Symmetric ladders and G-biliaison

0.18 MB·

by  Elisa Gorla

SYMMETRIC LADDERS AND G-BILIAISON
ELISAGORLA
0
1
0 Abstract. We study the familyof ideals generated by minors of mixed size
2 contained inaladder ofasymmetricmatrixfromthepointof viewofliaison
n theory. We prove that they can be obtained from ideals of linear forms by
a ascendingG-biliaison. Inparticular,theyareglicci.
J

5
2

]
C Introduction
A
Ideals generated by minors have been studied extensively. They are a central
. topic in commutative algebra, where they have been investigated mainly using
h
t Gr¨obner bases and combinatorial techniques (see among others [10], [18], [1], [2],
a
[21], [17]). They are also relevant in algebraic geometry, since many classical vari-
m
etiessuchastheVeroneseandtheSegrevarietyarecutoutbyminors. Degeneracy
[
loci of morphisms between direct sums of line bundles overprojective space have a

2 determinantal description, as do the Schubert varieties.
v In this paper, we study ideals of minors in a symmetric matrix from the point
7
of view of liaison theory. In particular, we consider ideals generated by minors of
3
mixed size which are contained in a symmetric ladder. Cogenerated ideals in a
3
4 ladder of a symmetric matrix belong to the family that we study. The family of
. cogenerated ideals is a natural one to study from the combinatorial point of view
9
(see [7] or [8]). However,fromthe point ofview of liaisontheory it is more natural
0
8 to study a larger class of ideals, as they naturally arise during the linkage process.
0 We call them symmetric mixed ladder determinantal ideals.
: InSection1wesetthe notationanddefinesymmetricmixedladderdeterminan-
v
i tal ideals (Definition 1.3). In Example 1.5 (3) we discuss why cogenerated ladder
X
determinantal ideals of a symmetric matrix are a special case of symmetric mixed
r ladder determinantal ideals. In Proposition 1.7 we show that symmetric mixed
a
ladder determinantal ideals are prime and Cohen-Macaulay. In Proposition1.8 we
express their height as the cardinality of a suitable subladder.
In Section2 we review the notionofG-biliaison, stating the definition and main
result in the algebraic language (see Definition 2.2 and Theorem 2.3). In Theo-
rem2.4weprovethatsymmetricmixedladderdeterminantalidealscanbeobtained
from ideals of linear forms by ascending G-biliaison. In particular, they are glicci
(Corollary 2.5).
1991 Mathematics Subject Classification. 14M06,13C40,14M12.
Key words and phrases. G-biliaison,Gorensteinliaison,minor,symmetricmatrix,symmetric
ladder,completeintersection,Gorensteinideal,Cohen-Macaulayideal.
The author was supported by the “Forschungskredit der Universit¨at Zu¨rich” (grant no.
57104101) andbytheSwissNationalScienceFoundation(grants no. 107887andno. 123393).
1
2 ELISAGORLA
1. Ideals of minors of a symmetric matrix
Let K be an algebraically closed field. Let X = (x ) be an n×n symmetric
ij
matrix of indeterminates. In other words, the entries x with i ≤ j are distinct
ij
indeterminates, and x =x for i>j. Let K[X]=K[x |1≤i≤j ≤n] be the
ij ji ij
polynomialringassociatedtothematrixX. Inthispaper,westudyidealsgenerated
bytheminorscontainedinaladderofagenericsymmetricmatrixfromthepointof
view of liaisontheory. Throughoutthe paper, we only consider symmetric ladders.
Thiscanbedonewithoutlossofgenerality,sincetheidealgeneratedbytheminors
ina ladder ofa symmetric matrixcoincides with the idealgeneratedby the minors
in the smallest symmetric ladder containing it.
Definition 1.1. A ladder L of X is a subset of the set X = {(i,j) ∈ N2 | 1 ≤
i,j ≤n} with the following properties :
(1) if (i,j)∈L then (j,i)∈L (i.e. L is symmetric), and
(2) if i<h,j >k and (i,j),(h,k)∈L, then (i,k),(i,h),(h,j),(j,k)∈L.
We do not make any connectedness assumption on the ladder L. For ease of
notation,wealsodonotassumethatX isthesmallestsymmetricmatrixcontaining
L. Let
X+ ={(i,j)∈X |1≤i≤j ≤n} and L+ =L∩X+.
Since L is symmetric, L+ determines L and vice versa. We will abuse terminology
and call L+ a ladder. Observe that L+ can be written as
L+ ={(i,j)∈X+ |i≤c or j ≤d for l =1,...,r and
l l
i≥a or j ≥b for l=1,...,u}
l l
for some integers 1 ≤ a < ... < a ≤ n, n ≥ b > ... > b ≥ 1, 1 ≤ c < ... <
1 u 1 u 1
c ≤ n, and n ≥ d > ... > d ≥ 1, with a ≤ b for l = 1,...,u and c ≤ d for
r 1 r l l l l
l=1,...,r.
The points (a ,b ),...,(a ,b ) are the lower outside corners of the lad-
1 2 u−1 u
der, (a ,b ),...,(a ,b ) are the lower inside corners, (c ,d ),...,(c ,d ) the
1 1 u u 2 1 r r−1
upper outside corners, and (c ,d ),...,(c ,d ) the upper inside corners. If
1 1 r r
a 6= b , then (a ,a ) is a lower outside corner and we set b = a . Similarly,
u u u u u+1 u
if c 6= d then (d ,d ) is an upper outside corner, and we set c = d . See also
r r r r r+1 r
Figure 1. A ladder has at least one upper and one lower outside corner. Moreover,
(a ,b )=(c ,d ) is both an upper and a lower inside corner.
1 1 1 1
The upper border of L+ consists of the elements (c,d) of L+ such that either
c ≤c≤c and d=d , or c=c and d ≤d≤d for some l. See Figure 2.
l l+1 l l l l−1
All the corners belong to L+. In fact, the ladder L+ corresponds to its set of
lower and upper outside (or equivalently lower and upper inside) corners. The
upper corners of a ladder belong to its upper border.
Given a ladder L we set L = {x ∈ X | (i,j) ∈ L+}, and denote by K[L] the
ij
polynomial ring K[x |x ∈L]. For t a positive integer, and 1≤α ≤...≤α ≤
ij ij 1 t
n, 1≤β ≤...≤β ≤n integers, we denote by [α ,...,α ;β ,...,β ] the t-minor
1 t 1 t 1 t
det(x ). We let I (L) denote the ideal generated by the set of the t-minors of
αi,βj t
X whichinvolve only indeterminates of L. In particularI (X) is the ideal of K[X]
t
generated by the minors of X of size t×t.
Inthisarticle,westudytheG-biliaisonclassofalargefamilyofidealsgenerated
by minors in a ladder of a symmetric matrix.
SYMMETRIC LADDERS AND G-BILIAISON 3
(a ,b ) = (c ,d ) (c ,d ) (n,n)
1 1 1 1 2 1
(c ,d )
r+1 r
(c ,d )
r r
(a ,b )
2 2
(a ,b )
1 2
(a ,b )
u u
(1,1)
Figure 1. An example of ladder with tagged lower and upper corners.
L+
Figure 2. The upper border of the same ladder.
4 ELISAGORLA
Notation 1.2. Let L+ be a ladder. For (c,d)∈L+ let
L+ ={(i,j)∈L+ |i≤c, j ≤d}, L ={x ∈X |(i,j)∈L+ }.
(c,d) (c,d) ij (c,d)
See also Figure 3. Notice that L+ is a ladder according to Definition 1.1 and
(c,d)
L+ = L+
(c,d)
(c,[d)∈U
where U denotes the set of upper outside corners of L+.
(v,w)
L+
(v,w)
Figure 3. The ladder L+ with a shaded subladder L+ .
(v,w)
Definition 1.3. Let {(v ,w ),...,(v ,w )} be a subsetof the upper borderof L+
1 1 s s
whichcontainsalltheupperoutsidecorners. Weorderthemsothat1≤v ≤...≤
1
v ≤ n and n ≥ w ≥ ... ≥ w ≥ 1. Let t = (t ,...,t ) be a vector of positive
s 1 s 1 s
integers. Denote L by L . The ideal
(vk,wk) k
I (L)=I (L )+...+I (L )
t t1 1 ts s
isasymmetricmixedladderdeterminantalideal. DenoteI (L)byI (L).
(t,...,t) t
We call (v ,w ),...,(v ,w ) distinguished points of L+.
1 1 s s
Remarks 1.4. (1) Let M ⊇ L be two ladders of X, and let M,L be the
corresponding sets of indeterminates. We have isomorphisms of graded
K-algebras
K[L]/It(L)∼=K[M]/It(L)+(xij |xij ∈M\L)∼=K[X]/I2t(L)+(xij |xij ∈X\L).
SYMMETRIC LADDERS AND G-BILIAISON 5
Here I (L) is regarded as an ideal in K[L],K[M], and K[X] respectively.
t
Thenthe heightofthe idealI (L)andthe propertyofbeingprime,Cohen-
t
Macaulay, Gorenstein, Gorenstein in codimension ≤ c (see Definition 2.1)
donotdependonwhetherweregarditasanidealofK[L],K[M],orK[X].
(2) We can assume without loss of generality that for each l = 1,...,s there
exists a k ∈{1,...,u−1} such that
t ≤min{v −a +1,w −b +1}
l l k l k+1
In fact, if t > min{v −a +1,w −b +1} for all k, then I (L ) = 0.
l l k l k+1 tl l
If that is the case, replace L by M := ∪ L , eliminate (v ,w ) from the
i6=l i l l
distinguished points and remove the l-th entry of t to get a new vector m.
Thenweobtainanewladderforwhichtheassumptionissatisfiedandsuch
that I (M)=I (L).
m t
(3) We can assume that
w −w <t −t <v −v , for k =2,...,s.
k k−1 k k−1 k k−1
Infact,ifv −v ≤t −t ,bysuccessivelydevelopingat -minorofL
k k−1 k k−1 k k
withrespecttothefirstv −v rowsweobtainanexpressionoftheminor
k k−1
asacombinationofminorsofsize t −(v −v )≥t thatinvolveonly
k k k−1 k−1
indeterminates from L . Therefore I (L ) ⊇ I (L ). Similarly, if
k−1 tk k tk−1 k−1
w −w ≥ t −t , by developing a t -minor of L with respect
k k−1 k k−1 k−1 k−1
to the last w −w columns we obtain an expression of the minor as a
k−1 k
combination of minors of size t −(w −w ) ≥ t that involve only
k−1 k−1 k k
indeterminates from L . Therefore I (L ) ⊆ I (L ). In either case,
k tk−1 k−1 tk k
we can remove a part of the ladder and reduce to the study of a proper
subladder that corresponds to the same symmetric ladder determinantal
ideal.
(4) We can always find k ∈ {1,...,s} such that v > v and w > w .
k k−1 k k+1
In fact, the two inequalities are satisfied if and only if (v ,w ) is an upper
k k
outside corner. Notice that if we have distinguished points (v ,w ) and
k k
(v ,w ) on the same row or column, then one of the following holds:
k+1 k+1
• either v =v and t >t ,
k k+1 k k+1
• or w =w and t >t .
k k+1 k+1 k
In particular, we can find k ∈ {1,...,s} such that t ≥ 2, v > v and
k k k−1
w >w , unless t =1 for all k.
k k+1 k
The followingareexamplesofdeterminantalidealsofasymmetricmatrixwhich
belong to the class of ideals that we study.
Examples 1.5. (1) If t = (t,...,t) then I (L) is the ideal generated by the
t
t-minors of X that involve only indeterminates from L. These ideals have
been studied in [4], [5], and [6].
(2) If L = X, then according to Remarks 1.4 we can assume that w = n for
l
all l = 1,...,s and v = n. From Remark 1.4 (3), we have t > t and
s l l−1
v > v for all l. Then I (L) is generated by the t -minors of the first
l l−1 t 1
v rows, the t -minors of the first v rows, ..., the t -minors of the whole
1 2 2 s
matrix. This is a simple example of a cogeneratedideal.
(3) The family of symmetric mixed ladder determinantal ideals contains the
familyofcogeneratedidealsinaladderofasymmetricmatrix,asdefined
in [4]. We follow the notation of [4], and assume for ease of notation that
6 ELISAGORLA
(1,n)isaninsidecornerofL(i.e.,thatX isthesmallestmatrixcontaining
L). If α = {α ,...,α }, then I (L) = I (L) where {(v ,w ),...,(v ,w )}
1 t α τ 1 1 s s
consists of the upper outside corners of L, together with the points of the
upper border of L which belongs to row α −1, for all l for which such an
l
intersection point is unique (if for some l the intersectionof the row α −1
l
with the upper border of L consists of more than one point, then L has an
upper outside corner on the row α −1 and we do not add any extra point
l
to the set). For each k =1,...,s, we let τ =min{l|α >v }.
k l k
(4) Let X be a matrix of size m×n, m≤n, whose entries are indeterminates.
Assume that X contains a square symmetric submatrix of indeterminates,
and that all the other entries of X are distinct indeterminates. In block
notation
M N
X =
(cid:18) S P (cid:19)
where S is a symmetric matrix of indeterminates and M,N,P are generic
matrices of indeterminates. Let t∈Z . Then I (X) is a symmetric ladder
+ t
determinantal ideal generated by the minors of size t ×t contained in a
symmetric ladder of
Y M N
 Mt S P 
Nt Pt Z
 
where Y,Z are symmetric matrices of indeterminates, and Mt denotes the
transpose of M. This was observed by Conca in [4].
In this section we establish some properties of symmetric mixed ladder deter-
minantal ideals. It is known ([4]) that cogenerated ideals are prime and Cohen-
Macaulay. InthesequelweshowthattheresultofConcaeasilyextendstosymmet-
ric mixed ladder determinantal ideals. We exploit a well known localization tech-
nique (see [3], Lemma 7.3.3). The same argument was used to prove Lemma 1.19
in[12]. Forcompleteness,westateitforthe caseofaladderofasymmetricmatrix
and we outline the proof. We use the notation of Definitions 1.1 and 1.3. From
Remark 1.4 (4) we know that we can always find k ∈ {1,...,s} such that t ≥ 2,
k
v >v and w >w , unless t=(1,...,1).
k k−1 k k+1
Lemma 1.6. Let L be a ladder of a symmetric matrix X of indeterminates. L has
a set of distinguished points {(v ,w ),...,(v ,w )}∈L+ and t=(t ,...,t )∈Zs.
1 1 s s 1 s +
Let I (L) be the corresponding symmetric mixed ladder determinantal ideal. Let
t
k ∈{1,...,s} such that t ≥2, v >v and w >w .
k k k−1 k k+1
Let t′ = (t ,...,t ,t − 1,t ,...,t ) and let L′ be the ladder obtained
1 k−1 k k+1 s
from L by removing the entries (v +1,w ),...,(v −1,w ),(v ,w ),(v ,w −
k−1 k k k k k k k
1)...,(v ,w +1) and the symmetric ones. Let
k k+1
(v ,w ),...,(v ,v ),(v −1,w −1),(v ,w ),...,(v ,w )
1 1 k−1 k−1 k k k+1 k+1 s s
be the distinguished points of L′.
Then there is an isomorphism between K[L]/I (L)[x−1 ] and
t vk,vk
K[L′]/It′(L′)[xvk−1+1,wk,...,xvk−1,wk,x±vk1,wk,xvk,wk−1,...,xvk,wk+1+1].
SYMMETRIC LADDERS AND G-BILIAISON 7
Proof. Undertheassumptionofthelemma,L′isaladderandIt′(L′)isasymmetric
mixed ladder determinantal ideal. Let
A=K[L][x−1 ]
vk,wk
and
B =K[L′][x ,...,x ,x±1 ,x ,...,x ].
vk−1+1,wk vk−1,wk vk,wk vk,wk−1 vk,wk+1+1
Define a K-algebra homomorphism
ϕ:A −→ B
x +x x x−1 if i6=v ,j 6=w and (i,j)∈L
xi,j 7−→ (cid:26) xii,,jj i,wk vk,j vk,wk otherwikse. k (vk,wk)
The inverse of ϕ is
ψ :B −→ A
x −x x x−1 if i6=v ,j 6=w and (i,j)∈L′
xi,j 7−→ i,j i,wk vk,j vk,wk k k (vk,wk)
(cid:26) xi,j otherwise.
It is easy to check that ϕ and ψ are inverse to each other. Since
ϕ(I (L )A)=I (L′ )B
tk (vk,wk) tk−1 (vk−1,wk−1)
we have
ϕ(It(L)A)=It′(L′)B hence A/It(L)A∼=B/It′(L′)B.
(cid:3)
Using Lemma 1.6 we can establish some properties of symmetric mixed ladder
determinantal ideals.
Proposition 1.7. Symmetric mixed ladder determinantal ideals are prime and
Cohen-Macaulay.
Proof. Let I (L) be the symmetric mixed ladder determinantal ideal associated to
t
the ladder L with distinguished points (v ,w ),...,(v ,w ) and t = (t ,...,t ).
1 1 s s 1 s
Let tmax = max{t1,...,ts}. If tmax = 1 then It(L) is generated by indeterminates,
hence it is prime and Cohen-Macaulay. Therefore assume that tmax ≥ 2 and let L
be the ladder with the same lower outside corners as L, and upper outside corners
b
(vk+tmax−tk,wk+tmax−tk)fork=1,...,s. Noticethatthe cornersaredistinct,
and the inequalities of Definition 1.1 are satisfied by Remark 1.4 (3). In other
words, for each k =2,...,s we have
wk+tmax−tk <wk−1+tmax−tk−1 and vk+tmax−tk >vk−1+tmax−tk−1.
Let L = {x ∈ X | (i,j) ∈ L, i ≤ j} and let M = L\L. Denote by I (L) the
ij tmax
idealbgenerated by the minorbs of size tmax which invbolve only indeterminatesbin L.
By Lemma 1.6 there exists a subset {z ,...,z } of M such that
1 m b
K[L]/Itmax(L)[z1−1,...,zm−1]∼=K[L]/It(L)[M][z1−1,...,zm−1].
The ring K[Lb]/I (Lb) is a Cohen-Macaulay domain by Theorem 1.13 in [4].
tmax
Therefore K[L]/I (L)[M][z−1,...,z−1] is a Cohen-Macaulay domain. Since M
b t b 1 m
is a set of indeterminates over the ring K[L]/I (L) and z ,....,z ∈ M, then
t 1 m
K[L]/I (L)[M] is a Cohen-Macaulay domain. Hence I (L) is prime and Cohen-
t t
Macaulay. (cid:3)
8 ELISAGORLA
Astandardargumentallowsustocomputetheheightofsymmetricmixedladder
determinantal ideals. These heights have been computed by Conca in [4] for the
family of cogenerated ideals. The arguments in [4] are of a more combinatorial
nature, and the height is expressed as a sum of lengths of maximal chains in some
subladders. Our formula for the heightis very simple. The proof is independent of
the results of Conca, and it essentially follows from Lemma 1.6. We use the same
notation as in Definitions 1.1 and 1.3, and Lemma 1.6. An example is given in
Figure 4.
Proposition 1.8. Let L be a ladder with distinguished points (v ,w ),...,(v ,w )
1 1 s s
and let
H+ ={(i,j)∈L+ |i≤v −t +1 or j ≤w −t +1 for k =2,...,s,
k−1 k−1 k k
j ≤w −t +1, i≤v −t +1}.
1 1 s s
Let H=H+∪{(j,i)|(i,j)∈H+}. Then H is a symmetric ladder and
htI (L)=|H+|.
t
L+
H+
Figure 4. Anexample ofL+ with three distinguishedpoints and
t=(3,6,4). The corresponding H+ is shaded.
Proof. Observe that by Remark 1.4 (3)
v −t +1>v −t +1, and w −t +1<w −t +1.
k k k−1 k−1 k k k−1 k−1
ThereforeH is a ladder with upper outside corners{(v −t +1,w −t +1)|k =
k k k k
1,...,s} and the same lower outside corners as L. Let H = {x | (i,j) ∈ H+}.
i,j
We argue by induction on τ = t +...+t ≥ s. If τ = s, then t = ... = t = 1,
1 s 1 s
and L=H. Hence
I (L)=(x |x ∈L)=I (H)=|H+|.
1 ij ij 1
SYMMETRIC LADDERS AND G-BILIAISON 9
Assume now that the thesis holds for τ −1 ≥ s and prove it for τ. Since τ > s,
by Remark 1.4 (4) there exists k ∈ {1,...,s} such that t ≥ 2, v > v and
k k k−1
w > w . By Lemma 1.6 we have an isomorphism between K[L]/I (L)[x−1 ]
k k+1 t vk,wk
and
K[L′]/It′(L′)[xvk−1,wk,...,xvk−1,wk,x±vk1,wk,xvk,wk−1,...,xvk,wk+1+1].
Since xvk,wk does not divide zero modulo It′(L′) and It(L), we have
htIt(L)=htIt′(L′).
The thesis follows by the induction hypothesis, observing that the same ladder H
computes the height of both It′(L′) and It(L). (cid:3)
2. G-biliaison of symmetric mixed ladder determinantal ideals
In this section we study symmetric mixed ladder determinantal ideals from the
point of view of liaison theory. We prove that they belong to the G-biliaison class
ofa complete intersection. Inparticular,they are glicci. This is yetanotherfamily
of ideals of minors for which one can perform a descending G-biliaison to an ideal
inthe samefamily,insuchawaythatoneeventuallyreachesanidealgeneratedby
linearforms. Otherfamiliesofidealsthatweretreatedwithananalogoustechnique
are ideals generated by maximal minors of a matrix with polynomial entries [16],
minorsofasymmetricmatrixwithpolynomialentries[11],minorsofamatrixwith
polynomial entries [13], minors of mixed size in a ladder of a generic matrix [12],
and pfaffians of mixed size in a ladder of a generic skew-symmetric matrix [9].
In [14], [15], [16] Hartshorne developed the theory of generalizeddivisors, which
is a useful language for the study of Gorenstein liaison via the study of G-biliaison
classes. In[15]itwasshownthatevenCI-liaisonandCI-biliaisongeneratethesame
equivalenceclasses. In[19]Kleppe,Migliore,Miro´-Roig,NagelandPetersonproved
thataG-biliaisononanarithmeticallyCohen-Macaulay,G schemecanberealized
1
viatwo G-links. The resultwas generalizedin[16] by Hartshorneto G-biliaisonon
an arithmetically Cohen-Macaulay, G scheme.
0
In Proposition 1.7 we saw that symmetric mixed ladder determinantal ideals
areprime, hencethey define reducedandirreducible,projectivealgebraicvarieties.
Sincewewishtoworkinthealgebraicsetting,westatethedefinitionofG-biliaison
andthemaintheoremconnectingG-biliaisonandG-liaisoninthelanguageofideals.
Definition 2.1. Let R=K[L] and let J ⊆R be a homogeneous, saturated ideal.
We say that J is Gorenstein in codimension ≤ c if the localization (R/J) is
P
a Gorenstein ring for any prime ideal P of R/J of height smaller than or equal to
c. We often say that J is G . We call generically Gorenstein, or G , an ideal J
c 0
which is Gorenstein in codimension 0.
Definition 2.2. ([16], Sect. 3) Let R = K[X] and let I and I be homogeneous
1 2
ideals in R of pure height c. We say that I is obtained by an elementary G-
1
biliaison of height h from I if there exists a Cohen-Macaulay, generically Goren-
2
stein ideal J in R of height c−1 such that J ⊆ I ∩I and I /J ∼= [I /J](−h) as
1 2 1 2
R/J-modules. If h>0 we speak about ascending elementary G-biliaison.
The following theorem gives a connection between G-biliaison and G-liaison.
10 ELISAGORLA
Theorem2.3. [Kleppe,Migliore, Mir`o-Roig,Nagel,Peterson[19];Hartshorne[16]]
Let I be obtained by an elementary G-biliaison from I . Then I is G-linked to I
1 2 2 1
in two steps.
We now show that symmetric mixed ladder determinantal ideals belong to the
G-biliaison class of a complete intersection. The idea of the proof is as follows:
starting from a symmetric mixed ladder determinantal ideal I, we construct two
symmetric mixed ladder determinantal ideals I′ and J such that J is contained in
I∩I′ and htI =htI′ =htJ +1. We show that I can be obtained from I′ by an
elementary G-biliaison of height 1 on J.
Theorem 2.4. Any symmetric mixed ladder determinantal ideal can be obtained
from anideal generatedbylinear forms byafinitesequenceofascending elementary
G-biliaisons.
Proof. Let I (L) be a symmetric mixed ladder determinantal ideal associated to a
t
ladder L+ with distinguished points (v ,w ),...,(v ,w ). Let L =L , then
1 1 s s k (vk,wk)
I (L)=I (L )+···+I (L )⊆K[L].
t t1 1 ts s
As discussed in Remark 1.4 (1) we will not distinguish between symmetric mixed
ladder determinantal ideals and their extensions. Therefore, all ideals will be in
R = K[L]. If t = ... = t = 1 then I (L) is generated by linear forms. Hence let
1 s t
t = max{t ,...,t } ≥ 2. From Remark 1.4 (3) we have that w −w < 0 <
k 1 s k+1 k
v −v . In particular (v ,w ) is an upper outside corner.
k k−1 k k
Let L′+ be the ladder with distinguished points
(v ,w ),...,(v ,w ),(v −1,w −1),(v ,w ),...,(v ,w ).
1 1 k−1 k−1 k k k+1 k+1 s s
Observe that L′+ is obtained from L+ by removing the entries
(v +1,w ),...,(v −1,w ),(v ,w ),(v ,w +1),...,(v ,w −1).
k−1 k k k k k k k k k+1
Lett′ =(t1,...,tk−1,tk−1,tk+1,...,ts),andletIt′(L′)betheassociatedsymmetric
mixed ladder determinantal ideal. It is easy to check that L′+ and t′ satisfy the
inequalities of Definition 1.3 and of Remarks 1.4. By Proposition 1.8
htIt(L)=htIt′(L′)=|H+|
where
H+ ={(i,j)∈L+ |i≤v −t +1 or j ≤w −t +1 for k =2,...,s,
k−1 k−1 k k
j ≤w −t +1, i≤v −t +1}.
1 1 s s
Let J+ be the ladder obtained from L+ by removing (v ,w ), and let
k k
(v ,w ),...,(v ,w ),(v −1,w ),(v ,w −1),(v ,w ),...,(v ,w )
1 1 k−1 k−1 k k k k k+1 k+1 s s
be its distinguished points (see Figure 5). Let u=(t ,...,t ,t ,t ,t ,...,t ).
1 k−1 k k k+1 s
Then
I (J)=I (L )+...+I (L )+I (J )+I (J )+
u t1 1 tk−1 k−1 tk (vk−1,wk) tk (vk,wk−1)
+I (L )+...+I (L ).
tk+1 k+1 ts s
In other words, I (J) is the ideal generated by the minors of I (L) that do not
u t
involve the indeterminate xvk,wk. We claim that Iu(J)⊆It(L)∩It′(L′). It is clear
that Iu(J) ⊆ It(L). The inclusion Iu(J) ⊆ It′(L′) follows from Itk(L(vk−1,wk))+
I (L )⊂I (L′ ).
tk (vk,wk−1) tk−1 (vk−1,wk−1)