Finite determinacy of matrices over local rings. Tangent modules to the miniversal deformation for R-linear group actions

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by  Genrich Belitskii

Finite determinacy of matrices over local rings.
Tangent modules to the miniversal deformation for R-linear group actions.
GenrichBelitskiiandDmitryKerner
6
1 Abstract. Weconsidermatriceswithentriesinalocalring,Matm×n(R). Fixagroupaction,G(cid:8)Matm×n(R),and
0 asubsetof alloweddeformations, Σ⊆Matm×n(R). Thestandard question inSingularityTheoryisthe finite-(Σ,G)-
2 determinacy of matrices. Finite determinacy implies algebraizability and is equivalent to a stronger notion: stable
algebraizability.
c
e In our previous work this determinacy question was reduced to the study of the tangent spaces T(Σ,A), T(GA,A),
D and their quotient, the tangent module to the miniversal deformation, T(1Σ,G,A) = T(Σ,A(cid:30))T(GA,A). In particular, the
orderofdeterminacyiscontrolledbytheannihilatorofthistangent module,ann(T1 ).
5 (Σ,G,A)
2 In this workwe study this tangent module forthe group action GL(m,R)×GL(n,R)(cid:8)Matm×n(R) and various
naturalsubgroupsofit.

We obtain ready-to-use criteria of determinacy for deformations of (embedded) modules, (skew-)symmetric forms,
]
G filteredmodules,filteredmorphismsoffilteredmodules,chainsofmodulesetc.
A
h. Contents
t
a 1. Introduction 1
m
2. The main results 3
[ 3. Backgroundand Preparations 7

4. Proofs, examples and corollaries 15
3
v References 21
8
6
1
7
0 1. Introduction
.
1 1.1. Setup. Let R be a (commutative, associative) local ring over some base field of zero characteristic. (The
0 simplest examples are [[x]], C{x}, and C∞(Rp,0).) Denote by m ⊂ R the maximalkideal and by Mat (R) the
5 k m×n
R-module of m×n matrices with entries in R. We always assume m≤n, otherwise one can transpose the matrix.
1
:
v
Various groups act on Mat (R), we consider mostly the following actions.
i m×n
X
Example 1.1. i. The left multiplications G := GL(m,R), the right and the two-sided multiplications G :=
l r
r GL(n,R), G := G ×G . Matrices considered up to G -transformations correspond to the embedded mod-
a lr l r r
ules, Im(A) ⊂ R⊕m. Matrices considered up to G -transformations correspond to the non-embedded modules,
lr
coker(A)= R⊕m(cid:30) .
Im(A)
ii. The congruence, G = GL(m,R) (cid:8) Mat (R), acts by A → UAUt. Matrices considered up to the
congr m×m
congruence correspond to the bilinear/symmetric/skew-symmetricforms. The conjugation, G =GL(m,R)(cid:8)
conj
Mat (R), acts by A→UAU−1. This action is important in representationtheory.
m×m
k k
iii. Fix some ordered compositions, m = m , n = n . Take the corresponding block-structure on the matrix,
i j
i=1 j=1
i.e. A = {A } , A ∈ Mat P(R). DenoPte by Matup (R) the set of the upper-block-triangular
ij 1≤i,j≤k ij mi×nj {mi}×{nj}
matrices,i.e. A =Ofori>j,thisisafreedirectsummandofMat (R). Accordinglyweconsiderthegroups
ij m×n
(1) Gup :=G ∩Matup (R), Gup :=G ∩Matup (R) and Gup :=Gup×Gup.
l l {mi}×{mj} r r {ni}×{nj} lr l r
The upper-block-triangularmatrices consideredup to the Gup-equivalence correspondto the submodules of R⊕m
r
filtered by chains ofdirect summands. The upper-block-triangularmatrices consideredup to the Gup-equivalence
lr
correspond to filtered homomorphisms of filtered free modules. (See §4.3 for more detail.)
Date:December 28,2016.
2000 Mathematics Subject Classification. Primary58K40,58K50Secondary32A19,14B07,15A21.
Keywordsandphrases. DeformationsofMatrices/Modules/QuadraticForms/SymplecticForms,Miniversaldeformations,MatrixFam-
ilies,FiniteDeterminacy,MatrixSingularities,SufficiencyofJets,Algebraization.
D.K.wassupportedbytheIsraelScienceFoundation (grantNo. 844/14).
2 GenrichBelitskiiandDmitryKerner
We study deformations of matrices. In applications one often deforms a matrix not inside the whole Mat (R)
m×n
butonlyinsidea“deformationsubspace”(asubsetofprescribeddeformations),A A+B,A+B ∈Σ⊆Mat (R).
m×n
In this paper the subset Σ−{A} ⊆ Mat (R) is a submodule. Besides the trivial choice Σ = Mat (R), we
m×n m×n
mostly work with the following submodules.
Example 1.2. i. The congruence of (skew-)symmetric matrices, A Gc∼ongr UAUT, U ∈ GL(m,R), preserves the
(skew-)symmetry. Thus it is natural to deform the matrix by the (skew-)symmetric matrices only, i.e. Σ =
Matsym (R) or Σ=Matskew−sym(R).
m×m m×m
ii. Similarly, it is natural to deform an upper-block-triangular matrix only inside Matup (R).
{mi}×{nj}
iii. Sometimes one deforms only by “higher order terms”, e.g. Σ = {A}+Mat (J) or Σ = {A}+Matsym (J),
m×m m×m
for some ideal J ⊂R.
RecallthatoveralocalringanymatrixisG -equivalenttoablock-diagonal,AG∼lr 1I⊕A,wherealltheentriesofA
lr
lieinthemaximalidealm,i.e. vanishattheoriginofSpec(R). Similarstatementsholdfor(skew-)symmetricmatrices
with respect to G , see proposition3.1. This splitting is natural in various senses andeis standardin commutativee
congr
algebra,singularitytheoryandother fields. Oftenit is Athatcarriesallthe essentialinformation. Thereforewe often
assume A| =O, i.e. A∈Mat (m).
0 m×n
e
1.2. The tangent spaces. Fix an action G (cid:8) Mat (R), a deformation space Σ ⊆ Mat (R), and a matrix
m×n m×n
A∈Σ. We assume that the orbit GA and the deformation space possess well defined tangent spaces at A, which are
R-modules. The general definitions/conditions are in [B.K.], for the examples relevant to our paper see §3.8.
The standard approachof deformationtheory is to establish the existence of the miniversal(semi-universal) defor-
mationand,whenthe laterexists,tounderstand/tocompute itstangentcone. Accordinglyonepassesfromthe study
of the germs (GA,A)⊆(Σ,A), to the study of the tangent spaces, T ⊆T .
(GA,A) (Σ,A)
Much of the information about the deformation problem is encoded in the quotient module
T (cid:30)
(2) T1 := (Σ,A) .
(Σ,G,A) T
(GA,A)
This R-module is the tangent space to the miniversal deformation, when the later exists and is smooth. It often
controls the deformation theory of modules, (skew-)symmetric forms and other objects. In Singularity Theory such a
module is known as the Tjurina algebra for the contact equivalence, and the Milnor algebra for the right equivalence.
This module T1 is the main object of our study.
(Σ,G,A)
1.3. (In)finite determinacy. The (Σ,G)-order of determinacy of A is the minimal number k ≤ ∞ satisfying: if
A,A1 ∈ Σ and jetk(A) = jetk(A1) then A1 ∈ GA. Here jetk is the projection Matm×n(R) j→etk Matm×n(R(cid:30)mk+1).
More precisely:
Definition 1.3. ordΣ(A):=min k| Σ∩ {A}+Mat (mk+1) ⊆GA ≤∞.
G m×n
n (cid:0) (cid:1) o
(We assume that the minimum is taken here over a non-empty set.)
If ordΣ(A) < ∞ then A is called finitely-(Σ,G)-determined. In words: A is determined (up to G-equivalence) by
G
a finite number of terms of its Taylor expansion at the origin. An immediate consequence is the algebraization: A is
G-equivalent to a matrix of polynomials. Moreover,the order of determinacy gives an upper bound on the degrees of
polynomials. We show that the finite determinacy is equivalent to the stable algebraizability (a strengthening of the
ordinary algebraizability),see §2.8.5 and §3.12.
If ordΣ(A) = ∞ the matrix is called infinitely determined. This condition is empty when m∞ := ∩ mk = {0}.
G
k>0
But for some rings m∞ 6= {0} ⊂ R, then even the infinite determinacy is non-trivial property. (The main example
is the ring of germs of smooth functions, R = C∞(Rp,0), and its sub-quotients.) In this case A is determined (up
to G-equivalence) by its image under the m-adic completion, A ∈ Mat (R), i.e. its full Taylor expansion at the
m×n
origin.
b b
The(in)finite determinacyisthefundamentalnotionofSingularityTheory. ForAlgebraicGeometry/Commutative
Algebra the finite determinacy means (roughly) “the deformation theory is essentially finite dimensional”. More
generally,as the determinacy expressesthe “minimalstability”,it is importantin anyareadealing withmatrices over
rings (or matrix families or matrices depending on parameters).
See §2.8 for further motivation and §2.9 for a brief overview and the relation of our work to the known results.
3
In [B.K.] we havereduced the study ofdeterminacy to the understanding of the support/annihilatorofthe module
T1 . ThematrixAis(in)finitelydeterminediffmNT ⊆T forsomeN ≤∞,alternativelymNT1 =
(G,Σ,A) (Σ,A) (GA,A) (G,Σ,A)
{0} or mN ⊆ ann(T1 ). The order of determinacy is fixed by the annihilator ann(T1 ), see §3.11 for more
(G,Σ,A) (G,Σ,A)
detail.
1.4. Contents of the paper. The results of [B.K.] reduce the study of (in)finite determinacy to the computation of
the support of T1 , i.e. the annihilator ideal ann(T1 ). In the current paper we study the module T1
(Σ,G,A) (Σ,G,A) (Σ,G,A)
and its support for the R-linear actions G(cid:8)Mat (R) of example 1.1.
m×n
One could begin from a completely general (set-theoretic) action G (cid:8) Mat (R). However, if an action G (cid:8)
m×n
Mat (R) is R-linear (though does not necessarily preserve the row/column structure) and sends the degenerate
m×n
matrices to degenerate (quite a reasonable assumption!) then G⊆G . This property is proved in §3.6, it belongs to
lr
theareaof“preserverproblemsonmatrices”,exhaustivelystudiedformatricesoverfields,butlessknownformatrices
over rings. Therefore example 1.1 contains the main “reasonable” cases.
We compute (or at least bound) the annihilator ann(T1 ), the main results are stated in §2. This gives the
(Σ,G,A)
ready-to-usecriteria to determine (or at least to bound) ordΣ(A) for various group-actions. We observe the standard
G
dichotomy: for a given data (m,n,dim(R),G(cid:8)Σ⊆Mat (R)) either there are no finitely determined matrices in
m×n
Mat (m)∩Σ, or the generic finite determinacy holds.
m×n
The proofs of these results, corollaries/examplesand further developments are in §4. In many cases we strengthen
(generalize and quantify) both the classical and the relatively recent results.
In §3 we collect the needed background material: determinantal ideals and Pfaffian ideals, annihilator-of-cokernel,
the integral closure of ideals and modules, the matrix preservers, tangent spaces to the group orbits, approximation
properties and the main result of [B.K.].
1.5. Acknowledgements. Many thanks to R.-O. Buchweitz, V. Grandjean, G.-M. Greuel, V. Kodiyalam and M.
Leyensonforhelpful/informativediscussionsandtoL.Molna´r,V.V.Sergeichukforthe muchneededreferencesonthe
“matrix preservers” problems.
2. The main results
Below we state the main theorems and corollaries. Their proofs and examples are in §4.
2.1. Notations and Conventions. Denote by I (A) the determinantal ideal of all the j × j minors of A. De-
j
note by ann.coker(A) the annihilator-of-cokernel ideal of the homomorphism of free modules R⊕n →A R⊕m. The
properties/relation/computabilityof these ideals are discussed in §3.2.
We often use the ideal quotient, I :J ={f ∈R| fJ ⊆I}, and the integral closure of ideals I ⊆I, see §3.4.
Sometimes we take the m-adic completion, R→R, denote by A the image of A under the completion map.
Suppose J ⊇ m∞. The Loewy length, ll (J) ≤ ∞b, is the minibmal number N ≤ ∞ that satisfies: J ⊇ mN. This
R
number also equals the degree ofthe socle of the quotient module R(cid:30). For R= [[x]] we have yetanother expression,
J
k
via the Castelnuovo-Mumfordregularity, ll (J)=reg(R(cid:30))+1, see [Eisenbud, exercise 20.18].
R J
For Artinian rings the finite determinacy is not interesting, e.g. if mN+1 ={0} then any matrix over R is trivially
N-determined. Therefore in this paper we usually assume that R is not Artinian, i.e. mN 6= {0} for any N < ∞.
Geometrically this means: dim(Spec(R))>0.
In some places we assume that the ring R is unique factorization domain (UFD), then we state this explicitly.
2.1.1. Generic properties. In this paper the genericity is always used in the following sense of [Tougeron]. Fix some
“parameterspace” M, typically anR-module, suppose the jetd-projections{Rj→etd R(cid:30)md+1}d∈N induce the projections
M j→etd jet (M). Weassumethatalljet (M)arealgebraicvarietiesover . ApropertyP issaidtoholdgenericallyin
d d
k
M ifitholdsforallpointsofM lyinginthecomplementtoasubsetX ⊂M whichis“ofinfinitecodimension”. Namely,
foranydtheprojectionofX liesinsidesomealgebraicsubset,jet (X)⊆Y ⊂jet (M),suchthatcodim (Y )→
d d d jetd(M) d
∞.
Definition 2.1. We say that the generic finite determinacy holds for a given action G (cid:8) Σ if for any A ∈ Σ, any
number N < ∞ and the generic matrix B ∈ Mat (mN), such that A+ B ∈ Σ, the matrix A+B is finitely
m×n
determined.
4 GenrichBelitskiiandDmitryKerner
Equivalently: the m-adic closure of the set of finitely-determined matrices is the whole space Σ.
In terms of singularity theory the finite determinacy means the finiteness of the Tjurina number, for a given
(Σ,G,A).
Thusthegenericityoffinitedeterminacymeans: thestratumofmatriceswhoseTjurinanumberisinfinite,ΣG ⊂
τ=∞
Σ, is itself of infinite codimension.
2.2. The criteria for the G , G , G actions of example 1.1. In this subsection Σ=Mat (R), with m≤n.
r l lr m×n
Theorem 2.2. Let A∈Σ=Mat (R), with m≤n.
m×n
⊕n
(1) T1 = coker(A) and ann(T1 )=ann.coker(A).
(Σ,Gr,A) (Σ,Gr,A)
(2) If m<n the(cid:16)n ann T1(cid:17) ={0}. If m=n then ann T1 =ann.coker(A).
(Σ,Gl,A) (Σ,Gl,A)
(3) ann T(1Σ,Glr,A) ⊇a(cid:0)nn.coker(cid:1)(A). If R is Noetherian then(cid:0) ann T(1Σ(cid:1),Glr,A) ⊆Im(A):Im−1(A).
In particu(cid:0)lar, when(cid:1)R is UFD and matrices are square (m = n),(cid:0)we get: a(cid:1)nn T1 = ann.coker(A), see
(Σ,Glr,A)
equation (10). For m<n the bounds are often close. In fact, in §4.1 we prove a slightly stronger statement:
(cid:0) (cid:1)
(3) ann T1 ·I (A)⊆I (A).
(Σ,Glr,A) m−1 m
(cid:0) (cid:1)
Theorem 2.2 bounds ann T1 in terms of ann.coker(A) and I (A) : I (A). These ideals are usually
(Σ,Glr,A) m m−1
rathersmall,thus finite-G -determinacyplacessevererestrictionsonthe ringR. Forthe applicationsto the(in)finite
lr (cid:0) (cid:1)
determinacy we assume:
(4) either R is Noetherian or the m-adic completion map is surjective, R։R, and R is Noetherian.
(The ring C∞(Rp,0) satisfies this condition, see §3.10.)
b b
Proposition 2.3. Suppose R satisfies condition (4). Fix Σ=Mat (R).
m×n
(1) Ifann.coker(A)⊇m∞ thenllR ann.coker(A) −1≤ordΣGr(A)≤llR ann.coker(A|mR⊕n) −1.Inparticular:
i. If dim(R)>(n−m+1) t(cid:16)hen no matrix(cid:17)in Matm×n(m) is finit(cid:16)ely-Gr-determined. (cid:17)
ii. If dim(R)≤(n−m+1) then the generic G -finite determinacy holds.
r
(2) If m<n then no matrix A∈Mat (m) is finitely-G-determined. If m=n then ordΣ (A)=ordΣ (A).
(3) i. If dim(R)>(n−m+1) then nmo×mnatrix in Mat l(m) is finitely-G -determined,Gr Gl
m×n lr
ii. If R is Noetherian then llR Im(A):Im−1(A) −1≤ordΣGlr(A)≤llR ann.coker(A|mR⊕n) −1.
iii. If R satisfies (4) and ann.(cid:16)coker(A)⊇m∞ th(cid:17)en (cid:16) (cid:17)
llRb Im(A):Im−1(A) −1≤ordΣGlr(A)≤llR ann.coker(A|mR⊕n) −1.
(cid:16) (cid:17) (cid:16) (cid:17)
Here ann.coker(A|mR⊕n) debnotes the abnnihilator-of-cokernelof the restricted map m ·R⊕n →A R⊕m.
Note that the bounds are rather tight: llR ann.coker(A) ≤ llR ann.coker(A|mR⊕n) ≤ llR ann.coker(A) +1
andalsothe idealsI (A):I (A), ann.coker(A)arequitecloseandoftencoincide. The proofs,examplesandother
m m−1 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
corollaries are given in §4.1.
We emphasize that part 1 of the proposition implies also criteria for the infinite-G -determinacy, see corollary 4.8
r
for the C∞(Rp,0) case.
2.3. The criteria for G . Through this subsection we assume m>1.
congr
Recallthatforskew-symmetricmatricesofevensizeI (A)=(det(A))=(Pf(A))2,the squareofthe Pfaffianideal
m
of A. Moreover,ann.coker(A)⊇Pf(A), see proposition 3.4.
For skew-symmetric matrices of odd size: det(A)=0 and ann.coker(A)={0}. As the measure of non-degeneracy
of such matrices we use the (m−1)’st Pfaffian ideal
m
(5) Pf (A):= (Pf(A )).
m−1 i
b
i=1
X
Here A is the (m−1)×(m−1) block of A, obtained by erasing the i’th row and column. (Note that A is skew-
i i
symmetbric and Pf(A )2 =I (A ).) See §3.3 for more detail. b
i m−1 i
b b
Theorem 2.4. Suppose I (A)6={0}.
m−1
(1) Let Σ=Mat (R), where R satisfies the condition (4). Suppose dim(R)>0, i.e. R is not Artinian. Then
m×m
ann(T1 )⊆m∞. In particular, if m∞ ={0} then ann(T1 )={0}.
(Σ,Gcongr,A) (Σ,Gcongr,A)
(2) Suppose R is Noetherian and A∈Σ=Matsym (R), then
m×m
I (A):I (A)⊆ann(T1 )⊆I (A):I (A).
m m−1 Σ,Gcongr,A) m m−1
5
(3) Suppose R is Noetherian and A∈Σ=Matskew−sym, then
m×m
i. If m is even then I (A):I (A)⊆ann(T1 )⊆I (A):I (A).
m m−1 (Σ,Gcongr,A) m m−1
ii. If m is odd then Pf (A)⊆ann(T1 )⊆I (A):I (A).
m−1 (Σ,Gcongr,A) m−1 m−2
As in the case of G , if R is UFD then in parts 2 and 3.i. holds: ann(T1 )=I (A):I (A).
lr (Σ,Gcongr,A) m m−1
The bounds on ann(T1 ) give immediate applications to the finite determinacy:
(Σ,Gcongr,A)
Proposition 2.5. 1. If dim(R)>1 then no A∈Matsym (m) is finitely-(Matsym (R),G )-determined.
m×m m×m congr
Similarly, no A∈Matskew−sym(m) with m-even is finitely-(Matskew−sym(R),G )-determined.
m×m m×m congr
1’. If m is odd and dim(R)>3 then no A∈Matskew−sym(m) is finitely-(Matskew−sym(R),G )-determined.
m×m m×m congr
1”. If dim(R)>0 then no A∈Mat (m) is finitely-(Mat (R),G )-determined.
m×m m×m congr
2. Suppose R is Henselian and satisfies condition (4).
i. Suppose dim(R) = 1 and either Σ = Matsym (R) or Σ = Matskew−sym(R) for m-even. Let A ∈ Σ and
m×m m×m
suppose I (A)⊇m∞. Then
m
ll I (A):I (A) −1≤ordΣ (A)≤ll I (A):I (A) .
Rb m m−1 Gcongr R m m−1
(cid:16) (cid:17) (cid:16) (cid:17)
In particular, A is finitely-(Σ,G )-determined iff det(A) is not a zero divisor in R. In particular, the generic finite
cobngr b
determinacy holds.
ii. Suppose dim(R)≤3 and A∈Σ=Matskew−sym(R), for m-odd. Suppose moreover I (A)⊇m∞. Then
m×m m−1
ll I (A):I (A) −1≤ordΣ (A)≤ll (Pf (A)).
Rb m−1 m−2 Gcongr R m−1
(cid:16) (cid:17)
In particular, A is finitely-(Σ,G )-determined iff either I (A) = R or I (A) ⊂ R is of height= dim(R), as
congr b b m−1 m−1
expected. In particular, the generic finite determinacy holds.
The proofs, examples and further corollaries are in §4.2.
For the criteria of infinite determinacy of C∞(Rp,0)-valued (skew-)symmetric forms see corollary 4.11.
2.4. Upper-block-triangular matrices. For an upper-block-triangular matrix A ∈ Matup (R) take the
{mi}×{nj}
block-structure, {A } , as in example 1.1. The context/motivation for this case, the definitions of T1 ,
ij 1≤i,j≤k (Σ,Gurp,A)
T1 , their roles, the proofs and corollaries are in §4.3.
(Σ,Gup,A)
lr
Theorem 2.6. Let R be a local ring and A∈Σ=Matup (R).
{mi}×{nj}
(1) ann(T1 )=ann.coker(A). In particular:
(Σ,Gurp,A)
k k
i. ann.coker(A )⊆ann(T1 )⊆ ∩ ann.coker(A ).
i=1 ii (Σ,Gurp,A) i=1 ii
ii. QIf m=n then ann(T1 )=I (A):I (A).
(Σ,Gurp,A) m m−1
iii. If n>m and height(I (A))=(n−m+1) then ann(T1 )=I (A).
m (Σ,Gurp,A) m
k k
(2) Suppose R is Noetherian then ann.coker(A ) ⊆ ann.coker(A) ⊆ ann(T1 ) ⊆ ∩ I (A ) :
i=1 ii (Σ,Gulrp,A) i=1 mi ii
(cid:16)
I (A ) . Q
mi−1 ii
(cid:17)
2.5. The non-finite determinacy for conjugation. The conjugation action is defined in example 1.1.
Theorem 2.7. Suppose R satisfies condition (4) and let A∈Mat (R).
m×m
(1) ann(T1 )⊆m∞.
(Matm×m(R),Gconj,A)
(2) In particular, if dim(R)>0 then there are no finitely-G -determined matrices.
conj
2.6. Finite determinacy of chains of free modules. As an immediate application of our methods we consider
boundedchains(or complexes)offree modules. Takesucha chain, ···φ→i+1 F →φi F φ→i−1 ···,fix some basesof{F },
i i−1 i
so that the maps are represented by some matrices. The deformations of such a chain are taken up to isomorphisms,
induced by the action of the product of groups GL(F ,R). We obtain the bounds on the determinacy in §4.5,
i
i
quantifying and generalizing theorem 5.8 of [CutkQosky-Srinivasan].
2.7. Relativedeterminacy/admissibledeformations. Asweseeintheorems2.2,2.4,2.6,theconditionann(T1 )⊇
(Σ,G,A)
mN can be quite restrictive. For example:
• for the action G (cid:8)Mat (R) it means: ann.coker(A)⊇mN, i.e. the module coker(A) is of finite length.
lr m×n
Geometrically: the module is supported only at one point, the origin, i.e. is a skyscraper.
• forthe actionG (cid:8)Matsym (R)itmeans: Aandits associatedquadraticformarenon-degenerateonthe
congr m×m
punctured neighborhood of the origin.
6 GenrichBelitskiiandDmitryKerner
However, one often needs to deform A not inside the whole Mat (R), or Σ, but only by elements of Mat (J)
m×n m×n
(forsome idealJ ⊂R)orΣ(J) :=Σ∩ {A})+Mat (J) . Forexample,one studiesthe determinacyforthe action
m×n
Glr (cid:8)Matm×n(J) or Gcongr (cid:8)Matsmy(cid:16)×mm(J). (cid:17)
Similarly, for a given groupaction, G(cid:8)Mat (R), one often considers the subgroupof transformationsthat are
m×n
identities modulo some (prescribed) ideal I (R. More precisely:
Mat (R(cid:30))
(6) G(I) := g ∈G| g·Mat (I)=Mat (I) and [g]=[Id](cid:8) m×n .
m×n m×n Mat (I)
m×n
n o
For example, for I =mk the group G(mk) consists of elements that are identities up to the order (k−1).
Working with Σ(J) and G(I) leads to the notion of relative determinacy/admissible deformations. For com-
plex analytic non-isolated hypersurface singularities this was studied in [Pellikaan88], [Pellikaan90], [Siersma83],
[Siersma00], [de Jong-van Straten], [Jiang]. For the C∞(Rp,0)-version see [Grandjean00], [Grandjean04], [Thilliez],
[Cutkosky-Srinivasan].
Our criteria directly adapt to this formulation, see proposition 4.19 in §4.6.
2.8. Remarks.
2.8.1. Theorems 2.2,2.4,2.6 and their corollariesare absolutely explicit, ready-to-usecriteria. Their role is similar to
the classical Mather theorem (for = ¯), the transition from:
k k
• “the hypersurface germ is finitely determined iff its miniversal deformation is finite dimensional”
to
• “the hypersurface germ is finitely determined iff it has at most an isolated singularity”.
Moreover,wedonotjustestablishthefinitedeterminacy,butgiverathertightboundsfortheorderofdeterminacy.
Toemphasize,inmostcasesthemoduleT1 doesnotadmitanyexplicit/shortpresentation,e.g. forT1
(Σ,G,A) (Matm×n(R),Glr,A)
one cannot write anything more explicit than Ext1(coker(A),coker(A)). But the bounds we get are quite explicit
R
and tight, the upper and lower bounds often have the same integral closure.
2.8.2. “Negativity” of the results. As our results read, for many actions there are no finitely determined matrices in
Mat (m). For example, there are no finitely determined m-valued quadratic forms when dim(R)>1 (proposition
m×n
2.5). For the action G (cid:8)Mat (R) the situation is even worse, theorem 2.7.
conj m×m
Sometimes one insists on having enough finitely determined objects, e.g. one wants the generic finite determinacy
in the sense of definition 2.1. The two ways to achieve this are:
• Eithertoenlargethegroup,e.g. tousealsothelocalcoordinatechanges/automorphismsofthering,Aut (R).
Then one considers the action of G ⋊Aut(R), G ⋊Aut(R), etc. k
lr congr
• Or to restrict the possible deformations, e.g. to deform in a way that preserves the determinantal ideals,
{I (A)}, or to preserve the characteristic polynomial for square matrices.
j
We do this in [B.K.-2].
2.8.3. A“theoretical”remark. FixsomeactionG(cid:8)Σ⊆Mat (R),weassumethatT isafreeR-module. (This
m×n (Σ,A)
holds in our cases.) Take the generating matrix A of the module T so that Im(A ) = T ⊆ T .
G,A (GA,A) G,A (GA,A) (Σ,A)
Suppose A is of size m ×n , here m =rank(T ), then
G,A 1 1 1 (Σ,A)
T (cid:30)
(7) T1 = (Σ,A) =coker(A ), while T1 =coker(A )⊕n1.
(Σ,G,A) T(GA,A) G,A (Matm1×n1(R),Gr,AG,A) G,A
(Here on the right G =GL(n ,R).) In particular, ann(T1 )=ann(T1 ).
r 1 (Σ,G,A) (Matm1×n1(R),Gr,AG,A)
Therefore the study of T1 for any G action is “embedded” into the study of T1 for the G action, on the space of
r
bigger size matrices. And the same “embedding” holds for the determinacy problems.
2.8.4. Finite determinacy of (non-)embedded modules and (skew-)symmetric forms. The notion of finite determinacy
and our results can be stated more algebraically as follows.
TheprojectionRj→etk R(cid:30)mk+1 inducesthefunctor,mod(R)→mod(R(cid:30)mk+1),definedbyM → M(cid:30)mk+1M. Thisfunctor
is trivially surjective on the objects, by the restriction of scalars. But the functor is not injective on the objects, e.g.
if the presentation matrix of M has all its entries in mk+1 then M(cid:30)mk+1M is a free R(cid:30)mk+1 module. Our results give
the regions of parameters (the size of presentation matrix, dim(R)) for which the functor is ”generically injective on
the objects”.
Let M ⊆ R⊕m be a finitely generated R-submodule. Fix some set of generators of M, combine them into the
matrix A.
• The finite-Gr-determinacy of A means: “if M,N ⊆ R⊕m and jetkM = jetkN for R j→etk R(cid:30)mk+1, then
M =N ⊆R⊕m”.
• The finite-Glr-determinacy of A means: “if M,N ⊆ R⊕m and jetkM = jetkN for R j→etk R(cid:30)mk+1, then
M =φ(N)⊆R⊕m for some φ∈GL(m,R)”.
7
Similarly for (skew-)symmetric forms the finite determinacy means that the form is determined (up to the change
of generators) by its k’th jet for k ≫0.
2.8.5. Algebraization. A trivial consequence of finite determinacy is the algebraization: the G-orbit of a given object
containsmany algebraicrepresentativesandthey areobtained by just cutting the “Taylor-tails”. Moreover,the order
of determinacy gives an upper bound on the possible degrees of polynomials. As we show in §3.12,finite determinacy
is equivalent to stable algebraizability: any higher order deformation of a givenobject is G-equivalent to an algebraic
family.
Even if A is not finitely determined, i.e. ann(T1 ) does not contain any mN, one can consider some ideal J
(Σ,G,A)
such that J +ann(T1 ) contains a power of the maximal ideal. Then one gets the “algebraization modulo J”,
(Σ,G,A)
or the “algebraization with respect to some subset of variables”. Therefore our results generalize those of [Elkik],
[Kucharz] and many others.
2.9. Relation to Singularity Theory. The determinacy is a classicalnotionofSingularityTheory,wellstudiedfor
functions and maps over the “classical”rings, when =R or C, and R= [[x]], {x}, [x] , C∞(Rp,0), [AGLV-1],
(x)
k k k k
[Damon84], [Wall81], etc.
The generic finite determinacy has been considered in [Wall79], [du Plessis1982], [du Plessis1983].
(For a short discussion of the results on the determinacy of functions/maps see [B.K., §2].)
2.9.1. Some known results on “matrix singularities”. Some necessary and sufficient conditions for finite determinacy
of modules over the ring C∞(Rp,0) were obtained in [Tougeron].
The square matrices over R = {x ,...,x }, for ∈R,C, and G=G ⋊Aut(R), were considered in [Haslinger],
1 p lr
k k
[Bruce],[Bruce-Tari],andfurtherstudiedin[Bruce-Goryunov-Zakalyukin],[Bruce],[Goryunov-Mond],[Goryunov-Zakalyukin].
In particular, the generic finite determinacy was established and the simple types were classified.
In [Cutkosky-Srinivasan] they study the finite determinacy of matrices for the actions of G , G , G ⋊Aut (R),
r lr r
G ⋊Aut (R) on matrices over complete local rings. They obtain the qualitative results for the finite determinkacy.
lr
k
In [Damon-Pike.1], [Damon-Pike.2] they study the vanishing topology of matrix singularities, relating these to the
free divisors. In [Fru¨hbis-Kru¨ger.Zach] they study the vanishing topology of determinantal singularities.
The study of determinacy in positive characteristic has been initiated in [Greuel-Pham], [Pham].
We emphasizethatmostofthe previousworksaddressedonlythegeneralcriteriaforthe finite determinacy. Inour
work we directly compute (or at least bound) the order(s) of determinacy.
2.9.2. In our results we see the standard dichotomy: either the finite determinacy is the generic property or there are
no finitely determined matrices in Mat (m)∩Σ. The later case is the analogue of the “bad dimensions” for the
m×n
determinacy of maps Maps ( n,0),( m,0) , [AGLV-1, III.1].
k k
(cid:16) (cid:17)
3. Background and Preparations
We denote the zero matrix by O, the identity matrix by 1I.
3.1. The relevant canonical forms for matrices. Let (R,m) be a local ring over a field , char( ) = 0. (As
a trivial case we allow R = , with m = {0}.) Fix the embedding ⊂ R by the decomposiktion of vkector spaces
R= ⊕m. Any element of Rk\m is invertible. k
k
Given A ∈ Mat (R) we sometimes “compute it at the origin”, i.e. take its image over the residue field,
m×n
A| ∈ Mat ( ). In particular, A| = O means A ∈ Mat (m). As A| is a matrix over a field, we take its
0 m×n 0 m×n 0
k
classical rank, r=rank(A| ). Recall that the rank of a skew-symmetric matrix is necessarily even.
0
0 1
We denote by E = ⊕ the canonical skew-symmetric matrix, the size (i.e. the number of summands)
−1 0
i
(cid:18) (cid:19)
should be clear from the context.
Proposition 3.1. (1) LetA∈Mat (R)thenAG∼lr 1I ⊕A,wherer =rank(A| )andA∈Mat (m).
m×n r×r 0 (m−r)×(n−r)
(2) Let A∈Matsym (R) then AGc∼ongr (⊕λ 1I )⊕A, where {λ ∈R\m} and A∈Matsym (m).
m×m i i i ei i (me−r)×(m−r)
(3) Let A∈Matmsk×ewm−sym(R) then AGc∼ongr ⊕λiEei ⊕A, where {λi ∈R\m}ieand A∈Mat(smke−wr−)×sy(mm−r)(m).
i
(cid:16) (cid:17)
Part 1 is known in Commutative Algebra e.g. as “passineg to the minimal resolutioneof the module coker(A)”.
Similarly,inparts2,3wesplita(skew-)symmetricformintoitsregularpart(inthecanonicalform)andthecomplement
- the ‘purely degenerate’ part.
(Part 2. is proved e.g. in [Birkhoff-MacLane, theorem 3, page 345], for part 3 see [Birkhoff-MacLane, exercise 9,
page 347].)
8 GenrichBelitskiiandDmitryKerner
Let R be a discrete valuation ring (DVR) over . In particular R is local, regular and dim(R) = 1. The simplest
k
examples of such rings are the formal power series, R = [[t]], or the analytic series, R = {t}, when is a normed
k k k
field. Note that the ring of real-valued smooth function-germs, C∞(R1,0), is not a DVR. Matrices over DVR often
have good canonical forms.
Proposition 3.2. Suppose R is a DVR over .
k
λ 0 ... ... 0
1
(1) For any A∈Matm×n(R) holds: AG∼lr .0.. .λ.2. .0.. ...... ......, where R⊇(λ1)⊇(λ2)⊇···⊇(λm).
 0 ... λm 0 
 
(2) For any A∈Matsym (m) holds: AGc∼ongr ⊕λ 1I , where R⊇(λ )⊇(λ )⊇···⊇(λ ).
m×m i i 1 2 m
i
(3) For any A ∈ Matskew−sym(m) holds: A Gc∼ongr ⊕λ E ⊕O, where R ⊇ (λ ) ⊇ (λ ) ⊇ ···, while O is the
m×m i i 1 2
i
zero matrix of an appropriate size. (cid:16) (cid:17)
Here part 1 is the Smith normal form. Considering A as a presentation matrix of the R-module coker(A), the
statement is: “every module over a DVR is a direct sum of cyclic modules”. Parts 2,3 read: every (skew-)symmetric
form over a DVR splits into the direct sum of rank-one forms.
Whileparts2,3arestandardfacts,wedidnotfindanexactreference,thuswesketchaproof. Lettbeauniformizing
parameter of R. Denote by p the minimal among the vanishing orders of the entries of A. Then t−pA is a (skew-
)symmetricmatrixoverRandt−pA| 6=O. Applyproposition3.1toget,inthesymmetriccase,AGc∼ongr tp (⊕ λ 1I )⊕
0 i i i
A , (or A Gc∼ongr tp (⊕ λ E )⊕A in the skew-symmetric case). Thus we get the reduction in size and(cid:16)one repeats
i i i
fo(cid:17)r A. (cid:16) (cid:17)
e e
Tehe ideals {(λ )} are called the invariant ideals of the matrix. They are expressible through the determinantal
i
ideals of A in the obvious way.
3.2. Determinantalidealsandtheannihilator-of-cokernel. [Eisenbud,§20]For0<j ≤mandA∈Mat (R)
m×n
denote by I (A)⊂R the j’th determinantal ideal, generated by all the j×j minors of A. By definition I (A)=R
j j≤0
and I (A) = {0}. When the size of A is not stated explicitly we denote the ideal of maximal minors by I (A).
j>m max
The chain of ideals R=I (A)⊇I (A)⊇···⊇I (A) is invariant under the G -action.
0 1 m lr
φ φ
Determinantalidealsbehavenicelyunderhomomorphismsofrings,i.e. foranyR→SandtheinducedMat (R)→
m×n
Mat (S) one has: I (φ(A))=φ(I (A)).
m×n j j
IfA| =O,i.e. A∈Mat (m), thenthe heightofI (A)isatmostmin (m+1−j)(n+1−j),dim(R) andthis
0 m×n j
boundisachievedgenerically. Moreprecisely,foranyA∈Mat (m),anyN(cid:16)>0andanygenericB ∈Mat(cid:17) (mN)
m×n m×n
the height of the ideal I (A+B) equals min (m+1−j)(n+1−j),dim(R) .
j
For A ∈ Matsym (m) the expected heigh(cid:16)t of I (A) is min m−j+2 ,dim(cid:17)(R) . For A ∈ Matskew−sym(m) and
m×m j 2 m×m
j even the expected height of I (A) is min m−j+2 ,dim(R) .(cid:16)(cid:0)For A(cid:1)∈ Matske(cid:17)w−sym(m) and j odd the expected
j 2 m×m
height of I (A) is min m−j+1 ,dim(R) . (cid:16)(cid:0) (cid:1) (cid:17)
j 2
A matrix A∈Mat(cid:16)(cid:0) (m) c(cid:1)an be con(cid:17)sidered as a homomorphism of free modules, its cokernel is an R-module as
m×n
well:
(8) R⊕n →A R⊕m →coker(A)→0.
Then one takes the annihilator-of-cokernelideal:
(9) ann.coker(A)={f ∈R| f ·R⊕m ⊆Im(A)}=annR⊕m(cid:30) ⊂R.
Im(A)
This ideal is G -invariant and is a refinement of the determinantal ideal I (A).
lr m
The annihilator-of-cokernel is a rather delicate invariant, for example it does not transform nicely under the ring
homomorphisms, ann.coker(φ(A)) 6= φ(ann.coker(A)). Yet, this ideal is controlled by the determinantal ideals as
follows:
ann.coker(A)m ⊆I (A)⊆ann.coker(A)⊆ I (A),
m m
and ∀ m>j >1: ann.coker(A)·I (A)⊆I (A) [Eisenbud, proposition 20.7]
(10) j−1 p j
If m=n then ann.coker(A)=I (A):I (A).
m m−1
If m<n and height(I )=(n−m+1) then ann.coker(A)=I (A), [Eisenbud, exercise 20.6]
m m
In particular, for one-row matrices, m=1, or when I (A) is a radical ideal, I (A)=ann.coker(A).
m m
9
We use the following properties:
Lemma 3.3. (1) (Block-diagonal case) ann.coker(A⊕B)=ann.coker(A)∩ann.coker(B).
(2) If m = n, i.e. A is a square matrix, then ann.coker(A) = ann.coker(AT). If moreover R is a unique
factorization domain (UFD) then ann.coker(A) is a principal ideal.
(3) If m>n then ann.coker(A)={0}.
(4) Let A(cid:3) ⊂A be a submatrix of A obtained by erasing some rows of A. Then ann.coker(A)⊆ann.coker(A(cid:3)).
Here statements 1 and 4 are immediate.
The firstpartof Statement2 followsdirectly, e.g. fromann.coker(A)=I (A):I (A) ofequation(10). Forthe
m m−1
secondpartof2note thatforasquarematrixthe heightofann.coker(A)⊂R isone. IfRis UFD thenann.coker(A)
is generated by just one element.
Statement3: inthiscasethesubmoduleIm(A)⊂R⊕m isofrank≤n<m. Thusthemodulecoker(A)issupported
on the whole Spec(R) and is not annihilated by any elements of R.
3.3. Skew-symmetric matrices and their Pfaffian ideals. SupposeAis asquarematrixthenI (A)=(det(A)).
m
If moreover A ∈ Matskew−sym(R), with m-even, then the determinant is a full square, det(A) = Pf(A)2. Here the
m×m
Pfaffian polynomial is explicitly written in terms of the entries of A, [Heymans], [Eisenbud, exercise A.2.11], [Greub].
In particular, the Pfaffian polynomial transforms under the homomorphism of rings: Pf(φ(A))=φ(Pf(A)).
The transformation of the Pfaffian under the congruence is: Pf(UAUT)=det(U)Pf(A).
If m is odd then det(A) = 0. In this case, as a measure of degeneracy, we need the following refinement of the
Pfaffian ideal:
m
(11) Pf (A):= Pf(A ) .
m−1 i
b
i=1
X(cid:0) (cid:1)
Here A is the (m−1)×(m−1) block of A, obtained by erasing the i’th row and column. (Note that A is skew-
i i
symmetbric and I (A )= Pf(A )2 .) b
m−1 i i
We use the followingbpropertiesbof these ideals:
(cid:0) (cid:1)
Lemma 3.4. Let R be a local ring.
1. Let A∈Matskew−sym(R) for m-even. There exists the “Pfaffian-adjugate” matrix, adjPf(A)∈Matskew−sym(R),
m×m m×m
that satisfies A·adjPf(A)=Pf(A)1I=adjPf(A)·A. In particular, ann.coker(A)⊇Pf(A).
1’. Moreover: Span (AU +UTA) ⊇Matskew−sym(Pf(A)).
R U∈Matm×m(R) m×m
2. Let A∈Matskew−sym(R) for m>1, odd. Then Span (AU +UTA) ⊇Matskew−sym(Pf (A)).
m×m R U∈Matm×m(R) m×m m−1
Proof. 1. We recallthe relevantexterioralgebra,[Eisenbud, exerciseA.2.11]. Let V be a free R-module ofrank n. Its
2
dual, V∗, is the module of one-forms. Associate to A the skew-symmetric form w := a dx ∧dx ∈ ∧V∗.
A ij i j
1≤i<j≤m
The top exterior power of w gives the Pfaffian: P
A
(12) ∧m2w =m!·Pf(A)dx ∧···∧dx ∈m∧V∗.
A 1 m
Now, using the pairing ∧2V∗ ⊗m∧−2V∗ → m∧V∗, we associate to the form m2∧−1w an element of ∧2V∗, which we call
A
m−1
adjPf(A). Then the relation w ∧(2∧ w )=Pf(A)dx ∧···∧dx transforms into A·adjPf(A)=Pf(A)1I.
A A 1 m
1’. This is now immediate, one takes U in the form adjPf(A)U˜, where U˜ ∈Matskew−sym(R).
m×m
2. By proposition 3.1 A is congruent to the block-diagonal matrix E ⊕A, where E is a non-degenerate skew-
symmetric matrix over R (i.e. det(E) is invertible), while A| = O. Moreover, this splitting is stable, any family
0
A is congruent to E ⊕ A . Therefore it is enough to prove the statemenet for A and from now on we assume
t t t
A∈Matskew−sym(m). e
m×m
As the Pfaffian, Pf(A),eis an explicit polynomial in the entries of the matrix AeT = −A, it is functorial under
the change of rings. Therefore it is enough to prove the statement for the skew-symmetric matrix of indeterminates,
A=(a ), over the ring R = [[{a }]]. In this case the ideal Pf (A) is Gorenstein and of height 3. Therefore we
ij ij m−1
k
can use the classical result, [Buchsbaum-Eisenbud, page 457] (for m odd):
(13) If B ∈Matskew−sym(R) and Pf (B)=Pf (A), then AGc∼ongr B.
m×m m−1 m−1
Our statement is just the infinitesimal version of this result, and the proof goes as follows.
For m=3 the statement is checked directly, thus we assume m>3.
Let ∆ ∈ Matskew−sym(Pf (A)). We claim that the two ideals coincide: (Pf (A)) = (Pf (A + ∆)).
m×m m−1 m−1 m−1
Indeed, for any block A take the standard expansion of the Pfaffian polynomial Pf(A ) in the entries of A . Then
i i i
we get: b b b
(14) Pf(A +∆ )=Pf(A )+ (...) .
i i i
b b b
∈m·(Pf(A ))
bi
|{z}
10 GenrichBelitskiiandDmitryKerner
Therefore for the ideals we have: Pf(A +∆ ) ⊆ Pf(A ) . The converse inclusion follows by the expansion
i i i
b b b
(cid:16) (cid:17) (cid:16) (cid:17)
(15) Pf (A −∆ )+∆ =Pf(A −∆ )+··· .
i i i i i
b b b b b
(cid:0) (cid:1)
Thus Pf(A +∆ ) = Pf(A ) . As this holds for any i we get: Pf (A+∆) = Pf (A) .
i i i m−1 m−1
b b b
Now(cid:16), by (13), we(cid:17)get(cid:16)A+∆ G(cid:17)c∼ongr A. Finally, take the infinites(cid:16)imal family of s(cid:17)uch c(cid:16)ongruences(cid:17), {A+ǫ∆ Gc∼ongr
A} . And take the first order terms in ǫ, to get: ∆∈T =Span (UA+AUT) .
ǫ∈k[ǫ]/(ǫ2) (GcongrA,A) R U∈Matm×m(R)
3.4. Integral closure of ideals. The integral closure of an ideal I ⊂R is defined as
d
(16) I¯= f ∈R fd+ a fd−j =0, for some d∈N and some {a ∈Ij} .
j j j=1,...,d
n (cid:12) Xj=1 o
(cid:12)
This subset is an ideal in R and is itself integrally closed, i.e. I¯¯=I¯.
Example 3.5. [Huneke-Swanson, §1.4]FixamonomialidealI ⊂R= [[x ,...,x ]]. Takethe setofallitsexponents:
1 p
S ={d|xd ∈I}. Thisleadstothediagramofinitialexponents(theNewktondiagram),Γ :={d|d∈Conv(S +Rp )}.
I I I ≥0
Then the integral closure is also a monomial ideal, I¯={xd| d∈Γ }.
I
Example 3.6. Let R= [[x(cid:30)]] , where f is in the Weierstraßform, f(x ,...,x )=xd+xd−1a +···+a , with the
k f(x) 1 p p p 1 d
coefficients a = a (x ,...,x ) ∈ (x ,...,x )j. Then obviously x ∈ (x ,...,x ), i.e. (x ,...,x ) = m. In
j j 1 p−1 1 p−1 p 1 p−1 1 p−1
particular, for the plane curve case, p=2, we have x ∈(x ).
2 1
We mention several useful properties:
• if the height of I is defined then height(I)=height(I);
• I :J ⊆I :J, [Huneke-Swanson, page 7];
• if R is a domain and I 6={0} then I ·J :I =J, [Huneke-Swanson, Corollary 6.8.7];
• if R is local, Noetherian and regular then f ·J =(f)·J.
In general the computation of I¯ is rather involved. Things simplify due to the following “geometric” criterion,
Theorem 6.8.3 of [Huneke-Swanson].
(17) If R is Noetherian then I¯= φ−1Sφ(I)
R→φ\SDVR
(Here the intersection goes over all the homomorphisms to the discrete valuation domains.) In words:
φ
(18) f ∈I iff for any homomorphism to a DVR, R→S, φ(f)∈Sφ(I).
Geometrically this criterion (initially for R=C{x}, [Teissier, 1.3.4]) reads:
(19) f ∈I iff for any map of the smooth curve-germ, (C,0)→ν Spec(R), the pullbacks satisfy: ν∗(f)∈ν∗(I).
3.5. Integral closureofmodules. Forthegeneraldefinitionoftheintegralclosureofmodulessee[Huneke-Swanson,
definition16.1.1]. InourcaseallthemodulesareembeddedintoR⊕m,theringisNoetherianoverafield,char( )=0,
k
φ
thus the definition simplifies. Note that any ring homomorphism R → S induces the homomorphism of embedded
modules R⊕m ⊇M →φ φ(M)⊆S⊕m.
Definition 3.7. [Huneke-Swanson, §16.1] The integral closure of a module M ⊂Rm is M := φ−1Sφ(M).
φ
R→TSDVR
φ
In words: z ∈M iff foranyhomomorphismto a discretevaluationdomain, R→S ,there holds φ(z)∈Sφ(M).
DVR
We will use the property, [Huneke-Swanson, Remark 1.3.2]:
(20) J ·J ⊆J ·J ⊆J ·J ,
1 2 1 2 1 2
More generally:
Lemma 3.8. Let R be local, Noetherian, N ⊆M ⊆R⊕m embedded modules, J ⊂R an ideal.
(1) J ·M ⊆J ·M ⊆J ·M.
(2) annM(cid:30) ⊇annM(cid:30) .
N N