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1. [ 43 ]
ON THE NOTION OF A FIRST NEIGHBOURHOOD RING
WITH AN APPLICATION TO THE AF+B® THEOREM
BY D. G. NORTHCOTT
Received 5 April 1956
Introduction. In an earlier paper (3) the author developed a theory of the neighbour-
hoods of a local domain, in which the concept of the first neighbourhood ring played
a central role. This notion has since proved useful in connexion with certain one-
dimensionalproblems, but it has emerged, in the process, that a considerable advantage
would be gained if the theory could be freed from the assumption that the basic ring
was to be without zero-divisors. This parallels the situation in the geometry of plane
curves, where it is desirable, so far as is possible, that results and methods should apply
with equal facility to reducible as well as to irreducible curves. Accordingly Part I of
the present paper is devoted to a fresh account of the first neighbourhood ring from
a more general standpoint than that used previously. Besides the greater generality
thus obtained, the theory is extended by the addition ofsome new results. Furthermore,
the proof of one of the main results of the original paper ((3), Theorem 10) has been
considerably simplified. Of course, the ideas of (3) reappear here in a modified form,
but, to spare the reader the irritating fragmentation of the subject which would other-
wise be necessary, the revised account has been made independent of the earlier one.
To this extent, Part I is self-contained.
Part II, on the other hand, represents an addition to the abstract theory of infinitely
near points presented by the author in (5), the results of this paper being quoted
without proof. Here the main result is the analogue of what is known to geometers as
the general form of the Af+B<p theorem.* It is included here because the method of
proof provides a particularly good illustration of how the results of Part I may be
applied.
PART I. THE FIRST NEIGHBOURHOOD RING
1. Notation used in Part I. Throughout the whole of Part I, Q will denote a one-
dimensional local ring with maximal ideal nt and residue field K = Qjm. It will not be
assumed that Q and K have the same characteristic, nor that K contains infinitely many
elements. It will, however, be assumed that not every element of m is a zero-divisor or
(equivalently) that all the prime ideals belonging to the zero ideal are of dimension unity.
One consequence of this assumption is that an ideal a 4= (1) is m-primary if and only if
it contains at least one element which is not a zero-divisor.
Let (uvu2, ...,ud) be a fixed but otherwise arbitrary set of generators for the
maximal ideal m, so that in particular there may be elements which are superfluous
among the ut. Let n denote the form ideal of the zero ideal of Q constructed with
respect to this base; then n is the homogeneous ideal in K[XX, X2,..., Xd] generated by
the leading forms of the zero element and its afiine dimension is unity.
* See, for example, Severi(8), Ch. 8, §111.

2. 44 D. G. NORTHCOTT
In future, when we say that <f>(XltX2, •••,Xd) is a form then it is to be understood
that this means a form in the d variables Xv X2, ...,Xd with coefficients in Q. If the
number of variables is not indicated explicitly, then it is to be taken that the number
intended is d. Further, when <j> is a form then <f> will be used to denote the same form
read modulo rrt. Thus to give one example of the use of these conventions, a form 0 of
degree s will satisfy <f> e n if and only if <j>(u) = 0 (mod m8+1
)
It will be convenient to denote by n0 the isolated component of n determined by its
one-dimensional prime ideals. There exists therefore a smallest integer c^ 0 with the
property that _ .„ „ „ .„ __ _ ., ,,
J
nor,(Z1,Z2,...,Zd)c
sncno. (1-1)
Finally, we use e to denote the multiplicity of the local ring Q, this being meant in the
sense of Samuel ((7), Ch. 2, § 2). Since n and n0 are of affine dimension unity and have
the same one-dimensional components, their Hilbert functions x(r
> n)an(
i X (r
> So) are
>
for large values ofr, both constant and equal. But we can say more, for, by Theorem 7
°f ( 2 ) >
X(r,no) = X(r,n) = e (r large). (1-2)
2. Superficial elements. The starting point of the theory of the first neighbourhood
ring is
Theorem 1. Let a em8
(s ^ 0) and suppose that a = <fr(u), where <j> is aform of degree s.
Then the following three assertions are equivalent:
(i) no:(0) =n0;
(ii) am" = m"+s
for all large v*
(iii) nV+s
: (a) = m" for all large v.f
Remark. In the course of the proof, we shall establish ratner more than is stated in
the theorem. We shall in fact show that if no:(0) = n0 then am'"8
= m" for all
v > max (se, c), where c is the integer which occurs in (1-1). This fact will be used later to
establish Lemma 4 which has particular significance for Part II.
Proof. It will be shown first that (i) and (ii) are equivalent and then that (i) and (iii)
are equivalent.
Assume that (i) is true. Then (n0, <f>) is aprimaryidealbelonging to (Xls X2,..., Xd) and
T
' "o ]
~ X[r
~ S
' "o ] {S
**r)>
consequently, by (1*2), if I is large enough
2 X?, (n0, ?)] = se.
r=0
But, for large values of I,
2 XX> ("o. ?)] = iength (n0,3);
r=0
accordingly length (n0, <f>) = se and therefore
* In the terminology of Northcott and Rees(6), (ii) asserts that (a) is a reduction of m of
type s. Some of the arguments of (6) are reproduced here.
f The equivalence of (i) and (iii) is a development of a result of Samuel(7).

3. Notion of a first neighbourhood ring 45
Let v be any integer which satisfies v > max (se, c) and let p{X) be a power-product of
the Xt of degree v then we can find a form i/r of degree v — s such that
p(X)^(X)f(X) (modn0).
But v^c; hence, by (1-1), p{X) = lj>{X)f{X) (modit).
which shows that p(u) = afr(u) (mod m"*1
) so that p(u) e am"~s
+ m"+1
. But
was any power-product of degree v, consequently m" £ am"~s
+ mv+1
and therefore
m" £ am'"8
. But the opposite inclusion is obvious, so we have
m" = a m " (v > max (se, c))
and, in particular, this establishes that (i) implies (ii).
Assume that (ii) is true and choose v so that am" = m"+s
. If p(X) is any power-product
of degree v + s then p(u) = <j>(u)ft(u),where ^ is a form of degree v, consequently
and therefore p(X) e (n0, (f>). It follows from this that (n0, <j>) is a primary ideal belonging
to (XVX2, •••,Xd) and hence that we must have no:(0) = n0. The equivalence of (i)
and (ii) has therefore been established.
Once again assume that (i) is true. By what has already been proved, (a) 2 am" = ttt"+8
for large values of v. This shows that (a) is an m-primary ideal, consequently a is not
a zero-divisor (see §1). It is therefore possible to choose an integer I such that
m": (a) £ mc
for v ^ 1. Suppose that v ^ I. It will be shown that m"+s
: (a) = m". Clearly
this will follow if we show that m"+s
:(a)£m". Let bemv+s
:(a) and suppose that
b i m". Let r < v be the integer such that b e mr
, b £ mr+1
and write b = i/r(u) where ijr is
a form of degree r. Then r^c and
<j)(u) f(u) = abe mv+s
£ mr+s+1
,
consequently ^(X)^r
(X)ett£n0 and therefore i/r(X)en0. But the degree r of ^ is at
least equal to c, hence tjr{X) e n which shows that b = i/r(u) = 0 (mod mr+1
). But this is
a contradiction and so it has been established that (i) implies (iii).
Finally, assume that (iii) is true so that m"+s
: (a) = m" for (say) v ^ v0, and let ^r be any
form such that ijreno:(<f>). Put h = max(c, v0) and suppose that p(X) is a power-
product of degree h. Then
and sop(u)ai/r(u)em8+t+h+1
, where t is the degree of xjr. But t + h+l^v0, consequently
p{u) ijr(u) e rnt+h+1
and therefore p(X) ifr{X) e n £ n0. Nowp(X) was any power-product
of degree h, accordingly ^ ^ ^XJl f{X) g ^
and therefore i/r(X)en0. It follows that no:(^) = n0 and so it has been shown that
(iii) implies (i). The proof of Theorem 1 is thus complete.
Definition. An element b will be called a ' superficial element of degree s' (s > 0) if b e nv*
and m"+s
: (b) = m"for all large values of v.
Remarks. By Theorem 1, if b e m" then b will be a superficial element of degree s if
and only if 6m" = m"+s
for all largevalues of v. So far as the present paper is concerned,

4. 46 D. G. NORTHCOTT
this is a more important property than that which occurs in the definition, but the
definition is none the less appropriate because it is the one which lends itself to an
extension to local rings of higher dimension (see Samuel (7), Ch. 2, § 1). The reader
should note the following facts.
(a) The superficial elements of degree zero are the units of Q.
(b) Ifb and c are superficial elements ofdegrees s and t respectively then be is a superficial
element of degree s + t.
If now we denote by Q the set of integers which occur as the degrees of superficial
elements, then, by (a) and (6), Oe Q, and Q is closed under addition. We shall call Q the
semi-group of superficial degrees. Note that when the residue field of Q is infinite,
Q. consists of all the non-negative integers for it is then possible to find a linear form
0 for which n0: (<fi) = n0. This means, of course, that 1 e Q.. So far as the general case is
concerned, we have
THEOREM 2. The semi-group Q. of superficial degrees contains all the non-negative
integers with at most a finite number of exceptions.
Proof. Let p1,p2, •••>pr ^e
*n e
prime ideals belonging to n0. For each i, l^i^r,
choose a form 0f so that ^)iep1... pi-ip^ ... pr, <j>itpi, and also choose a coordinate
variable, X^) say, so that Xa<i)£p.i
Let ht denote the degree of (j>i and let n be any
integer satisfying n>max (hx,h2,...,hr). Then <j> = 2 -^"(T)7
"'Pi is a
f°r m
of degree n
i=l _
which is not in any of plt p2,..., pr and which therefore satisfies n0: {(j>) = n0. Theorem 1
now shows that neQ. and this completes the proof.
3. The first neighbourhood ring. If b is a superficial element of degrees, then, if s = 0,
b is a unit, while if s > 0 the relation 6m" = m"+s
(v large) shows that (6) is an m-primary
ideal. Thus, in either case, b is not a zero-divisor. We use this fact to make the
DEFINITION. The set of all quotients ab, where a em8
, b em8
and b is a superficial
element of degree s (s > 0 being a variable integer) forms an extension ring of Q. This ring
will be called the 'first neighbourhood ring' of Q and it will be denoted by 3ft.
Remarks. The fact that 3d is a ring follows from the general observations already
made concerning superficial elements. 3? is, of course, a subring of the full ring of
quotients of Q and, by Theorem 1, it consists of all quotients <fi(u)li/r(u), where <j> and ijr
are forms of the same degree and n0: (i/r) = n0.
THEOREM 3. Let v belong to m8
. Then vis a superficial element of degree s if and only if
Proof. If v is a superficial element of degree s then m8
S 3fh> (by the definition of 9?)
and «e3ftm8
; consequently "Siv — SRm8
. Now assume that fftv = 3ftms
. Let p(X) be a
power-product of Xlt X2,...,Xd of degree s, and write v = ft(u) where rjr is a form of
degree s. Since p(u) e SRm",
where <f>0 and i/r0 are forms of the same degree and n0: (ijr0) = n0. Let <j> be an arbitrary
form such that (p&no:(tjr). Then ^ e n 0 and y ( I ) f o - ^ o f € n £ n o consequently
p(X) ty$ e n0 and therefore^(X) ^ e UQ. It follows that (Xv X2,..., Xd)3
$ £ n0 and hence

5. Notion of a first neighbourhood ring 47
that 0 e n0. This shows that n0: (ijr) = n0 and now we see, from Theorem 1, that v is
a superficial element of degree s.
THEOBEM 4. The first neighbourhood ring 5ft is a finite Q-module.
Proof. We first show that 5R is an integral extension of Q. Let z e 5ft then x = b/c
where, for a suitable integer s, b e ms
and c is a superficial element of degree s. By (ii)
of Theorem 1, we can choose v so that cm" = ms+
" hence, if m" = (7i,y2> •••>7<)>
t
byi e ms+
" = cm" and therefore byi = 2 CQayp where q^ e Q. It follows that if Sii is the
usual Kronecker symbol and A is the determinant | xSi:j — qi:j |, then Ayf = 0 for
1 < i < t. Thus Am" = 0 and, since m" is not composed exclusively of zero-divisors
(see § 1), this implies that A = 0 and hence that x is integral with respect to Q.
It has now been shown that 5ft is an integral extension of Q. We shall complete the
proof by establishing
LEMMA 1. Let vbea superficial element of degree a and let rrf = {<xv a2,..., at). Then
v v
Proof. It is clear from the definition of 91 that
v v
Let x € 5ft, then x = bjc where, for a suitable integer s, b e tns
, ce ms
and c is a superficial
element of degree s. Then x = {bc"~1
jvs
) (v'jc"), Vs
and c0
' are superficial elements of
degree sex and (6c<r
~"1
/us
) e Q — , — ,...,— . Accordingly if we show that c^jv8
, which
[_v v v J
belongs t o Q — , — ,...,— L i s a unit in this ring it will follow that x e Q — , — , ...,—
and this will complete the proof. But this is clear. For if c^/v8
were not a unit, then it
would be contained in a proper prime ideal of Q —, — ,...,— which itself would be
[" v v
the contraction of a proper prime ideal in the integral extension fft. In other words,
(fjv8
would be a non-unit in 5ft and this is clearly false.
We consider now under what circumstances Q and 5R will coincide. The situation
here is more complicated than in (3), because Q may have zero-divisors and Q/m may
be finite, but in fact we obtain the same result, namely
THEOREM 5. The first neighbourhood ring SR coincides with Q ifandonly if Q is regular.
Two auxiliary results will be needed.
LEMMA 2. Q is regular if and only if the multiplicity eofQ is unity.
Remark. Samuel (7) proves a similar result for a complete equidimensional local ring
of arbitrary dimension provided (i) the ring has the same characteristic as its residue
field, and (ii) the residue field is infinite. His conditions, however, do not apply in the
present instance.

6. 48 D. G. NOBTHCOTT
Proof. We assume that e = 1 and deduce that Q is regular. The converse is obvious.
By (1-2), x(r,n0) = 1 for all large r consequently n0 has only one primary component.
It is therefore possible to choose i so that n0: (X^) = n0 and then, by Theorem 1, there
will exist an integer v such that ^m" = m"+1
. This implies that u^xn" = m"+r
and hence
that mr
2 {ur
i)^mv+r
for all r ^ 0. Accordingly
length mr
< length {u) ^ length m"+r
.
But ut is not a zero-divisor (it generates an m-primary ideal), consequently
length (u) = r length (ut)
and therefore - length mr
^ length (uj < - length m"+r
,
and so, letting r->co, we see that length (wj = 1 or (ut) — m. This proves that Q is
regular.
LEMMA 3. / / some positive power of mis a principal ideal then Q is regular.
Proof. Suppose that ms
= (6), where s > 0; then, for any integer n, msn
is a principal
ideal and therefore msn
lxnsn+1
is a one-dimensional vector space over Qfxn. It follows
that length (m3n+1
) — length (msn
) = 1. But, for all large values of v,
length (m"+1
) - length (m") = e;
hence e = 1 and now Lemma 3 follows from Lemma 2.
Proof of Theorem 5. If Q is regular then it is, in particular, an integrally closed
integral domain*; hence, by Theorem 4, Q and fft must coincide. Assume now that
Q = fR. By Theorem 2, we can find a superficial element v of strictly positive degree s
(say) and then, by Theorem 3,
Qms
= dim8
= dtv = Qv.
Thus rrt* is a principal ideal and therefore Q is regular by Lemma 3.
LEMMA 4. Assume that the following conditions are satisfied:
(1) Q has an infinite residue field;
(2) m can be generated by two elements, say m = (u, v);
(3) the form ideal n of the zero ideal of Q is unmixed.
Then the conductor cfrom Q to its first neighbourhood ring 91 is me
~1
, where e is the multi-
plicity of Q.
Remark. We recall that the conductor c consists of all elements x of 91 such that
Proof. Since n is unmixed, fi = n0 and the integer c of (1 • 1) is zero. Further, because
Qlm is infinite, we can find a superficial element b of degree unity; consequently, by the
remark following Theorem 1, bm"'1
= nt" for all v^e. In particular, 6me
~1
= me
and
therefore frm^1
= me
~1+s
for all s S= 0. Let Ae 91. By Lemma 1, for a suitable integer
s, A = ajb8
, where a em8
; consequently
Am6
"1
c - me
-1+s
= xxf-1
.
0s
This shows that m6
"1
£ c.
* See Krull(l), Th. 6.

7. Notion of a first neighbourhood ring 49
Again, since m = (u, v), we can take it = n0 to be an unmixed one-dimensional ideal
in K[X, Y]. Consequently we can find a form <j>(X, Y) such that
n = n0 = @(X, Y)}
and since, by (1-2), %(r, n) = e for large values of r it follows that <?>(X, Y) is of degree e.
Suppose now that q e c, q € m' and q £ m/+1
. The proof will be complete if we establish
that qeme
~1
. Assume the contrary; then t<e—l. We now write q = ijr{u,v), where ^is
a form of degree t, and choose £, rj in Q so that E,X + yY is a non-zero linear form not
dividing either <f>{X, Y) or ty(X, Y). This choice is possible because Qjm is infinite,
= n0,
£,11 + yv is a superficial element of degree unity and therefore uJ{E,u + yv) and vj(£,u + yv)
are both in 91. But q = ijr(u, v) is in c, consequently
ui]/{u, v) e Q(E,u + yv) and vi/r(u,v)eQ(£,u + yv).
Now n:(£X + y~Y) = n, consequently* Q(£u + yv) has the form ideal
Thus Xf{X, Y)€{4>,lX + yY) and
But Xi/r(X, Y) and YiJr(X, Y) are both of degree t + 1 < e, consequently both must be
divisible by %X + y~Y. This implies, however, that £X + y~Y divides ft(X, Y) and thus
we have a contradiction. The proof is now complete.
4. The ring K[X; Q.]. Let D. be the semi-group of superficial degrees. We shall say
that a polynomial g(XvX2, ...,Xd) = g(X), with coefficients in K = Q/rn, belongs to
K[Xlt ...,Xd Q] = KX Q.] if the degrees of all its non-zero homogeneous con-
stituents belong to Q. It is clear that K[X; Q] is a ring and it is easy to see that it is
Noetherian. Of course, if there exist superficial elements of degree one then
KX £2] = K[X]. In particular, K[X; Q] = K[X] whenever K is a infinite field. Put
no = nonK[X;Q]; (4-1)
then n0 is a homogeneous ideal in K[X; Q] all of whose prime ideals are of (affine)
dimension unity.
It will now be shown that the proper idealsf of 9t which contain 9im are very
faithfully represented in KX; Q], by the unmixed homogeneous ideals of (affine)
dimension unityj which contain n0. A similar situation was analysed in (3) but,
because the hypotheses of the present paper necessitate a considerable number of
modifications to the original arguments, it is proposed to describe the whole process
rather concisely but completely.
Let c be a proper ideal of 9? containing 9tm and let x e c. Since x € 9t, we can express x
(invariousways) intheform x = </>(u)/ip~ (u), where (f>, ijrareforms ofthesame degrees (say)
* See KruU(l), Theorem 12.
t By a proper ideal is meant one which is not the whole ring.
% To prevent misunderstandings over the use of the word unmixed, it should be noted that
these ideals can also be described as being homogeneous and having every belonging prime ideal
of dimension unity.
4 Camb. Philos. 53, 1

8. 50 D. G. NORTHCOTT
andno:(V^) = n0. By Theorem 1, seQ and therefore lfieK[X; Q]. Keeping c fixed and
varying x so that it always remains in c, we construct all forms <f> which can be obtained
in the manner just described. These forms will generate in KX; Q.~ a homogeneous
ideal f, which will be called the homogeneous ideal in K[X; Q] corresponding to c. It is
now a straightforward matter to establish the following result concerning c and Z.
LEMMA 5. Let <p(u)/i/r(u) belong to SR, where <p, r]r are forms of degree s {say) and
t v W = "o- Then seQ. and tfeK[X; Q]. Further, we have (j>(u)ijr(u)e.z if and only
It follows at once that different ideals c have different ideals corresponding to them.
LEMMA 6. Let <p(u)lip-(u) belong to 8t, where (j>, xjr areforms of the same degree s (say) and
ttV(^") = n0. If now ^€tt0 then <f>(u)li/r(u) eflim.
Proof. By Theorem 2, we can choose a form ^0 of degree s0 ^ c so that n0: (^o) = n0.
Then ffi0efx0n (Xx,X2,...,Xdf £ n and therefore <j>(u)fto(u)= 0(modm8+s
<>+1
). But
ijr{u) i/ro{u) is a superficial element of degree s + s0; consequently
=
f{u) f(u) ijro{u)
as required.
It follows from Lemmas 5 and 6, that if c is an ideal satisfying 9tnt £ c c fR then the
corresponding ideal contains n0.
LEMMA 7. Let c be an ideal of 9t such that 5R => C2 9tm and let c be the corresponding
homogeneous ideal in KX; Q]. Then each prime ideal, p which belongs to c, belongs
also to n0.
Proof. Since ito £i!ep it will suffice to prove that the dimension of p is not zero.
Assume the contrary, then, because p mil be homogeneous,
V = (X1,X2,...,Xd)nK[X;&].
Choose a form jr0 of positive degree such that n0: (ijr0) = n0. Let 0 be an arbitrary form
such that <f> is not null and, in K[X; Ci], ^ec: {i/r0). The degree s (say) of <j> will then
belong to Q consequently we can find a form ijr, also of degree s, such that n0: {>Jr) = n0.
We now have <t>ijr0 e c consequently, by Lemma 5,
~ WwW&)=
°{ m o d c )
and therefore, by the same lemma, 0eC. This establishes that C = C:(^0) and now we
have a contradiction because ^"oep.
It follows, from Lemma 7, that C is an unmixed ideal. We are now ready to bring
together and complete these various observations in
THEOREM 6. There is a 1-1 correspondence between the proper ideals co/jR which
contain SRm and the homogeneous unmixed one-dimensional ideals c of K[X; Q] which
contain n0 = n0 n K[K; Q.]. The connexion between two corresponding ideals c and c is as
follows. If<fi(u)/i/r(u) belongs to 9t (where $, r]r areforms ofthe same degree and n0: (x{r) = n0)
then <fi(u)lfr(u) belongs to c if and only i/^ec.

9. Notion of a first neighbourhood ring 51
Proof. Let e be a homogeneous unmixed one-dimensional ideal in K[X; Q.] con-
taining n0. After what has so far been said, it is enough to show that c arises from some
ideal between 9ft and Dftm. Let 0 be a form of degree s (say), where se. Q, such that
0 e c and let b be a superficial element of degree s. Then 0(u)jb e 9ft. Denote by cx the set
of all elements of 9ft which can be obtained in this way. It is now easy to verify that
Cx is an ideal with all the required properties.
The correspondence between the ideals c and c has several important properties
which are set out in Corollaries 1-5. These are almost immediate consequences of what
is stated in the theorem taken in conjunction with certain basic facts about homo-
geneous ideals.*
n n
COROLLARY 1. Let ct and ct be corresponding idealsfor l^i^n. Then C Ct and f) fy
i=l i=l
are corresponding ideals.
It is clear that if 9ft:D
c12C23 9ftm then the ideal corresponding to cx contains the
ideal corresponding to c2. It follows that we have
COROLLARY 2. The ideal in K[X; Q], which corresponds to 9ftm, is fi0.
COROLLARY 3. Let c and c be corresponding ideals. Then if one is prime so is the other,
while if one is primary so is the other.
The proof of this is a simple verification and we can at once derive the following
supplementary result.
COROLLARY 4. Let c and Z be corresponding primary ideals. Then the prime ideals to
which they belong also correspond.
Finally, if c and C are corresponding ideals, then each of them has a unique normal
decomposition into primary components. In view of Corollaries 1, 3 and 4 this yields.
COROLLARY 5. Let c and Z be corresponding ideals. Then the correspondence extends to
their respective primary components and also to the prime ideals which belong to them.
PART II. THE AF + B<& THEOREM
5. Notation used in Part II. As stated in the introduction, the treatment of the
AF + B& Theorem given here has a natural place in the abstract theory of infinitely
near points as developed in (5). Accordingly, we begin by recalling some of the
terminology and results of that paper. For this it will be convenient to drop the
notational conventions that were laid down in § 1 and to set up new conventions.
From now on, Q will denote a fixed two-dimensional regular local ring with maximal
ideal m and residue field Q/m = K. It will be assumed throughout that K is an infinite
field but it will not be assumed that Q and K have the same characteristic. Since Q is regular
and two-dimensional, tn can be generated by two elements. In what follows we
suppose that m = (u, v).
For geometric convenience, we associate with Q an object o which will be referred
to as the origin. The points p in the first neighbourhood of o then correspond to the
* The reader will find all that is needed here set out in (4).
4-2

10. 52 D. G. NORTHCOTT
irreducible formsp{X, Y) in KX, Y], two forms being regarded as essentially the same
if each is a constant multiple ofthe other. With each pointp, in thefirstneighbourhood
of 0, there is associated a ring Qp. By its definition ((5), §3), Qp consists of all quotients
$(u, v)/i/r(u, v) where <f>, tjr are forms ofthe same degree (with coefficients in Q) andwhere
i/r(X, Y)£{p(X, Y)}. Of course by rjr(X, Y) is to be understood the result of reading
i/r(X, Y) modulo m. It is shown in (5) that (in an appropriate sense) Qp does not depend
on the choice ofthe base (u, v) ofm. It is also shown that Qp is a two-dimensional regular
local ring containing Q and that different points p have different rings Qp associated
with them.* In view of the fact that Qp has all the properties which were assumed to
hold for Q, we can repeat the construction indefinitely. This leads us to the points in
the higher neighbourhoods of the origin and to the regular two-dimensional local rings
associated with them. However, before we proceed to say anything detailed about
these it will be convenient to make available, in a new form, some of the results of
Parti.
Let g =j= 0 be a non-unit in Q. As in (5), the multiplicity of o on (g) is taken to be the
integer s such that gem", g£ms+1
or, equivalently, the multiplicity of the local ring
Ql(g). Now m does not belong to the ideal (g). This is most easily seen by noting that
because Q is regular it is integrally closed (Krull(i), Th. 6). Accordingly, Qj(g) satisfies
the general assumptions of Part I as set out in § 1. Again, Q/(g) has an infinite residue
field and its maximal ideal xnl(g) can be generated by two elements. Further, the form
ideal of the zero ideal of Q/(g) is the same as the form ideal of (g) in Q. This, however,
is simply the principal ideal generated by the leading form of g and this, we note, is
unmixed. We may therefore apply Lemma 4 and so obtain
THEOREM 7. Let gr =|= 0 be a non-unit in the ring Q and suppose that the origin has
multiplicity s on (g). Then the conductor from Qj{g) to its first neighbourhood ring is
6. The AF +B<& theorem. Let/=)=O belong to Q and suppose that the origin o has
multiplicity r (r ^ 0) on (/). Let p be a point in the first neighbourhood of o then
p corresponds to an irreducible form p(X, Y). By the element fp which corresponds to
fin Qp is meant a generator ofthe ideal Qpfr~T
where T is an element ofm whose leading
form is linear andprime top{X, Y). Thus/p is uniquelydetermined towithin multiplica-
tion by a unit of Qp. We use the symbol p to denote the origin of Qp (in the same sense
that o denotes the origin of Q) and then by the multiplicity ofp on (/) we understand the
multiplicity ofp on (fp).
These notions extend, by an inductive construction, to the higher neighbourhoods
of 0. In dealing with these, however, it is convenient to use a different notation. This
is done by letting 0a denote a typical point infinitely near to the original origin which is
now denoted by 0. We suppose a to range over a set A of symbols, and 0a may belong
to a neighbourhood of 0 of arbitrarily high order. The two-dimensional regular local
ring associated with Ox is then denoted by Qa.
Suppose now that/=(= 0, g 4= 0 are elements of Q and that (/, g) is an m-primary ideal.
For each a, there will be elements fa and ga which, we say, correspond to / and g
* See (5), Theorem 1.

11. Notion of a first neighbourhood ring 53
respectively. These elements are uniquely determined up to multiphcation by a unit
of Qa. The infinitely near point 0a is said to lie on (/) if fa is a non-unit of Qx. The
multiplicity of 0a on (/) is defined to be the integer ra ^ 0 such that fa e m£«, fa £ m£°+1
and this is zero if and only if 0a does not he on (/). Here ma denotes the maximal ideal
of Qa. Finally, it is known ((5), Theorem 8, Cor.) that there are only a finite number of
points 0a which he on (/) and (g) simultaneously.
We are now in a position to formulate the AF + B<!) theorem in a rather crude form
which, however, will be refined shortly.
THEOREM 8. Letf^Q, gr + O be elements of Q which generate an m-primary ideal and
leth + O also belong to Q. Denote by ra, sa and ta the multiplicities of a typical point 0a
(infinitely near to 0) on (/), (g) and (h) respectively. Suppose now that ta^ra + sa-l
whenever Ox lies on both (/) and (g). Then he (f, g).
The hypotheses of this theorem require that the actual multiplicity of Oa on (h) be
at least ra + sa - 1 whenever Oa lies on both (/) and (g). But there is another sense in
which h may be regarded as having multiplicities at least equal to ra + sa — 1 at such
points. This alternative sense allows for some kind of compensation so that an
unnecessarily high multiplicity near 0 can make up for one which is too small in amore
remote neighbourhood. In spite of the considerable abuse of language involved, the
idea, which is of course familiar to geometers, is a most valuable one. Accordingly, we
proceed to elaborate it and postpone, for the time being, the proof of Theorem 8.
Let A be a non-empty finite set of points infinitely near to 0, with the property that
whenever 0a belongs to A so do all the points (including 0) which are intermediate
between it and 0. Such a set will be called a root system relative to 0. In a special case,
A may consist solely of the point O. If, however, A contains other points we can pick
out those, say 0x, 02,..., 0d, that are in the first neighbourhood of 0. Denote by A4 the
set of all points of A which he in neighbourhoods of 0t. Then A consists of 0 together
with the sets Ax, A2,..., Ad and, by construction, Af is a root system of points relative
To clarify the further discussion we make two definitions.
DEFINITION. The largest integer m such that the root system A, relative to 0, contains
a point in the mth neighbourhood of O will be called the 'degree' of A.
Before making the next definition, let us note that m = 0 when and only when A
reduces to the single point 0.
DEFINITION. Let A be a root system, relative to 0, of degree m~£ and let Olt O2,..., 0d
be the points of A that are in the first neighbourhood of 0. By ' a general element of m with
respect to A' will be meant an element r whose leadingform is linear and prime to each of
the forms associated with the points Ox, 02,..., 0d.
Suppose now that with each point 0a of A we associate a non-negative integer pa.
It will be convenient to denote the integer associated with 0 by p. Let h be an element
of Q. We shall now, by means of an inductive construction, give a precise meaning to
the statement:
the element h has, in a modified sense, multiplicity at least equal to pa at each point
(6-1)

12. 54 D. G. NORTHCOTT
If the degree m of A is zero then this statement is to mean simply that h belongs to mp
.
Note that if the statement holds for h then it also holds for eh where e is any unit of Q.
IfTO= 1 then (6-1) is to be taken to mean that we have simultaneously
(A) h belongs to mp
(B) whenever T is a general element of m with respect to A then, for each value of
i(l < i < d), hr~P has, in the modified sense, multiplicity at least equal to pa at each point
Oa of the root system Af.
It will be clear that we need only verify that hr~P satisfies (B) for a single general
element T (with respect to A) since, by what has been said for the case m = 0, it will
follow then that (B) holds for all such general elements. It is clear too (still supposing
that m = 1) that if the statement (6-1) holds for h then it also holds for eh where e is
any unit of Q. But, without further modification, (A) and (B) allow us to proceed
successively to the cases m = 2, 3,4,... and it will continue to be true that the condition
(B) will be satisfied by all general elements of m with respect to A if it is satisfied by
one of them.
THEOREM 9. Let A be a root system ofpoints relative to 0 and let a non-negative integer
pa be associated with eachpoint Oa of A. Then the set q ofelements which have, in the modified
sense,, multiplicity at least equal topa at each point Oa of A, is an ideal of Q. Further, if
the ' actual' multiplicity of an element of Q at each point Oa of A is at least equal to pa then
the element will belong to q.
The proof of the theorem is an easy induction on the degree of A and we omit the
details.
Let us now observe that if (/, g) is an m-primary ideal, then the set consisting of the
points Oa which lie on both (/) and (g) is not only finite (as already observed) but also
a root system relative to 0. In view of this we can now state the AF + SO in its final
form. That the new theorem includes Theorem 8 follows at once from Theorem 9.
THEOREM 10. Letfj= 0, g # 0 be elements of Q which generate an m-primary ideal and
let h also belong to Q. Denote by ra, sa the. multiplicities of a typical point Oa (infinitely
near to O) on (f) and (g) respectively, and suppose that h has, in the modified sense,
multiplicity at least equal to ra + sa— 1 at each point Oa which lies in both (/) and (g).
Then h e (/, g). Further, ifh = £/+ t]g, where £ and v are in Q, then £ and v have (in the
modified sense) multiplicities at least equal to sx — 1 and ra—l respectively at each point
Oa which lies on both (/) and (g).
Proof. Let A be the root system composed of all points Oa which lie on both (/) and
(g) and let the degree of A be m. Once again we argue by induction on m but, on this
occasion, it will be more convenient to begin with the inductive step. Accordingly we
shall suppose that m ^ 1 and that the theorem has been proved in all cases where the
set of common points has smaller degree than m—but we shall observe, in passing,
that the arguments used to accomplish the inductive step also suffice to establish the
truth of the theorem when in = 0.
Let d(X, Y) be the leading form of g and let
6(X, Y) = pftX, YM*(X, Y) ...p&(X, Y), (6-2)

13. Notion of a first neighbourhood ring 55
where Pi(X, Y) are essentially distinct irreducible forms. Each form Pi{X, Y) deter-
mines a pointy = 0f in the first neighbourhood of o = 0, and these points are precisely
the points in the first neighbourhood of o which lie on (g) ((5), Theorem 2). Without loss
of generality we may suppose that all of OltO2, ...,Ot and none of Oi+1,Os+2, ...,0d
belong to A. For convenience we use Qi where 1 < i ^ d, to denote the regular local ring
associated with pt.
Let T be an element of m so chosen that its leading form is linear and prime to all the
Pi(X, Y) for 1 < i ^ d, then, in particular, r will be a general element of m with respect
to A. Further, let r and s be the multiplicities of 0 on (/) and (g) respectively.
For 1 < i sg Swe can apply the inductive hypothesis to//rr
, g/T8
and hlTr+s
~1
considered
as elements of Qi and so obtain
while if d+l^i^d then fjrr
is a unit in Qt and the same relation holds trivially.
Accordingly heQJr^ + Qigr^ <!<•«*) (6-3)
and we note too that the same reasoning shows that (6-3) is also true in the case m = 0.
By virtue of the definition of Qi we may therefore write
jri(u,vy ft(v,v)
where (j>t, <pf, i/rt are forms of the same degree (with coefficients in Q) and
UX,Y)i{Pi(X,Y)}.
Further it is clear that we can arrange that the forms ty^X, Y), for 1 ^ i ^ d, have the
same degree I (say); consequently
iri(u,v)hefxn1+8
-1
+ gmt+r
-1
(l<i<d). (6-4)
But K = Qjm is an infinite field and therefore we can find av a2,..., ad in Q such that if
, Y) = axfx(X, Y) + aa#t{X, Y)+...+adfd(X, Y)
then, for each value of i in the range l^i^d, i/r(X, Y)£{pi{X, Y)}. Moreover it
follows, from (6-4), that ,, ,
Put Q* = Ql(g) and let us use an asterisk to denote residues modulo (g). Then
^*(tt*,z>*)A*€/*(m*)'+8
-1
where m* = m/fa). But f(X, T)i{pt{X, Y)} and f(X, Y)
is the same as ft*(X, Y) read modulo m*. This shows that i/r*(X, Y), read modulo m*,
is prime to the form ideal {B(X, Y)} of the zero ideal of Q*. Accordingly
£ 8 t
i/r*(u*, v*)
where 9fJ* is the first neighbourhood ring of Q*. We see now that h* e 9ft*/*(m*)8
-1
.
But, by Theorem 7, (m*)8
"1
is the conductor from Q* to 9t*; consequently
h*ef*(m*y-1
cQ*f* (6-5)
and so h e(f, g)

14. 56 D. G. NOETHCOTT
Now let h = gf+yg where g, TJ are in Q. Then £*/* = fc*€/*(m*)«-1
. But/ is not
contained in any prime ideal belonging to (g), consequently / * is not a zero-divisor
and therefore £*e (tn*)8
"1
. It follows, since ms
~1
2tns
2 (g), that ijetn8
-1
and, by sym-
metry, we must also have TjexxV"1
. The theorem is thus established when m = 0. It
remains only to complete the details of the inductive step which have to do with the
final sentence in the statement of the theorem. Suppose that 1 ^ i ^ S. Then since
h
= f £ g
Vyr+8—1
is an equation which holds in Q{, it follows (by the inductive hypothesis) that gis
~x
and
7)lTr
~1
have (in the modified sense) multiplicities at least equal to sa — 1 and ra—
respectively at each point Oa which is infinitely near to O{ and lies on both (fjrr
) and
(glrs
). But as this holds for each i in the range 1 ^ i ^ 8 the proof of the theorem is
complete.
REFERENCES
(1) KRULX., W. Dimensionstheorie in Stellenringen. J. reine angew. Math. 179 (1938), 204-26.
(2) NOKTHCOTT, D. G. Hubert's function in a local ring. Quart. J. Math. (2) 4 (1953), 67-80.
(3) NOBTHCOTT, D. G. The neighbourhoods of a local ring. J. Lond. Math. Soc. 30 (1955),
360-75.
(4) NOBTHCOTT, D. G. On homogeneous ideals. Proa. Glasg. Math. Ass. 2 (1955), 105-11.
(5) NOBTHCOTT, D. G. Abstract dilatations and infinitely near points. Proc. Camh. Phil. Soc.
52 (1956), 176-97.
(6) NOBTHCOTT, D. G. and RBES, D. A note on reductions of ideals with an application to the
generalized Hilbert function. Proc. Camb. Phil. Soc. 50 (1954), 353-9.
(7) SAMCTEL, P. La notion de multiplicite en algebre et en geome'trie (Thesis, Paris, 1951).
(8) SEVBBI, F. Trattato di geometria algebrica, 1 (Bologna, 1926).
THE UNIVERSITY
SHEFFIELD