We describe the cone of Betti tables of all finitely generated graded modules over the homogeneous coordinate ring of three non-collinear points in the projective plane. We also describe the cone of Betti tables of all finite length modules

Gheorghita, Iulia

Sam, Steven V.

Caltech Authors - Main

The cone of Betti tables over three non-collinear points in the plane

JOURNAL OF COMMUTATIVE ALGEBRA
Volume 8, Number 4, Winter 2016
THE CONE OF BETTI TABLES OVER THREE
NON-COLLINEAR POINTS IN THE PLANE
IULIA GHEORGHITA AND STEVEN V SAM
ABSTRACT. We describe the cone of Betti tables of
all nitely generated graded modules over the homogeneous
coordinate ring of three non-collinear points in the projective
plane. We also describe the cone of Betti tables of all nite
length modules.
1. Introduction. The idea of studying the cone of Betti tables of
nitely generated graded modules over a polynomial ring originates in
the work of Boij and Soderberg [3]. To summarize, they conjectured
that the cone of Betti tables derived from Cohen-Macaulay modules
(of a xed codimension) is spanned by Betti tables derived from pure
resolutions. This conjecture was proven by Eisenbud and Schreyer
[7] and later improved to get a statement for the cone of all nitely
generated modules [4]. For a survey of these developments, we point
the reader to [8, 9].
A crucial step was the construction of the dening inequalities of
the cone of Betti tables. Eisenbud and Schreyer found a surprising
connection with the cone of cohomology tables of vector bundles on
projective space. A conceptual understanding of this connection was
proposed by Eisenbud and Erman [5].
In a dierent direction, one can replace the polynomial ring with a
singular ring and try to generalize these results. The main dierence
is that all nitely generated modules over polynomial rings have nite
length free resolutions, whereas this property fails for singular rings.
The next class of rings in terms of complexity of resolutions are the
hypersurface rings: the minimal free resolutions of nitely generated
2010 AMS Mathematics subject classication. Primary 13C05, 13D02.
Keywords and phrases. Boij-Soderberg theory, Betti tables, free resolutions.
The rst author was supported by the Caltech SURF program. The second
author was supported by a Miller research fellowship.
Received by the editors on January 31, 2015, and in revised form on July 27,
2015.
DOI:10.1216/JCA-2016-8-4-537 Copyright c2016 Rocky Mountain Mathematics Consortium
537
538 IULIA GHEORGHITA AND STEVEN V SAM
modules become periodic (of period 2). The cone of Betti tables over
hypersurface rings of low embedding dimension are described in [1].
Another perspective, when the ring is Cohen-Macaulay, is that the
long-term behavior of a minimal free resolution is reduced to the
study of minimal free resolutions of maximal Cohen-Macaulay modules.
From this point of view, the next level of complexity of rings are
the Cohen-Macaulay rings with nitely many isomorphism classes of
indecomposable maximal Cohen-Macaulay modules. These have been
classied in [6]; the homogeneous coordinate rings of rational normal
curves are one family, and the cone of Betti tables over this ring has
been described in [12].
In this paper, we describe the cone of Betti tables over another
example of such a Cohen-Macaulay ring: the homogeneous coordinate
ring of three non-collinear points in the projective plane. Up to
isomorphism, these points are [1 : 0 : 0], [0 : 1 : 0] and [0 : 0 : 1]. A
pleasant feature of this example is that all of the dening inequalities
of the cone of Betti tables have concrete interpretations. This is similar
to the examples considered in [1], but one crucial dierence is that our
ring is not a hypersurface ring, so that this example bridges ideas from
[1, 12].
Now we x notation. Throughout, k is a xed eld. The Cohen-
Macaulay ring in question, i.e., the coordinate ring of the three points
[1 : 0 : 0], [0 : 1 : 0] and [0 : 0 : 1], is
B = k[x; y; z]=(xy; yz; xz):
Let V be the Q-vector space of tables (vi;j), where i  0 and j 2 Z,
with the property that, for each i, there are only nitely many j such
that vi;j 6= 0. Given a nitely generated graded B-module M , dene
B(M)i;j = dimk Tor
B
i (M;k)j ;
which is an element of V. Our main object of study is the Q0-span of
B(M) in V as M varies over all nitely generated (respectively, nite
length) B-modules. Call these cones BQ(B) and B
f
Q(B), respectively.
The main results of this paper are Theorem 3.1 and Corollary 3.6
which completely describe the cone of Betti tables of all nitely gen-
erated B-modules, and the cone of Betti tables of all nite length
BETTI TABLES OVER THREE NON-COLLINEAR POINTS 539
B-modules, respectively. We give two descriptions: one in terms of
generators, and one in terms of linear inequalities.
In Section 2, we begin with some general results on MCM modules
and pure resolutions, and in Section 3, we prove the main results and
give a local version of the results.
2. Generalities on B-modules.
2.1. Notation. Given a B-module M , its Hilbert series
HM (t) =
X
d
(dimkMd)t
d
is of the form p(t)=(1   t) for some polynomial p(t). We dene
e(M) = p(1). From generalities on Hilbert polynomials, we obtain
dimkMd = e(M) for d 0:
In particular, e(M)  0, and M is of nite length if and only if
e(M) = 0.
The (reduced) syzygy module 
(M) of M is the kernel of a surjec-
tion,
F  !M  ! 0;
where F is a free module whose basis maps to a minimal generating set
of M .
Finally, if M is a graded module, then M( d) denotes the same
module with a grading shift:
M( d)e =Me d:
2.2. MCMmodules. There are eight indecomposable maximal Cohen-
Macaulay (MCM) B-modules (see [13, Remark 9.16] for the com-
pletion of B; to compare this to the graded case we can use [13,
Lemma 15.2.1]): the free module B, the canonical module !B and
the quotients
M1 = B=(x); M2 = B=(y); M3 = B=(z);
M1;2 = B=(x; y); M2;3 = B=(y; z); M1;3 = B=(x; z):
540 IULIA GHEORGHITA AND STEVEN V SAM
Their Hilbert series are as follows:
HB(t) =
1 + 2t
1  t ; H!B (t) =
2 + t
1  t ;
HMi(t) =
1 + t
1  t ; HMi;j (t) =
1
1  t :
An MCM module M is pure if M is (up to a grading shift) of one of
the following four forms:
Br; !rB ;
Mr11 Mr22 Mr33 ; Mr11;2 Mr21;3 Mr32;3 :
Implicit in this denition is that M is generated in a single degree.
We treat each of the modules Mi the same because their numerical
invariants (such as Betti numbers) are the same, and similarly with
Mi;j . We use the convention that Mi may refer to any of M1, M2 or
M3, and similarly for Mi;j . To clarify, if we say that M is a direct sum
of Mi, we mean that it is a direct sum of the copies of M1, M2 or M3,
and similarly for Mi;j .
Proposition 2.1. The syzygy modules are as follows:

(!B) = (M1;2 M2;3 M1;3)( 1);

(Mi) =Mf1;2;3gni( 1);

(Mi;j) = (Mj;k Mi;k)( 1);
where, in the last line, k = f1; 2; 3g n fi; jg.
Proof. Let A = k[x; y; z]. The minimal free resolution of B over A
is
0  ! A2
 z 0
y  y
0 x

       ! A3

xy xz yz

           ! A  ! B  ! 0:
In particular, !B has the following minimal presentation over A (and
B):
A3
 z y 0
0  y x

        ! A2  ! !B  ! 0:
The image of the three basis vectors of A3 give, respectively, copies of
M1;2, M1;3 and M2;3.
The last two equalities are straightforward calculations. 
BETTI TABLES OVER THREE NON-COLLINEAR POINTS 541
2.3. Herzog-Kuhl equations.
Denition 2.2. A B-module M has a pure resolution if it is Cohen-
Macaulay and there is an exact sequence of the form,
0  ! 
(M)  ! F0  !M  ! 0;
where 
(M) is a pure MCM module and F0 is a free B-module
generated in a single degree. IfM is a nite length module, we say that
its type is (d0; d1), where d0 is the common degree of all generators of
F0 and d1 is the common degree of all generators of 
(M).
The Betti numbers of a pure resolution satisfy certain relations which
we now describe. Over a polynomial ring, such relations were worked
out by Herzog and Kuhl [11, Theorem 1]. IfM is a nite length module
with pure resolution of type (d0; d1) with d0 < d1, then it has an exact
sequence of the form,
0  ! N( d1)1  ! B( d0)0  !M  ! 0;
where N is an MCM module: B, !B, Mi or Mi;j . The latter three
modules have the following minimal free resolutions over B:
   ! B( m)32m 1 !   ! B( 2)6 ! B( 1)3 ! B2 ! !B ! 0;
   ! B( m)2m 1 !   ! B( 2)2 ! B( 1)! B !Mi ! 0;
   ! B( m)2m !   ! B( 2)4 ! B( 1)2 ! B !Mi;j ! 0:
To prove this, one can use Proposition 2.1. In particular, in all cases,
if
    ! Fi  ! Fi 1  !     ! F1  ! F0  !M  ! 0
is the minimal free resolution of M over B, then each Fi is generated
in a single degree di; let i be its rank. We see that di = di 1 + 1 for
i  2. Also, for i  2, we get a relation
i = 2
i 2c(N)1(2.1)
where c(N) depends on the isomorphism type of N and is 0; 1; 3=2; 2 if
N is B;Mi; !B;Mi;j , respectively.
Proposition 2.3. Let M be a nite length B-module with a pure
resolution of type (d0; d1). Write 
(M) = N
r where N is one of
542 IULIA GHEORGHITA AND STEVEN V SAM
B;Mi; !B;Mi;j, and set
c = c(N) 2

0; 1;
3
2
; 2

:
Then the Betti numbers of M are a multiple of the entries of the vector
1;
3
3  c ;
3c
3  c ; 2
3c
3  c ; 4
3c
3  c ; 8
3c
3  c ; : : : ; 2
i 2 3c
3  c ; : : :

Proof. Using equation (2.1), we obtain
HM (t) = HB(t)

0t
d0   1td1 + 1ctd1+1
X
i2
( 1)i2i 2ti 2

=
1 + 2t
1  t

0t
d0   1td1 + 1ct
d1+1
1 + 2t

=
(1 + 2t)(0t
d0   1td1) + 1ctd1+1
1  t :
SinceM is of nite length, HM (t) is a polynomial, and so the numerator
of the above expression is divisible by 1 t. Thus, the substitution t 7! 1
gives the relation
3(0   1) + 1c = 0:
Taking 0 = 1 gives us the desired statement. 
Remark 2.4. The Betti numbers do not depend on the actual degree
sequence, just on the isomorphism class of the MCM module. For
convenience, we list here the rst lattice point on the rays spanned by
the vectors from Proposition 2.3:
v1 = (1; 1; 0; 0; 0; : : :);
v2 = (2; 3; 3; 6; 12; : : :);
v3 = (1; 2; 3; 6; 12; : : :);
v4 = (1; 3; 6; 12; 24; : : :):
Note that 2v3 = v1+v4 and 2v2 = 3v1+v4, so that the cone generated
by v1;v2;v3;v4 is minimally generated by v1 and v4.
In particular, the Betti diagrams of !B and Mi are positive linear
combinations of the Betti diagrams of B and Mi;j . Finally, the Betti
BETTI TABLES OVER THREE NON-COLLINEAR POINTS 543
diagram of Mi;j is
2
3
v4 +

1
3
; 0; 0; : : :

:
3. The cone of Betti diagrams.
3.1. Main result. Each pure resolution has an associated degree
sequence which documents in what degree each term of the resolution
is generated. If the resolution is nite, we pad the degree sequence with
the symbol 1, so the possible degree sequences are:
(d0;1;1; : : :); (d0; d1;1; : : :); (d0; d1; d1 + 1; d1 + 2; : : :);
where d0 < d1.
For each degree sequence d dene d 2 V as follows. First,
(d)0;j = 1 for j = d0 and 0 otherwise. If
d = (d0;1;1; : : :);
i;j = 0 for all i  1 and all j. If
d = (d0; d1;1; : : :);
1;j = 1 for j = d1 and 0 otherwise. If
d = (d0; d1; d1 + 1; d1 + 2; : : :);
i;j = 3  2i 1 for i  1 and j = di and 0 otherwise. Recall that these
are the Betti numbers (up to scalar multiple) of a nite length module
whose syzygy module is a direct sum of copies of Mi;j and that we do
not use the others because of the relations in Remark 2.4.
For each v 2 V, dene
i;j(v) = vi;j ; k(v) = 21;k(v)  2;k+1(v);
k(v) =
X
jk
(30;j(v)  31;j+1(v) + 2;j+2(v)):
We also dene 1 by summing over all j. Note that, while k involves
an innite sum, only nitely many nonzero terms of v are involved so
it is well-dened.
Theorem 3.1. The following cones are equal :
544 IULIA GHEORGHITA AND STEVEN V SAM
(i) The cone BQ(B) spanned by the Betti diagrams of all nitely
generated B-modules.
(ii) The cone D spanned by d for all B-degree sequences d.
(iii) The cone F dened to be the intersection of the halfspaces
fi;j  0g for all i; j  0, fk  0g for all k 2 Z, fk  0g for
all k 2 Z, and the subspaces f2i;j = i+1;j+1g for i  2; j 2 Z.
The proof of Theorem 3.1 follows by establishing the inclusions
D  BQ(B)  F  D;
which are the content of the next three lemmas.
Lemma 3.2. D  BQ(B).
Proof. It suces to show that, for each degree sequence d, there
exists a B-module M with B(M) = d. We assume d0 = 0 for simpli-
city of notation.
 When d = (0;1;1; : : :), take M = B.
 When d = (0; d1;1; : : :), take M = B=((x+ y + z)d1).
 When d = (0; d1; d1+1; d1+2; : : :), takeM = B=(xd1 ; yd1 ; zd1).

Lemma 3.3. BQ(B)  F .
Proof. For a nitely generated gradedB-moduleM , we want to show
that the inequalities dening F are nonnegative on B(M). There are
no negative entries in B(M), so
i;j(
B(M)) = Bi;j(M)  0
for all i; j. The inequalities
fk  0 j k 2 Zg
and the equalities
f2i;j = i+1;j+1 j i  2; j 2 Zg
hold for B(M) by our discussion in subsection 2.3.
BETTI TABLES OVER THREE NON-COLLINEAR POINTS 545
It remains to show that k(
B(M))  0 for all k 2 Z. Let
    ! F1  ! F0  !M  ! 0
be a minimal free resolution of M . Suppose that

(M) = B0  !1B M2i M3i;j :
Let i be the ith Betti number of M (ignoring the grading). From
subsections 2.2 and 2.3, we can determine the relations
e(
(M)) = 30 + 31 + 22 + 3;
1 = 0 + 21 + 2 + 3;
2 = 31 + 2 + 23:
In particular, e(
(M)) = 31   2. From the short exact sequence,
0  ! 
(M)  ! F0  !M  ! 0;
we get
e(M) = e(F0)  e(
(M)) = 30   31 + 2;
and e(M) is a nonnegative quantity, so
30   31 + 2  0:(3.1)
If F is a free B-module, let k(F) be the free module generated by
minimal generators of degree  k. For a general B-module M , dene
k(M) to be the quotient of
k+1(F1)  ! k(F0):
In subsection 2.3, we established that the dierentials Fi ! Fi 1 are
linear for i  2, and so we conclude that the following is a minimal free
resolution of k(M):
    ! k+i(Fi)  ! k+i 1(Fi 1)  !   
 ! k+1(F1)  ! k(F0)  ! k(M)  ! 0:
In particular, applying equation (3.1) to k(M), we conclude that
k(
B(M))  0, as desired. 
Lemma 3.4. F  D.
546 IULIA GHEORGHITA AND STEVEN V SAM
Proof. The proof is formally the same as the proofs of [1, Lemmas
2.7, 2.8]. 
Remark 3.5. Following [1], dene a partial order on the degree
sequences by d  d0 if d0  d00 and d1  d01 where either inequality
is strict, or if d0 = d
0
0, d1 = d
0
1 and dn  d0n for all n  2. Then the
proofs of [1, Lemmas 2.7, 2.8] show that BQ(B) has a triangulation
coming from the simplicial cones spanned by the rays corresponding to
the elements of maximal chains in this partial order.
From Theorem 3.1, we immediately obtain a result about the cone
of nite length B-modules:
Corollary 3.6. The following cones are equal :
(i) The cone BfQ(B) spanned by the Betti diagrams of all nite length
B-modules.
(ii) The cone Df spanned by d for all B-degree sequences d with
d1 <1.
(iii) The cone F f dened to be the intersection of the halfspaces
fi;j  0g for all i; j  0, fk  0g for all k 2 Z, fk  0g
for all k 2 Z, and the subspaces f2i;j   i+1;j+1 = 0g for i  2,
j 2 Z and 1 = 0.
Proof. In each of three items, we have added one extra condition
to the cones dened in Theorem 3.1, so we just have to verify that
these conditions are the same. From the proof of Lemma 3.3, we get
that 1(B(M)) = e(M). Since e(M) = 0 if and only if M has nite
length, this shows that the extra conditions in (i) and (iii) coincide.
From Proposition 2.3, 1(d) = 0 if d2 < 1. If d1 < 1 and d2 = 1,
it is immediate from the denition that 1(d) = 0. Finally, for the
last case with d1 =1, we have 1(d) = 1, so the extra conditions in
(ii) and (iii) coincide. 
3.2. Local version. Following [2], we now give a local version of our
main result by considering minimal free resolutions over the completionbB = k[[x; y; z]]=(xy; yz; xz):
BETTI TABLES OVER THREE NON-COLLINEAR POINTS 547
In this case, the relevant numerical invariant is the Betti sequence (M)
of a module M which records the number of generators of each term in
a minimal free resolution of M .
As discussed above, the only relevant information are the rst three
Betti numbers (0; 1; 2) since i = 2i 1 for i > 2. The following
results are easy to prove from what we have already discussed.
Proposition 3.7. The following cones are equal :
(i) The cone spanned by the Betti sequences of all nitely generatedbB-modules.
(ii) The cone spanned by (1; 0; 0; : : :), (1; 1; 0; : : :) and (1; 3; 6; : : :).
(iii) The cone dened by the inequalities 2  0, 30 + 2  31 and
21  2, and the equalities i = 2i 1 for i > 2.
Proposition 3.8. The following cones are equal :
(i) The cone spanned by the Betti sequences of all nite length bB-
modules.
(ii) The cone spanned by (1; 1; 0; : : :) and (1; 3; 6; : : :).
(iii) The cone dened by the inequalities 2  0 and 21  2, and the
equalities 30 + 2 = 31 and i = 2i 1 for i > 2.
Acknowledgments. The software system Macaulay2 [10] was help-
ful for carrying out this research. In particular, we made use of the
FourierMotzkin package written by Gregory Smith.
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theory and invariants of free complexes, arxiv:1205.0449v2.
548 IULIA GHEORGHITA AND STEVEN V SAM
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Cohen-Macaulay rings of nite representation type, Math. Ann. 280 (1988), 347{
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California Institute of Technology, Pasadena, CA 91106 and Boston Col-
lege, Chestnut Hill, MA 02467
Email address: gheorgiu@bc.edu
University of California, Berkeley, CA 94720 and University of Wisconsin,
Madison, WI 53706
Email address: svs@math.wisc.edu


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THE CONE OF BETTI TABLES OVER THREE
NON-COLLINEAR POINTS IN THE PLANE
IULIA GHEORGHITA AND STEVEN V SAM
Abstract. We describe the cone of Betti tables of all finitely generated graded modules
over the homogeneous coordinate ring of three non-collinear points in the projective plane.
We also describe the cone of Betti tables of all finite length modules.
1. Introduction
The idea of studying the cone of Betti tables of finitely generated graded modules over
a polynomial ring originates in the work of Boij and Söderberg [BS1]. To summarize, they
conjectured that the cone of Betti tables coming from Cohen–Macaulay modules (of a fixed
codimension) is spanned by Betti tables coming from pure resolutions. This conjecture was
proven by Eisenbud and Schreyer [ES1] and later improved to get a statement for the cone
of all finitely generated modules [BS2]. For a survey of these developments, we point the
reader to [ES2, Fl].
A crucial step was the construction of the defining inequalities of the cone of Betti tables.
Eisenbud and Schreyer found a surprising connection with the cone of cohomology tables
of vector bundles on projective space. A conceptual understanding of this connection is
proposed by Eisenbud and Erman [EE].
In a different direction, one can replace the polynomial ring with a singular ring and try
to generalize these results. The main difference is that all finitely generated modules over
polynomial rings have finite length free resolutions, whereas this property fails for singular
rings. The next class of rings in terms of complexity of resolutions are the hypersurface rings:
the minimal free resolutions of finitely generated modules become periodic (of period 2). The
cone of Betti tables over hypersurface rings of low embedding dimension are described in
[BBEG].
Another perspective, when the ring is Cohen–Macaulay, is that the long-term behavior of a
minimal free resolution is reduced to the study of minimal free resolutions of maximal Cohen–
Macaulay modules. From this point of view, the next level of complexity of rings are the
Cohen–Macaulay rings with finitely many isomorphism classes of indecomposable Cohen–
Macaulay modules. These have been classified in [EH]; one family are the homogeneous
coordinate rings of rational normal curves, and the cone of Betti tables over this ring have
been described in [KS].
In this paper, we describe the cone of Betti tables over another example of such a Cohen–
Macaulay ring: the homogeneous coordinate ring of 3 non-collinear points in the projective
plane. Up to isomorphism, these points are [1 : 0 : 0], [0 : 1 : 0], and [0 : 0 : 1]. A pleasant
feature of this example is that all of the defining inequalities of the cone of Betti tables have
concrete interpretations. It is similar to the examples considered in [BBEG], but one crucial
Date: December 22, 2014.
2010 Mathematics Subject Classification. 13D02; 13C05.
1
2 IULIA GHEORGHITA AND STEVEN V SAM
difference is that our ring is not a hypersurface ring, so that this example bridges ideas from
[BBEG] and [KS].
Now we fix notation. Throughout, k is a fixed field. The Cohen–Macaulay ring in
question, i.e., the coordinate ring of the 3 points [1 : 0 : 0], [0 : 1 : 0], [0 : 0 : 1], is
B = k[x, y, z]/(xy, yz, xz).
Let V be the Q-vector space of tables (vi,j) where i ≥ 0 and j ∈ Z with the property
that for each i, there are only finitely many j such that vi,j 6= 0. Given a finitely generated
graded B-module M , define βB(M)i,j = dimk Tor
B
i (M,k)j, which is an element of V. Our
main object of study is the Q≥0-span of β
B(M) in V as M varies over all finitely generated
(respectively, finite length) B-modules. Call these cones BQ(B) and B
f
Q(B), respectively.
The main results of this paper are Theorem 3.1 and Corollary 3.7 which completely de-
scribe the cone of Betti tables of all finitely generated B-modules, and the cone of Betti
tables of all finite length B-modules, respectively. We give two descriptions: one in terms of
generators, and one in terms of linear inequalities.
In §2 we begin with some general results on MCM modules and pure resolutions, and in
§3 we prove the main results and give a local version of the results.
Acknowledgements. The software system Macaulay2 [M2] was helpful for carrying out
this research. In particular, we made use of the FourierMotzkin package written by Gregory
Smith. Iulia Gheorghita was supported by the Caltech SURF program. Steven Sam was
supported by a Miller research fellowship.
2. Generalities on B-modules
2.1. Notation. Given a B-module M , its Hilbert series HM (t) =
∑
d(dimk Md)t
d is of the
form p(t)/(1 − t) for some polynomial p(t). We define e(M) = p(1). From generalities on
Hilbert polynomials, we get dimk Md = e(M) for d ≫ 0. In particular, e(M) ≥ 0, and M is
of finite length if and only if e(M) = 0.
The (reduced) syzygy module Ω(M) of M is the kernel of a surjection F → M → 0 where
F is a free module whose basis maps to a minimal generating set of M .
Finally, if M is a graded module, then M(−d) denotes the same module with a grading
shift: M(−d)e = Me−d.
2.2. MCM modules. There are 8 indecomposable maximal Cohen–Macaulay (MCM) B-
modules (see [Yo, Remark 9.16] for the completion of B; to compare this to the graded case
we can use [Yo, Lemma 15.2.1]): the free module B, the canonical module ωB, and the
quotients
M1 = B/(x), M2 = B/(y), M3 = B/(z),
M1,2 = B/(x, y), M2,3 = B/(y, z), M1,3 = B/(x, z).
Their Hilbert series are as follows:
HB(t) =
1 + 2t
1− t
, HωB(t) =
2 + t
1− t
, HMi(t) =
1 + t
1− t
, HMi,j (t) =
1
1− t
.
An MCM module M is pure if M is (up to a grading shift) of one of the following four
forms:
B⊕r, ω⊕rB , M
⊕r1
1 ⊕M
⊕r2
2 ⊕M
⊕r3
3 , M
⊕r1
1,2 ⊕M
⊕r2
1,3 ⊕M
⊕r3
2,3 .
Implicit in this definition is that M is generated in a single degree.
THE CONE OF BETTI TABLES OVER THREE NON-COLLINEAR POINTS IN THE PLANE 3
We treat each of the modules Mi the same because their numerical invariants (such as
Betti numbers) are the same, and similarly with Mi,j . We use the convention that Mi may
refer to any of M1,M2,M3 and similarly for Mi,j. To clarify, if we say that M is a direct
sum of Mi, we mean that it is a direct sum of copies of M1,M2,M3, and similarly for Mi,j.
Proposition 2.1. The syzygy modules are as follows:
Ω(ωB) = (M1,2 ⊕M2,3 ⊕M1,3)(−1),
Ω(Mi) = M{1,2,3}\i(−1),
Ω(Mi,j) = (Mj,k ⊕Mi,k)(−1),
where in the last line, k = {1, 2, 3} \ {i, j}.
Proof. Let A = k[x, y, z]. The minimal free resolution of B over A is
0 → A2




−z 0
y −y
0 x




−−−−−−−−→ A3
(
xy xz yz
)
−−−−−−−−−−→ A → B → 0.
In particular, ωB has the following minimal presentation over A (and B):
A3


−z y 0
0 −y x


−−−−−−−−−−−→ A2 → ωB → 0.
The image of the 3 basis vectors of A3 give, respectively, copies of M1,2, M1,3, and M2,3.
The last two equalities are straightforward calculations. 
2.3. Herzog–Kühl equations.
Definition 2.2. A B-module M has a pure resolution if it is Cohen–Macaulay, and there
is an exact sequence of the form
0 → Ω(M) → F0 → M → 0
where Ω(M) is a pure MCM module and F0 is a free B-module generated in a single degree.
If M is a finite length module, we say that its type is (d0, d1) where d0 is the common degree
of all generators of F0 and d1 is the common degree of all generators of Ω(M). If M is a
pure MCM module, then its degree sequence is simply (d0) where d0 is the common degree
of its generators. 
The Betti numbers of a pure resolution satisfy certain relations which we now describe.
Over a polynomial ring, such relations were worked out by Herzog and Kühl [HK, Theorem
1]. If M is a finite length module with pure resolution of type (d0, d1) with d0 < d1, then it
has an exact sequence of the form
0 → N(−d1)
β1 → B(−d0)
β0 → M → 0
where N is an MCMmodule: B, ωB, Mi, orMi,j. The latter three modules have the following
minimal free resolutions over B:
· · · → B(−m)3·2
m−1
→ · · · → B(−2)6 → B(−1)3 → B2 → ωB → 0,
· · · → B(−m)2
m−1
→ · · · → B(−2)2 → B(−1) → B → Mi → 0,
· · · → B(−m)2
m
→ · · · → B(−2)4 → B(−1)2 → B → Mi,j → 0.
4 IULIA GHEORGHITA AND STEVEN V SAM
To prove this, one can use Proposition 2.1. In particular, in all cases, if
· · · → Fi → Fi−1 → · · · → F1 → F0 → M → 0
is the minimal free resolution of M over B, then each Fi is generated in a single degree di;
let βi be its rank. We see that di = di−1 + 1 for i ≥ 2. Also, for i ≥ 2, we get a relation
βi = 2
i−2c(N)β1(2.3)
where c(N) depends on the isomorphism type of N and is 0, 1, 3
2
, 2 if N is B,Mi, ωB,Mi,j,
respectively.
Proposition 2.4. Let M be a finite length B-module with a pure resolution of type (d0, d1).
Write Ω(B) = N⊕r where N is one of B,Mi, ωB,Mi,j and set c = c(N) ∈ {0, 1,
3
2
, 2}. Then
the Betti numbers of M are a multiple of the entries of the vector
(1,
3
3− c
,
3c
3− c
, 2
3c
3− c
, 4
3c
3− c
, 8
3c
3− c
, . . . , 2i−2
3c
3− c
, . . . )
Proof. Using (2.3), we get
HM (t) = HB(t)(β0t
d0 − β1t
d1 + β1ct
d1+1
∑
i≥2
(−1)i2i−2ti−2)
=
1 + 2t
1− t
(β0t
d0 − β1t
d1 +
β1ct
d1+1
1 + 2t
)
=
(1 + 2t)(β0t
d0 − β1t
d1) + β1ct
d1+1
1− t
.
Since M is finite length, HM(t) is a polynomial, and so the numerator of the above expression
is divisible by 1− t. So the substitution t 7→ 1 gives the relation 3(β0−β1)+β1c = 0. Taking
β0 = 1 gives us the desired statement. 
Remark 2.5. The Betti numbers do not depend on the actual degree sequence, just on the
isomorphism class of the MCM module. For convenience, we list here the first lattice point
on the rays spanned by the vectors from Proposition 2.4:
v1 = (1, 1, 0, 0, 0, . . . ),
v2 = (2, 3, 3, 6, 12, . . . ),
v3 = (1, 2, 3, 6, 12, . . . ),
v4 = (1, 3, 6, 12, 24, . . . ).
Note that 2v3 = v1 + v4 and 2v2 = 3v1 + v4, so that the cone generated by v1,v2,v3,v4 is
minimally generated by v1 and v4. 
3. The cone of Betti diagrams
3.1. Main result. Each pure resolution has an associated degree sequence which documents
in what degree each term of the resolution is generated. If the resolution is finite, we pad
the degree sequence with the symbol ∞, so the possible degree sequences are
(d0,∞,∞, . . . ), (d0, d1,∞, . . . ), (d0, d1, d1 + 1, d1 + 2, . . . )
where d0 < d1.
For each degree sequence d define πd ∈ V as follows. First, (πd)0,j = 1 for j = d0 and
0 otherwise. If d = (d0,∞,∞, . . . ), πi,j = 0 for all i ≥ 1 and all j. If d = (d0, d1,∞, . . . ),
THE CONE OF BETTI TABLES OVER THREE NON-COLLINEAR POINTS IN THE PLANE 5
π1,j = 1 for j = d1 and 0 otherwise. If d = (d0, d1, d1 + 1, d1 + 2, . . . ), πi,j = 3 · 2
i−1 for i ≥ 1
and j = di and 0 otherwise. Recall that these are the Betti numbers (up to scalar multiple)
of a finite length module whose syzygy module is a direct sum of copies of Mi,j and that we
do not use the others because of the relations in Remark 2.5.
For each v ∈ V, define
ǫi,j(v) = vi,j, αk(v) = 2ǫ1,k(v)− ǫ2,k+1(v), γk(v) =
∑
j≤k
(3ǫ0,j(v)− 3ǫ1,j+1(v) + ǫ2,j+2(v)).
We also define γ∞ by summing over all j. Note that while γk involves an infinite sum, only
finitely many nonzero terms of v are involved so it is well-defined.
Theorem 3.1. The following cones are equal:
(i) The cone BQ(B) spanned by the Betti diagrams of all finitely generated B-modules.
(ii) The cone D spanned by πd for all B-degree sequences d.
(iii) The cone F defined to be the intersection of the halfspaces {ǫi,j ≥ 0} for all i, j ≥ 0,
{αk ≥ 0} for all k ∈ Z, {γk ≥ 0} for all k ∈ Z, and the subspaces {2ǫi,j = ǫi+1,j+1}
for i ≥ 2, j ∈ Z.
The proof of Theorem 3.1 follows by establishing the inclusions D ⊆ BQ(B) ⊆ F ⊆ D,
which are the content of the next three lemmas.
Lemma 3.2. D ⊆ BQ(B).
Proof. It suffices to show that for each degree sequence d there exists a B-module M with
βB(M) = πd. We assume d0 = 0 for simplicity of notation.
• When d = (0,∞,∞, . . . ), take M = B.
• When d = (0, d1,∞, . . . ), take M = B/((x+ y + z)
d1).
• When d = (0, d1, d1 + 1, d1 + 2, . . . ), take M = B/(x
d1 , yd1, zd1). 
Lemma 3.3. BQ(B) ⊆ F .
Proof. For a finitely generated graded B-module M , we want to show that the inequali-
ties defining F are nonnegative on βB(M). There are no negative entries in βB(M), so
ǫi,j(β
B(M)) = βBi,j(M) ≥ 0 for all i, j. The inequalities {αk ≥ 0 | k ∈ Z} and the equalities
{2ǫi,j = ǫi+1,j+1 | i ≥ 2, j ∈ Z} hold for β
B(M) by our discussion in §2.3.
It remains to show that γk(β
B(M)) ≥ 0 for all k ∈ Z. Let · · · → F1 → F0 → M → 0 be a
minimal free resolution of M . Suppose that Ω(M) ∼= B⊕α0 ⊕ ω⊕α1B ⊕M
⊕α2
i ⊕M
⊕α3
i,j . Let βi
be the ith Betti number of M (ignoring the grading). From §2.2 and §2.3, we can read off
the relations
e(Ω(M)) = 3α0 + 3α1 + 2α2 + α3,
β1 = α0 + 2α1 + α2 + α3,
β2 = 3α1 + α2 + 2α3.
In particular, e(Ω(M)) = 3β1 − β2. From the short exact sequence
0 → Ω(M) → F0 → M → 0,
we get e(M) = e(F0)− e(Ω(M)) = 3β0 − 3β1 + β2, and e(M) is a nonnegative quantity, so
3β0 − 3β1 + β2 ≥ 0.(3.4)
6 IULIA GHEORGHITA AND STEVEN V SAM
If F is a free B-module, let τ≤k(F) be the free module generated by minimal generators of
degree ≤ k. For a general B-module M , define τ≤k(M) to be the quotient of τ≤k+1(F1) →
τ≤k(F0). In §2.3, we established that the differentials Fi → Fi−1 are linear for i ≥ 2, and so
we conclude that the following is a minimal free resolution of τ≤k(M):
· · · → τ≤k+i(Fi) → τ≤k+i−1(Fi−1) → · · · → τ≤k+1(F1) → τ≤k(F0) → τ≤k(M) → 0.
In particular, applying (3.4) to τ≤k(M), we conclude that γk(β
B(M)) ≥ 0, as desired. 
Lemma 3.5. F ⊆ D.
Proof. The proof is formally the same as the proofs of [BBEG, Lemmas 2.7, 2.8]. 
Remark 3.6. Following [BBEG], define a partial order on the degree sequences by d ≤ d′ if
d0 ≤ d
′
0 and d1 ≤ d
′
1 where either inequality is strict, or if d0 = d
′
0, d1 = d
′
1, and dn ≤ d
′
n for all
n ≥ 2. Then the proofs of [BBEG, Lemmas 2.7, 2.8] show that BQ(B) has a triangulation
coming from the simplicial cones spanned by the rays corresponding to the elements of
maximal chains in this partial order. 
From Theorem 3.1, we immediately get a result about the cone of finite length B-modules:
Corollary 3.7. The following cones are equal:
(i) The cone BfQ(B) spanned by the Betti diagrams of all finite length B-modules.
(ii) The cone Df spanned by πd for all B-degree sequences d with d1 < ∞.
(iii) The cone F f defined to be the intersection of the halfspaces {ǫi,j ≥ 0} for all i, j ≥ 0,
{αk ≥ 0} for all k ∈ Z, {γk ≥ 0} for all k ∈ Z, and the subspaces {2ǫi,j−ǫi+1,j+1 = 0}
for i ≥ 2, j ∈ Z and γ∞ = 0.
Proof. In each of three items, we have added one extra condition to the cones defined in
Theorem 3.1, so we just have to verify that these conditions are the same. From the proof
of Lemma 3.3, we get that γ∞(β
B(M)) = e(M). Since e(M) = 0 if and only if M has finite
length, this shows that the extra conditions in (a) and (c) coincide. From Proposition 2.4,
γ∞(πd) = 0 if d2 < ∞. If d1 < ∞ and d2 = ∞, it is immediate from the definition that
γ∞(πd) = 0. Finally, for the last case with d1 = ∞, we have γ∞(πd) = 1, so the extra
conditions in (b) and (c) coincide. 
3.2. Local version. Following [BEKS], we now give a local version of our main result by
considering minimal free resolutions over the completion B̂ = k[[x, y, z]]/(xy, yz, xz). In this
case, the relevant numerical invariant is the Betti sequence β(M) of a module M which
records the number of generators of each term in a minimal free resolution of M .
As discussed above, the only relevant information are the first 3 Betti numbers (β0, β1, β2)
since βi = 2βi−1 for i > 2. The following results are easy to prove from what we have already
discussed.
Proposition 3.8. The following cones are equal:
(i) The cone spanned by the Betti sequences of all finitely generated B̂-modules.
(ii) The cone spanned by (1, 0, 0, . . . ), (1, 1, 0, . . . ), and (1, 3, 6, . . . ).
(iii) The cone defined by the inequalities β2 ≥ 0, 3β0 + β2 ≥ 3β1, and 2β1 ≥ β2, and the
equalities βi = 2βi−1 for i > 2.
Proposition 3.9. The following cones are equal:
(i) The cone spanned by the Betti sequences of all finite length B̂-modules.
THE CONE OF BETTI TABLES OVER THREE NON-COLLINEAR POINTS IN THE PLANE 7
(ii) The cone spanned by (1, 1, 0, . . . ), and (1, 3, 6, . . . ).
(iii) The cone defined by the inequalities β2 ≥ 0 and 2β1 ≥ β2, and the equalities 3β0+β2 =
3β1 and βi = 2βi−1 for i > 2.
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California Institute of Technology, Pasadena, CA
E-mail address : igheorgh@caltech.edu
University of California, Berkeley, CA
E-mail address : svs@math.berkeley.edu
URL: http://math.berkeley.edu/~svs/


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The cone of Betti tables over three non-collinear points in the plane

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