COHOMOLOGY OF NUMERICAL MONOIDS
GENERATED BY THREE ELEMENTS
Arne B. Sletsjøe
Abstract.
We calculate the dimension of the tangent space of the deformation functor for numer-
ical monoid algebras generated by three elements. Combining this with similar results of Buchweitz
we obtain an implicit formula for the Frobenius number of the monoid.
0. Introduction
A numerical semigroup S is a subsemigroup of the natural numbers N (0 included). We
shall assume that S contains N shifted by some n ≥ 0, i.e. n + N ⊂ S. The least number
with this property is called the conductor of S, denoted by c. A numerical semigroup is
called symmetric if there exists m ∈ Z such that for all s ∈ Z we have s ∈ S if and only
if m − s ∈
/ S. It is easy to see that this m is the greatest gap, i.e. the greatest positive
nonnumber of S, and that c = m + 1. The number m is often referred to as the Frobenius
number of S. Let H = {n ≥ 0 | n ∈
/ S} be the set of gaps. The cardinality of H is called
the genus of S. All numerical semigroups S =< p, q >, generated by two elements are
symmetric with m = pq − p − q. If g = |H| is the genus of S, a simple computation shows
that for a symmetric numerical semigroup 2g = m + 1.
In this paper we shall mainly be concerned with numerical semigruops generated by three
elements, nicely described by Herzog in [H]. In this case we put S =< g1 , g2 , g3 >. Let
R(S) = {(z1 , z2 , z3 ) ∈ Z3 | z1 g1 + z2 g2 + z3 g3 = 0}
be the set of relations between the generators. A relation (v1 , v2 , v3 ) is minimal of type
i, i = 1, 2, 3, if for all (z1 , z2 , z3 ) ∈ R(S) such that either zi > 0 and zj ≤ 0, j 6= i or zi < 0
and zj ≥ 0, j 6= i, we have |vi | ≤ |zi |. A relation is minimal if it is minimal of any type.
P3
The degree of v is given by deg(v) = i=1 g1 max(o, vi ).
A symmetric numerical monoid generated by three elements g1 , g2 , g3 has exactly to minimal relations of degree r1 , r2 and the conductor is given by c = r1 + r2 − g1 − g2 − g3 + 1.
To find the moduli space of k[S] we compute the tangent space T 1 (k[S]) and the obstruction space T 2 (k[S]). If S is generated by three elements the associated k-algebra k[S] has
embedding dimension 3, and T 2 (k[S]) = 0 (see e.g. [B]). Thus the moduli space is smooth
of dimension dimT 1 (k[S]) = 2g − 1 + |M|, where M is the set of “maximal” gaps, i.e.
M = {h ∈ H | h + S+ ⊂ S}. In the symmetric case |M| = 1 and we have
dimk T 1 (k[S]) = 2g = c
Herzog ([H]) gives the formula m = r1 + r2 − g1 − g2 − g3 for the Frobenius number of S.
Thus we get the formula
dimk T 1 (k[S]) = r1 + r2 − g1 − g2 − g3 + 1
which is verified in this paper by quite different methods.
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The non-symmetric case is somewhat more complicated. Then we have three minimal
relations and the formula for the Frobenius number is no more valid. Buchweitz ([B]) gives
a formula for the dimension of the moduli space
dimk T 1 (k[S]) = 2g − 1 + |M|
We prove that in the non-symmetric, three-generator case |M| = 2 and it follows that this
dimension is 2g + 1. On the other hand we also prove the formula
dimk T 1 (k[S]) = r1 + r2 − g1 − g2 − g3 + |M| − |N |
where r1 , r2 are the smallest minimal relations and N ⊂ H is a set of gaps, defined via the
order relation of S. We also prove the formula
2g = m1 + 1 + (2m2 − m1 )ˆ
where M = {m2 < m1 } and γ̂ = (γ − S) ∩ N. This gives a formula involving the Frobenius
number
m1 + (2m2 − m1 )ˆ+ |N | = r1 + r2 − g1 − g2 − g3
The paper is divided into two parts. In the first we recall some basic and important facts
about cohomology of monoid-like sets and in the second part we do the computations in
the three-generator case.
1. Cohomology
A numerical semigroup S has the structure of an ordered set given by
s1 ≤ s2 ∈ S
if
∃t ∈ S
such that s1 + t = s2
Let L ⊂ S be some sub-ordered set.
DEFINITION (1.1). L ⊂ S is said to be a monoid-like ordered set if for all
(S-)relations s1 ≤ s2 in L there exists t ∈ L such that s1 + t = s2 .
Let L ⊂ S be a monoid-like set and let k be a field of characteristic zero. Let Cn (L) be
n
the vector space on
Pnthe set {(λ1 , . . . , λn ) ∈ L | w(λ) ∈ L} where w is the weight-function
given by w(λ) = i=1 λi ∈ S. The symmetric group Σn acts on the basis of Cn (L) by
σ(λ1 , . . . , λn ) = (λσ−1 (1) , . . . , λσ−1 (n) )
The group Σn also acts on the dual groups C n (L) = Homk (Cn (L), k), by(φ · σ)(λ) =
φ(σ(λ)) for φ ∈ C n (L) and σ ∈ Σn .
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Denote by Shn (L) the subspace of Cn (L) generated by all shuffle-products, i.e. the subPn−1
module sn · Cn (L), where sn = i=1 si,n−i is the sum of all (i, n − i)-shufflings and let
CSn (L) = Homk (Cn (L)/Shn (L), k).
The inhomogenous differential
δ n : C n−1 (L) −→ C n (L)
is defined by
δ n ξ(λ1 , . . . , λn ) =ξ(λ2 , . . . , λn ) +
n−1
X
(−1)i ξ(λ1 , . . . , λi + λi+1 , . . . λn )
i=1
n
+ (−1) ξ(λ1 , . . . , λn−1 )
For n=1 we put δ 1 ξ = 0.
Notice that the differential acts as a graded derivation with respect to the shuffle product.
DEFINITION (1.2). The inhomogenous Harrison cohomology HAn (L, k) of the ordered set L is the cohomology of the cocomplex CS• (L) with the inhomogenous differential
δ.
In this paper we have fixed a ground field k and without any confusion we may skip the k
in the expression HAn (L, k).
There is also a relative version of Harrison cohomology. Let L0 ⊂ L ⊂ S and suppose L0
is full in L, i.e. if γ ∈ L, γ0 ∈ L0 and γ ≥ γ0 , then γ ∈ L0 . Then S − L0 is a monoid-like
set. The relative Harison cocomplex is given by
CSn (L − L0 , L) = {φ ∈ C n (L − L0 , L) | φ(Shn (L)) = 0}
where
C n (L − L0 , L) = {φ ∈ C n (L) | φ(λ) = 0 for ω(λ) ∈ L − L0 }
and relative Harrison cohomology, HAn (L − L0 , L) is cohomology of the relative complex.
There is a long-exact sequence
0 −→ HA1 (L − L0 , L) −→HA1 (L) −→ HA1 (L − L0 )
−→ HA2 (L − L0 , L) −→ HA2 (L) −→ . . .
relating Harrison cohomology of the ordered sets L and L − L0 with relative cohomology.
There is a close relation between Harrison cohomology of a numerical semigroup and the
graded parts of algebra cohomology of the associated semigroup algebra. The associated
semigroup algebra is a curve with an isolated singularity at the origin and the cohomology
groups Harrn (S, k[S]) for n ≥ 2 have finite dimension. Thus we have an isomorphism
X
∼
Harrn,λ (S, k[S]) → Harrn (S, k[S])
λ∈Z
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for all n ≥ 1. The graded cohomology group Harrn,λ (S, k[S]) is obtained from the subcocomplex of homogenous cochains of degree λ. (See [Sl] for details.)
The relation between graded Harrison cohomology groups and Harrison cohomology of
ordered sets is stated in the next theorem and proved in [Sl].
THEOREM (1.3). With the notation as above there is an isomorphism in cohomology
∼
Harrn,λ (S, k[S]) → HAn (S+ − S−λ , S+ )
n≥1
where Sλ = (λ + S) ∩ S+ , and S+ = S − {0}.
Proof. See [Sl].
For n = 1 it is easy to see that
HA1 (S+ − S−λ , S+ ) ≃ Harr1,λ (S, k[S]) ≃ Der1,λ (S, k[S])
The dual of the cohomology group Harr2 (S, k[S]) equals the tangent space T 1 (k[S]) of the
deformation functor for the k-algebra k[S]. Thus we have
dimk T 1 (k[S]) =
X
HA2 (S+ − S−λ ; S+ )
λ∈Z
In the next section we shall give an explicit formula for the dimension of this group for an
arbitrary numerical monoid generated by three elements.
2. The case of three generators
In this section we study numerical monoids generated by three elements as described in
the introduction.
PROPOSITION (2.1). The first relative Harrison cohomology group is given by
1
HA (S+ − Sλ , S+ ) ≃
k
0
when −λ ∈ S ∪ M
when −λ ∈
/ S ∪ M¸
where M = {λ ∈ H | λ + S+ ⊂ S} is the set of “maximal” gaps.
Proof. Since S ⊂ N+ it is easy to see that dim Der1,λ (S, k[S]) ≤ 1. It follows that
1,λ
Der (S, k[S]) 6= 0 if and only if there exists a graded derivation D such that D(gi ) 6= 0
for all i = 1, 2, 3, i.e. gi + λ ∈ S.
It is easy to see that HA1 (S+ ) ≃ k and from [Sl] we know that HAn (S+ ) = 0 for n ≥ 2.
Thus in this case the long-exact sequence reduces to the exact sequence
0 −→ HA1 (S+ − Sλ , S+ ) −→ k −→ HA1 (S+ − Sλ ) −→ HA2 (S+ − Sλ , S+ ) −→ 0
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Combining this with Prop. (2.1) we get
dim HA2 (S+ − Sλ , S+ ) = dim HA1 (S+ − Sλ ) + χS∪M (−λ) − 1
where χS∪M is the characteristic function of S ∪ M, i.e.
χS∪M (λ) =
n
1
0
λ∈S∪M
λ∈
/ S∪M
To compute the dimension of the tangent space we need some more information about
HA1 (S+ − Sλ , S+ ) for various λ ∈ Z.
It is easily seen that for a monoid-like subset K ⊂ S of a monoid S generated by three
elements, we have
dim HA1 (K) = α0 (K) − α1 (K)
where α0 (K) is the number of generators of S inside K and α1 (K) is the number of minimal
relations of S inside K, upward bounded by 2. We shall compute the number of generators
and minimal relations of S inside K = S+ − Sλ for various λ ∈ Z. In fact what we shall
compute the sum of the numbers
dim HA2 (S+ − Sλ , S+ ) = α0 (S+ − Sλ ) − α1 (S+ − Sλ ) + χS∪M (−λ) − 1
when λ ranges over all of Z.
Observe that for λ << 0 we have S+ − Sλ = ∅, −λ ∈ S ∪ M and therefore
dim HA2 (S+ − Sλ , S+ ) = 0 − 0 + 1 − 1 = 0
In the other end, for λ >> 0 we have α0 (S+ − Sλ ) = 3, α1 (S+ − Sλ ) = 2, −λ ∈
/ S∪M
and consequently
dim HA2 (S+ − Sλ , S+ ) = 3 − 2 + 0 − 1 = 0
The vanishing outside a bounded set also follows from the fact that k[S] is an isolated
singularity.
So consider λ ∈ [−n, n] where n is big enough. A simple computation shows that for any
element γ ∈ S+ we have γ ∈ S+ − Sλ if and only if −λ ∈
/ S − γ. Suppose we have given
γ ∈ S+ . The number of λ such that −λ ∈
/ S − γ equals the cardinality of the complement
(inside [−n, n]) of (−γ + S) ∩ [−n, n] = S ∩ [−n + γ, n + γ]. This number is easily seen to
be (2n + 1) − (n + γ + 1 − g) = n − γ + g. On the other hand the number of λ ∈ [−n, n]
such that χS∪M (−λ) 6= 0 is given by n + 1 − g + |M|.
LEMMA (2.2). For a symmetric monoid S we have M = {m}, the maximal gap.
Proof. For a symmetric monoid the set of gaps H = (m − S) ∩ Z+ = m̂, thus for all
h ∈ H − {m} there is a λ ∈ S+ such that h + λ = m ∈
/ S.
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Recall that a symmetric monoid generated by three elements has to minimal relations.
The following theorem is due to Buchweitz ([B]) and the formula c = r1 +r2 −g1 −g2 −g3 is
proved by Herzog in [H]. We give an alternative proof for the dimension formula, involving
cohomology of monoid-like sets as described above.
THEOREM (2.3 (Buchweitz)). For a symmetric monoid S =< g1 , g2 , g3 > with
minimal relations R1 , R2 of degree r1 , r2 the dimension of the moduli space is given by
dimk T 1 (k[S]) = r1 + r2 − g1 − g2 − g3 + 1 = c
Proof. From the discussion above we get
X
dimk T 1 (k[S]) =
HA2 (S+ − Sλ , S+ )
λ∈Z
and thus
dimk T 1 (k[S]) = (n − g1 + g) + (n − g2 + g) + (n − g3 + g)
− (n − r1 + g) − (n − r2 + g) + (n + 1 − g + |M |) − (2n + 1)
= r1 + r2 − g 1 − g 2 − g 3 + 1 = c
For the non-symmetric case the situation is somewhat more complicated.
We shall compute the cardinality of M when S is non-symmetric, generated by three
elements. Let S =< g1 , g2 , g3 > be a numerical semigroup generated by three elements
and let s 6= 0 be an element of S. Let B(s) = {b0 , . . . , bs−1 } be the Apéry set with respect
to s , i.e. b0 = s and for i ≥ 0 bi is the least integer in S having s-residue distinct from
those of b0 , . . . , bi−1 . For the set B(s) we define
B̃(s) = {b ∈ B(s) | b 6= b0 , b + bi ∈
/ B(s) for every i = 0, . . . , s − 1}
PROPOSITION (2.4 (Cavahiere-Niesi)). Let S be a numerical monoid generated
by three elements. Then the cardinality |B̃(s)| of B̃(s) is independent of choice of s and
we have
1 if S is symmetric
|B̃(s)| =
2 if S is non-symmetric
Proof. see [CN] (2.6 and 5.1)
Let x ∈ B(s) and suppose x − s ∈ S. Then x, x − s have the same s-residue, contrary to
the definition of B. Thus (B̃(s) − s) ∩ S = ∅.
We can sharpen the result of the Cavahiere-Niesi-proposition, using the set B̃(s) − s.
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PROPOSITION (2.5). The set B̃(s) − s is independent of choice of s and we have
B̃(s) − s = M
Proof. Let x ∈ B̃(s) ⊂ B(s) and suppose x − s ∈ S. Then x, x − s have the same
s-residue, contrary to the definition of B(s). Thus
(B̃(s) − s) ∩ S = ∅
Let s = g1 + g2 + g3 . Then g1 , g2 , g3 ∈ B(s). For x ∈ B̃(s) we have x + g1 , x + g2 , x + g3 ∈
/
B(s), by definition. Thus, for each i = 1, 2, 3, there is a ti > 0 such that
x − ti s + g i ∈ S
Now suppose ti > 2. Then ti − 1 > 0 and we get
x − s = (x − ti s + gi ) + ((ti − 1)s − gi ) ∈ S
since s−gi ∈ S. But this contradicts the fact that x ∈ B(s). Thus ti = 1 and x−s+gi ∈ S
∀i = 1, 2, 3 and x − s + S+ ⊂ S.
On the other hand, if h ∈
/ S and h + S+ ⊂ S we have h + s ∈ S and consequently
h + s ∈ B(s). Obviously h + s 6= s. Since h + b ∈ S for all b ∈ B(s) we have h + b + s ∈ S
and the minimality property ensures that h + b + s ∈
/ B(s). Thus h + s ∈ B̃(s) and
h ∈ B̃(s) − s, proving that the two sets are equal.
COROLLARY (2.6). If S is a numerical semigroup generated by three elements, and
M is as defined above, then we have
1 S symmetric
|M| =
2 S is not symmetric
Proof. Combine the two last propositions.
In the non-symmetric case there are three minimal relations R1 , R2 , R3 of degree r1 , r2 , r3 .
Using the same argument as above we obtain the formula
dimk (⊕λ∈Z HA2 (S+ − Sλ , S+ ))
= r1 + r2 − g1 − g2 − g3 + |M| − (n − r3 + g) + |K ′ |
where K ′ = {λ ∈ Z | r1 , r2 , r3 ∈ S+ − Sλ }.
Now γ ∈ S+ − Sλ if and only if λ ∈ γ − H̃, where H is the extended set of gaps, i.e.
H̃ = Z − S. Thus we can write
K ′ = (r1 − H̃) ∩ (r2 − H̃) ∩ (r3 − H̃)
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or more convenient as a disjoint union
K ′ = K ∪ [r3 + 1, n]
where K = (r1 − H̃) ∩ (r2 − H̃) ∩ (r3 − H) and |K ′ | = |K| + n − r3 . There is a one-to-one
correspondance between the set K and
L = H ∩ (r3 − r1 + H̃) ∩ (r3 − r2 + H̃)
given by a 7→ r3 − a. Let
N = H ∩ [(r3 − r1 + S) ∪ (r3 − r2 + S)]
Then |N | + |K| = g. The minimality of the relation R3 implies r3 − r1 , r3 − r2 ∈ H.
With this notation we have
dimk T 1 (k[S]) = r1 + r2 − g1 − g2 − g3 + |M| − |N |
The next lemma gives a generalization of the fact that for a symmetric monoid H̃ = m−S,
where m is the maximal gap.
LEMMA (2.7).
H̃ = M − S =
[
(m − S)
m∈M
Proof. Since M ⊂ H it is obvious that m − S ⊂ H̃.
On the other hand, for each h ∈ H̃ we have h + S+ ⊂ S, or h + S+ 6⊂ S. In the first case
h ∈ M and h = m − 0 ∈ M − S. In the other case there exists γ ∈ S+ such that h + γ ∈ H̃:
Repeat the argument until h + s + S+ ⊂ S, where s = λ1 + . . . + λk . Then h + s ∈ M and
we get H̃ ⊂ M − S.
Using the S-order relation on Z, i.e. n1 ≤ n2 if n2 − n1 ∈ S we recognize the set N as the
elements lying in between {m1 , m2 } and {r3 − r1 , r3 − r2 }.
Thus we have the following theorem.
THEOREM (2.8). For a non-symmetric monoid S =< g1 , g2 , g3 > with minimal
relations R1 , R2 , R3 of degree r1 < r2 < r3 the dimension of the tangent space of the
deformation functor equals the conductor c of the monoid, i.e.
dimk T 1 (k[S]) = r1 + r2 − g1 − g2 − g3 + |M| − |N |
where N is the set
N = [(m1 − S) ∪ (m2 − S)] ∩ [(r3 − r1 + S) ∪ (r3 − r2 + S)]
Proof. Follows from the above discussion.
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We can give another formula for this dimension, using the result of [B], stating that the
dimension equals 2g + 1.
Let M = {m1 > m2 }. It is easily seen that the set [0, m1 ] is contained in the disjoint
union
[0, m1 ) ⊂ S ∪ m̂1 ∪ (m̂2 − m̂1 )
Counting the elements inside [0, m1 ] we get the formula 2g + |(m̂2 − m̂1 )|. Because of the
maximality property of m2 we have m1 −m2 ∈ H. Furthermore we have m1 −(m1 −m2 ) =
m2 ∈
/ S and Lemma 2.7 gives m1 − m2 ∈ m2 − S, and thus 2m2 − m1 ∈ S.
LEMMA (2.9). There is a one-to-one-correspondance
(2m2 − m1 )ˆ→ m̂2 − m̂1
given by s 7→ s + (m1 − m2 ).
Proof. Let u ∈ S such that m2 − u ∈ m̂2 . Suppose m2 − u ∈
/ m̂1 , i.e. m1 − m2 + u ∈
/ S.
Since m2 ∈
/ S we have m1 − m2 + u ∈
/ m1 − S and we must have m1 − m2 + u ∈ m2 − S,
say m1 − m2 + u = m2 − t, t ∈ S. But the we have (m2 − u) − (m1 − m2 ) = t ∈ S, and
we have proved that m1 − m2 ≤ m2 − u ≤ m2 or 0 ≤ (m2 − u) − (m1 − m2 ) = 2m2 − m1 .
For 0 ≤ γ ≤ 2m2 − m1 , γ ∈ S we have m1 − m2 ≤ γ + m1 − m2 ≤ m2 . Suppose
γ + m1 − m2 ≤ m1 , i.e. m1 − (γ + m1 − m2 ) = m2 − γ ∈ S. Then m2 ∈ S which is obvious
a contradiction. It follows that γ + m1 − m2 ∈ m̂2 − m̂1 .
Using this lemma we get a formula relating the maximal elements (including the Frobenius
number m1 )
m1 + (2m2 − m1 )ˆ+ |N | = r1 + r2 − g1 − g2 − g3
References
[AS] Altmann,K.,Sletsjøe,A.B., “André-Quillen Cohomology of Monoid Algebras”
J. Algebra 210, pp. 708-718 (1998).
[B] Buchweitz,R.-O., “On deformations of monomial curves”
Springer Lect. Notes in Math. 777, pp. 205-220 (1980).
[CN] Cavahiere,M.P., Niesi,G., “On monomial curves and Cohen-Macauly type”
Manus. Math 42 pp. 147-159 (1983).
[F] Frøberg,R., “The Frobenius number of some semigroups”
Com. in alg. 22(14) pp. 6021-6024 (1994) .
[H] Herzog,J., “Generators and relations of abelian semigroups and semigroup rings”
Manus. Math. 3 pp. 175-193 (1970).
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[Sk] Skaar,L.M., “Deformasjoner av numeriske monoide-algebraer”
Masters thesis, University of Oslo (1996).
[Sl] Sletsjøe,A.B., “Cohomology of monoid algebras”
J. Algebra 161(1) pp. 102-128 (1993).
MATEMATISK INSTITUTT
UNIVERSITETET I OSLO
Pb. 1053 BLINDERN
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