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Some Properties of Finite Morphisms on Double Points

Article in Compositio Mathematica · March 2000

DOI: 10.1023/A:1001865413625

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Some Properties Of Finite Morphisms
On Double Points

Hassan Haghighi (hashag@vax.ipm.ac.ir)

Department of Mathematics, University of Tehran

Joel Roberts (roberts@math.umn.edu)

School of Mathematics, University of Minnesota
Minneapolis, MN 55455, USA

Rahim Zaare-Nahandi (rahimzn@kharazmi.ut.ac.ir)

University of Tehran, and Cent. Theo. Phys. Math.,
P.O.Box 11365-8486, Tehran, Iran

Abstract. For a finite morphism f : X → Y of smooth varieties such that f

maps X birationally onto X 0 = f (X), the local equations of f are obtained at
the double points which are not triple. If C is the conductor of X over X 0 , and
D = Sing(X 0 ) ⊂ X 0 , ∆ ⊂ X are the subschemes defined by C, then D and ∆
are shown to be complete intersections at these points, provided that C has ”the
expected” codimension. This leads one to determine the depth of local rings of X 0
at these double points. On the other hand, when C is reduced in X, it is proved that
X 0 is weakly normal at these points, and some global results are given. For the case
of affine spaces, the local equations of X 0 at these points are computed.

Keywords: generic projections, double points, singular locus, conductor, complete

intersection, weak normality, seminormality, local equations

Mathematics Subject Classification (1991): 14E40, 14M05, 14M10

0. Introduction

While generic projections enjoy significant properties, part of these

properties may be derived from rather mild assumptions on arbitrary
finite morphisms. This was the philosophy of the second author to initi-
ate this work. Let X be a projective smooth variety of dimensions r and
let π : X → Pm be a strongly generic projection with r + 1 ≤ m ≤ 2r.
Outside a closed subset of X of low dimension, the local canonical form
of π is known (see [18], sec. 12). These canonical forms enable one to
study the local properties of π, the local structure of π(X) and that of
the singular locus of π(X). As an example, most of the approaches to
deal with the Andreotti-Bombieri-Holm conjecture, which claims that
π(X) is a weakly normal variety, have been of local nature, i.e., based
on the local canonical forms of π (see [6], Theo. 3.7, [1], Theo. 2.7, [25],
Theo. 3.2). Thus it is tempting to look for rather general situations

c 2005 Kluwer Academic Publishers. Printed in the Netherlands.

2 Haghighi, Roberts, and Zaare-Nahandi

when one still may get canonical forms of morphisms.

Another related feature, is the structure of the singular loci of maps.

An old result due to Enriques ([5], page 8), states that if a surface has
a pinch point, its singular locus is smooth at this point. This has been
generalized as follows: if π : X → Pm is a strongly generic projection,
and if y = π(x) is a pinch point, then Sing(π(X)) is smooth at y (see
[25], Prop. 2.14). Again one can study this question for rather general
morphisms. On the other hand, even for strongly generic projections,
Sing(π(X)) at triple points is not Cohen-Macaulay unless m = r + 1,
i.e., π(X) is a hypersurface (see [21], Prop. 4.6, [26], Cor. 2.7). Since
the hypersurface case is extensively studied(see [19], [9]), we will not
deal with this case, and will only study the double points which are
not triple.

We will consider a finite morphism f : X → Y of smooth varieties

of dimensions r and m respectively, where r + 1 ≤ m ≤ 2r. It is known
that every irreducible component of Sing(f (X)) has dimension at least
2r − m (e.g., see [2], Theo. 5.1). As for generic projections, one of the
basic ingredients to derive certain results is that the singular locus
has the least dimension, a similar hypothesis in the case of arbitrary
finite morphism leads to some nice results on Sing(f (X)) at the double
points. Primarily one needs a suitable scheme structure on Sing(f (X)).
A useful scheme structure in this situation is the structure given by the
conductor of X in f (X). This provides a scheme structure not only for
D = Sing(f (X)) but also for ∆ = f −1 (D) (see [9]). Throughout this
paper we will use the scheme structure on D and ∆ given by the conduc-
tor. One of our results is that when D or ∆ has the expected dimension,
they are both complete intersections at the double points of f . These
results meet Kleiman’s criterion of not imposing any hypothesis other
than the appropriateness of the dimension of the singular locus.

The conductor happens to have an important role in dealing with

weak normality too. By a result due to C. Traverso ([22], Lemma 1.3),
if X 0 = f (X) is weakly normal, then the conductor is a reduced sub-
scheme of X. The converse is not true in general. But it turns out that
for the double points which are not triple, this condition is sufficient to
guarantee weak normality of X 0 (see Theo. 4.8). The weak normality
property needs to be checked at ”depth one primes” ([6], Cor. 2.7).
In view of this, we have shown that actually, the reducedness of the
conductor at only one double point on each irreducible component of
the singular locus will still imply the weak normality of X 0 (see Theo.
4.9). Then using a result of Vitulli, we have given a necessary and
 Some Properties of Finite Morphisms on Double Points 3

sufficient version of Hartshorne’s depth-connectivity result for the local

rings on X 0 (see Prop. 4.10). The local defining ideals of double and
triple singularities of strongly generic projections and finite presenta-
tions related to them are studied in ([21]). We have obtained similar
results for the double points provided that the canonical forms of f are
given by polynomials rather than formal power series (see Theo. 5.2).

1. Preliminaries

Let B be an integral over-ring of A, the seminormalization of A in B

is defined to be

+ A = {b ∈ B | ∀x ∈ Spec(A), bx ∈ Ax + R(Bx )}
B

where R(Bx ) is the Jacobson radical of Bx , while the weak normaliza-

tion of A in B is defined to be
n
∗ A = {b ∈ B | ∀x ∈ Spec(A), ∃n ∈ N : (bx )p ∈ Ax + R(Bx )}
B

If A = + A, we say A is seminormal in B and if A = ∗ A, A is called

B B
weakly normal in B. If B is the integral closure of A in its total ring
of quotients, then we say A is SN (resp. WN) if it is SN (resp. WN) in B.

Since + A is contained in ∗ A (and they coincide if char k = 0),

B B
thus if A is WN in B, it is also SN in B.

A more geometric interpretation of seminormality and weak normal-

ity is the following:
The ring + A is always SN in B and is the largest among the subrings
B
A0 of B containing A such that:
i)∀x ∈ Spec(A), there exists exactly one x0 ∈ Spec(A0 ) lying over x,
ii) the canonical homomorphism k(x) → k(x0 ) is an isomorphism.

In the same way ∗ A is always WN in B and is the largest among

B
the subrings A0
of B containing A such that;
i)∀x ∈ Spec(A), there exists exactly one x0 ∈ Spec(A0 ) lying over x,
ii) the canonical homomorphism k(x) → k(x0 ) makes k(x0 ) a purely
inseparable extension of k(x) (see [11], [6]).

The following two results will be used in section 4 to check seminor-

mality and weak normality of certain schemes.
4 Haghighi, Roberts, and Zaare-Nahandi

PROPOSITION 1.1. For an integral extension A ⊂ B the following

statements are equivalent
1) A is SN in B.
2) For each b in B, the conductor of A in A[b] is a radical ideal of
A[b].
3) A contains each element b of B such that bn , bn+1 ∈ A for some
positive integer n.
4) For a fixed pair of relatively prime integers e > f > 1, A contains
each element b of B such that be , bf ∈ A.

Proof. (see [10], Prop. 1.4). 2

PROPOSITION 1.2. Let A ⊂ B be as above, then the following are

equivalent
1) A is WN in B.
2) A is SN in B and every element b in B which satisfies bp ∈ A
and pb ∈ A for some prime integer p, belongs to A.

Proof. (see [24], Theo. 1). 2

A scheme is called SN (resp. WN), if the stalks are all SN (resp.

WN) rings. By a complete intersection, we mean a local ring whose
completion is a quotient of a complete regular local ring by a regular
sequence. A scheme is locally complete intersection, if the stalks are all
complete intersections. (see [12] sec. 21, [8], Ch. II, Remark 8.22.2).

We now will recall some results in algebraic geometry which will be

used freely in this paper.

Let k be an algebraically closed field and let Z be a variety over

k, OZ,z the local ring at z ∈ Z. It is known that O bZ,z is reduced,
hence (0) is the intersection of prime ideals P1 , · · · , Pd in ObZ,z . These
minimal prime ideals are defined to be the branches of Z at z. If P is a
branch of Z at z, we say that P is simple if O bZ,z /P is a regular ring,
otherwise it is singular. Let X be the normalization of Z, ν : X → Z the
canonical morphism. The branches of Z at z are in 1-1 correspondence
with points x ∈ X such that ν(x) = z. Namely, x corresponds to the
kernel of the homomorphism O bZ,z → O
bX,x (see [15], Theo. A, or [14],
(37.6)). Let X and Y be smooth varieties over k of dimensions r and
m respectively, where r + 1 ≤ m ≤ 2r. Let f : X → Y be a finite
morphism which is birational onto X 0 = f (X). Let y ∈ X 0 , f −1 (y) =
{x1 , · · · , xd } and let OX,f −1 (y) be the semilocal ring along the fibre,
which is the ring of germs of functions which are regular at xi ; i =
 Some Properties of Finite Morphisms on Double Points 5
f∗
1, · · · , d. Consider the local homomorphism OY,y → OX,f −1 (y) where
OY,y is the local ring at y. If my is the maximal ideal of OY,y , by
finiteness of f , OX,f −1 (y) /f ∗ (my )OX,f −1 (y) is a semilocal Artinian ring,
therefore it has finite length as an OY,y -module. This length is called
”the multiplicity of f at y”. Since X is the normalization of X 0 , we
may call this length ”the multiplicity of X 0 at y”. Since any semilocal
Artinian ring is a product of Artinian local rings,
d
∼
Y
∗
OX,f −1 (y) /f (my )OX,f −1 (y) = (OX,xi /fi∗ (my )OX,xi )
i=1

where fi∗ : OY,y → OX,xi is the natural homomorphism. Thus the

length of the first module is the sum of the lengths of the factors on
the right hand side above. Since any Artinian local ring is complete,
we get the isomorphism
d
OX,f −1 (y) /f ∗ (my )OX,f −1 (y) ∼
Y
= bX,x /f ∗ (my )O
(O bX,x ). (?)
i i i
i=1

The length of each factor in the above may be interpreted as ”the mul-
tiplicity of a branch of X 0 ”. Therefore we can say that the multiplicity
of X 0 at y is the sum of the multiplicities of the branches of X 0 at y.

From now on, we will only consider the points on X 0 with mul-
tiplicity 2. Since the fi∗ ’s are local homomorphisms, the length of
each factor of the product in (?) is at least one. Thus, either d = 2
and each of the two factors in (?) is isomorphic to k, or, d = 1 and
length(O bX,x /f ∗ (my )O bX 0 ,y ∼
bX,x ) = 2. In the first case, O =ObY,y /ker(fb∗ )∩
1
∗
ker(fb2 ), where, O bY,y /ker(fb ) ∼
∗
= ObX,x for i = 1, 2, thus X 0 has two sim-
i i
ple branches at y. In the latter case, X 0 has an analytically irreducible
double point at y.

The morphism f determines an exact sequence of sheaves on X 0

0 → OX 0 → f∗ OX → f∗ OX /OX 0 → 0.

The conductor of X over X 0 is the annihilator of f∗ OX /OX 0 as an

OX 0 -module. This is a sheaf of ideals in OX 0 and naturally lifts to
a sheaf of ideals in OX . We will use C for both of these sheaves of
ideals in OX 0 and OX . Thus C determines a closed subscheme D of
X 0 . Hence, x ∈ D if and only if OX 0 ,x is not normal. Since X 0 has
nonsingular normalization, x ∈ D if and only if x is a singular point of
X 0 . The closed subscheme of X defined by C is denoted by ∆. Thus,
6 Haghighi, Roberts, and Zaare-Nahandi

the underlying set of ∆ is f −1 (Sing(X 0 )), when C is considered as a

sheaf of ideals in OX (see [19]).

The following is an special case of a general result due to D. Rees

([17], also see [12], page 265). We will use its corollary in section 4.
LEMMA 1.3. Let (S, m) be an algebro-geometric local ring. If S is re-
duced, then S is analytically unramified, i.e., its completion is reduced.

Proof. Consider the inclusion S ,→ d1 S/Pi , where (0) = P1 ∩· · ·∩Pd

Q
is the prime decomposition of (0) in S. The last ring is semilocal and
its Jacobson radical lies over m. The completion
Qd of this semilocal ring
with respect to its Jacobson radical is 1 (S/Pi )b (see [12], Theo. 8.15).
By Artin-ReesQ lemma, the m-adic topology on S is induced from the
topology of d1 (S/Pi ). By flatness of completion, the homomorphism
Sb ,→ d1 (S/Pi )b is injective. Since each factor (S/Pi )b is reduced, Sb is
Q
reduced. 2

COROLLARY 1.4. Let ∆ be reduced at some point x ∈ X. Then ∆ is

analytically unramified at x. Similar statement is true for D.

Proof. Since O∆ = OX /C, O∆,x is an algebro-geometric local ring.

Thus by Lemma 1.3, ∆ is analytically unramified at x. The proof of
the claim for D is similar. 2

2. Double points with simple branches

In this section we assume that y is a double point of X 0 at which X 0 has

two simple branches. Identifying O bY,y by R = k[[u1 , · · · , um ]] and O
bX,x
i
by B = k[[t1 , · · · , tr ]], we arrive to a homomorphism ϕ : R → B × B.
Let πi : B × B → B ; i = 1, 2, be the projections. If Pi = ker(πi ◦ ϕ),
then kerϕ = P1 ∩ P2 . Thus O bX 0 ,y = R/P1 ∩ P2 . Since f is finite, R/Pi
is r-dimensional, so Pi has height m − r. It is known that (see [3], sec.
7; or [16], Lemma 3), in this case, the conductor is (P1 + P2 )/P1 ∩ P2 .
If we assume that the conductor has codimension m − r in R/P1 ∩ P2 ,
then P1 + P2 will be of codimension (m − r) + (m − r) = 2(m − r) in
R. We will show that under this assumption, D and ∆ are complete
intersections at points under consideration.

LEMMA 2.1. Let P1 and P2 be prime ideals in R and let R/P1 , R/P2
be regular. Then the conductor of R/P1 × R/P2 is reduced in R/P1 ∩ P2
if and only if it is reduced in R/P1 × R/P2 .
 Some Properties of Finite Morphisms on Double Points 7

Proof. The integral closure of R/P1 ∩ P2 is R/P1 × R/P2 . The

conductor in this ring is (P1 + P2 )/P1 × (P1 + P2 )/P2 . Then
(R/P1 ∩ P2 )/C ∼= R/(P1 + P2 ),
(R/P1 ×R/P2 )/((P1 +P2 )/P1 ×(P1 +P2 )/P2 ) ∼
= R/(P1 +P2 )×R/(P1 +P2 ),
thus the assertion follows. 2

The following is well-known, but we state it for further reference in

this paper.
LEMMA 2.2. With the hypothesis and notation as above, under some
change of variables, it can be assumed that P1 = (u1 , · · · , um−r ).

Proof. Consider ϕ1 : R → B which is surjective by assumption. Let

ϕ1 (fi ) = ti ; i = 1, · · · , r. Observe that fi ’s are of order one, since ϕ1
is a homomorphism. We claim that the linear parts of fi ; i = 1, · · · , r,
are linearly independent . Since otherwise, ord(λ1 f1 + · · · + λr fr ) > 1
for some λi ’s in k, but then ϕ1 (λ1 f1 + · · · + λr fr ) = λ1 t1 + · · · + λr tr
will be of order greater than one, which is a contradiction. Thus by
([27], Vol II, Ch. VII, Cor.2 of Lemma 2), the change of variables Ui =
fi ; i = 1, · · · , r, is an isomorphism. Therefore we may assume that
ϕ1 (Ui ) = ti ; i = 1, · · · , r. Now observe that ϕ1 (ur+i ) = gi (t1 , · · · , tr ) =
ϕ1 (gi (U1 , · · · , Ur )) for i = 1, · · · , m−r, and hence ur+i −gi (U1 , · · · , Ur ) ∈
ker ϕ1 . Again, the change of variables Ur+i = ur+i − gi (U1 , · · · , Ur ) is
an isomorphism. Thus (Ur+1 , · · · , Um ) ⊂ ker ϕ1 . But since ker ϕ1 is of
height m − r, it follows that P1 = ker ϕ1 = (Ur+1 , · · · , Um ). 2

LEMMA 2.3. With the assumption as above, let the singular locus of
X 0 be of local dimension 2r − m at y = f (x1 ) = f (x2 ). Then D is a
complete intersection at y, and ∆ is a complete intersection at x1 and
x2 .

Proof. By Lemma 2.2, we may assume that P1 = (u1 , · · · , um−r ).

Now we apply Lemma 2.2 for ϕ2 : R → B. It follows that under
similar changes of variables, P2 = ker ϕ2 = (U1 , · · · , Um−r ). Changing
the variables back to ones used for P1 , we see that P2 = (h1 , · · · , hm−r )
for some hi ∈ R. Therefore, as an ideal in R, the conductor is P1 +P2 =
(u1 , · · · , um−r , h1 , · · · , hm−r ) = (u1 , · · · , um−r , k1 , · · · , km−r ) where ki ∈
k[[um−r+1 , · · · , um ]]. Since the height of this ideal is assumed to be 2(m−
r), the unmixedness theorem implies that u1 , · · · , um−r , k1 , · · · , km−r
form a regular sequence and hence R/C is a complete intersection.
The claim about ∆ follows by the fact that
b∆,x ∼
O 2
i = R/P1 + P2 ; i = 1, 2.
8 Haghighi, Roberts, and Zaare-Nahandi

3. Analytically irreducible double points

Let f : X → Y be as introduced in section 1. Assume that y =

f (x) is an analytically irreducible double point of X 0 = f (X), i.e.,
f −1 (f (x)) = {x}, then O
bX 0 ,y is an integral domain. Let ϕ : R → B be
the homomorphism induced by the local homomorphism OY,y → OX,x ,
where R and B are the completions of these rings respectively. Assume
that y is a double point which is not triple. Then by definition, the
B-module B/ϕ(mR )B has length 2. In this section, we will give the
”canonical form” of ϕ. Using this result we will compute the conductor
at y. Then assuming that the conductor has codimension m − r, we
will show that D and ∆ are complete intersections at y. This, together
with the similar result from section 2, will enable us to compute the
depth of the local ring at any double point.

LEMMA 3.1. Under the assumptions as above, the maximal ideal of

B/ϕ(mR )B is principal.

Proof. If the maximal ideal of B/ϕ(mR )B is generated by more than

one elements: ā1 , · · · , ā` , using the chain of submodules of B/ϕ(mR )B

(1̄) ⊃ (ā1 , · · · , ā` ) ⊃ (ā1 , · · · , ā`−1 ) ⊃ . . . ⊃ (ā1 ) ⊃ (0̄) ,

one concludes that B/ϕ(mR )B has length greater than 2, which is a

contradiction. 2

PROPOSITION 3.2. Let R and B be as above, and let B/ϕ(mR )B

be of length two. There exist automorphisms of R and B such that if
we identify these two rings with their images, then ϕ has the following
form :
i) ϕ(ui ) = ti ; i = 1, · · · , r − 1,
ii) ϕ(ur ) = t2r if char(k) 6= 2,
ii0 ) if char(k) = 2, then ϕ(ur ) ≡ t2r mod (t1 , · · · , tr−1 , t3r ), and,
t2r = ϕ(ur ) + g0 (t1 , · · · , tr−1 , ϕ(ur ))tr , where g0 is a non-unit power
series in r variables.
iii) ϕ(ur+i ) = gi (t1 , · · · , tr−1 , ϕ(ur ))tr ; i = 1, · · · , m − r,
where for each i, gi is a non-unit power series in r variables. Thus,
if char(k) 6= 2, then ϕ(ur+i ) = gi (t1 , · · · , tr−1 , t2r )tr ; i = 1, · · · , m − r,
where for each i, gi is a power series in r variables.

Proof. Since ϕ is the completion of a continuous homomorphism

(with respect to the adic topologies), ϕ is continuous. Therefore ϕ is
a substitution map ([27] Vol. II, page 136). Thus, ϕ(h(u1 , · · · , um )) =
h(ϕ(u1 ), · · · , ϕ(um )) for every h in R, and hence it suffices to determine
 Some Properties of Finite Morphisms on Double Points 9

ϕ(u1 ), · · · , ϕ(um ). Since the B-module B/ϕ(mR )B has length two, and
since ϕ is a local homomorphism, we may assume that ϕ(mR )B =
(t1 , · · · , tr−1 , t2r ). Thus by some linear change of variables in R, it can
be assumed that
ϕ(ui ) = ti ; i = 1, · · · , r − 1,
ϕ(ur ) ≡ t2r mod(t1 , · · · , tr−1 , t3r ). (?)
In particular we get i) and the first part of ii0 ). Let A0 = k[[u1 , · · · , ur ]]
and let m0 be the maximal ideal of A0 . We first claim that B is a finite
A0 -module. Observe that
ϕ(ur ) ≡ (1 + h(tr ))t2r mod(ϕ(m0 )B),
where h is a non-unit power series in tr . Thus t2r ∈ ϕ(m0 )B, i.e.,
ϕ(m0 )B is primary for mB . Therefore B is a finite A0 -module (see
[27] Vol. II, page 211). While B/ϕ(m0 )B is now generated by 1̄ and
t̄r , by Nakayama’s lemma (or equivalently, by [4], Theo. 30.6), B is
generated by 1 and tr as an A0 -module. Therefore, t2r = ϕ(g)tr + ϕ(h)
for some non-unit power series g, h ∈ A0 , which gives a relation of
integral dependence of tr over A0 . Now we claim that we may assume
that h = ur . Observe that by (?),
ϕ(ur ) = (1 + a(t1 , · · · , tr−1 ))t2r + b(t1 , · · · , tr−1 )tr
+ c(t1 , · · · , tr−1 ) + d(t1 , · · · , tr )t3r ,
where a, b, c and d are power series and a is non-unit. By the change
of variable ur − c(u1 , · · · , ur−1 ) to ur , we may assume that c = 0. By
changing ur /(1 + a(u1 , · · · , ur−1 )) to ur , it can be assumed that a = 0.
Thus we have
ϕ(ur ) = t2r + b(t1 , · · · , tr−1 )tr + d(t1 , · · · , tr )t3r , (??)
where b is non-unit, because otherwise, tr ∈ ϕ(m0 )B ⊂ ϕ(mR )B, so
that ϕ(mR )B = mB which contradicts Lemma 3.1. Now consider the
relation ϕ(h) = t2r − ϕ(g)tr , which may be written as
h(t1 , · · · , tr−1 , ϕ(ur )) = t2r − g(t1 , · · · , tr−1 , ϕ(ur ))tr .
Substituting from (??), since b is non-unit, the coefficient of t2r on the
right hand side of the above equality is a unit. Since b2 will also be
a non-unit, in order to get a unit coefficient for t2r on the left hand
side of the above equality, the coefficient of ur in the linear part of
h(u1 , · · · , ur ) must be nonzero. Consequently, we may change h to ur
to arrive to
ϕ(ur ) = t2r + ϕ(g0 )tr = t2r + g0 (t1 , · · · , tr−1 , ϕ(ur ))tr .
10 Haghighi, Roberts, and Zaare-Nahandi

This settles ii0 ). If char(k) 6= 2, then by ”completing the square”

above, i.e., by changing tr − ϕ(g0 )/2 into tr , ii) follows. Now since
B is generated by 1 and tr as an A0 -module, we get
ϕ(ur+i ) = ϕ(fi (u1 , · · · , ur )) + ϕ(gi (u1 , · · · , ur ))tr ; i = 1, · · · , m − r,
for non-unit power series fi and gi in r variables. Indeed, since comple-
tion is an exact functor, by ([21], Lemma 2.7), B is free A0 -module, so
that fi ’s and gi ’s are unique. Replacing ur+i − fi (u1 , · · · , ur ) by ur+i
for i = 1, · · · , m − r , we may assume that
ϕ(ur+i ) = ϕ(gi (u1 , · · · , ur ))tr = gi (t1 , · · · , tr−1 , ϕ(ur ))tr ,
concluding iii). 2

Let A = ϕ(R). For simplicity, we identify A0 with its image ϕ(A0 ) ⊂

B. Thus ui = ti for i = 1, · · · , r − 1, and ur = t2r if char(k) 6= 2 (resp.
ur ≡ t2r mod (t1 , · · · , tr−1 , t3r ) in general). Hence A0 is identified with
the subring k[[t1 , · · · , tr−1 , ϕ(ur )]] ⊂ B. Observe that regardless of the
characteristic, we have B = A0 ⊕ A0 tr . Let C be the conductor of B in
A = ϕ(R). It is the annihilator of B/A as an A-module which is also
an ideal in B. Since gi (t1 , · · · , tr−1 , ϕ(ur )) and gi (t1 , · · · , tr−1 , ϕ(ur ))tr
belong to A = ϕ(R), we see that gi ∈ C.
PROPOSITION 3.3. Let the hypotheses and notation be as in Propo-
sition 3.2. Then, the conductor C as an ideal in B is generated by
g1 , · · · , gm−r and as an ideal in R is generated by g1 , · · · , gm−r , g1 tr , · · · ,
gm−r tr .

Proof. Let f ∈ C and let b ∈ B, then

f = f0 (t1 , · · · , tr−1 , ur ) + f1 (t1 , · · · , tr−1 , ur )tr ,
b = p(t1 , · · · , tr−1 , ur ) + q(t1 , · · · , tr−1 , ur )tr ,
for some power series f0 , f1 , p and q in r variables. For every b ∈ B we
have f b ∈ ϕ(R). In particular, for every p, q ∈ A0 we have:
f p = f0 p + f1 ptr ∈ A, f qtr = f1 qt2r + f0 qtr ∈ A . (? ? ?)
Observe that ϕ(R) = A0 ⊕ Γtr where Γ = (g1 , · · · , gm−r )A0 . As an
element of the direct sum A0 ⊕ Γtr , both f p and f qtr have unique
representations. Hence by (? ? ?), f1 p ∈ Γ for every p ∈ A0 . Thus
f1 ∈ Γ. Now if char(k) 6= 2, (? ? ?) implies that f0 qtr ∈ Γtr and hence
f0 q ∈ Γ for every q ∈ A0 . It follows that f0 ∈ Γ. If char(k) = 2, we
substitute from Prop. 3.2 ii0 ) into (? ? ?) to obtain:
f qtr = f1 qϕ(ur ) + (f0 + g0 f1 )qtr ,
 Some Properties of Finite Morphisms on Double Points 11

so that f0 + g0 f1 ∈ Γ. Since f1 ∈ Γ, it follows again that f0 ∈ Γ.

Therefore C = (g1 , · · · , gm−r )B. Since B = A + Atr , we get

C = (g1 , · · · , gm−r , g1 tr , · · · , gm−r tr )A

as an ideal in A. 2

COROLLARY 3.4. With the notation as in Prop. 3.3 and its proof:
i) C = Γ ⊕ Γtr ,
ii) B/C ∼= A0 /Γ ⊕ (A0 /Γ)tr ,
iii) A/C ∼= A0 /Γ, and
∼
iv) B/A = (A0 /Γ)tr .
In particular, B/A is a free A/C-module of rank 1.

Proof. i) Follows from Prop. 3.3 and the fact B is a free A0 -module
generated by 1, tr .
ii) Follows from this latter fact and i).
iii) This is a direct consequence of Prop. 3.3.
iv) This follows from ii) and iii).
The statement about B/A follows from iii). 2

COROLLARY 3.5. With the assumption as in Prop. 3.3, if further-

more Sing(X 0 ) is of local dimension 2r − m at y, then ∆ is a complete
intersection at x.

Proof. Observe that D and ∆ have the same dimensions as Sing(X 0 ).

Using the same notation as in Prop. 3.3, ∆ is locally given by the ideal
C = (g1 , · · · , gm−r ) ⊂ B. By assumption, C has height r − (2r − m) =
m−r. Thus C is unmixed, and since B is Cohen-Macaulay, g1 , · · · , gm−r
form a regular sequence. Consequently B/C is a complete intersection.
2

REMARK 3.6. By the conductor, one usually means (ϕ(R) : B).

Here the inverse image of the this ideal in R will also be called the
conductor.

LEMMA 3.7. Let ϕ : T → S be any surjective homomorphism of

commutative rings. Let I = (s1 , · · · , sm ) be an ideal in S. Assume that
ϕ(fi ) = si ; i = 1, · · · , m, for some elements fi ∈ T . Then

ϕ−1 (I) = kerϕ + (f1 , · · · , fm ).

In particular if ker ϕ ⊂ (f1 , · · · , fm ), then ϕ−1 (I) = (f1 , · · · , fm ).

12 Haghighi, Roberts, and Zaare-Nahandi

∼
=
Proof. Let ϕ̄ : T /kerϕ → S be the isomorphism induced by ϕ. Thus
ϕ̄−1 (I) = (f¯1 , · · · , f¯m ), and hence ϕ−1 (I) = (f1 , · · · , fm ) + kerϕ. 2

LEMMA 3.8. Let ϕ : R → B be the homomorphism defined in Prop.

3.2, then ker ϕ ⊂ (ur+1 , · · · , um , g1 , · · · , gm−r ).

Proof. Let ϕ̃ : R[tr ] → B be the homomorphism extending ϕ by the

identity on tr . Then

kerϕ̃ = (ur+i − gi (u1 , · · · , ur )tr ; i = 1, · · · , m − r, ur − ϕ(ur )),

and

kerϕ = kerϕ̃∩R ⊂ (ur+i , gi (u1 , · · · , ur ); i = 1, · · · , m−r, ur −ϕ(ur ))∩R

= (ur+i , gi (u1 , · · · , ur ); i = 1, · · · , m − r). 2

Although the following is a consequence of Cor. 3.4 iii), in order

to prove it, we prefer to use the generating set of the conductor as an
ideal in R.

COROLLARY 3.9. With the notation as above,

ϕ−1 (C) = (ur+i , gi ; i = 1, · · · , m − r).

In particular A/C is a complete intersection. Therefore D is a complete

intersection at analytically irreducible double points of X 0 = f (X).

Proof. The first claim follows from Prop. 3.3, Lemmas 3.7 and 3.8.
Since g1 , · · · , gm−r form a regular sequence in k[[u1 , · · · , ur ]], it is clear
that g1 , · · · gm−r , ur+1 , · · · , um will also form a regular sequence in R.
Therefore R/ϕ−1 (C) ∼ = A/C is a complete intersection. 2

LEMMA 3.10. Let t̄r be the class of tr in B/A. Then the conductor as
an ideal in A, is equal to AnnA (t̄r ).

Proof. Since B is generated by 1, tr , as an A-module, B/A is gener-

ated by t̄r as an A-module. Thus by definition,

C = AnnA (B/A) = AnnA (t̄r ). 2

REMARK 3.11. In Cor. 3.9, we have seen that A/C is a complete

intersection, hence it is Cohen-Macaulay. A similar result is not true
for analytically irreducible triple points, even with the hypothesis that
 Some Properties of Finite Morphisms on Double Points 13

f is an strongly generic projection. For example, it is shown in ([21],

Theo. 4.12) that
depthA/C = 3r − 2m + 1 ,
at an analytically irreducible triple point where:

r = 2n + 2 , m = 3n + 2 ,

so that the above number is 3. Since

dimA/C = 2r − m,

we see that A/C is Cohen-Macaulay if and only if

3r − 2m + 1 = 2r − m,

i.e., if and only if m = r+1. Consequently, A/C is not Cohen-Macaulay

at y if m > r + 1. For B/C there is a similar situation.

PROPOSITION 3.12. Let f : X → Y be a finite morphism as specified

in section 1. Let y ∈ X 0 = f (X) be a point of multiplicity 2. Assume
that Sing(X 0 ) is of local dimension 2r − m at y. Then

depth OX 0 ,y = 2r − m + 1.

Proof. If m = r + 1, then X 0 is a hypersurface and depth OX 0 ,y = r.

Thus we work out the case m > r+1. First assume that y is analytically
irreducible. Let A = OX 0 ,y , B = OX,x . Thus B is the integral closure of
A. Let C be the conductor of B in A. By Cor. 3.4, B/A is a free A/C-
module, thus it is a Cohen-Macaulay A/C-module. Therefore B/A is a
Cohen-Macaulay A-module, so that

depthA (B/A) = dim(B/A) = 2r − m.

Now consider the exact sequence of A-modules

0 → A → B → B/A → 0.

Since B is Cohen-Macaulay and finite over A,

depthA B = depth B = r.

By assumption A has R1 property (regular in codimension 1), thus A

is not Cohen-Macaulay. Because otherwise A will be integrally closed
which is a contradiction. Therefore

depth A < depthA B.

14 Haghighi, Roberts, and Zaare-Nahandi

By the behavior of depth on exact sequences (see [2], Lemma 1.4),

depthA (B/A) = depth A − 1,

and hence the assertion follows.

Now assume that y is a double point at which X 0 has two simple

branches. Using the same notation as in Lemma 2.1, let A = R/P1 ∩P2 .
Then B = R/P1 × R/P2 is the integral closure of A in its total ring of
quotients. If C is the conductor, by the proof of Lemma 2.1,

A/C ∼
= R/(P1 + P2 ),

and,
B/C ∼
= R/(P1 + P2 ) × R/(P1 + P2 ).
Consequently, the exact sequence of A/C-modules

0 → A/C → B/C → B/A → 0,

splits. Hence B/A is a projective A/C-module. While A/C is local,

B/A is a free A/C-module. Now as the previous case, using the depth
relation for a similar exact sequence, the assertion follows. 2

Let X ⊂ PN be a projective smooth variety with no trisecant lines.

The projection of X from any point outside X into PN −1 has no triple
point. The study of these varieties is a classical problem in algebraic
geometry. The case of space curves goes back to G. Castelnuovo. The
general problem is still far from being settled. The surfaces in P4 and
P5 with no trisecant lines are characterized and the list of surfaces
in P6 with this property has been conjectured by S. Di Rocco and
K. Ranestad (see [28]). However, it is known that if π : X → Pm is
a generic projection, for 3r
2 ≤ m ≤ 2r, π has no triple point ([18], sec.
12). For finite morphisms with a similar property we have the following.

THEOREM 3.13. Let f : X → Y be a finite morphism specified as in

section 1. Assume that f has no triple point. If the singular locus of
X 0 = f (X) is of dimension 2r − m, then D and ∆ are locally complete
intersections.

Proof. This follows by Lemma 2.3, Cor. 3.5 and Cor. 3.9. 2
 Some Properties of Finite Morphisms on Double Points 15

4. Weak Normality

In this section we will keep the notations used in the previous sections.
We will assume that char(k) 6= 2. We will check seminormality and
weak normality of the varieties and schemes introduced in the earlier
sections at the double points. In particular, we will prove that when the
conductor, as a subscheme of X, is reduced, then X 0 = f (X) is WN at
double points. This implies the global result that when ∆ is reduced,
X 0 is WN provided that it has no triple point. Assuming that Sing(X 0 )
has the expected dimension, this result is strengthened.

LEMMA 4.1. Let y be a double point of X 0 at which X 0 has two simple

branches. If OX 0 ,y is SN, then it is also WN.

Proof. By assumption O bX 0 ,y ∼
= R/P1 ∩ P2 , where R/P1 and R/P2
are regular. Since OX 0 ,y is SN, OX 0 ,y is SN. To see OX 0 ,y is WN, it is
b
sufficient to show that ObX 0 ,y is WN ([11], II,3). Let A = R/P1 ∩P2 , then
the integral closure of A is B = R/P1 ×R/P2 ([4], Ch. V, Prop. 9). So by
Prop. 1.2, we need to check that for b ∈ B, if pb, bp ∈ A, for some prime
integer p then b ∈ A. Let b = (α, β) ∈ B, then pb = (pα, pβ) = (γ, γ) for
some γ ∈ R. If p 6= chark, since k ⊂ O bX 0 ,y , 1 γ ∈ A which is mapped to
p
b = ( p1 γ, p1 γ). If p = char k , and bp ∈ A, then (αp , β p ) = (δ, δ) for some
δ ∈ R, thus αp − δ ∈ P1 , β p − δ ∈ P2 and hence αp − β p = (α − β)p is in
P1 +P2 . Since A is SN, P1 +P2 is radical in A, therefore α−β ∈ P1 +P2 .
Let α − β = g + h with g ∈ P1 , h ∈ P2 . Thus the class of α − g = β + h
in A maps to (α, β), i.e., b ∈ A. 2

LEMMA 4.2. Let y ∈ X 0 , be as in Lemma 4.1. Furthermore assume

that D is reduced at y. Then OX 0 ,y is SN.

Proof. Using the same notation as above, O bX 0 ,y ∼

= R/P1 ∩ P2 and
its integral closure is R/P1 × R/P2 . By Cor. 1.4, D is analytically
unramified at y. This means that the conductor of R/P1 ∩P2 in R/P1 ×
R/P2 is a radical ideal in R/P1 ∩ P2 , because the completion of the
conductor of two rings is the conductor of the completion of these rings
([27], Vol. II, Ch. VIII, Cor. 8 to Theo. 11). This implies seminormality
of ObX 0 ,y and hence that of OX 0 ,y by ([16],Theo. 2.3(d)). 2

COROLLARY 4.3. Under the assumptions as in Lemma 4.2, OX 0 ,y is

WN.

Proof. This follows by Lemmas 4.1 and 4.2. 2

16 Haghighi, Roberts, and Zaare-Nahandi

PROPOSITION 4.4. Let y = f (x) be an analytically irreducible double

point of X 0 , and let B = OX,x , A = OX 0 ,y , C = (A : B). Assume that
∆ is reduced and char k 6= 2. Then A/C is SN in B/C, and OX 0 ,y is SN.

Proof. As indicated in the proof of Lemma 4.2, C b = (A : B) ⊗

A = (A : B). Since seminormality descends under completion, we may
b b b
assume that B = O bX,x , A = ObX 0 ,y and C = (A : B). Recall that by
Cor. 3.4 iii), B/C = A/C ⊕ (A/C)tr . Let b = α + βtr ∈ B/C with
α, β ∈ A/C and let b2 , b3 ∈ A/C. Then b2 = (α2 + β 2 t2r ) + (2αβ)tr ∈
A/C. Since the sum A/C ⊕(A/C)tr is direct, 2αβtr = 0, and since char
k 6= 2, αβtr = 0. On the other hand, b3 = (α3 +3αβ 2 t2r )+3α2 βtr +β 3 t3r ∈
A/C. Since α2 βtr = α(αβtr ) = 0 again, β 3 t3r ∈ (A/C) ∩ (A/C)tr = 0,
thus β 3 t3r = 0 in B/C. Since B/C is reduced, βtr = 0 , and hence
b = α ∈ A/C. Therefore A/C is SN in B/C. Now by ([6], Lemma 2.5
(VI)), A is SN in B. In other words, OX 0 ,y is SN. 2

PROPOSITION 4.5. With the assumption of Prop. 4.4, OX 0 ,y is WN.

Proof. By Prop. 4.4, OX 0 ,y is SN. By Prop. 1.2, we need to verify that

for all b ∈ B, if pb, bp ∈ A, for some prime p, then b ∈ A. In our case B
is generated by 1, tr and by Lemma 3.10, C is the annihilator of tr in
B/A as an ideal of A. Assume that b = a+a0 tr ∈ B, pb = pa+pa0 tr ∈ A,
so pa0 tr ∈ A and hence pa0 ∈ C. Now assume that bp = (a + a0 tr )p =
ap + pa0 c + (a0 tr )p ∈ A, where c ∈ B. Then pa0 c ∈ A and (a0 tr )p ∈ A.
0 0
For p > 2, p − 1 is even and a p tpr ∈ A, hence a p tp−1
r ∈ C, i.e., a0 tr is
nilpotent in B/C. Since a tr ∈ B and B/C is reduced, a0 tr ∈ C ⊂ A.
0

Therefore b = a + a0 tr ∈ A. If p = 2, then 2a0 ∈ C, and since char

k 6= 2, a0 ∈ C, thus a0 tr ∈ C ⊂ A, and hence b ∈ A. 2

COROLLARY 4.6. Under the assumptions of Prop. 4.4, A/C is WN

in B/C.

Proof. (see [24], Prop. 3). 2

The following result is a partial generalization of ([20], Prop. 4.1).

PROPOSITION 4.7. Under the assumptions of Prop. 4.4, if X 0 is SN

at y and ∆ is SN at x, then D is SN at y.

Proof. Let A = OX 0 ,y , B = OX,x , C = (A : B). Since A is SN in

B ([6], Prop. 2.5(VI)), A/C is SN in B/C. By assumption, B/C is SN
in (B/C)¯. Thus by transitivity of seminormality, A/C is SN in (B/C)¯.
 Some Properties of Finite Morphisms on Double Points 17

Since (A/C)¯ ⊂ (B/C)¯ (see [20], the proof of Prop. 4.1), by ([6], Prop.
1.5(b)), A/C is SN. 2

In the next two results we will consider a finite morphism f : X → Y

as specified in sec. 1. We will assume that f has no triple point.

The following generalizes ([1], Theo. 2.7), ([10], Prop. 3.5) and ([19],
Prop. 4.4).
THEOREM 4.8. With the notation as above, if ∆ is a reduced scheme,
then X 0 = f (X) is WN.

Proof. This follows from Cor. 4.3 and Prop. 4.5. 2

In the following, by a point, we mean a scheme-theoretic point, i.e.,

it is not necessarily a closed point.

Theorem 4.8 can be strengthened in the following sense.

THEOREM 4.9. With the notation as above, let r + 1 ≤ m ≤ 2r − 1.
Assume that :
i) Sing(X 0 ) has dimension 2r − m,
ii) X 0 has no triple point,
iii) On each irreducible component of Sing(X 0 ) there is at least one
point where X 0 has two simple branches, on which D is reduced.
Then X 0 is WN.

Proof. We may restrict the problem to the case when X = SpecB,

X 0 = SpecA are affine varieties. Let C be the conductor. Let P ⊂ A be
a prime ideal containing C such that its image is of height h in A/C. It
corresponds to a point ξ ∈ D. We first show that depthOX0 ,ξ = h + 1.
For m = r + 1, OX0 ,ξ ia a Cohen-Macaulay ring, hence,
depth OX 0 ,ξ = dim OX 0 ,ξ = h + 1.
Assume that m > r + 1. As it was seen in the proof of Prop. 3.12, B/A
is a Cohen-Macaulay A-module. Thus (B/A)P is a Cohen-Macaulay
AP -module. Now apply the method of the proof of Prop. 3.12 to the
exact sequence
0 → AP → BP → (B/A)P → 0.
Since m > r + 1, by i), A is regular in codimension one, thus AP is reg-
ular in codimension one. Since C ⊂ P , AP is not normal. Consequently,
AP is not Cohen-Macaulay. Thus as AP -modules,
depth(AP ) < depth(BP ).
18 Haghighi, Roberts, and Zaare-Nahandi

Hence, by the behavior of depth on exact sequences,

depth OX 0 ,ξ = depth(B/A)P + 1 = h + 1.

If y is the double point of X 0 on which D is reduced, then by Lemmas

4.2 and 4.3, OX 0 ,ξ is WN. Thus for h = 0, OX 0 ,ξ is WN. For h > 1,
depthOX 0 ,ξ ≥ 2. Thus by ([25], Prop. 2.10), X 0 is SN. Therefore, by
Lemma 4.1 and Prop. 4.5, X 0 is WN. 2

For a local ring (A, m), let Spex(A) = Spec(A) \ {m}. A result of M.
Vitulli together with Theo. 4.9, yield the following necessary and suf-
ficient version of Hartshorne’s depth-connectivity result (see [7], Prop.
2.1).

PROPOSITION 4.10. With the assumption of Theo. 4.9, let ξ be a

singular point of X 0 . Then Spex(O
bX0 ,ξ ) is connected if and only if ξ is
0
not a generic point of Sing(X ).

Proof. By Theo. 4.9, the complete local ring O bX 0 ,ξ is SN. Let ξ be

0
a singular point of X . By assumption i) of Theo. 4.9, dim(O bX 0 ,ξ ) ≥ 2.
By the proof of Theo. 4.9, depth (O bX 0 ,ξ ) ≥ 2 if and only if ξ is not a
0
generic point of Sing(X ). Therefore by ([23], Cor. 3.4), to prove the
claim, it is sufficient to show that O bX 0 ,ξ has rational normalization, i.e.,
its residue field is equal to the residue field of its normalization. But this
is immediate since O bX 0 ,ξ has finite normalization and k is algebraically
closed. 2

5. The case of affine spaces

In this section we consider ”the affine model” of the analytically irre-

ducible double points, studied in section 3. For simplicity, we will as-
sume that char(k) 6= 2. More precisely, assume that R = k[u1 , · · · , um ],
B = k[t1 , · · · , tr ] and let g1 (t1 , · · · , tr−1 , t2r ), · · · , gm−r (t1 , · · · , tr−1 , t2r ) be
polynomials in B which form a regular sequence. Let ϕ : R → B be
defined similar to Prop. 3.2, namely,
ϕ(ui ) = ti ; i = 1, · · · , r − 1,
ϕ(ur ) = t2r ,
ϕ(ur+i ) = gi tr ; i = 1, · · · , m − r.
Let X = Ark , Y = Am k and let f : X → Y be the morphism correspond-
ing to ϕ. It follows that f is a finite morphism which is birational onto
X 0 = f (X) and X 0 has a double point at the origin which is not a triple
 Some Properties of Finite Morphisms on Double Points 19

point.

We give a finite presentation for B as an R-module. This in particu-

lar gives the Fitting ideals of B as an R-module. The Fitting ideals are
not usually radical (see [13], discussion after Prop. 1.5). It turns out
that in this situation, the 0-th Fitting ideal is a prime ideal and indeed
it is the defining ideal of X 0 . While in general, the first Fitting ideal is
contained in the conductor of B in A = ϕ(R) (see [13], Theo. 3.4), in
this case equality holds.
PROPOSITION 5.1. The sequence of R-moules
R2(m−r) → R2 → B → 0,
where the first map is defined by the matrix
 
ur+1 g1
 ur g1 ur+1 
 
 ur+2 g2 
 
 ur g2 ur+2
M = ,

 .. .. 

 . . 

 um gm−r 
ur gm−r um
and the second map is defined as
(a, a0 ) 7→ a.1 + a0 .tr ,
is exact, i.e., it gives a finite presentation for B as an R-module.

Proof. As it was seen in the proof of Prop. 3.2, B is generated by 1

and tr as an R-module. So we only need to show that if a.1 + a0 tr = 0,
then (a, a0 ) is generated by the rows of the matrix M . We subtract
certain multiples of the rows of M from (a, a0 ) to arrive to zero. Let
s = Max{degree of ur+1 in a , degree of ur+1 in a0 }.
Multiplying (ur+1 , g1 ) by an appropriate factor and subtracting it from
(a, a0 ), we can reduce the degree of ur+1 in a by at least 1, and using
(ur g1 , ur+1 ) in a similar manner, we can reduce the degree of ur+1 in
a0 by at least 1. Thus, we can reduce s by at least one. Repeating
this process, we arrive to the case where a, a0 are independent of ur+1 .
The same argument applies to ur+2 , using the third and fourth rows
of M . Continuing this for remaining rows, we arrive to an element
(a, a0 ) where a and a0 are independent of ur+1 , · · · , um . Since B is a
free module over k[u1 , · · · , ur ], a.1 + a0 tr = 0 implies that a = a0 = 0.
2
20 Haghighi, Roberts, and Zaare-Nahandi

THEOREM 5.2. Let F0 be the 0-th Fitting ideal of B as an R-module.

The defining ideal of X 0 is equal to F0 . In other words, the defining
ideal of X 0 is generated by the maximal minors of M .

Proof. It is known that X 0 = V (F0 ). On the other hand, ker ϕ is

the defining ideal of the closure of X 0 = f (X) which is closed as f is
a finite morphism. Therefore rad(F0 ) = ker ϕ. In particular, F0 ⊂ ker
ϕ. Let g belong to ker ϕ. Then (g, 0) is a syzygy of (1, tr ). By Prop.
5.1, the module of syzygies of (1, tr ) is generated by the rows of M .
Therefore there exist a1 , · · · , a2(m−r) in R such that

g = a1 ur+1 + a2 ur g1 + · · · + a2(m−r) ur gm−r ,

and,
0 = a1 g1 + a2 ur+1 + · · · + a2(m−r) um .
Since g1 , · · · , gm−r form a regular sequence, and they are polynomials in
u1 , · · · , ur , the sequence g1 , ur+1 , g2 , ur+2 , · · · , gm−r , um is also regular,
i.e., the second column of M form a regular sequence. By a well known
result in commutative algebra, the module of syzygies of a regular
sequence is generated by the so called ”trivial syzygies”. These are,
by definition, syzygies of the form

(ur+1 , −g1 , 0, · · · , 0), (g2 , 0, −g1 , 0, · · · , 0), · · · , (0, · · · , 0, um , −gm−r ) .

Taking (a1 , · · · , a2(m−r) ) to be equal to the first syzygy, we see that

g = (ur+1 )2 − ur g12 ,

which is a maximal minor of M . Similarly, the other trivial syzygies

give rise to maximal minors of M . Thus g is generated by the maximal
minors of M , i.e., g ∈ F0 (M ). 2

COROLLARY 5.3. With the notation as above, the 1-st Fitting ideal
of B as an R-module is equal to the conductor of B in A (as an ideal
in R, see Remark 3.6).

Proof. Similar to the proof of Cor. 3.9, the conductor is generated

by ur+1 , · · · , um , g1 , · · · , gm−r as an ideal in R. By Prop. 5.1, F1 is
generated with the entries of M which gives the same generators as
above. 2

Acknowledgement. All of the authors are grateful for the opportu-

nity of this cooperation, and they would like to thank everyone who
has helped them to overcome impediments to the completion of this
 Some Properties of Finite Morphisms on Double Points 21

project. The third author would like to thank IMPA, and the Third
World Academy of Science for their financial support during his visit
of IMPA in July-August 1995. He also likes to thank B. Teissier for
some discussions on weak normality.

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