RIGID IDEALS BY DEFORMING QUADRATIC LETTERPLACE
IDEALS

arXiv:1605.07417v1 [math.AG] 24 May 2016

GUNNAR FLØYSTAD AND AMIN NEMATBAKHSH
Abstract. We compute the deformation space of quadratic letterplace ideals L(2, P )
of finite posets P when its Hasse diagram is a rooted tree. These deformations are unobstructed. The deformed family has a polynomial ring as the base ring. The ideal J(2, P )
defining the full family of deformations is a rigid ideal and we compute it explicitly. In
simple example cases J(2, P ) is the ideal of maximal minors of a generic matrix, the
Pfaffians of a skew-symmetric matrix, and a ladder determinantal ideal.

1. Introduction
Monomial ideal theory has much developed into a branch of its own. But before that
one studied polynomial ideals in general. Monomial ideals came about since they are
specializations, typically initial ideals, of such ideals. One should then ask: Can monomial ideal theory give something back? Can one start with monomial ideals and derive
interesting classes of polynomial ideals in general? Yes one can, and here we do this for a
reasonably large class of monomial ideals. We get a full understanding of the polynomial
ideals which specialize to the monomial ideals we start out from.
The ideals we work with. More precisely we consider quadratically generated letterplace
ideals L(2, P ) associated to a finite poset P . These are precisely the edge ideals of CohenMacaulay bipartite graphs. Its generators are the monomials x1,p x2,q where p ≤ q in
the poset P . That edge ideals of Cohen-Macaulay bipartite graphs have this form, is an
astonishing discovery of J.Herzog and T.Hibi [13]. This class of ideals were generalized
in [9] and further studied and generalized in [10] were they were called letterplace ideals,
see Section 2.
Results. When the Hasse diagram of P has the form of a rooted tree we get a complete
algebraic understanding of all ideals which are deformations of the quadratic letterplace
ideals L(2, P ). This is all the more unusual and suprising for the following reason: Monomial ideals are degenerations of polynomial ideals. Thus whenever a monomial ideal is on
a Hilbert scheme, it tends to be a singular point on the Hilbert scheme. Its infinitesimal
deformations are then obstructed and it is a very hard and messy task to compute the
space of all deformations.
However for the letterplace ideals L(2, P ) we consider, it turns out that every nice thing
one could wish for, actually happens:
• The ideals L(2, P ) are unobstructed, i.e. every infinitesimal deformations lifts. In
particular whenever this ideal is on a Hilbert scheme, it is a smooth point.
• The full deformation space which a priori is defined only over a complete local
ring, acutally lifts to a deformation over a polynomial ring. This follows from our
computation of the first cotangent cohomology of L(2, P ), Corollary 6.11, and the
explicit family we give in Section 3.
Date: May 25, 2016.
1

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GUNNAR FLØYSTAD AND AMIN NEMATBAKHSH

• The full family of deformations over this polynomial ring is defined by a rigid ideal
J(2, P ), Corollary 8.3. So deformations of L(2, P ) come from a coordinate change
in J(2, P ).
• We explicitly compute the ideal J(2, P ) by a simple recursive procedure, see Section
3 and Equation 3.1.
Deforming Borel-fixed ideals. As said monomial ideals are usually obstructed. Any ideal
can be specialized to a Borel-fixed monomial ideal, which in characteristic zero is the same
as a strongly stable ideal. One could then envision a path to classify ideals by deforming
strongly stable ideals. Unfortunately this is rather hopeless since strongly stable ideals
typically are very obstructed. (A notable exception to this is the lexsegment ideal. When
the polynomial ring is given the standard grading the lexsegment ideal is a smooth point
on the Hilbert scheme [17], thus giving a distinguished component of the Hilbert scheme
for every Hilbert function h : Z → N.)
Deforming Stanley-Reisner ideals. Deformation theory applied to Stanley-Reisner ideals
has been developed by K.Altman and J.Christophersen. In [2] they give the basic deformation theory for Stanley-Reisner rings. In [3] and [5] they consider triangulations of
spheres, which deform to Calabi-Yau manifolds, and triangulations of tori, which deform
to Abelian varieties. For classes of triangulations they compute the versal deformation
space (base space) of the (infinitesimal) deformation functor. This space is typically not
smooth, i.e. typically not a power series ring, but they give equations for the relations
of this space, and give a detailed description of it. Here, for quadratic letterplace ideals,
we find that the base space both is smooth and global and that we can give explicit
equations for the whole family of deformations. Recently [1] applied the theory developed
by Christophersen and Altman to investigate when monomial ideals are rigid. For edge
ideals they develop a number of results for when this holds. They also classify the (few)
letterplace ideals which are rigid.
Rigid ideals. The notion of a rigid ideal occurs in (infinitesimal) deformation theory.
Although it is a well-known notion, we have not been able to find many examples of rigid
ideals in the literature. Classically determinantal ideals of generic matrices are known
to be rigid, [4]. Recently [6] shows that the coordinate rings of Grassmannians for the
Plücker embedding are rigid ideals. As mentioned above [1] also gives classes of rigid
monomial ideals. With the present article we therefore believe we make a substantial
contribution to the known classes of rigid ideals.
Multigraded Hilbert schemes. While rigidity is an infinitesimal notion, one obtains global
families of deformations, the (multigraded) Hilbert schemes, when one endows the ambient
polynomial ring with a grading by an abelian group A [11]. The A-graded infinitesimal
deformations are a subset of all infinitesimal deformations. For global families there is then
a situation close to rigidity. An ideal I ⊆ k[X] may not be rigid, but there nevertheless
is a rigid ideal J in a larger polynomial ring k[Y ] such that any deformation of I comes
from a coordinate change in J and then restricting to k[X]. J.Kleppe [15] shows that
this is the case when I is a determinantal ideal associated to a matrix of linear forms,
but where there may be dependencies between the linear entries. We show that the same
phenomenon happens here when we consider the letterplace ideals I = L(2, P ) and their
A-graded deformations, Theorem 9.5 and the applications after it.
Organization of the paper. In Section 2 we recall letterplace ideals as defined in [10]. In
this article we are concerned with the quadratic letterplace ideals L(2, P ) where the Hasse

RIGID IDEALS BY DEFORMING QUADRATIC LETTERPLACE IDEALS

3

diagram of P is a rooted tree. In Section 3 we give an explicit recursive procedure for
computing the family J(2, P ) of deformations of L(2, P ). Section 4 contains examples of
these deformed families for posets P of cardinality 3 and 4, and also for two other simple
classes of posets. In the first cases we get the ideal of two-minors of a 2 × 5-matrix and
the ideal of Pfaffians of a skew-symmetric 5 × 5-matrix. Section 5 shows that the family of
ideals J(2, P ) is very finely graded, by a free abelian group of cardinality 2|P |. Section 6
investigates the deformation theory of the ideals L(2, P ) and we compute the non-trivial
first order deformations of L(2, P ) for any finite poset P . These are given by the first
cotangent cohomology group. This module turns out to have an extremely nice set of
generators. For each generator there is a single monomial x1,p x2,p mapping to another
monomial while all other monomials map to zero. Section 7 shows flatness of J(2, P )
over the base polynomial ring. In Section 8 we show rigidity of J(2, P ) (an infinitesimal
notion). Section 9 considers global families of deformations. We show that the letterplace
ideals L(2, P ) are smooth points on the Hilbert schemes and that the general point on the
Hilbert scheme comes from the ideal J(2, P ) by a coordinate change and then restricting.
Sections 8 and 9 are developed in a general setting, and the results concerning L(2, P )
are just particular instances of general results. In the end we give Conjecture 9.8, that
the results of this article holds for any finite poset P and not just for posets whose Hasse
diagram is a rooted tree.
Acknowledgement. We thank Jan Christophersen for useful discussions which significantly
influenced the form of this paper.
2. Letterplace ideals of posets
Let k be a field. If R is a set, denote by k[xR ] the polynomial ring k[xi ]i∈R . For a
natural number n let the chain poset be [n] = {1 < 2 < · · · < n}, so [2] = {1, 2}.
Given a finite poset P . We shall in this paper be concerned with the monomial ideal
L(2, P ) in k[x[2]×P ] ] generated by quadratic monomials x1,p x2,q where p ≤ q. These ideals
are by [13] precisely the edge ideals of Cohen-Macaulay bipartite graphs, see Section 9 of
[14] for more on this. The ideals L(2, P ) are a special case of letterplace ideals L(n, P )
introduced in [9] and [10], generated by monomials
x1,p1 x2,p2 , · · · , xn,pn
where p1 ≤ · · · ≤ pn are weakly increasing chains in P . By [9, Corollary 2.5], see also [10,
Corollary 2.4], L(n, P ) is a Cohen-Macaulay ideal of codimension equal to the cardinality
|P |. The multiplicity of L(2, P ) is the cardinality of the distributive lattice of poset ideals
of P , see Section 2 of [8]. This is the same as the degree of the algebraic subscheme of
the affine space A2|P | defined by L(2, P ).
For ease of notation we shall rather write a variable xi,p as pi . Thus L(2, P ) is generated
by quadratic monomials p1 q2 where p ≤ q.
For more on letterplace ideals L(n, P ) and their Alexander duals L(P, n) and the omnipresence of these monomial ideals, see [10]. In [8] we compute the Betti tables of the
letterplace ideals L(n, P ), in particular of L(2, P ).
3. The family of deformations
We here describe the main object of study in this article: the ideals J(2, P ) which are
deformations of the letterplace ideals L(2, P ). The generators of L(2, P ) are monomials
p1 q2 where p ≤ q. We shall deform each such generator, and the ideal J(2, P ) will be

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GUNNAR FLØYSTAD AND AMIN NEMATBAKHSH

the ideal generated by these deformations. We do this for the situation that the Hasse
diagram of the poset P is a rooted tree. Except for Section 5, this is our assumption
throughout the paper.
The root of P is at the bottom. If an element b covers a we say that a is a parent of
b and we write a ≺ b. Two elements b and c are called siblings if they have the same
parent.
For each pair q, p where the meet of q and p is the parent of p, we introduce a variable
uq,p. Let b and c be distinct siblings. We define
X
Tc (b) = −
q2 uq,b.
q≥c

If a is a parent of b we let
Ta (b) = −a2 ua,b .
We also define
T (b) = Tb (b) = −Ta (b) −

X

Tc (b),

c

where we sum over siblings c of b, distinct from b. If ρ is the root of P we define
T (ρ) = u∅,ρ .
The rationale for introducing these variables will become clear in Subsection 6.4 where
we compute the cotangent cohomology of the ring k[x[2]×P ]/L(2, P ), Corollary 6.11. Let
B be the polynomial ring in these variables
B = k[u∅,ρ , uq,p ]
ranging over all pairs (q, p) such that the meet of q and p is the parent of p. This will be
the base ring for our family of deformations. Let B(2, P ) be the ring B ⊗k k[x[2]×P ]. This
is the ring where the ideal of the full deformation family lives.
For an element p in the poset P we define the depth of p, depth(p) to be the length
of the longest chain upwards, starting from p. Thus if p is a maximal element, then
depth(p) = 0.
We now define the following.
• If a ≺ b so a is the parent of b, we shall define determinants D(a)b lying in B(2, P ),
as well as determinants D(a)a .
• If a ≤ b we shall define polynomials Sa (b2 ) lying in B(2, P ). Then Sa extends
uniquely to a linear map on linear combinations of elements b2 where b ≥ a. For
short we shall often write Sa (b) for Sa (b2 ).
We shall do this inductively on depth(a).
Given these definitions, we also define the following:
• If b and c are distinct siblings let
Sc Tc (b) = Sc (Tc (b)).
This definition will appear by induction on depth(c) as we define Sc .
We also define the following:
• Sb Tb (b) = b1 .
• If a ≺ b so a is the parent of b, let Sa Ta (b) = −ua,b .

RIGID IDEALS BY DEFORMING QUADRATIC LETTERPLACE IDEALS

5

Note that these last two definitions are not compositions, i.e. Sb Tb (b) is not Sb (Tb (b)).
Rather we think of these definitions as symbolic expressions.
Now let us start the inductive definitions. If a is maximal, that is depth(a) = 0 we
define Da (a) = 1 and Sa (a) = 1.
Otherwise let b1 , . . . , bm be the children of a. For uniformity we denote b0 = a. We
form the m × (m + 1) matrix M(a) = [Sbi Tbi (bj )] where the column index i = 0, . . . , m
and the row index j = 1, . . . , m.


Sa Ta (b1 ) Sb1 Tb1 (b1 ) · · · Sbm Tbm (b1 )
 Sa Ta (b2 ) Sb1 Tb1 (b2 ) · · · Sbm Tbm (b2 ) 


..
..
..


.
.
.

i

Sa Ta (bm ) Sb1 Tb1 (bm ) · · · Sbm Tbm (bm )


b11
· · · Sbm Tbm (b1 )
−ua,b1
 −ua,b2 Sb1 Tb1 (b2 ) · · · Sbm Tbm (b2 )

=
..
..

 ...
.
.
m
m
b1
−ua,bm Sb1 Tb1 (b ) · · ·

Let M(a)b be the matrix obtained by deleting column i. Define the signed determinant
i

i

D(a)i = D(a)b = (−1)i |M(a)b |.
Note that in order to define this, we need to have defined Sp for all p strictly bigger than
a.
For a ≤ b define R(a, b) = 1 if a = b and if a < b define
Y
R(a, b) =
D(p)q
a≤p≺q≤b

where the product is over all covering relations p ≺ q between a and b.
Now define
Sa (b2 ) = R(a, b)D(b)b .
As said before this definition extends in a natural way to linear combinations of variables
b2 such that b ≥ a. Also we shall often for short write Sa (b) for Sa (b2 ). This completes
the inductive definitions. Now we give the ideal defining the full family of deformations
of L(2, P ).

Definition 3.1. Let J(2, P ) be the ideal in B(2, P ) generated by
p1 q2 − T (p)Sp (q)
for all p ≤ q.
The following shows the various entities defined, in the simplest situation. It is useful
in the next section where we give examples of the ideals J(2, P ).
Lemma 3.2. Suppose a ∈ P has a single child b. Then:
a. Sa (a) = b1 ,
b. D(a)a = b1 ,
c. D(a)b = ua,b ,
d. T (b) = a2 ua,b .


Proof. In this case the matrix M(a) = −ua,b b1 .



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GUNNAR FLØYSTAD AND AMIN NEMATBAKHSH

We shall in the Section 7 show that the ring B(2, P )/J(2, P ) is a deformation of the
letterplace ring k[2, P ]/L(2, P ), flat over the base ring B.
4. Examples
We consider here four examples of posets and give the deformed equations explicitly.
We also identify the variety they define.
4.1. Determinantal variety. Let P be the totally ordered poset [n] = {1 < 2 < · · · <
n}. For simplicity we assume n = 4 and write P = {a < b < c < d}.
The deformations of monomials p1 p2 for p ∈ P are
p1 p2 − T (p)Sp (p).
Since in this case p has one or none child, and also no sibling, we apply Lemma 3.2 and
these deformations are:
a1 a2 − u∅,a b1
b1 b2 − a2 ua,b c1
c1 c2 − b2 ub,c d1
d1 d2 − c2 uc,d
Furthermore we have the deformed polynomials:
a1 b2 − u∅,a ua,b c1
b1 c2 − a2 ua,b ub,c d1
c1 d2 − b2 ub,c uc,d.

a1 c2 − u∅,a ua,b ub,c d1
b1 d2 − a2 ua,b ub,c uc,d

a1 d2 − u∅,a ua,b ub,c uc,d

These binomials are the 2-minors of the following 2 × 5 matrix


a1 b1 ua,bc1 ua,b ub,c d1 ua,b ub,c uc,d
,
u∅,a a2
b2
c2
d2
after we localize by inverting ua,b and ub,c . This matrix can also be written as:


a1 Sa (a) Sa (b) Sa (c) Sa (d)
.
u∅,a
a2
b2
c2
d2
4.2. The Pfaffians of 5 × 5 skew-symmetric matrices. Consider the star poset P
b1 b2

bm

a
We will show the deformations when the root has two or three children. First suppose
we have two children so the poset is
b

c

a

RIGID IDEALS BY DEFORMING QUADRATIC LETTERPLACE IDEALS

7

The matrix M(a) is:

 

Sa Ta (b) Sb Tb (b) Sc Tc (b)
−ua,b
b1
−uc,b
=
.
Sa Ta (c) Sb Tb (c) Sc Tc (c)
−ua,c −ub,c
c1

There are five generating monomials in L(2, P ) and the deformed polynomials are:
b1 b2 + T (b) · 1
c1 c2 + T (c) · 1
a1 b2 + T (a)Sa (b)
a1 c2 + T (a)Sa (c)
a1 a2 + T (a)Sa (a)
which are:
b1 b2 − a2 ua,b − c2 uc,b
c1 c2 − a2 ua,c − b2 ub,c
a1 b2 − u∅,a ua,c uc,b − u∅,a ua,b c1
a1 c2 − u∅,a ua,b ub,c − u∅,a ua,c b1
a1 a2 − u∅,a b1 c1 + u∅,a uc,bub,c
These are (after diving by u∅,a , the Pfaffians of the following skew-symmetric matrix:


0
ub,c
c2
b1
a1
−ub,c
0
ua,c u−1
c1 
∅,a a1


 −c2
.
−u
0
u
b
a,c
a,b
2


−1
 −b1 −u a1 −ua,b
0
uc,b
∅,a
−a2
−c1
−b2 −uc,b 0

Setting u∅,a = 1, they are also the Plücker relations defining the Grassmann variety G(2, 5)
embedded in projective space P9 .
Now let us consider the case of the star poset P with three childen:
b
c
d

a
The matrix M(a) is:




−ua,b
b1
−uc,b −ud,b
−ua,c −ub,c
c1
−ud,c  ,
−ua,d −ub,d −uc,d
d1
and the D(a)x are the signed maximal minors of this matrix. There are seven generating
monomials in L(2, P ). Their deformations are the following:
b1 b2 − a2 ua,b − c2 uc,b − d2 ud,b
c1 c2 − a2 ua,c − b2 ub,c − d2 ud,c
d1 d2 − a2 ua,d − b2 ub,d − c2 uc,d
(1)

a1 b2 − u∅,a D(a)b
a1 c2 − u∅,a D(a)c

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GUNNAR FLØYSTAD AND AMIN NEMATBAKHSH

a1 d2 − u∅,a D(a)d
a1 a2 − u∅,a D(a)a
Question 4.1. Is there a natural description of the variety defined by these equations?
4.3. Ladder determinantal varieties. Consider now the poset P :
br

cs
br−1

cs−1
b1

c1

a
1
1
where r, s ≥ 2. We let br+1
= 1 = cs+1
1
1 , and for short write b = b and c = c . The matrix
M(a) is:



−ua,b
b1
Sc Tc (b)
.
−ua,c Sb Tb (c)
c1
The monomials in L(2, P ) deform to the following polynomials generating J(2, P ):

(2)

bi1 bj2

−

bi−1
2 (

j
Y

ubk−1 ,bk )bj+1
1

2≤i≤j≤r

ubk−1 ,bk )cj+1
1

2≤i≤j≤s

k=i

ci1 cj2

−

ci−1
2 (

j
Y
k=i

(3)

b11 bj2

− (a2 ua,b1 +

s
X
j=1

c11 cj2

− (a2 ua,c1 +

s
X
j=1

(4)

a1 bj2 − u∅,a D(a)b (

j
Y

cj2 ucj ,b1 )(

j
Y

ubk−1 ,bk )bj+1
1

1≤j≤r

uck−1 ,ck )cj+1
1

1≤j≤r

k=2

bj2 ubj ,c1 )(

j
Y

k=2

ubk−1 ,bk )bj+1
1

1≤j≤r

uck−1 ,ck )cj+1
1

1≤j≤s

k=2

a1 cj2

c

− u∅,a D(a) (

j
Y

k=2

(5)

a1 a2 − u∅,a D(a)a

RIGID IDEALS BY DEFORMING QUADRATIC LETTERPLACE IDEALS

9

Write T = a2 ua,b ua,c + ua,b Tb (c) + ua,c Tc (b). We claim that the ideal J(2, P ) they
generate is precisely the ideal of two-minors of the ladder:
..
..
..
.
.
.
2
2
0
0
Sc (c )
c2
Sc (c1 )
c12
0
0
· · ·,
b
1
2
D(a)
T
b2
b2
···
c
1
2
u−1
a
D(a)
S
(b
)
S
(b
)
···
b
b
∅,a 1
if we localize by inverting u∅,a , ua,b and ua,c . Let us call the part of the above ladder
starting from the column with b12 , the left leg. The minors of the left leg are precisely the
equations (2). The minors formed by taking the first column (only the two lowest entries)
and a column of the left leg, gives the equations (4). Now by multiplying the column of bi2
with ua,b ubi ,c and subtracting the sum of all these scaled columns from the column with
T , we obtain a column


ua,c Tc (b) + a2 ua,b ua,c
.
−ua,c b1
Taking the determinant of this column and the columns of the left leg, we obtain the
equations (3) after inverting ua,c .
Lastly we want to obtain the equation (5). Take the determinant of the lower left 2 × 2
matrix in the ladder. This is (after mulitplying with u∅,a ):
a1 T − u∅,a D(a)b D(a)c
=ua,b ua,c a1 a2 + a1 ua,b Tb (c) + a1 ua,c Tc (b)
+u∅,a [ua,b ua,c b1 c1 − c1 u2a,b Sb Tb (c) − b1 u2a,c Sc Tc (b) + ua,b ua,c Sb Tb (c)Sc Tc (b)].
By subtracting the following polynomial, obtained as a linear combination of the r first
equations in (4):
ua,b [a1 Tb (c) − u∅,a D(a)b Sb Tb (c)]
and the polynomial, obtained as a linear combination of the s second equations in (4):
ua,c [a1 Tc (b) − u∅,a D(a)c Sc Tc (b)],
we obtain the last equation (5).
4.4. Variety of complexes. For definition of variety of complexes refer to [7]. Let P be
the poset with Hasse diagram:
cs
cs−1
c1

b

a
where s ≥ 2. We write c = c1 and define cs+1
= 1. We have
1


Qj
Ps
uck−1 ,ck
−ua,b
b1
− j=1 ucj ,b cj+1
1
k=2
M(a) =
−ua,c −ub,c
c1

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GUNNAR FLØYSTAD AND AMIN NEMATBAKHSH

The deformed relations of L(2, P ) are the following:
(6)

ci1 cj2

−

ci−1
2 (

j
Y

uck−1,ck )cj+1
1

2≤i≤j≤s

k=i

(7)

c11 cj2

− (a2 ua,c + b2 ub,c )

j
Y

uck−1 ,ck cj+1
1

1≤j≤s

k=2

(8)

b1 b2 − a2 ua,b −

s
X

cj2 ucj ,b

j=1

(9)

a1 cj2

c

− u∅,a D(a) (

j
Y

uck−1 ,ck )cj+1
1

1≤j≤s

k=2

(10)

a1 b2 − u∅,a D(a)b

(11)

a1 a2 − u∅,a D(a)a

The 2-minors of the matrix


Qs
uck−1 ,ck
a1
c11 u∅,a
c21 u∅,a ua,c1 · · · cs+1
1 u∅,a ua,c1
k=2
X=
−1
···
cs2
c12
b1 + ua,b u−1
a,c ub,c a2 + b2 ua,c ub,c

except the one associated to the first two columns give us the relations (6), (7) and (9).
Now define the matrix Y as


b2 −ua,b −uc1 ,b · · · −ucs ,b

The two entries of the matrix XY give the relations (8) and (10). If we multiply equation
(10) with u−1
a,c ub,c and subtract it from the minor of the first two columns of X, we get
equation (11).
In the appendix we give a larger poset P and the generators of J(2, P ). As we see these
polynomials grow quickly in size.
5. The fine positive grading on the ideal J(2, P )

In this section we show that the ideal J(2, P ) is very finely graded. In fact it is graded
by a free abelian group on 2|P | free generators. We show that this grading is positive in
the sense of [11]. This enables us to state a simple criterion for flatness of homogeneous
ideals, which we will apply in Section 7 to conclude that the quotient ring B(2, P )/J(2, P )
is flat over its base ring B = k[u∅,ρ , uq,p].
5.1. The grading on J(2, P ). Let Z([2] × P ) be the free abelian group of order 2|P |
generated by the p1 ’s and p2 ’s for p ∈ P . The ideal J(2, P ) lives in the polynomial ring
B(2, P ) = k[x[2]×P ] ⊗k k[u∅,ρ , uq,p].
The pairs (q, p) are all pairs such that the meet of q and p is the parent of p. They,
together with u∅,ρ correspond to the minimal generators of the first cotangent module
T 1 (k[x[2]×P ]/L(2, P )) by Corollary 6.11. For an element p ∈ P let b1 , . . . , bm be its children. Denote by p̂ the degree p2 − b11 − b21 − · · · − bm
1 in Z([2] × P ).
Now define a grading on the B(2, P ) by letting the variable xi,p (which we write as pi )
have degree (with some abuse of notation) pi in the abelian group Z([2] × P ). Also define
the degree of uq,p to be p1 − q2 + p̂.

RIGID IDEALS BY DEFORMING QUADRATIC LETTERPLACE IDEALS

11

Proposition 5.1. The ideal J(2, P ) is homogeneous for this Z([2]×P )-grading. Moreover:
(1) T (p) is homogeneous of degree p1 + p̂.
(2) If p ≤ q then Sp (q) is homogeneous of degree q2 − p̂.
(3) When p and q are siblings, Sq Tq (p) is homogeneous of degree p1 + p̂ − q̂.
(4) Let q be a child of p. The determinant D(p)q is homogeneous of degree q̂ − p̂. The
determinant D(p)p is homogeneous of degree p2 − p̂.
Proof. (1) easily follows from the definitions. Note that if ρ is the root then deg(T (ρ)) =
deg(u∅,ρ ) = ρ1 + ρ̂.
We prove (2) and (4) simultaneously since they are both dependent on each other. (3)
follows from (2) by definition. Let m be the cardinality of P and η : P → [m] a linear
extension.
We prove (2) and (4) by descending induction on η(p). Suppose η(p) = m, the maximal
possible. Then D(p)p = Sp (p) = 1 has degree p2 − pb = 0.
Now we show that if for any two elements p ≤ q such that η(p) = k + 1 we have
deg D(p)p = p2 − p̂,

deg D(p)q = q̂ − p̂,

deg Sp (q) = q2 − p̂

then the above statement is also true for any two elements p ≤ q with η(p) = k.
So suppose η(p) = k and let b1 , . . . , bm be children of p. These have all η-value ≥ k + 1.
By assumption we have
m
X Y

p
Sbi Tbi (bσ(i) )
deg(D(p) ) = deg
=

m
X
i=1

σ∈Sm i=1
m
X

bi1 +

i=1

bbi −

m
X
i=1

bbi = p2 − p̂.

where Sm is symmetric group on m letters.
Similarly when q is a child of p, we have deg(D(p)q ) = q̂ − p̂. For p ≤ q in P let
p = x0 , . . . , xm = q be the maximal chain between p and q in P . Then

1
2
deg(Sp (q)) = deg D(p)x D(x1 )x · · · D(xm−1 )q D(q)q
ˆ ) + (q2 − q̂)
= (xb1 − p̂) + (xb2 − xb1 ) + · · · + (q̂ − xm−1
= q2 − p̂,

This completes the proof of (2)-(4). The main assertion now follows from (1) and (2). 

5.2. Positive gradings by an abelian group. Let Y be a finite-dimensional vector
space graded by an abelian group A. This gives an A-graded polynomial ring k[Y ]. This
grading is positive if the only elements in k[Y ] of degree 0 are the constants in k. By [16,
Prop. 8.6], this is equivalent to each graded piece k[Y ]a , a ∈ A being a finite-dimensional
vector space over k. The following is another characterization.
Lemma 5.2. The grading by A on k[Y ] is positive, iff there is a homomorphism A → Z
such that the Z-degree of any A-homogeneous element of Y is positive, i.e. is in Z>0 .
Proof. The if direction is clear. The only if direction follows by [16, Cor. 8.7] by applying
this to the torsion free part of A, which must be nonzero. (Note that the definition of
positive grading in [16, Chap.8] requires A to be torsion-free. We follow the convention of
[11] which does not have this requirement.)


12

GUNNAR FLØYSTAD AND AMIN NEMATBAKHSH

Proposition 5.3. The grading on B(2, P ) given in Proposition 5.1 is positive.
d

Proof. Define a map from Z([2] × P ) −→ Z by letting every d(p2 ) = 1 and define
P d(p1 )
inductively on depth(p) such that the d(p) is positive and larger than the sum b d(b)
where we sum over the children b of p. Then it is seen that all variables pi have positive
d-values, as well as all variables uq,p .

Let J ⊆ k[Y ] be an ideal which is homogeneous for a positive A-grading on Y . Let
U ⊆ Y be a homogeneous subspace for this grading, and I ⊆ k[Y /U] be the ideal such
that
k[Y /U]/I = (k[Y ]/J) ⊗k[Y ] k[Y /U].
Let f1 , . . . , fk be generators of J and let f1 , . . . , fk denote their images in I, which will
generate I. The following is the criterion we use, in Theorem 7.6, to show that J(2, P ) is
a flat deformation of L(2, P ).
Proposition 5.4. Suppose the A-grading on k[Y ] is positive. If every relation of f1 , . . . , fk
lifts to a relation for f1 , . . . , fk , then k[Y ]/J is flat over k[U].
Proof. We apply the local criterion of flatness [18, Thm.A.5] and the criterion of lifting
of relations [18, Thm.A.10], to the algebra homomorphism k[Y ]/J → k[Y /U]/I. The situation in [18, Thm.A.5] is a local homomorphism of local noetherian rings. Our situation
is a graded homomorphism of positively graded rings. This situation works just as well
for the arguments.

6. First order deformations and the cotangent cohomology
This sections presents the basic general deformation theory that we need. General
references for deformation theory are [18, Chap.3], [12, Chap.1,3] or [19, Sec.3]. We
compute explicitly the first cotangent cohomology T 1 (k[x[2]×P /L(2, P )) for any quadratic
letterplace ideal of a poset P . As we shall see the elements of this module are remarkably
simple in form.
6.1. The deformation functor. We consider a k-algebra R and an ideal I in R. Let B
be another k-algebra with a distinguished k-point b ∈ Spec B corresponing to a morphism
B → k. A deformation over B of the ideal I ⊆ R, is an ideal J ⊆ R ⊗k B such that:
• (R ⊗k B)/J is flat over B,
• The natural map R ⊗k B → R induces a map (R ⊗k B)/J → R/I which becomes
an isomorphism
∼
=

(R ⊗k B)/J ⊗B k → R/I.
Let Set be the category of sets, and k − Art be the category of local artinian k-algebras
with residue field k.
The functor of infinitesimal deformations of I
DefI : k − Art → Set,
is given by letting DefI (A) be the set of such deformations J over the ring A.

RIGID IDEALS BY DEFORMING QUADRATIC LETTERPLACE IDEALS

13

6.2. The tangent space. Let k[ǫ] = k[t]/(t2 ). The deformations over this ring are called
first order deformations. The tangent space of the deformation functor is DefI (k[ǫ]) (see
below). This space identifies as HomR (I, R/I). Let k − vect be the category of finitedimensional vector spaces. For a vector space V denote by V ∗ its dual space. There is a
functor
k − vect → k − Art
sending V to k[V ∗ ]/(V ∗ )2 . By abuse of notation we get a restricted functor DefI from
k − vect where DefI (V ) = DefI (k[V ∗ ]/(V ∗ )2 ). We also have a functor
Homk (−, HomR (I, R/I)) : k − vect → Set,
sending V to the set of linear maps V → HomR (I, R/I). The following is standard, see
[18] Proposition 3.2.1 and Definition 3.2.3.
Proposition 6.1. There is a natural isomorphism of functors between the two functors
DefI , Homk (−, HomR (I, R/I)) : k − vect → Set.
The upshot is that the tangent space DefI (k[ǫ]) of the functor DefI on k − Art identies
as HomR (I, R/I)). We describe the isomorphism of functors in more detail.
1. Choose a basis v1 , . . . , vr for V . We obtain a dual basis v1∗ , . . . , vr∗ for V ∗ . A linear map
φ : V → HomR (I, R/I) gives an ideal
X
J = {f +
vi∗ φ(vi )(f ) | f ∈ I}.
i

This is the flat deformation corresponding to φ.
Alternatively φ gives an element of V ∗ ⊗k HomR (I, R/I) and so a map
φ : I → I ⊗k V ∗ ⊗k HomR (I, R/I) → V ∗ ⊗k R/I.

Then J is the ideal
J = {f + φ(f ) | f ∈ I}.
2. Conversely given a deformation J ⊆ R ⊗k k[V ∗ ]/(V ∗ )2 of I ⊆ R, flat over k[V ∗ ]/(V ∗ )2 .
Note first that the inclusion k → k[V ∗ ]/(V ∗ )2 induces R → R ⊗k k[V ∗ ]/(V ∗ )2 , and so the
ideal I embeds as a linear subspace of the right ring. There is a short exact sequence
0 → R ⊗k V ∗ → R ⊗k k[V ∗ ]/(V ∗ )2 → R → 0.
Tensoring this with −⊗R⊗k[V ∗ ]/(V ∗ )2 J we obtain, recall that J is flat, a short exact sequence
π

0 → V ∗ ⊗k[V ∗ ]/(V ∗ )2 J → J −→ J/(V ∗ · J) = I → 0.
The term V ∗ ⊗k[V ∗ ]/(V ∗ )2 J identifies as V ∗ · J = V ∗ ⊗k I. An f ∈ I lifts by π to an
fˆ = f + p where p ∈ R ⊗k V ∗ . Any two liftings differ by an element of V ∗ ⊗k I, and so
we get a well-defined R-module map I → V ∗ ⊗k (R/I) sending
f 7→ p ∈ V ∗ ⊗k (R/I).
But such an R-module map corresponds to a linear map V → HomR (I, R/I).
Now consider the situation of a subset T of Homk (V ∗ , R) = V ⊗k R. It gives a map
V → R ⊗k T ∗ . This gives a map of algebras:
∗

τ : R ⊗k k[V ∗ ]/(V ∗ )2 → R ⊗k R ⊗k k[T ∗ ]/(T ∗ )2 → R ⊗k k[T ∗ ]/(T ∗ )2 ,
where in the last map we have used the multiplication on R.

14

GUNNAR FLØYSTAD AND AMIN NEMATBAKHSH

Lemma 6.2. Let J ⊆ R ⊗k k[V ∗ ]/(V ∗ )2 be a flat deformation of I ⊆ R over k[V ∗ ]/(V ∗ )2 .
Then the image J ′ = τ (J) is a flat deformation of I over k[T ∗ ]/(T ∗ )2 . If the deformation
J corresponds to the linear map V → HomR (I, R/I), then the deformation J ′ corresponds
to the linear map T → V ⊗k R → HomR (I, R/I), where to define the latter map we have
used the R-module structure on HomR (I, R/I).
Proof. Let the vj ’s form a basis for V and the ti ’s form a basis for T . If the map T → V ⊗k R
is given by
X
vj ⊗k rij ,
ti 7→
j

then the map V ∗ → T ∗ ⊗k R is given by

X

vj∗ 7→

t∗i ⊗k rij .

i

Let the deformation J be given by
f+

X

vj∗ fj ,

f ∈ I.

j

The linear map V ∗ → T ∗ ⊗k R gives the deformation
XX
f+
(t∗i ⊗k rij )fj , f ∈ I
j

and the ideal J ′ consists of

f+

i

XX
j

t∗i rij fj ,

f ∈I

i

But this is the deformation corresponding to the linear map that sends each ti to the
R-module map
X
rij vj : I → R/I
j

f 7→

X

rij fj ,

j

and so is a flat deformation.



6.3. Deformations over polynomial rings. Now assume the ring R is a polynomial
ring k[X] where X is a finite-dimensional vector space.
A first order deformation of I over k[ǫ] is trivial if it is the image of
I ⊗k k[ǫ] ⊆ k[X] ⊗k k[ǫ]
by an (infinitesimal) coordinate change in k[X] ⊗k k[ǫ], meaning a linear map
X → k[X] ⊗k k[ǫ]
lifting the canonical inclusion X → k[X]. This linear map induces the infinitesimal
coordinate change, an algebra homomorphism
k[X] ⊗k k[ǫ] → k[X] ⊗k k[ǫ].

RIGID IDEALS BY DEFORMING QUADRATIC LETTERPLACE IDEALS

15

Example 6.3. Let k[X] = k[x, y] and I = (xy). The map
x 7→ x + ǫy 3 ,

y 7→ y + ǫxy

is an infinitesimal coordinate change. The ideal I is mapped to the ideal (xy + ǫx2 y + ǫy 4)
by this coordinate change.
Let Derk (k[X]) be the derivations of k[X]. If the xi form a basis for X, this is a free
k[X]-module generated by the derivatives ∂/∂xi for i = 1, . . . , n. There is a map
(12)

δ ∗ : DerK (k[X]) → Homk[X] (I, k[X]/I)

which sends ∂ to the homomorphism sending fi to ∂fi + I. The image of the homomorphism δ ∗ consists of all the trivial infinitesimal deformations of k[X]/I.
There is a diagonal map X → X ⊕ X inducing
τ : k[X] → k[X ⊕ X] = k[X] ⊗k k[X] → k[X] ⊗k k[X]/(X)2 .
Example 6.4. Let X have basis x1 , x2 and let the other copy of X have basis t1 , t2 . The
image by τ of the ideal (x21 x2 ) in k[x1 , x2 ] is then the ideal generated by
(x1 + t1 )2 (x2 + t2 ) = x21 x2 + t1 2x1 x2 + t2 x21 .
The following characterizes the images of the partial derivatives in a coordinate free
way.
Lemma 6.5. Let I ⊆ k[X] be an ideal. Then I˜ = τ (I) is a flat deformation of I over
k[X]/(X 2 ). The corresponding linear map X ∗ → Hom(I, k[X]/I) sends a dual basis
element x∗i to the derivation ∂/∂xi .
Proof. By tensoring with k[X] we get a flat (trivial) deformation I ⊗k k[X] in k[X]⊗k k[X]
over k[X]. Denote the generators of the algebra k[X] ⊗k k[X] = k[X ⊕ X] by xi ⊕ 0 and
0 ⊕ ti . We now perform the coordinate change xi 7→ xi ⊕ ti and ti 7→ ti . Note that this
coordinate change is a k[X]-module map for the inclusion xi 7→ 0 ⊕ ti . The image of
˜ Since
I ⊗k k[X] by this coordinate change is denoted I.
(k[X]/I) ⊗k k[X] → k[X ⊕ X]/I˜
is a k[X]-module isomorphism, the right side will be flat over k[X]. The result now follows
by the base change k[X] → k[X]/(X)2 .

The cokernel Coker δ ∗ of the map δ ∗ in (12) is called the first cotangent module of k[X]/I
and is denoted by T 1 (k[X]/I). Its elements are in one-to-one correspondence with the
non-trivial infinitesimal deformations of k[X]/I.
Definition 6.6. The ideal I is a rigid ideal if T 1 (k[X]/I) vanishes or equivalently the
map δ ∗ is surjective.
This means that every deformation of I comes from a coordinate change. We show in
Section 8 that J(2, P ) ⊆ B(2, P ) is a rigid ideal.

16

GUNNAR FLØYSTAD AND AMIN NEMATBAKHSH

6.4. The first cotangent cohomology for letterplace ideals. We consider a finite
poset P . Let S be the polynomial ring k[x[2]×P ]. Recall that we denote the variable xi,p
as pi . The letterplace ideal L = L(2, P ) is generated by all monomials p1 q2 where p ≤ q.
We shall compute the first cotangent cohomology group T 1 (S/L), or rather its minimal
generating set as an S-module. The general theory for doing this for Stanley-Reisner rings
is developed in [2]. The results for T 1 (k[x[2]×P /L(2, P )) are however so simple that we do
it from first principles, without recalling technicalities in loc.cit.
If J and U are subsets of P we say that U is an upper bound set for J if for every p ∈ J
there is a q ∈ U with p ≤ q. Similarly if F and D are subset of P , we say D is a lower
bound set for F if for every p ∈ F there is a r ∈ D with r ≤ p. For p ∈ P let J(< p)
(resp. J(≤ p)) be the order ideal of P consisting of all r ∈ P with r < p (resp. r ≤ p),
and let F (> p) (resp. F (≥ p)) be the order filter consisting of all q ∈ P with q > p (resp.
q ≥ p).
The first cotangent cohomology T 1 (S/L) is the quotient of HomS (L, S/L) by the derivations ∂/∂xi,p as we range over the variables of S. The following describes the minimal
generators of T 1 (S/L). It is remarkable in that they come from maps in HomS (L, S/L)
where only one monomial is mapped to something nonzero. Also all monomials p1 q2 ∈ I
where p < q are always mapped to zero.
Theorem 6.7. Given p ∈ P . Let U be a an inclusion minimal subset of P − F (≥ p)
which is an upper bound set for J(< p), and let D be an inclusion minimal subset of
P − J(≤ p) which is a lower bound set for F (> p). Suppose also that whenever r ∈ D
and s ∈ U, we do not have r ≤ s. Then the map
φ : L → S/L
sending
p1 p2 7→ Π r1 Π s2
r∈D

s∈U

and all other minimal monomial generators in L to zero, is a nonzero map of S-modules.
Moreover as we vary over p ∈ P and sets D and U with this property, the maps φ form
a minimal generating set of the cotangent cohomology T 1 (S/L).
Note that D and U above will both be antichains. We denote the map above as
φ = (p1 p2 7→ Π r1 Π s2 ).
r∈D

s∈U

Before proving the above theorem we develop some lemmata.
Lemma 6.8. Let φ ∈ HomS (L, S/L). Modulo Im δ ∗ the homomorphism φ is equal to a
homomorphism φ′ with the property that for any p ≤ q in P , φ′ (p1 q2 ) can be written as a
linear combination of monomials m not divisible by p1 or q2 .
Proof. Suppose φ(p1 q2 ) contains a nonzero term of the form cp1 m for some monomial m
and constant c. We show that modulo Im δ ∗ we can eliminate this term. More precisely
we show that for any generator of L of form r1 q2 , for some r ≤ q, φ(r1 q2 ) contains the term
cr1 m. Therefore by subtracting cm ∂q∂2 from φ we can eliminate the terms cr1 m without
adding any extra terms to φ(r1 q2 ).
Suppose r1 q2 is a generator of L that contains q2 . The syzygy r1 (p1 q2 )−p1 (r1 q2 ) induces
the relation r1 φ(p1 q2 ) − p1 φ(r1 q2 ). The term cr1 p1 m either belongs to L or for some m′
in φ(r1 q2 ) it cancels out by the term cp1 m′ . If r1 p1 m is in L then r1 m ∈ L. If it cancels
out then m′ = cr1 m. Therefore in either case φ(r1 q2 ) contains cm ∂q∂2 (r1 q2 ). Note that if

RIGID IDEALS BY DEFORMING QUADRATIC LETTERPLACE IDEALS

17

for some generator g of L, q2 ∤ g then m ∂q∂2 (g) = 0. The proof for the case where φ(p1 q2 )
contains a term of the form mq2 is similar.

For any homomorphism φ ∈ HomS (L, S/L), the following lemma shows which monomials can appear in the image of generators of L.
Lemma 6.9. Let φ ∈ HomS (L, S/L). For p ≤ q let p1 q2 be a generator of L. We may
write φ(p1 q2 ) as a linear combination of monomials m that are not in L.
(1) If p 6= q then gcd(p1 q2 , m) 6= 1.
(2) If p = q then either gcd(p1 p2 , m) 6= 1 or, for all r < p there is t ≥ r such that t2 |m
and for all r ′ > p, there is s ≤ r ′ such that s1 |m.
Proof. 1. Suppose p 6= q and gcd(p1 q2 , m) = 1. We have a relation p2 φ(p1 q2 ) − q2 φ(p1 p2 )
in S/L since p2 (p1 q2 ) − q2 (p1 p2 ) is a syzygy of L. If p2 m is canceled with some monomial
in q2 φ(p1 p2 ) then q2 |m, against our initial assumption above.
Hence p2 m ∈ L, and this implies that there is some s ≤ p such that s1 |m. In a similar
manner the syzygy q1 (p1 q2 ) − p1 (q1 q2 ) shows that there is some t ≥ q such that t2 |m. So
s1 t2 |m for s ≤ t which is a contradiction. The upshot is that we must have gcd(p1 q2 , m)
must be nontrivial.
2. Suppose p = q and gcd(p1 p2 , m) = 1. Let x(p1 p2 ) − y(p′1 q2′ ) be a relation involving the
generator p1 p2 , so y contains at least one of p1 or p2 (and x does not). If xm is going to
be canceled by some monomial in yφ(p′1q2′ ) then p1 p2 and m can not be relatively prime.
Therefore if gcd(p1 p2 , m) = 1 then for any such syzygy xm belongs to L. Note that if
x(p1 q2 ) − y(p′1 q2′ ) is not a linear syzygy then xm belongs to L, so we only need to consider
linear syzygies. For any r < p the linear syzygy r1 (p1 p2 ) − p1 (r1 p2 ) implies r1 m ∈ L.
Hence for some t ≥ r, t2 divides m. Similarly for any r ′ > p, m should be divided by a
variable s1 for some s ≤ r ′.

Proof of Theorem 6.7. Let φ ∈ Hom(L, S/L) be a homomorphism. By Lemmata 6.8 and
6.9 we can decompose φ as φ = φ1 + φ2 where φ1 belongs to the image of δ ∗ and for any
p ∈ P , all the monomials in φ2 (p1 p2 ) are relatively prime to p1 p2 and also for any p < q,
φ2 (p1 q2 ) = 0. Now let m be a nonzero monomial in φ(p1 p2 ), let ψ be a map that sends
p1 p2 to m and any other generator of L to zero. Since m is relatively prime to p1 p2 the
second part of proof of 6.9 shows that this map satisfies all the relations of L. Hence
it is a well-defined homomorphism. By Lemma 6.9 (2), there exists some U and D as
in 6.7 such that Πr∈D r1 Πs∈U s2 |m. Note that if U contains an element s in F (≥ p) then
D contains an element r such that r ≤ s and m belongs to L which is a contradiction.
Hence U ⊆ P − F (≥ p). Similarly D ⊆ P − J(≤ p). Therefore the homomorphisms in
6.7 generate T 1 (S/L).

Example 6.10. Let P be a poset with following Hasse diagram.
e

c

d

a

b

18

GUNNAR FLØYSTAD AND AMIN NEMATBAKHSH

For the point a, J(< a) = ∅ so the empty set is the only inclusion minimal subset of
P − F (≥ a) which is an upper bound for J(< a). We also have F (> a) = {c, d, e} and
the inclusion minimal subsets of P − J(≤ a) that are lower bounds for F (> a) are {c, d}
and {b}. Therefore for a, we have two first order deformations corresponding to maps
a1 a2 7→ c1 d1 ,

a1 a2 7→ b1 .

Consider the point c. J(< c) = {a, b} and the minimal upper bounds in P − F (≥ c) are
{a, b} and {d}. We also have F (> c) = {e} and the minimal lower bounds in P − J(≤ c)
are {d} and {e}. Note that since d1 d2 is in L(2, P ) we only have 3 maps corresponding
to the point c of P .
c1 c2 7→ a2 b2 d1 ,

c1 c2 7→ a2 b2 e1 ,

c1 c2 7→ d2 e1 .

Finally, one can show that the first cotangent cohomology module is minimally generated
by the following 11 maps.
a1 a2 7→ c1 d1 ,
b1 b2 7→ c1 d1 ,
c1 c2 7→ a2 b2 d1 ,
d1 d2 7→ a2 b2 c1 ,
e1 e2 7→ c2 d2 .

a1 a2 7→ b1 ,
b1 b2 7→ a1 ,
c1 c2 7→ a2 b2 e1 ,
c1 c2 7→ a2 b2 e1 ,

c1 c2 7→ d2 e1 ,
c1 c2 7→ c2 e1 ,

Corollary 6.11. Suppose the Hasse diagram of P is a rooted tree. Let p in P and
b1 , . . . , bm its children.
• Let q be such
the meet of q and p is the parent of p. The map sending
Q that
i
1
p1 p1 7→ q2 m
b
and
all other monomials to zero,
i=1 1
Qmis ini T (S/L).
• Let ρ be the root of P . The map sending ρ1 ρ2 7→ i=1 b , and all other monomials
to zero, is in T 1 (S/L).

As q and p vary, these maps generate T 1 (S/L).

7. Flatness of deformation family
We show that the ring B(2, P )/J(2, P ) is a flat deformation of k(2, P )/L(2, P ) over the
base ring B = k[u∅,ρ , uq,p ]. We do this in Theorem 7.6, but before that we develop some
auxiliary results.
Proposition 7.1. Given element p, b, c ∈ P with p ≤ b and p ≤ c. Then
Sp (b)c2 − b2 Sp (c)
is in J(2, P ).
We shall prove this by induction on depth(p). For the below lemmata we assume the
above proposition holds for a given p, and the lemmata are consquences of this.
Lemma 7.2. Assume Proposition 7.1 holds for a given p. Let q be sibling of p (possibly
equal) and b ≥ p. Then
(13)
is in J(2, P ).

Sp Tp (q)b2 − Tp (q)Sp (b)

RIGID IDEALS BY DEFORMING QUADRATIC LETTERPLACE IDEALS

19

Proof. If p = q, then (13) is
p1 b2 − T (p)Sp (b)
P
and so is in J(2, P ) by definition. So assume p 6= q. Let Tp (q) = − c≥p c2 uc,q . Since
Sp (c)b2 − c2 Sp (b)

is in J(2, P ) we immediately get the lemma.



Lemma 7.3. Assume Proposition 7.1 holds for a given p. Let q, r and p be siblings (some
possibly equal). Then
Sp Tp (q)Tp (r) − Tp (q)Sp Tp (r)
is in J(2, P ).
Proof. If all three are equal this clearly holds.PAssume then that p is distinct from either
q or r, say distinct from r. Then Tp (r) = − b≥p b2 ub,r . The statement then follows by
Lemma (7.2) above.

Let b1 , . . . , bm be the children of a = b0 . Let bi , bj be elements of {b0 , . . . , bm } and bk
i j
element of {b1 , . . . , bm }. Let M(a)bbkb be the submatrix of M(a) obtained by deleting
columns i and j and row k, and define the signed determinant
(
i j
(−1)i+j+k |M(a)bbkb |
i<j
i
j
bb
D(a)bk =
i+j+k−1
bi bj
(−1)
|M(a)bk | i > j

More generally for a sequence of indices i : i0 , . . . , iℓ from 0, 1, . . . , m let |i| be its length
ℓ + 1 and σ(i) the sign of the permutation that puts them in strictly increasing order.
Similiarly for a sequence k : k1 , . . . , kℓ from 1, 2, . . . , m, and let
i

i

D(a)bbk = (−1)|i|+|k|+σ(i)+σ(k) |M(a)bbk |,
where the last matrix is obtained by deleting the columns from i and the rows from k. It
i
may be verified that we may expand D(a)bbk along a new row r as
X
i
i,c
D(a)bbk =
Sc Tc (r)D(a)bbk,r
c

and similarly when expanding along a column.
Suppose now Lemma 7.1 is proven for all p with depth(p) ≤ n.

Lemma 7.4. Let depth(a) ≤ n + 1, and let b, c, d be distinct children of a.
Pm
1. bx=b1 D(a)bc
x Td (x) is in J(2, P ).
Pbm
2. x=b1 D(a)bc
x Ta (x) is in J(2, P ).
Pbm
3. x=b1 D(a)ab
x Tc (x) is in J(2, P ).

Proof. 1. We expand the determinant Dxbc (a) by column d. We then get
m

Dxbc (a)Td (x) =

b
X

bcd
Dxy
(a)Sd Td (y)Td (x).

y=b1 ,y6=x

We now sum these over x. Thus for each pair (x, y) both xy and yx occur as indices of
the determinant. But these determinants will then be negatives of each other. We obtain
m

b
X

x=b1

D(a)bc
x Td (x) =

X
x<y

D(a)bcd
xy [Sd Td (y)Td (x) − Td (y)Sd Td (x)].

20

GUNNAR FLØYSTAD AND AMIN NEMATBAKHSH

By Lemma 7.3 the last bracket is in the ideal J(2, P ).
2. We do as above but expand along the column a. The sum in the statement is:
m

b
X

D(a)bc
x Ta (x) =

X

D(a)abc
xy [Sa Ta (y)Ta (x) − Ta (y)Sa Ta (x)].

x<y

x=b1

But the expression in the last bracket is ua,y a2 ua,x − ua,y a2 ua,x which is zero.
3. We now expand by column c. Again in the same way we get that the sum in the
statement is:
bm
X
X
D(a)ab
T
(x)
=
D(a)abc
x a
xy [Sc Tc (y)Tc (x) − Tc (y)Sc Tc (x)].
x<y

x=b1

By Lemma 7.3, again the last bracket is in the ideal J(2, P ).



We are now in a position to prove Proposition 7.1.
Proof of Proposition 7.1. We shall split into two cases. The first is when p < c and p < b
strictly, and the second is when p = b and p < c.
So consider the first case. Let q and r be chidren of p such that
p < q ≤ b,

p < r ≤ c,

so the depths of q and r are less than the depth of p. Then
Sp (b)c2 − b2 Sp (c) = D(p)q Sq (b)c2 − b2 D(p)r Sr (c).
Now we expand D(p)q by its column r, and we expand D(p)r by its column q. Then
the above is:
bm
bm
X
X
qr
(14)
Sq (b)
D(p)x Sr Tr (x)c2 − Sr (c)
D(p)rq
x Sq Tq (x)b2
x=b1

x=b1

Now we do a little trick by subtracting and adding the same terms, to make this:
(15)
bm
bm
X
X
qr
D(p)qr
D(p)x (Sr Tr (x)c2 − Tr (x)Sr (c)) + Sr (c)
Sq (b)
x (Sq Tq (x)b2 − Tq (x)Sq (b))
x=b1

x=b1

bm

+Sq (b)Sr (c)

X

D(p)qr
x (Tr (x) + Tq (x)).

x=b1

By Lemma 7.2 the first two summands are in J(2, P ). For the lower sum note that
X
Tr (x) + Tq (x) = −Tp (x) −
Tbi (x).
bi 6=q,r

By Lemma 7.4 this terms is also in the ideal J(2, P ).

Now assume p = b. The change to the above is that we first get
Sp (b)c2 − b2 Sp (c) = D(p)q · 1 · c2 − b2 D(p)r Sr (c),
so Sq (b) is replaced by 1. The sum (15) is replaced by
m

m

b
X

x=b1

D(p)pr
x (Sr Tr (x)c2

− Tr (x)Sr (c)) + Sr (c)

b
X

x=b1

D(p)qr
x (Sp Tp (x)p2 − Tp (x) · 1)

RIGID IDEALS BY DEFORMING QUADRATIC LETTERPLACE IDEALS

21

m

+Sq (b)Sr (c)

b
X

D(p)pr
x (Tr (x) + Tp (x)).

x=b1

The first and third summands are in J(2, P ) just as above. The paranthesis in the
second term is:
up,xp2 − p2 up,x
and so the second term vanishes. Thus this expression is also in J(2, P ).

Proposition 7.5. Let a ≤ b. Then
a1 T (b) − T (a)R(a, b)b1

(16)
is in the ideal J(2, P ).

Proof. Suppose p is the parent of b and let b = b1 , b2 , . . . , bm be the childern of p. Then
m

a1 T (b) − T (a)R(a, b)b1 = a1 p2 up,b −

b
X

a1 Tx (b) − b1 T (a)R(a, p)D(p)b

x=b2
b

Expanding D(p) by row b, it is

m

b

D(p) =

−up,bD(b)bp
b

+

b
X

Sx Tx (b)D(p)bx
b .

x=b2

We insert this in the above expression and it becomes:
m

m

a1 p2 up,b −

b
X

a1 Tx (b) −

b1 T (a)R(a, p)[up,bD(p)pb
b

+

Sx Tx (b)D(p)bx
b ]

x=b2

x=b2

Since

b
X

m

D(p)p =

b1 D(p)pb
b

+

b
X

Sx Tx (b)D(p)px
b ,

x=b2

the above equals

m

m

a1 p2 up,b −

b
X

p

a1 Tx (b) − T (a)R(a, p)up,bD(p) +

−

T (a)R(a, p)up,bSx Tx (b)D(p)px
b

x=b2

x=b2
bm
X

b
X

b1 T (a)R(a, p)Sx Tx (b)D(p)bx
b .

x=b1

Note that

up,b[a1 p2 − T (a)R(a, p)D(p)p ]
is in the ideal J(2, P ). Adding this to the above, the above will modulo J(2, P ) be:
(17)
bm
bm
bm
X
X
X
px
b1 T (a)R(a, p)Sx Tx (b)D(p)bx
T (a)R(a, p)Sx Tx (b)[up,b D(p)b ] −
a1 Tx (b) +
−
b
x=b2

x=b2

x=b2

Expanding D(p)x by row b it is:

m

x

D(p) =

−up,b D(p)xp
b

+

b
X

y=b2
y6=x

xb
Sy Ty (b)D(p)xy
b + b1 D(p)b .

22

GUNNAR FLØYSTAD AND AMIN NEMATBAKHSH

Using this we replace up,bDbpx in (17) and get
(18)

−

x=b2

m

m

m

b
X

a1 Tx (b) +

b
X

T (a)R(a, p)Sx Tx (b)[D(p)x − b1 D(p)xb
b −

b
X

Sy Ty (b)D(p)xy
b ]

y=b2
y6=x

x=b2

m

−

b
X

b1 T (a)R(a, p)Sx Tx (b)D(p)bx
b

x=b1

Here

a1 Tx (b) − T (a)R(a, p)Sx Tx (b)D(p)x = a1 Tx (b) − T (a)Sa (Tx (b))
is in the ideal J(2, P ). Furthermore the terms with xy superscripts cancels against the
terms with yx superscripts. Thus (18) is in the ideal J(2, P ).

Theorem 7.6. The ring B(2, P )/J(2, P ) is a flat deformation of of k[x[2]×P ]/L(2, P )
over the ring B = k[u∅,ρ , uq,p].
Proof. By Proposition 5.4 it is enough to show that all relations between the generators
of L(2, P ) lift to relations between the corresponding generators of J(2, P ).
The relations of L(2, P ) are the following.
1. c2 · a1 b2 − b2 · a1 c2 where a ≤ b and a ≤ c.
2. b1 · a1 c2 − a1 · b1 c2 where a ≤ b ≤ c.
The monomial a1 b2 deforms to a1 b2 − T (a)Sa (b2 ), and similarly for a1 c2 . Let mB be the
maximal ideal of B = k[u∅,ρ , uq,p ] generated by the u’s. For relations of type 1. it will be
enough to show that
(19)

c2 (a1 b2 − T (a)Sa (b2 )) − b2 (a1 c2 − T (a)Sa (c2 ))

is in J(2, P ) ∩ (mB · B(2, P )). But (19) is
T (a)[Sa (c)b2 − c2 Sa (b)].
By Proposition 7.1 the second factor is in J(2, P ) and since T (a) is in mB we are done.
Consider now relations of type 2. It is enough to show that
(20)

b1 (a1 c2 − T (a)Sa (c2 )) − a1 (b1 c2 − T (b)Sb (c2 ))

is in J(2, P ) ∩ (mB · B(2, P )). But (20) is
Sb (c2 )[a1 T (b) − T (a)R(a, b)b1 ].
By Proposition 7.5 below, the second factor is in J(2, P ). It is also in mB due to the T ’s
in the second factor, and so we are done.

Recall the positive Z([2] × P )-grading on B(2, P ) from Section 5.
Corollary 7.7. Let (L(2, P )) be the ideal generated by L(2, P ) in B(2, P ). The Z([2]×P )graded rings B(2, P )/J(2, P ) and B(2, P )/(L(2, P )) have the same Hilbert function h :
Z([2] × P ) → N.
Proof. We consider the situation in greater generality. We have an A-graded polynomial
ring S, and a homogeneous ideal I ⊆ S. Let B = k[uk ]k∈K be an A-graded polynomial
ring and J ⊆ S ⊗k B a flat deformaion of I over B. Suppose the A-grading on (S ⊗k B)/J
is positive. Then we show that (S ⊗k B)/J and (S ⊗k B)/(I) have the same Hilbert
function.

RIGID IDEALS BY DEFORMING QUADRATIC LETTERPLACE IDEALS

23

Let u1 be a variable in B, and B ′ = B/u1 ·B. Note that u1 is A-homogeneous. Consider
u1
0 → B −→
B → B ′ → 0. Tensoring this exact sequence with the rings above and taking
into account their flatness over B, we have exact sequences
u

1
0 → (S ⊗k B)/J −→(S
⊗k B)/J → (S ⊗k B ′ )/J ′ → 0

u

1
0 → (S ⊗k B)/(I) −→(S
⊗k B)/(I) → (S ⊗k B ′ )/(I) → 0

By induction on the number of u-variables, we may assume that (S ⊗k B ′ )/J ′ and (S ⊗k
B ′ )/(I) have the same Hilbert function. Then the same must be true for (S ⊗k B)/J and
(S ⊗k B)/(I).

8. Rigidity of the deformation family
We show that the ideal J(2, P ) ⊆ B(2, P ) is a rigid ideal, meaning that every deformation of J(2, P ) comes from a change of coordinates.
Let Y be a finite-dimensional vector space and U ⊆ Y a subspace. We also denote
Y /U = X. Fixing a splitting the reader may think of Y as X ⊕ U. We shall use such a
direct decomposition in the arguments but will not need it for our statements. The space
X may usually be thought of as the space generated by the variables p1 and p2 as p ranges
over P , and U as the space of variables uq,p and u∅,ρ from Section 3. Our basic situation
is an ideal J ⊆ k[Y ] such that k[Y ]/J is flat over k[U]. It is then a flat deformation of the
ideal I = J ⊗k[Y ] k[Y /U] in k[Y /U]. The situation we have in mind is when J = J(2, P )
and I = L(2, P ).
Now we take a base change k[U] → k[U]/(U)2 and get an ideal
J ⊗k[U ] k[U]/(U)2 ⊆ k[Y ]/(U)2
which is a flat deformation of I ⊆ k[Y /U] over k[U]/(U)2 . Hence by Proposition 6.1 it
induces a map
α
U ∗ −→ Homk[Y /U ] (I, k[Y /U]/I).
Recall the first cotangent cohomology group T 1 (k[Y /U]/I). It parametrizes the first
order non-trivial deformations of I. The following says that if the deformation of I over
k[U]/(U)2 encompasses all non-trivial deformations, then all first order deformations of I
come from infinitesimal coordinate changes of J ⊆ k[Y ].
Lemma 8.1. Suppose the composition
α

τ

U ∗ −→ Homk[Y /U ] (I, k[Y /U]/I) −→ T 1 (k[Y /U]/I)
maps U ∗ to a generating set for T 1 .
Then the image of the composition (the first map is the map from Lemma 6.5)
(21)

β

q

Y ∗ −→ Homk[Y ] (J, k[Y ]/J) −→ Homk[Y /U ] (I, k[Y /U]/I)

is a generating set for Homk[Y /U ] (I, k[Y /U]/I).
Proof. Let X = Y /U and fix a splitting Y = X ⊕ U. Let the ui ’s form a basis for U.
Consider elements in the ideal J written as
X
f = f0 +
ui fi + terms involving degree two monomial in the ui ,
i

where f0 ∈ I and the fi are in k[X]. Then the image of u∗i ∈ U ∗ ⊆ Y ∗ by β is the map
sending
f 7→ fi + terms of degrees ≥ 1 in the ui .

24

GUNNAR FLØYSTAD AND AMIN NEMATBAKHSH

The image of u∗i by α is the map sending
f0 7→ fi .
∗

Hence we see that restricted to U ⊆ Y ∗ there is a commutative diagram
=

U∗

β
y

U∗

α
y

−−−→

Homk[Y ] (J, k[Y ]/J) −−−→ Homk[Y /U ] (I, k[Y /U]/I)
(We do not get a commutative diagram if we replace U ∗ by Y ∗ at the upper left.) Consider
the map
β̂

k[X] ⊗k (X ∗ ⊕ U ∗ ) = k[X] ⊗k Y ∗ −→ Homk[X] (I, k[X]/I)
coming from the composing the maps in (21). We have established that the map on the
second factor identifies with α. The cokernel of
k[X] ⊗k X ∗ → Homk[X] (I, k[X]/I).
is T 1 (k[X]/I). Hence we get the lower row in the diagram below, and a commutative
diagram
k[X] ⊗k X ∗ −−−→ k[X] ⊗k (X ∗ ⊕ U ∗ ) −−−→ k[X] ⊗k U ∗




β̂
τ ◦α
y
y
y
τ

Im(k[X] ⊗k X ∗ ) −−−→ Homk[X] (I, k[X]/I) −−−→
T1
Since the left and right maps are surjective, the middle one is also by the snake lemma.


Theorem 8.2. Suppose k[Y ] has a positive grading by an abelian group and the subspace
U ⊆ Y and the ideal J are homogeneous for this grading. If the composition
U ∗ → Homk[Y /U ] (I, k[Y /U]/I) → T 1 (k[Y /U]/I)
is surjective, then J ⊆ k[Y ] is a rigid ideal.
Proof. By the lemma above we know that the composition
β̂

q

Y ∗ ⊗ k[Y ] −→ Homk[Y ] (J, k[Y ]/J) −→ Homk[Y /U ] (I, k[Y /U]/I)
is surjective. We shall show by induction on U that whenever the composition q ◦ β̂ is
surjective, then β̂ is surjective. This will prove that J is rigid.
Let V be a subspace of U of codimension one, homogeneous for the grading. Let J be
J ⊗k[Y ] k[Y /V ]. By base extension k[Y /V ]/J is flat over k[Y /V ]. We have a factorization
β̂

q1

r

Y ∗ ⊗k[Y ] −→ Homk[Y ] (J, k[Y ]/J) −→ Homk[Y /V ] (J, k[Y /V ]/J) −→ Homk[Y /U ] (I, k[Y /U]/I)
By induction, it is sufficient to prove that q1 ◦ β̂ is surjective. Let us rename Y /V as
Y . Note that U/V = (u) generated by one element. We must then show that if the
composition
β̂1

r

Y ∗ ⊗ k[Y ] −→ Homk[Y ] (J, k[Y ]/J) −→ Homk[Y /(u)] (I, k[Y /(u)]/I)
is surjective, then the map β̂1 is surjective. Let φ in Homk[Y ] (J , k[Y ]/J) map to zero in
Homk[Y /(u)] (I, k[Y /(u)]/I). We first show that φ = uψ for some ψ ∈ Homk[Y ] (J, k[Y ]/J).
P
Let {fi } be a generating set for J and suppose fi 7→ ugi by φ. Let i ri fi be a relation

RIGID IDEALS BY DEFORMING QUADRATIC LETTERPLACE IDEALS

25

P
between the fi . Then u i ri gi is zero in k[Y ]/J. But due to flatness of k[Y ]/J over k[u],
P
the element u is a nonzero divisor. Hence i ri gi is zero in J and so fi 7→ gi gives an
element ψ ∈ Homk[Y ] (J, k[Y ]/J). So there is an injection
Homk[Y ] (J, k[Y ]/J) ⊗k[Y ] k[Y /(u)] ֒→ Homk[Y /(u)] (I, k[Y /(u)]/I).
But this must also be a surjection since the composition r ◦ β̂1 is surjective. Hence the
above is an isomorphism, and so
Homk[Y ] (J, k[Y ]/J) ⊗k[Y ] k ∼
= Homk[Y /(u)] (I, k[Y /(u)]/I) ⊗k[Y /(u)] k.
We apply Nakayama’s lemma applied to the finitely generated module Homk[Y ] (J, k[Y ]/J)
over the positively graded ring k[Y ]. Since the image of Y ∗ generates Homk[Y /(u)] (I, k[Y /(u)]/I)

its image must then also generate Homk[Y ] (J, k[Y ]/J).
Corollary 8.3. The ideal J(2, P ) ⊆ B(2, P ) is a rigid ideal.
9. The multigraded Hilbert schemes
In the previous section we established that the ideals J(2, P ) were rigid. This is a
property of (infinitesimal) deformation theory, concerned with first order deformations.
This section is concerned with the global deformation family. The moduli spaces of
families of quotient rings of a polynomial ring, are the Hilbert schemes. We shall establish
that the family of quotient rings B(2, P )/J(2, P ) by coordinate chages maps dominantly
onto any component of the Hilbert scheme containing L(2, P ). In order to have a Hilbert
scheme, we need a grading on the ring B(2, P ). We shall follow the most general such
approach, that of Haiman and Sturmfels considering multigraded Hilbert schemes [11].
Continuing the setting of Section 8, we assume that Y is graded by an abelian group A.
Then A gives a grading on the polynomial ring k[Y ]. We assume that U is a homogeneous
subspace of Y , and that J is a homogeneous ideal for this A-grading. The grading on
k[Y /U]/I is admissible if for each degree a, the graded piece (k[Y /U]/I)a is a finitedimensional vector space.
9.1. The generic coordinate change. Choose a finite subspace T ⊆ Homk (Y, k[Y /U])0 .
We get a map
Y → k[Y /U] ⊗k T ∗ → k[Y /U] ⊗k k[T ∗ ],
giving a morphism of algebras
k[Y ] → k[Y /U] ⊗k k[T ∗ ].

(22)

Each t ∈ T corresponds to a point in Spec k[T ∗ ] or a map k[T ∗ ] → k. This induces from
(22) a map k[Y ] → k[Y /U]. This is the natural map coming from t ∈ Homk (Y, k[Y /U])0 .
The map (22) induces the map
∆

γ

β

α

τ : k[Y ] −→ k[Y ⊕Y ] −→ k[Y /U]⊗k k[Y ] −→ k[Y /U]⊗k k[Y /U]⊗k k[T ∗ ] −→ k[Y /U]⊗k k[T ∗ ],
where in the last map we have used the multiplication on k[Y /U]. Let J˜ = τ (J). Note
that the fiber over 0 ∈ Spec k[T ∗ ] is I ⊆ k[Y /U]. The ideal J˜ ⊆ k[Y /U] ⊗k k[T ∗ ] is the
ideal we get from J by performing a generic coordinate change of J and then restricting
to k[Y /U].

26

GUNNAR FLØYSTAD AND AMIN NEMATBAKHSH

Example 9.1. Let Y be hx1 , x2 , ui where x1 and x2 have degree 1 and u has degree 2.
The subspace U is generated by u. Let T = Hom(hx1 , x2 , ui, k[x1 , x2 , u])0 . The space T
is eight-dimensional. A basis of T are the maps
t11 : x1 7→ x1 ,
t12 : x1 7→ x2 ,
···
tu,22 : x1 7→ 0,

x2 7→ 0,
x2 7→ 0,
x1 7→ 0,

u 7→ 0
u 7→ 0
, u 7→ x22

The basis of T is (the meaning of the maps should be clear from the above)
t11 , t12 , t21 , t22 , tu,u , tu,11 , tu,12 , tu,22 .
and their dual elements give a basis for T ∗ . Denote the other copy of Y by hy1 , y2 , vi
We consider the homogeneous polynomial x21 x2 + ux1 in k[Y ]. Applying the map τ this
becomes
∆

x21 x2 + ux1 7→(x1 + y1 )2 (x2 + y2 ) + (u + v)(x1 + y1 )
α◦β◦γ

7→ (x1 + t∗11 x1 + t∗12 x2 )2 (x2 + t∗21 x1 + t∗22 x22 )
+(t∗u,11 x21 + t∗u,12 x1 x2 + t∗u,22 x22 )(x1 + t∗11 x1 + t∗12 x2 ).

This is the form we get by performing a generic coordinate change on the polynomial
x21 x2 + ux1 and then taking the image in the quotient ring k[Y /U] = k[x1 , x2 ]
Proposition 9.2. Suppose the A-grading on k[Y /U]/I is admissible. There is a localization k[T ∗ ]f with f (0) 6= 0 such that J˜f ⊆ k[Y /U] ⊗k k[T ∗ ]f is flat over k[T ∗ ]f .
Proof. Let X = Y /U and fix a splitting Y /U → Y such that Y = X ⊕ U. The elements
˜ a is a finitely
in T have degree 0. Note that for each degree a ∈ A then (k[X, T ∗ ]/J)
∗
generated k[T ]-module.
Let m be the maximal ideal in k[T ∗ ] corresponding to the origin. We obtain a localized
˜ m ⊆ k[X, T ∗ ]m . We first show that ((k[X, T ∗ ]/J)
˜ a )m is flat over the local ring
ideal (J)
∗
k[T ]m . By the local criterion of flatness, Theorem A.5 in [18, App.A], it will be enough
to show that k[X, T ∗ ]/((T ∗ )p + J˜)a is flat over k[T ∗ ]/(T ∗ )p for each natural number p.
(Strictly speaking this theorem applies to the situation of a local homomorphism of local
noetherian rings, but the argument works just as well when the codomain ring has positive
grading.)
Let the fk form a generating set for J. We write fk′ for their images in J/(U)p ·
J ⊆ k[X, U]/(U)p . Let fk denote the image in I ⊆ k[X], and fˆk the images of fk
∗ p
∗ p
˜
in ((T
the deformation over k[U]/(U)p , all relations
P ) + J)/(T ) . Due to flatness of P
0 = k rk fk in k[X] lift to a relation 0 = k rk′ fk′ in k[Y ]/(U)p . Substituting
X
X
xj 7→ xj +
t∗j,m m, uk 7→
t∗k,n n,
deg(m)=deg(xj )

we get a relation

0=

X

deg(uk )=deg(n)

rˆk fˆk in k[X, T ∗ ]/(T ∗ )p

k

∗ p
˜
Hence all relations for I lift to relations for ((T ∗ )p + J)/(T
) . By Corollary A.11 in
∗
∗ p
˜
[18, App.A] the quotient ring k[X, T ]/((T ) + J) is flat over k[T ∗ ]/(T ∗ )p . Hence J˜m ⊆
k[X, T ∗ ]m is a deformation of I, flat over k[T ∗ ]m .

RIGID IDEALS BY DEFORMING QUADRATIC LETTERPLACE IDEALS

27

In the following we use: Let M be a finitely generated module over an integral domain
R and p ∈ Spec R. i) If the localization Mp is a free Rp module of rank r, then there is
an open subset Spec Rf ⊆ Spec R containing p such that Mf is free of rank r on Rf . ii) If
for some p ∈ Spec R, the fiber Mk(p) is generated by r ′ elements, there is an open subset
Spec Rg ⊆ Spec R containing p such that Mg is generated on Rg by r ′ elements. Hence
r ′ ≥ r since Spec Rg and Spec Rf intersect nonempty. (They both contain the zero ideal).
We also use the following [11, Prop.3.2]:
Let k[X] be an A-graded polynomial ring, and h : A → N a Hilbert function. There is a
finite subset of degrees D ⊆ A such that the following holds. Let I ⊆ k[X] be a monomial
ideal which is i) generated in degrees D and ii) its Hilbert function hI has hI (a) = h(a)
for a ∈ D. Then hI (a) = h(a) for all a ∈ A.
Let h : A → N be the Hilbert function of k[X]/I, and let D be a finite set of degrees
˜ a )m is a free k[T ∗ ]m -module of finite
given by the above proposition. Every ((k[X, T ∗ ]/J)
∗
˜
rank h(a). We may find an elements fa ∈ k[T ∗ ] with
Q f (0) 6= 0 such that ((k[X, T ]/J)a )fa
is a free module of rank h(a) for a ∈ D. Let f = a∈D fa .
˜ a0 )f is not locally free. The
Suppose now there is some a0 ∈ A such that ((k[X, T ∗ ]/J)
˜ a0 )m has rank h(a0 ). There must then be t ∈ Spec k[T ∗ ]f
free k[T ∗ ]m -module ((k[X, T ∗ ]/J)
∗
˜ k(t) has dimension > h(a0 ) in degree a0 . We may write
such that the fiber (k[X, T ]/J)
∗
′
˜ k(t) = k(t)[X]/J where J ′ is the image of J˜k(t) in k(t)[X]. Fix a monomial
(k[X, T ]/J)
order on k(t)[X]. Let M be ideal generated by the initial monomials of the ideal of
J ′ ⊆ k(t)[X] in degrees D. Then hM (a) = h(a) for a ∈ D, but
hM (a0 ) ≥ dimk ((k(t)[X]/J ′ )a0 > h(a0 ).
This contradicts [11, Prop.3.2] given above. Hence ((k[X, T ∗ ]/J)f )a is a locally free k[T ∗ ]f
module of rank h(a) for every degree a ∈ A.

To the flat family J˜ ⊆ k[Y /U]⊗k k[T ∗ ]f we apply the base change k[T ∗ ]f → k[T ∗ ]/(T ∗ )2 .
Corollary 9.3. Let J ′ ⊆ k[Y /U] ⊗k k[T ∗ ]/(T ∗ )2 be the image of J by this base change.
The ideal J ′ is a flat deformation of I over k[T ∗ ]/(T ∗ )2 .
Note that by Lemma 6.2 the map corresponding to this deformation:
T → Homk[Y /U ] (I, k[Y /U]/I)
is the map obtained from the composition
Y ∗ → Homk[Y ] (J, k[Y ]/J) → Homk[Y /U ] (I, k[Y /U]/I)
by using the k[Y /U]-module structure on the right module.
9.2. The multigraded Hilbert scheme. We recall the multigraded Hilbert scheme as
introduced by Haiman and Sturmfels in [11]. As before A is an abelian group and X a finite
dimensional A-graded vectorspace over k, so the polynomial ring k[X] becomes A-graded.
The Hilbert schemes come about by introducing a Hilbert function h : A → N. There is
h
then a Hilbert scheme Hk[X]
parametrizing all ideals I ⊆ k[X] such that dimk (k[X]/I)a =
h
h(a) for a ∈ A. More precisely, let k − Alg be the category of k-algebras. Then Hk[X]
is
a k-scheme which represents the point functor
h
Ĥk[X]
: k − Alg → Set

28

GUNNAR FLØYSTAD AND AMIN NEMATBAKHSH

h
where Ĥk[X]
(R) is the set of all ideals J ⊆ R[X] such that (R[X]/J)a is a locally free Rmodule of rank h(a). In particular an ideal I ⊆ k[X] with Hilbert function h corresponds
p
h
to a k-point Spec k −→ Hk[X]
of the Hilbert scheme.
h
We may restrict Ĥk[X] to the category k − Art of local artinian k-algebras with residue
field k. If A → k is the augmentation map, it induces

(23)

h
h
Ĥk[X]
(A) → Ĥk[X]
(k).

h
Let ĤI⊆k[X]
(A) be the inverse image of the element (I ⊆ k[X]) in the right set of (23).
h
Thus ĤI⊆k[X] is a subfunctor of DefI , giving the A-graded deformations of I. Again we
h
may restrict ĤI⊆k[X]
to k − vect and we get the following specialization of Proposition
6.1.

Proposition 9.4. There is a natural isomorphism of functors between
h
ĤI⊆k[X]
, Homk (−, Homk[X] (I, k[X]/I)0) : k − vect → Set.
h
h
,the tangent space of this
Since Ĥk[X]
on k − Alg is represented by the Hilbert scheme Hk[X]
scheme at I is Homk[X] (I, k[X]/I)0 .

We now assume that U ⊆ Y is graded by the abelian group A, and that I and J are Agraded ideals in k[Y /U] resp. k[Y ]. We assume the A-grading is admissible on k[Y /U]/I
and so we have a Hilbert function:
h : A → N,

h(a) = dimk (k[Y /U]/I)a .

There is a map
Homk (Y, k[Y /U])0 → Homk (Y, k[Y /U]/I)0 .
The right Hom-space is a finite dimensional vector space due to the grading being admissible.
Theorem 9.5. Suppose the A-grading on k[Y /U]/I is admissible and that the composition
U ∗ → Homk[Y /U ] (I, k[Y /U]/I) → T 1 (k[Y /U]/I)
maps U ∗ to a generating set of T 1 . Let T be a finite dimensional subspace of Homk (Y, k[Y /U])0
which maps surjectively to Homk (Y, k[Y /U]/I)0 .
Then the induced morphism from Theorem 9.2
h
Spec k[T ∗ ]f → Hk[Y
/U ]

is surjective on tangent spaces at the origin 0 ∈ Spec k[T ∗ ]f .
Proof. Let m ⊆ k[T ∗ ]f be the maximal ideal corresponding to the origin in k[T ∗ ]f . The
morphism
h
Spec k[T ∗ ]f /m2 → Hk[Y
/U ]
is then
h
Spec k[T ∗ ]/(T ∗ )2 → Hk[Y
/U ] .
By Proposition 9.4 this corresponds to the map of tangent spaces
(24)

T → Homk[Y /U ] (I, k[Y /U]/I)0 .

By the comment after Corollary 9.3 and Lemma 6.2 this map is obtained from the composition
Y ∗ → Homk[Y ] (J, k[Y ]/J) → Homk[Y /U ] (I, k[Y /U]/I)

RIGID IDEALS BY DEFORMING QUADRATIC LETTERPLACE IDEALS

29

as
T → Hom(Y, k[Y /U]/I)0 = (Y ∗ ⊗ k[Y /U]/I)0 → Homk[Y /U ] (I, k[Y /U]/I)0 .
Since the right map above is surjective by Lemma 8.1 the map (24) on tangent spaces, is
surjective.

h
Corollary 9.6. The ideal I is a smooth point on the Hilbert scheme Hk[Y
/U ] . The mor∗
h
phism Spec k[T ]f → Hk[Y /U ] is dominant on the component of the Hilbert scheme conh
taining I. So there is an open subset of the Hilbert scheme Hk[Y
/U ] such that the ideals in
this open subset are obtained from J ⊆ k[Y ] by a coordinate change, and then restricting
to k[Y /U].

Let Y ⊆ Ŷ be an inclusion of finite-dimensional A-graded vector spaces. Consider the
ideal (J) ⊆ k[Ŷ ] generated by J. Note that this identifies as J ⊗k[Y ] k[Ŷ ] ⊆ k[Ŷ ]. Similarly
we get an ideal (I) ⊆ k[Ŷ /U]. Let ĥ denote the Hilbert function of the quotient ring, if
h is the Hilbert function of the quotient ring of I ⊆ k[Y /U]. The cotangent cohomology
T 1 (k[Ŷ /U]/(I)) = T 1 (k[Y /U]/I) ⊗k[Y ] k[Ŷ ].
The theorem and corollary above still applies to this situation.
Applications. Let Z([2] × P ) → A be a homomorphism of abelian groups. We take Y
to be the space generated by the linear forms of B(2, P ) = k[x[2]×P ] ⊗k k[u∅,ρ , uq,p ], so Y
is generated by the x and u-variables in this ring, and U the linear forms in k[u∅,ρ , uq,p].
1. If A gives an admissible grading on k[X]/L(2, P ), then for an open subset of the Hilbert
h
, the ideals in this open subset come from a change of coordinates in J(2, P )
scheme Hk[X]
and then restricting the ideal to k[X].
2. Let Y ⊆ Ŷ be such that Ŷ /Y ∼
= Y.
= U as A-graded spaces, or equivalently Ŷ /U ∼
(For instance we take a copy V of U and let Ŷ = Y ⊕ V .) Suppose k[Ŷ /U]/(L(2, P )) is
admissible for the A-grading (which by Corollary 7.7 is equivalent to k[Y ]/J(2, P ) being
admissible). Then there is an open subset of the Hilbert scheme Hk[ĥ Ŷ /U ] (which identifies
ĥ
as Hk[Y
] ) such that the ideals in this open subset are obtained from a coordinate change
of (J(2, P )) ⊆ k[Ŷ ] and then restricting to k[Ŷ /U]. But since (J(2, P )) is generated by
J(2, P ) ⊆ k[Y ] and Ŷ /U ∼
= Y , this identifies as a coordinate change of J(2, P ), and so
ĥ
the Hilbert scheme component of Hk[Y
] has an open subset consisting of ideals obtained
from J(2, P ) by coordinate changes.

3. Often the A-grading on k[Ŷ /U]/(L(2, P )) is not admissible, but there is a space
Y ⊆ Y + ⊆ Ŷ such that k[Y + /U]/(L(2, P )) is admissible. For instance take A = Z and
let the pi map to positive values in Z. Some of the u-variables typically map to negative
values. Then k[Y ] is infinite-dimensional in degree 0. Let Ŷ = Y ⊕ V where V is a copy
of U and let U + ⊆ U and U − ⊆ U be the variables of non-negative and negative degrees,
and similarly for V . Let
J ′ (2, P ) = J(2, P ) ⊗k[Y ] k[Y /U − ]
so we are setting the u-variables of negative degrees equal to zero. Let Y + = Y ⊕ V + .
Then there is an open subset of the Hilbert scheme of (L(2, P )) ⊆ k[Y + /U] where the
ideals in this open subset are obtained from (J(2, P )) ⊆ k[Y + ] by coordinate changes and
then restricting to k[Y + /U]. Since Y + /U only has elements of positive degree, all elements

30

GUNNAR FLØYSTAD AND AMIN NEMATBAKHSH

of negative degree in k[Y + ] map to zero. Also (J(2, P )) is generated by J(2, P ) ⊆ k[Y ].
Since Y /U − ∼
= Y + /U, this is then really simply a coordinate change of J ′ (2, P ).
Example 9.7. Let P be the star poset with unique minimal element a and 3 maximal
elements b, c, d. Consider the standard Z-grading on k[x[2]×P ]. Since deg(u∅,a ) = a1 + a2 −
b1 − c1 − d1 with respect the standard grading on k[x[2]×P ] we have deg(u∅,a ) = −1. For
any other uq,p , we have deg(uq,p ) = 1. The space U − is generated by u∅,a , the grading is
positive (and so admissible) on k[Y /V ] = k[x[2]×P , ua,b, ua,c , ub,c , uc,b]. Let J ′ ⊆ k[Y /U − ] be
the ideal obtained from J by setting u∅,a = 0. This is the ideal generated by the forms (1)
after setting u∅,a = 0. Note that Y + = Y ⊕ V + , and Y + /U ∼
= Y /U − as A-graded spaces.
Then there is an open subset of the Hilbert scheme component of (L(2, P )) ⊆ k[Y + /U],
such that the ideals in this open subset come from a coordinate change in J ′ ⊆ k[Y /U − ].
Now consider another Z-grading on k[x[2]×P ]. Let a1 have degree 2 and all the other
variables have degree 1. In this case deg(u∅,a ) = 0 and any homomorphism in T maps
u∅,a to a scalar. In this case there is open subset of the Hilbert scheme component of
(L(2, P )) ⊆ k[Ŷ /U] such that the ideals in this open subset come from a coordinate
change of J(2, P ) ⊆ k[Y ], where Y ∼
= Ŷ /U and u∅,a will always be sent to a scalar.
Conclusion The letterplace ideal I = L(2, P ) is usually not rigid, but we see that something nearly as good holds when the Hasse diagram of the poset P has tree structure.
There is a “lifting” to a rigid ideal J = J(2, P ), and for an open set of the Hilbert scheme
component of L(2, P ), all the ideals come from a coordinate change of J(2, P ).
We have also done computations investigating simple cases when P does not have tree
structure, f.ex. the four element diamond poset. It seems everything we show in this
article also goes through. We therefore make the following conjecture.
Conjecture 9.8. For any finite poset P , the letterplace ideal L(2, P ) ⊆ k[x[2]×P ] deforms
to a rigid ideal J(2, P ) in a ring k[x[2]×P ] ⊗k k[U], with the quotient ring by the ideal flat
over a polynomial base ring k[U]. The ring k[x[2]×P ] ⊗k k[U] is naturally positively graded
by Z([2] × P ) with J(2, P ) homogeneous for this grading.
10. Appendix
Example 10.1. Let P be a poset with following Hasse diagram.
g

f
e
c

b

d

a
f
e
We start from the outer branch
Since f is a maximal point we have

g
and compute the deformed relations.

f1 f2 − T (f ) = f1 f2 − ue,f e2 − ug,f g2

RIGID IDEALS BY DEFORMING QUADRATIC LETTERPLACE IDEALS

31

and similarly for g we get the relation g1 g2 − ue,g e2 − uf,g f2 . By definition


−ue,f
f1
−ug,f
M(e) =
−ue,g −uf,g
g1

and

e1 e2 − T (e)D(e)e = e1 e2 − ud,e d2 (f1 g1 − ug,f uf,g ).
Similarly we have e1 f2 − T (e)D(e)f = e1 f2 − g1 d2 ue,f ud,e − d2 ug,f ue,g ud,e . For d1 f2 we have
d1 f2 − T (d)D(d)e D(e)f = d1 f2 − (ua,d a2 + ub,d b2 + uc,dc2 )ud,e (g1 ue,f + ug,f ue,g )
For the deformed relation of a1 f2 we need the matrix M(a) for the branch point a. The
matrix M(a) is given by


−ua,b
b1
−uc,b −(ud,b e1 + ue,bud,e D(e)e + uf,bud,e D(e)e + uf,b ud,e D(e)f + ug,bud,e D(e)g )
−ua,c −ub,c
c1
−(ud,c e1 + ue,cud,e D(e)e + uf,c ud,e D(e)e + uf,b ud,e D(e)f + ug,c ud,e D(e)g )
−ua,d −ub,d −uc,d
d1
Now we have

a1 f2 − T (a)Sa (f ) = a1 f2 − u∅,a D(a)d D(d)e D(e)f D(f )f
= a1 f2 − u∅,a D(a)d ud,e (ue,g ug,f + ue,f g1 )
Analogously, we can compute the remaining deformed relations and get the following
flat family.
(1) a1 a2 −b1 c1 d1 u∅,a +b1 e1 u∅,a ud,c uc,d+b1 f1 g1 u∅,a ue,c uc,dud,e +b1 f1 u∅,a ug,c uc,d ud,e ue,g +b1 g1 u∅,a uf,c uc,dud,e ue,f −
b1 u∅,a ue,c uc,dud,e ug,f uf,g + b1 u∅,a uf,c uc,d ud,e ug,f ue,g + b1 u∅,a ug,c uc,dud,e ue,f uf,g + c1 e1 u∅,a ud,b ub,d +
c1 f1 g1 u∅,a ue,b ub,d ud,e +c1 f1 u∅,a ug,b ub,d ud,e ue,g +c1 g1 u∅,a uf,b ub,d ud,e ue,f −c1 u∅,a ue,b ub,d ud,e ug,f uf,g +
c1 u∅,a uf,b ub,d ud,e ug,f ue,g +c1 u∅,a ug,b ub,d ud,e ue,f uf,g +d1 u∅,a uc,b ub,c +e1 u∅,a uc,b ud,cub,d +e1 u∅,a ud,b ub,c uc,d +
f1 g1 u∅,a uc,b ue,c ub,d ud,e +f1 g1 u∅,a ue,b ub,c uc,d ud,e +f1 u∅,a uc,b ug,c ub,d ud,e ue,g +f1 u∅,a ug,b ub,c uc,d ud,e ue,g +
g1 u∅,a uc,b uf,c ub,d ud,e ue,f +g1 u∅,a uf,b ub,c uc,d ud,e ue,f −u∅,a uc,b ue,c ub,d ud,e ug,f uf,g +u∅,a uc,b uf,c ub,d ud,e ug,f ue,g +
u∅,a uc,b ug,c ub,d ud,e ue,f uf,g −u∅,a ue,b ub,c uc,dud,e ug,f uf,g +u∅,a uf,b ub,c uc,d ud,e ug,f ue,g +u∅,a ug,b ub,c uc,dud,e ue,f uf,g ,
(2) a1 b2 −c1 d1 u∅,a ua,b −c1 e1 u∅,a ud,b ua,d −c1 f1 g1 u∅,a ue,b ua,d ud,e −c1 f1 u∅,a ug,b ua,d ud,e ue,g −c1 g1 u∅,a uf,b ua,d ud,e ue,f +
c1 u∅,a ue,b ua,d ud,e ug,f uf,g − c1 u∅,a uf,b ua,d ud,e ug,f ue,g − c1 u∅,a ug,b ua,d ud,e ue,f uf,g − d1 u∅,a uc,b ua,c +
e1 u∅,a ua,b ud,cuc,d −e1 u∅,a uc,b ud,c ua,d −e1 u∅,a ud,b ua,c uc,d+f1 g1 u∅,a ua,b ue,c uc,d ud,e −f1 g1 u∅,a uc,b ue,c ua,d ud,e −
f1 g1 u∅,a ue,b ua,c uc,d ud,e +f1 u∅,a ua,b ug,c uc,d ud,e ue,g −f1 u∅,a uc,b ug,c ua,d ud,e ue,g −f1 u∅,a ug,b ua,c uc,d ud,e ue,g +
g1 u∅,a ua,b uf,c uc,d ud,e ue,f −g1 u∅,a uc,b uf,c ua,d ud,e ue,f −g1 u∅,a uf,b ua,c uc,d ud,e ue,f −u∅,a ua,b ue,c uc,d ud,e ug,f uf,g +
u∅,a ua,b uf,c uc,d ud,e ug,f ue,g +u∅,a ua,b ug,c uc,d ud,e ue,f uf,g +u∅,a uc,b ue,c ua,d ud,e ug,f uf,g −u∅,a uc,b uf,c ua,d ud,e ug,f ue,g −
u∅,a uc,b ug,c ua,d ud,e ue,f uf,g +u∅,a ue,b ua,c uc,d ud,e ug,f uf,g −u∅,a uf,b ua,c uc,d ud,e ug,f ue,g −u∅,a ug,b ua,c uc,dud,e ue,f uf,g ,
(3) b1 b2 − a2 ua,b − c2 uc,b − d2 ud,b − e2 ue,b − f2 uf,b − g2 ug,b ,
(4) a1 c2 −b1 d1 u∅,a ua,c −b1 e1 u∅,a ud,c ua,d −b1 f1 g1 u∅,a ue,c ua,d ud,e −b1 f1 u∅,a ug,c ua,d ud,e ue,g −b1 g1 u∅,a uf,c ua,dud,e ue,f +
b1 u∅,a ue,c ua,dud,e ug,f uf,g − b1 u∅,a uf,c ua,d ud,e ug,f ue,g − b1 u∅,a ug,c ua,d ud,e ue,f uf,g − d1 u∅,a ua,b ub,c −
e1 u∅,a ua,b ud,cub,d +e1 u∅,a ud,b ua,c ub,d −e1 u∅,a ud,b ub,c ua,d −f1 g1 u∅,a ua,b ue,c ub,d ud,e +f1 g1 u∅,a ue,b ua,c ub,d ud,e −
f1 g1 u∅,a ue,b ub,c ua,d ud,e −f1 u∅,a ua,b ug,c ub,d ud,e ue,g +f1 u∅,a ug,b ua,c ub,d ud,e ue,g −f1 u∅,a ug,b ub,c ua,d ud,e ue,g −
g1 u∅,a ua,b uf,c ub,d ud,e ue,f +g1 u∅,a uf,b ua,c ub,d ud,e ue,f −g1 u∅,a uf,b ub,c ua,dud,e ue,f +u∅,a ua,b ue,c ub,d ud,e ug,f uf,g −
u∅,a ua,b uf,c ub,d ud,e ug,f ue,g −u∅,a ua,b ug,c ub,d ud,e ue,f uf,g −u∅,a ue,b ua,c ub,d ud,e ug,f uf,g +u∅,a ue,b ub,c ua,d ud,e ug,f uf,g +
u∅,a uf,b ua,c ub,d ud,e ug,f ue,g −u∅,a uf,b ub,c ua,d ud,e ug,f ue,g +u∅,a ug,b ua,c ub,d ud,e ue,f uf,g −u∅,a ug,b ub,c ua,d ud,e ue,f uf,g ,
(5) c1 c2 − a2 ua,c − b2 ub,c − d2 ud,c − e2 ue,c − f2 uf,c − g2 ug,c ,
(6) a1 d2 − b1 c1 e1 u∅,a ua,d − b1 e1 u∅,a ua,c uc,d − c1 e1 u∅,a ua,b ub,d − e1 u∅,a ua,b ub,c uc,d − e1 u∅,a uc,b ua,c ub,d +
e1 u∅,a uc,b ub,c ua,d,
(7) d1 d2 − e1 a2 ua,d − e1 b2 ub,d − e1 c2 uc,d ,
(8) a1 e2 −b1 c1 f1 g1 u∅,a ua,d ud,e +b1 c1 u∅,a ua,d ud,e ug,f uf,g −b1 f1 g1 u∅,a ua,c uc,d ud,e +b1 u∅,a ua,c uc,d ud,e ug,f uf,g −
c1 f1 g1 u∅,a ua,b ub,d ud,e +c1 u∅,a ua,b ub,d ud,e ug,f uf,g −f1 g1 u∅,a ua,b ub,c uc,d ud,e −f1 g1 u∅,a uc,b ua,c ub,d ud,e +
f1 g1 u∅,a uc,b ub,c ua,d ud,e +u∅,a ua,b ub,c uc,dud,e ug,f uf,g +u∅,a uc,b ua,c ub,d ud,e ug,f uf,g −u∅,a uc,b ub,c ua,d ud,e ug,f uf,g ,
(9) d1 e2 − f1 g1 a2 ua,d ud,e − f1 g1 b2 ub,d ud,e − f1 g1 c2 uc,d ud,e + a2 ua,d ud,e ug,f uf,g + b2 ub,d ud,e ug,f uf,g +
c2 uc,dud,e ug,f uf,g ,
(10) e1 e2 − f1 g1 d2 ud,e + d2 ud,e ug,f uf,g ,

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GUNNAR FLØYSTAD AND AMIN NEMATBAKHSH

(11) a1 f2 −b1 c1 g1 u∅,a ua,d ud,e ue,f −b1 c1 u∅,a ua,d ud,e ug,f ue,g −b1 g1 u∅,a ua,c uc,d ud,e ue,f −b1 u∅,a ua,c uc,d ud,e ug,f ue,g −
c1 g1 u∅,a ua,b ub,d ud,e ue,f −c1 u∅,a ua,b ub,d ud,e ug,f ue,g −g1 u∅,a ua,b ub,c uc,d ud,e ue,f −g1 u∅,a uc,b ua,c ub,d ud,e ue,f +
g1 u∅,a uc,b ub,c ua,d ud,e ue,f −u∅,a ua,b ub,c uc,dud,e ug,f ue,g −u∅,a uc,b ua,c ub,d ud,e ug,f ue,g +u∅,a uc,b ub,c ua,d ud,e ug,f ue,g ,
(12) d1 f2 − g1 a2 ua,d ud,e ue,f − g1 b2 ub,d ud,e ue,f − g1 c2 uc,dud,e ue,f − a2 ua,d ud,e ug,f ue,g − b2 ub,d ud,e ug,f ue,g −
c2 uc,dud,e ug,f ue,g ,
(13) e1 f2 − g1 d2 ud,e ue,f − d2 ud,e ug,f ue,g ,
(14) f1 f2 − e2 ue,f − g2 ug,f ,
(15) a1 g2 −b1 c1 f1 u∅,a ua,d ud,e ue,g −b1 c1 u∅,a ua,d ud,e ue,f uf,g −b1 f1 u∅,a ua,c uc,dud,e ue,g −b1 u∅,a ua,c uc,d ud,e ue,f uf,g −
c1 f1 u∅,a ua,b ub,d ud,e ue,g −c1 u∅,a ua,b ub,d ud,e ue,f uf,g −f1 u∅,a ua,b ub,c uc,dud,e ue,g −f1 u∅,a uc,b ua,c ub,d ud,e ue,g +
f1 u∅,a uc,b ub,c ua,d ud,e ue,g −u∅,a ua,b ub,c uc,d ud,e ue,f uf,g −u∅,a uc,b ua,c ub,d ud,e ue,f uf,g +u∅,a uc,b ub,c ua,d ud,e ue,f uf,g ,
(16) d1 g2 − f1 a2 ua,d ud,e ue,g − f1 b2 ub,d ud,e ue,g − f1 c2 uc,d ud,e ue,g − a2 ua,d ud,e ue,f uf,g − b2 ub,d ud,e ue,f uf,g −
c2 uc,dud,e ue,f uf,g ,
(17) e1 g2 − f1 d2 ud,e ue,g − d2 ud,e ue,f uf,g ,
(18) g1 g2 − e2 ue,g − f2 uf,g ,

References
1. Klaus Altmann, Mina Bigdeli, Juergen Herzog, and Dancheng Lu, Algebraically rigid simplicial complexes and graphs, arXiv preprint arXiv:1503.08080 (2015). 2
2. Klaus Altmann and Jan Arthur Christophersen, Cotangent cohomology of Stanley-Reisner rings,
Manuscripta Mathematica 115 (2004), no. 3, 361–378. 2, 16
3.
, Deforming Stanley–Reisner schemes, Mathematische Annalen 348 (2010), no. 3, 513–537. 2
4. W Bruns and U Vetter, Determinantal Rings (Lecture Notes in Mathematics, 1327), 1988. 2
5. Jan Arthur Christophersen, Deformations of equivelar Stanley–Reisner abelian surfaces, Advances in
Mathematics 227 (2011), no. 2, 801–829. 2
6. Jan Arthur Christophersen and Nathan Owen Ilten, Vanishing cotangent cohomology for pl\” ucker
algebras, arXiv preprint arXiv:1409.3432 (2014). 2
7. Corrado De Concini and Elisabetta Strickland, On the variety of complexes, Advances in Mathematics
41 (1981), no. 1, 57 – 77. 9
8. Alessio D’Ali, Gunnar Fløystad, and Amin Nematbakhsh, Resolutions of letterplace ideals of posets,
preprint, arXiv (2016). 3
9. Viviana Ene, Jürgen Herzog, and Fatemeh Mohammadi, Monomial ideals and toric rings of Hibi type
arising from a finite poset, European Journal of Combinatorics 32 (2011), no. 3, 404–421. 1, 3
10. Gunnar Fløystad, Bjørn Møller Greve, and Jürgen Herzog, Letterplace and co-letterplace ideals of
posets, preprint, arXiv:1501.04523 (2015). 1, 2, 3
11. Mark Haiman and B Sturmfels, Multigraded Hilbert schemes, Journal of Algebraic Geometry 13
(2004), no. 4. 2, 10, 12, 25, 27
12. Robin Hartshorne, Deformation theory, vol. 257, Springer Science & Business Media, 2009. 12
13. Jürgen Herzog and Takayuki Hibi, Distributive lattices, bipartite graphs and alexander duality, Journal
of Algebraic Combinatorics 22 (2005), no. 3, 289–302. 1, 3
14.
, Monomial ideals, Springer, 2011. 3
15. Jan O Kleppe, Deformations of modules of maximal grade and the Hilbert scheme at determinantal
schemes, Journal of Algebra 407 (2014), 246–274. 2
16. Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, vol. 227, Springer Science &
Business Media, 2005. 11, 12
17. Alyson Reeves and Mike Stillman, Smoothness of the lexicographic point, Journal of Algebraic Geometry 6 (1997), no. 2, 235–246. 2
18. Edoardo Sernesi, Deformations of algebraic schemes, vol. 334, Springer Science & Business Media,
2007. 12, 13, 26
19. Jan Stevens, Deformations of singularities, Lecture Notes in Mathematics, vol. 1811, Springer-Verlag,
Berlin, 2003. 12
Universitet i Bergen, Matematisk Institutt, Postboks 7803, 5020, Bergen, Norway
E-mail address: gunnar@mi.uib.no
Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
E-mail address: nematbakhsh@ipm.ir

