be a minimal free resolution
•
of R/I. An i-syzygy h of R/I minimal if h is part of a minimal generating set of
kerφ . By the graded version of Nakayama’s lemma it follows that h is minimal if
i
and only if h cannot be written as an R-linear combination of i-syzygies of R/I of
strictly smaller A-degrees.
The next theorem examines the cardinality of the set of minimal i-syzygies of
a free resolution of R/I.
Theorem 2.6. Let I be an A-homogeneous ideal and let (F ,φ) be a mini-
•
mal free resolution of R/I. Let ≡ be the following equivalence relation among the
elements of F :
i
h≡h′ if and only if h=ch′,c∈k∗ ,
and let B be a basis of F . The set of equivalence classes of the i-syzygies of R/I
i i
that are minimal and simple with respect to B is finite.
i
Proof. We will show that the number of equivalence classes of the i-syzygies
that are simple and have A-degree equal to an (i+1)-Betti degree b of R/I is
finite. By [10, Theorem 3.8] if h,h′ ∈ kerφ are simple with respect to B and
i i
S (h)=S (h′) then h≡h′. Thus it is enough to show that there is only a finite
Bi Bi
number of candidates for S (h) when h ∈ F has deg (h) = b. We consider the
Bi i A
set
C ={xaE :deg (xaE )=b , E ∈B }.
t A t t i
We note that S (h) ⊂ P(C) where P(C) is the power set of C. The number of
Bi
basis elements E ∈B such that deg (E )≤b is finite. Moreoverfor such E the
t i A t t
number of monomials xa such that deg (xa)+deg (E ) = b is finite. It follows
A A t
that C and its power set P(C) are finite as desired. (cid:3)
In the next section we determine the cardinality of this set when I is a lattice
ideal and i=0. In other words we compute the number of the equivalence classes
of the minimal binomials of I.
SYZYGIES 5
3. The gcd-complex
For b∈A, we let C equal the fiber
b
C :=deg−1(b)=deg−1(b):={xu : deg (xu)=b}.
b A L A
Let I be a lattice ideal. The fiber C plays an essential role in the study of the
L b
minimalfree resolutionofR/I asis evidentfromseveralworks,see[3, 9, 10, 11,
L
19, 20].
We denote the supportofthe vectoru=(u )bysupp(u):={i:u 6=0}. Next
j i
werecallthedefinitionofthesimplicialcomplex∆ onnvertices,constructedfrom
b
C as follows:
b
∆ :={F ⊂supp(a): xa ∈C }.
b b
∆ has been studied extensively, see for example [2, 4, 5, 7, 8, 17, 18]. Its
b
homology determines the Betti numbers of R/I :
L
βi,b(R/IL)=dimkH˜i(∆b) ,
see [22] or [13] for a proof.
Inthissectionwepresentanothersimplicialcomplex,thegcd-complex∆ (b),
gcd
whoseconstructionisbaseduponthedivisibilitypropertiesofthemonomialsofC .
b
Definition 3.1. For a vector b ∈ A we define the gcd-complex ∆ (b) to
gcd
be the simplicial complex with vertices the elements of the fiber C and faces all
b
subsets T ⊂C such that gcd(xa : xa ∈T)6=1.
b
The example below compares graphically the two simplicial complexes in a
particular case.
Example1. LetR=k[a,b,c,d]andletAbethesemigroupA=h(4,0),(3,1),
(1,3),(0,4)i. For b = (6,10) we consider the fiber C = {bc3,ac2d,b2d2} and
(6,10)
the corresponding simplicial complexes. We see that
∆gcd(b) ∆b
a
b ac2d b
bbc3 b b2d2 c b b d
b
b
The main theorem of this section, Theorem 3.2 was proved independently in
[16].
Theorem 3.2. Let b∈A. The gcd complex ∆ (b) and the complex ∆ have
gcd b
the same homology.
Proof. First we consider the simplicial complex ∆ with vertices the elements
of the set S ={supp(a): xa ∈C } and faces all subsets T ⊂S such that
b
\ supp(a)6=∅ .
supp(a)∈T
We define anequivalencerelationamongthe verticesof∆ (b): we letxa ≡xa′ if
gcd
andonly if supp(a)=supp(a′). We note that the subcomplex A of ∆ (b) onthe
gcd
6 H.CHARALAMBOUSANDA.THOMA
vertices of an equivalence class is contractible. By the Contractible Subcomplex
Lemma[6] wegetthat the quotientmapπ :|∆ (b)|−→|∆ (b)|/|A| is ahomo-
gcd gcd
topy equivalence. A repeated application of the Contractible Subcomplex Lemma
yields that ∆ (b) and ∆ have the same homology.
gcd
Nextweconsiderthe familyF ofthe facetsof∆ andthe correspondingnerve
b
complex N(F). The vertices of N(F) correspond to the facets of ∆ , while the
b
faces of N(F) correspond to collections of facets with nonempty intersection. It
follows that N(F) is isomorphic to ∆. By [21, Theorem 7.26] the two complexes
∆ and ∆ have the same homology and the theorem now follows. (cid:3)
b
The following is now immediate:
Corollary 3.3. Let I be a lattice ideal.
L
βi,b(R/IL)=dimkH˜i(∆gcd(b)).
The connected components of ∆ (b) were used in [10] to determine certain
gcd
complexesassociatedtoasimpleminimalfreeA-homogeousresolutionofR/I ,see
L
[10, Definitions 4.7 and5.1]. InTheorem3.5 belowwe use the complex∆ (b) to
gcd
determine the number of equivalence classes of minimal binomial generators of I .
L
First we prove the following lemma:
Lemma 3.4. For b∈A, let I be the ideal generated by all binomials of I of
L,b L
A-degree strictly smaller than b. Let G(b) be the graph with vertices the elements
of C and edges all the sets {xu,xv} whenever xu−xv ∈I . A set of monomials
b L,b
in C forms the vertex set of a component of G(b) if and only if it forms the vertex
b
set of a component of ∆ (b).
gcd
Proof. We note that if xu,xv belong to the same component of ∆ (b)
gcd
then there exists a sequence of monomials xu = xu1,xu2,...,xus = xv such that
d=gcd(xui,xui+1)6=1. Therefore
xui xui+1
xui −xui+1 =d( − )∈IL,b .
d d
It follows that xu−xv ∈I and xu,xv belong to the same component of G(b).
L,b
For the converse we note that the binomials of degree b in I are spanned
L,b
by binomials of the form xa(xr −xs) where xa 6= 1. Moreover any such binomial
determines an edge from xa+r to xa+s in ∆ (b). Thus if xu, xv lie in the same
gcd
component of G(b) then any minimal expression of xu−xv as a sum of binomials
xa(xr−xs) results in a path from xu to xv in ∆ (b). (cid:3)
gcd
ThegraphG(b)wasfirstintroducedin[9]todeterminethenumberofdifferent
binomialgeneratingsetsofatoricidealI . Theresultsstatedfortoricidealsin[9]
L
hold more generally for lattice ideals with identical proofs. We choose an ordering
of the connected components of ∆ (b) and let t (b) be the number of vertices of
gcd i
the i-th component of ∆ (b).
gcd
Theorem 3.5. Let I be a lattice ideal and consider the equivalence relation
L
on R of Theorem 2.6. The cardinality of the set T of equivalence classes of the
minimal binomials of I is given by
L
|T|= XXti(b)tj(b).
b∈Ai6=j
SYZYGIES 7
Proof. In the course of the proof of [9, Theorem 2.6] applied to the lattice
ideal I it was shown that the minimal binomials of A-degree b are the difference
L
of monomials that belong to different connected components of G(b). Lemma 3.4
and a counting argument finishes the proof. (cid:3)
We remark that if b ∈ A is not a 1-Betti degree of R/I , then there is no
L
minimalbinomial generatorofA-degree b. It followsthat ∆ (b) has exactly one
gcd
connected component. The nontrivialcontributions to the formula of Theorem3.5
come from the 1-Betti degrees of R/I .
L
4. Indispensable syzygies
In this section we discuss the notion of indispensable complexes that first ap-
peared in [10, Definition 3.9]. Intrinsically an indispensable complex of R/I is
a based complex (F ,φ,B) that is “contained” in any based simple minimal free
•
resolution of R/I.
The indispensable binomialsofa lattice idealI are the binomials thatappear
L
in every minimal system of binomial generators of the ideal up to a constant mul-
tiple. They werefirstdefinedin[15]andtheirstudy wasoriginallymotivatedfrom
Algebraic Statistics; see [1, 14, 15, 24] for a series of related papers.
Theorem4.1. LetI bealatticeideal. TheindispensablebinomialsofI occur
L L
exactly in the minimal A-degrees b such that ∆ (b) consists of two disconnected
gcd
vertices.
Proof. Thistheoremwasprovedin[9]fortoricideals. Thesameproofapplies
to lattice ideals. (cid:3)
An immediate consequence of Theorem 4.1 is the following:
Corollary 4.2. Let I be a lattice ideal and S a minimal system of A-
L
homogeneous (not necessarily binomial) generators of I . If f is an indispensable
L
binomial of I then there is a c∈k∗ such that cf ∈S.
L
Proof. Let f be an indispensable binomial of I and b = deg f. Since
L A
H (∆ (b)) = 1 there is a unique element f′ in S of A-degree b. Since C is a
1 gcd b
set with exactly two elements it follows that f′ is a binomial. Since I contains no
L
monomials, it follows that f′ =cf for some c∈k∗. (cid:3)
It is clear that if (F ,φ,B) is a minimal free resolution of R/I and f is an
• L
indispensablebinomial,thenthereexistsanelementE ∈B andac∈k∗ suchthat
1
φ (E) = cf. We let the indispensable 0-syzygies of R/I to be the indispensable
1 L
binomials of I . We extend the definition for i≥0:
L
Definition 4.3. Let (F ,φ,B) be a based complex. We say that (F ,φ,B)
• •
is an indispensable complex of R/I if for each based minimal simple free resolution
(G ,θ,C) of R/I where C = {1}, there is an injective based homomorphism
• 0
ω :(F ,φ,B)→(G ,θ,C) such that ω =id . If B=(B ) and E ∈B we say
• • 0 R j i+1
that φ (E)∈F is an indispensable i-syzygy of R/I.
i+1 i
Itfollowsimmediatelyfromthedefinitionthatanindispensablei-syzygyofR/I
issimple. Moreoverif(F ,φ,B)isanindispensable complex ofR/I and(G ,θ,W)
• •
isaminimalsimple freeresolutionofR/I thenthe basedhomomorphismofDefini-
tion4.3 is unique, up to rearrangementofthe bases elements ofthe same A-degree
8 H.CHARALAMBOUSANDA.THOMA
and constant factors. In [10, Theorem 5.2] we showed that if I is a lattice ideal
L
then the generalized algebraic Scarf complex is an indispensable complex.
The next theorem examines when the Koszul complex of a lattice ideal gen-
erated by an R-sequence of binomials is indispensable. Let I be an ideal gener-
ated by an R-sequence f ,...,f and let (K ,φ) be the Koszul complex on the
1 s •
f . We denote the basis element e ∧ ···∧e of K by e where J is the or-
i j1 jt t J
dered set {j ,...,j } and let sgn[j ,J] = (−1)k+1. For each j ∈ J we write J
1 t k j
for the set J \ {j}. The canonical system of bases B = (B ,...,B ) consists
0 s
of the following: B = {1}, B = {e : i = 1,...,s} where φ (e ) = f and
0 1 i 1 i i
B ={e : J ={j ,...,j }, 1≤j <...<j ≤s} where
t J 1 t 1 t
φt(eJ)=Xsgn[j,J] fjeJj .
j∈J
In[10,Example3.7]itwasshownthat(K ,φ,B)isasimpleminimalfreeresolution
•
of R/I.
Theorem4.4. LetI =(f ,...,f )bealatticeidealwhere{f :i=1,...,s}is
L 1 s i
an R-sequence of binomials such that b =deg (f ) are incomparable. Let (K ,φ)
i A i •
be the Koszul complex on the f and let B be the canonical system of bases of K.
i
Then (K ,φ,B) is an indispensable complex of R/I .
• L
Proof. Let fi =xui −xvi. We note that if eJ ∈Bt then
degA(eJ)=XdegAfi
i∈J
and (K ,φ) is A-homogeneous. The incomparability assumption on the degrees of
•
the f shows that each b is minimal and that β (R/I)=1. It follows that f is
i i 1,bi i
an indispensable binomial, see [9, Corollary3.8]. We also note that for each i, C
bi
consists of exactly two monomials.
Let(G ,θ,W)beasimpleminimalresolutionofR/I whereW=(W ,...,W )
• L 0 s
and W = {1}. We let ω = id : K −→G . We prove that there is a based
0 0 R 0 0
isomorphism ω : (K ,φ,B)−→ (G ,θ,W) which extends ω by showing that if
• • 0
ω : K −→G has been defined for i ≤ k then ω can be constructed with the
i i i k+1
desired properties. Thus we assume that for each basis element e of B there
J k
exists c ∈k∗ and H ∈W such that ω (e )=c H . We note that if e ∈B
J J k k J J J L k+1
then ω φ (e ),i.e.
k k+1 L
Xsgn[j,L] fj cLj HLj
j∈L
isasimplek-syzygywithrespecttoW . Thisfollowsasintheproofof[9,Corollary
k
3.8]. We will define ω : K −→G by specifying its image in the basis
k+1 k+1 k+1
elements e of B so that the following identity holds:
L k
θ ω (e )=ω φ (e ) .
k+1 k+1 L k k+1 L
Since ω φ (e ) is a k-syzygy,it follows that
k k+1 L
t
ωkφk+1(eL)=Xθk+1(piHi)
i=1
where H ∈W and deg (p H )=deg (e ). We will show that t=1. First we
i k+1 A i i A L
notice that for some i
S (θ (p H ))∩S (ω (φ (e )))6=∅ .
Wk k+1 i i Wk k k+1 L
SYZYGIES 9
Without loss of generality we can assume that this is the case for i = 1 and we
write H in place of H . Moreover we can assume that
1
• L={1,...,k+1} and that
• xu1HL1 ∈SWk(θk+1(H))∩SWk(ωk(φk+1(eL))).
Letq be the coefficient ofH inθ (H). We havethatdeg (p q )=b . We
L1 L1 k+1 A 1 L1 1
will show that p q is a constant multiple of f . For t ∈ L we write L for the
1 L1 1 1 1,t
setL \{t}. Sinceθ θ (H)=0thecoefficientofH inθ θ (H)mustbezero
1 k k+1 L1,t k k+1
for any t ∈ L . The contributions to this coefficient come from the differentiation
1
ofthetermofθ (H)involvingH andallothertermsofθ (H)involvingH
k+1 L1 k+1 L′
where L′ \{t′} = L . Let X be the set consisting of such L′ and let q′ be the
1,t
coefficient of H when L′ ∈X. We get
L′
0=sgn[t,L1]qL1ft+ X q′sgn[t′,L′]ft′ .
L′∈X
Since f ,...,f is a complete intersection it follows that q ∈ hf : L′ ∈ Xi.
1 s L1 t′
Therefore b ≥ deg (q ) ≥ deg (f ) for at least one t′. By the incomparability
1 A L1 A t′
of the degrees of the f it follows that t′ = 1, and q is a constant multiple of f
i L1 1
and thus p ∈ k∗. Moreover we have shown that for each t in L there is a term
1 1
in θ (H) involving H . By a degree consideration it follows that the
k+1 {1,...,tˆ,...,k+1}
coefficient of this term has degree b and thus repeating the above steps we can
t
concludethatthecoefficientofthistermisaconstantmultipleoff . Itfollowsthat
t
S (ω φ (e ))⊂S (θ (H)) .
Wk k k+1 L Wk k+1
Since θ (H)) is simple it follows that
k+1
S (ω φ (e ))=S (θ (H)) ,
Wk k k+1 L Wk k+1
and θ (H)) = c ω φ (e ) where c ∈ k∗. We let H = H and c = c−1. It
k+1 k k+1 L L L
follows that the homomorphism ω :K −→G defined by setting
k+1 k+1 k+1
ω (e )=c H
k+1 L L L
has the desired properties. (cid:3)
Next we consider strongly indispensable complexes.
Definition 4.5. Let (F ,φ,B) be a based complex. We say that (F ,φ,B) is
• •
a strongly indispensable complex of R/I if for every based minimal free resolution
(G ,θ,C)ofR/I,(notnecessarysimple)withC ={1},thereisaninjectivebased
• 0
homomorphism ω : (F ,φ,B)−→(G ,θ,C) such that ω = id . If B = (B ) and
• • 0 R j
E ∈B we say that φ (E)∈F is a strongly indispensable i-syzygy of R/I.
i+1 i+1 i
Strongly indispensable complexes are indispensable. This is a strict inclusion
as[10,Example6.5]shows. WhenI isalattice ideal,the algebraicScarfcomplex
L
[19, Construction 3.1], is shown to be “contained” in the minimal free resolution
of R/I , [19, Theorem 3.2], and is a strongly indispensable complex. Moreover as
L
follows from Corollary 4.2 the strongly indispensable 0-syzygies of R/I coincide
L
with the indispensable 0-syzygies of R/I and are the indispensable binomials of
L
I . For higher homological degrees this is no longer the case. First we note the
L
following:
Theorem 4.6. Let I be a lattice ideal and let (F ,φ,B) be a strongly indis-
L •
pensable complex for R/I . Let B = (B ), E ∈ B and deg (E) = b. Then
L j i+1 A
dimkH˜i(∆gcd(b))=1 and b is a minimal i-Betti degree of R/IL.
10 H.CHARALAMBOUSANDA.THOMA
Proof. Suppose that dimkH˜i(∆gcd(b))>1 or that there is an i-Betti degree
b′ such that b′ < b. Let (G ,θ,C) be a minimal resolution of R/I where C =
• L
(C ), let ω :(F ,φ,B)−→(G ,θ,C) be the based homomorphism of Definition 4.5
i • •
and suppose that ω(E) = cH where H ∈ C and c ∈ k∗. By our assumptions
i+1
there exists H′ ∈ Ci+1 such that H′ 6= H and degA(H)≤ b. Let xa ∈Cb−b′. By
replacing H with H +xaH′ we get a new basis C′ of G and a new system of
i+1 i+1
bases C′ =(C′), where C′ =C for j 6=i+1. Let ω′ :(F ,φ,B)−→(G ,θ,C′) be
j j j • •
the based homomorphism of Definition 4.5: ω = ω′ for j ≤ i. Let H′′ ∈ C′ be
j j i+1
such that ω′ (E)=c′H′′ where c′ ∈k∗. Thus
i+1
θ (c′H′′)=θ (ω′ (E))=ω′φ (E)=ω φ (E)=
i+1 i+1 i+1 i i+1 i i+1
θ (ω (E))=cθ (cH) .
i+1 i+1 i+1
It follows that c′H′′ −cH ∈ kerθ . If H′′ 6= H +xaH′ then H′′ ∈ C and
i+1 i+1
we get a direct contradiction to the minimality of (G ,θ,C). If H′′ = H +xaH′
•
then θ ((c′−c)H +c′xaH′)=0. Examination of the two cases when (a) c′ 6=c,
i+1
and (b) c′ = c, leads again to a contradiction of the minimality of the resolution
(G ,θ,C). (cid:3)
•
Theorem 4.6 shows that the two conditions
(1) dimkH˜i(∆gcd(b))=1
(2) b is a minimal i-Betti degree
are necessary for the existence of a strongly indispensable i-syzygy in A-degree
b. The following example shows that these conditions are not sufficient for the
existenceofanindispensablei-syzygyandconsequentlyofastronglyindispensable
i-syzygy.
Example 2. Consider the lattice ideal I = hf ,f i where f = x − x ,
L 1 2 1 1 2
f = x −x and deg f = 1. Let (K ,φ) be the Koszul complex on the f . By
2 2 3 A i • i
considering the i-Betti numbers for i = 1,2 it is immediate that dimkH2(∆2) = 1
and 2 is a minimal 2-Betti degree. However there is no indispensable complex of
length greater than 0, since the generators of I are not indispensable binomials.
L
Generic lattice ideals are characterized by the condition that the binomials in
a minimal generating set have full support, [19]. In this case the Scarf complex is
aminimal free resolutionofR/I andeachofthe BettidegreesofR/I satisfy the
L L
conditions of Theorem4.6. We finish this section by giving the strongestresult for
the opposite direction of Theorem 4.6.
Theorem 4.7. Let I be a lattice ideal. The A-homogeneous minimal free
L
resolution (F ,φ,B) of R/I is strongly indispensable if and only if for each i-
• L
Betti degree b of R/IL, b is a minimal i-Betti degree and dimkH˜i(∆gcd(b))=1.
Proof. One direction of this theorem follows directly from Theorem 4.6. For
theotherdirectionweassumethatbisminimalwheneverbisani-Bettidegreeand
that dimkH˜i(∆gcd(b)) = 1 for all i. Let (G•,θ,D) be a minimal free resolution
of R/I . By assumption the A-degrees of the elements of D are distinct and
L i
incomparable. It follows that the A-homogeneous isomorphism ω : F−→G that
extends id :F −→G is a based homomorphism. (cid:3)
R 0 0
Acknowledgment
The authors wouldlike to thank EzraMiller for his essentialcomments onthis
manuscript.