Intersection Theory and the Homological Conjectures
in Commutative Algebra
Paul C. Roberts
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA
The subject of this article is a set of conjectures which have been a central
topic of investigation in Commutative Algebra for several years. Many of these
conjectures are closely related to questions in Intersection Theory, and in some
cases they arose directly from attempts to define intersection multiplicities in
an algebraic setting. We begin with a discussion of some of these questions in
Intersection Theory and their influence in Commutative Algebra. Next we discuss
a few of the main conjectures related to this topic. Finally, we show how recent
developments in Intersection Theory have made it possible to settle some of these
problems.
1. Introduction: Serre's Definition of Intersection Multiplicities
We begin this section with some background on the problem of defining intersection multiplicities. The general question is as follows : given a variety V, a point
p of V, and subvarieties X and Y of F such that p is an isolated point of the
intersection X n Y, we wish to to define the multiplicity of the intersection of X
and Y at the point p. We are interested here in an algebraic definition (which
does not involve topology, for instance), and in one which is defined locally at
the point.
As an illustration, we consider the case in which X and Y are curves in the
plane. Let A be the local ring at the point p; that is, the ring of rational functions
defined in a neighborhood of p. Let the curves X and Y be defined locally near p
as the sets of zeros of polynomials f(x,y) and g(x,y). Then the correct definition
for the multiplicity is the length of the quotient ring (A/(f,g)), where (f,g) is the
ideal generated by / and g (or, in the complex case, the dimension of A/(f,g)
as a vector space over (C). This definition agrees with intuition; if the curves are
smooth and not tangent at p it gives multiplicity one, and if they are tangent it
is greater than one. In addition, it satisfies Bézout's Theorem, which states that if
V is the projective plane, and if X and Y are subvarieties of V of degrees m and
n respectively, then the total number of intersections, counted with multiplicities,
is mn. This theorem is very important in enumerative geometry.
Before proceeding, we recall some basic definitions and notation. Let A be
a commutative Noetherian local ring with maximal ideal m and residue field
A/m = k. Let dim (A) = d denote the Krull dimension of A; this can be defined
Proceedings of the International Congress
of Mathematicians, Kyoto, Japan, 1990
© The Mathematical Society of Japan, 1991
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Paul C. Roberts
either as the maximum length of a chain of prime ideals or as the minimum
number d such that there are elements x\,...Xd G m which generate an ideal
primary to the maximal ideal. Such a sequence of elements is called a system
of parameters. The ring A is regular if the maximal ideal can be generated
by d elements; in the geometric situation, the local ring at a point is regular
when the variety is smooth at the point. The ring is Cohen-Macaulay when the
system of parameters form a regular sequence; that is, when axt G (xi,...,Xf_i)
implies a G (xi,...,Xj_i) for each i. We remark that the above definition of
multiplicities for curves works because A is Cohen-Macaulay and (f,g) is a
system of parameters in that case.
If A is an integral domain, we say that A is equicharacteristic if the characteristic of A is the same as the characteristic of k; otherwise A has mixed
characteristic.
We now return to the question of defining intersection multiplicities. Let A
be a local ring, and let / and J be ideals of A; in the geometric situation, J
and J will be the ideals of functions vanishing on subvarieties X and Y. The
straightforward generalization of the definition for curves would be to define the
multiplicity to be the length of the quotient A/(I + J). However, this will not
satisfy Bézout's Theorem (among other things).
The idea of Serre was to add extra correction terms. In addition, it is more
convenient to generalize from the quotients A/1 and A/J of the ring A to more
general finitely generated modules M and N.
Definition (Serre [28]). Let A be a regular local ring and let M and N be finitely
generated ^4-modules such that M ®A N is a module of finite length. Then the
intersection multiplicity of M and N is defined to be
X(M,N) = ^(-l)4ength(Tor f (M,iV)).
This definition makes sense since all modules over regular local rings have
finite projective dimension. Furthermore, it satisfies Bézout's Theorem, and it
agrees with the definition given above in the case of curves. Serre stated three
additional properties and proved them in the equicharacteristic case.
1. dim(M)+ dim(N) < dim(A).
2. (Vanishing) If dim(M)+ dim(iV) < dim(A), then x(M,N) = 0.
3. (Positivity) If dim(M)+ dim(iV) = dim(A), then x(M,N) > 0.
The first of these Serre proved in general; we note that it can be interpreted
as a statement on the topology of intersections in smooth varieties. The second
and third were proven in the equicharacteristic case, and were conjectured in the
general regular case.
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363
2. The Peskine-Szpiro Intersection Theorem
Attempts to prove Serre's conjectures in the ensuing years included various
generalizations of them to non-regular rings. The three statements listed above
make sense for any local ring as long as M is assumed to have finite projective
dimension (and, of course, M ®AN has finite length). In fact, the first statement
makes sense even without the assumption of finite projective dimension, but it is
false in this generality; whether it holds with this assumption is not known.
The next major advance was the proof of the "Intersection Theorem" in a
quite general situation by Peskine and Szpiro [16, 17]. This theorem states:
Theorem (Intersection Theorem). Let M be a module of finite projective dimension
and let N be a module such that M ®A N has finite length as a module. Then
dim N < proj dim M.
If one replaces projective dimension in this theorem by codimension (i.e.
dim(y4)— dim(M)), it becomes the first statement of Serre's conjectures, and, like
that statement, it can be considered to describe the topology of the intersection
of a variety which is the support of a module of finite projective dimension. The
importance of the theorem was demonstrated by the fact that it implies several
other conjectures (Peskine-Szpiro [17]):
Corollary (Bass's Conjecture). If A has a non-zero finitely generated module of
finite injective dimension, then A is Cohen-Macaulay.
Corollary (Zero-Divisor Conjecture). If M is an A-module of finite projective dimension and x in A is a zero-divisor on A, then x is a zero-divisor on M.
Shortly thereafter, a stronger version of this theorem was stated (PeskineSzpiro [18], Roberts [20]):
Theorem (New Intersection Theorem). Let
0 - • Fk -> ... - • F 0 -> 0
be a complex of finitely generated free modules with homology of finite length. Then
if Fm is not exact, we have dim(A) < k.
This theorem implies the original Intersection Theorem.
In the paper cited above, the Intersection Theorem was proven for rings of
positive characteristic and for rings whose completion is the completion of a ring
of finite type over a ring of characteristic zero. The method, which has since
been applied successfully to a wide variety of questions, was to reduce (using the
Artin Approximation Theorem) to the case of a ring of positive characteristic p,
and then to use iterations of the Frobenius map (the map which sends a to ap)
to prove the theorem in the positive characteristic case. The method of reducing
using the Artin Approximation Theorem was extended by Höchster [11] to the
general equicharacteristic case. These techniques did not extend to the case of
mixed characteristic.
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Paul C. Roberts
3. The Homological Conjectures
The conjectures of Serre and the Intersection Conjecture in mixed characteristic,
together with a number of others, became known as the Homological Conjectures. These conjectures dealt with intersection multiplicities, modules of finite
projective dimension, the existence of Cohen-Macaulay modules, and properties of systems of parameters, together with relations between these topics. A
summary of these conjectures in 1975 can be found in Höchster [12]; in this
monograph Höchster introduced big Cohen-Macaulay modules and proved their
existence (thus implying several other conjectures, including the Intersection Conjecture) in the equicharacteristic case. The proof made use of reduction to positive
characteristic.
In a somewhat different approach, in Roberts [21] the notion of CohenMacaulay complex is defined and proven to exist in the complex algebraic case
via the Grauert-Riemenschneider vanishing theorem. The connection between
the technique of reduction to positive characteristic and vanishing theorems in
cohomology coming from Hodge theory was further shown by a proof of the
Kodaira vanishing theorem by reduction to positive characteristic, conjectured
by Szpiro and proven recently by Deligne and Illusie [2].
Among these conjectures, as mentioned above, was the question of whether
Serre's vanishing and positivity conjectures held for arbitrary rings, when M
was assumed to have finite projective dimension. This conjecture was proved for
graded modules over a graded ring by Peskine and Szpiro [18] using computations
with Hilbert polynomials in a free resolution, and they showed that several other
conjectures followed from this one. These results led to specuation that an
appropriate theory of Chern classes and Riemann-Roch theorem over general
local rings could be used to prove the multiplicity conjectures, and hence many
others, in general. The idea was that a bounded complex of free modules F.
would define a sequence of Chern characters chj(F.), and that these would vanish
for i less than the codimension of the support of Fm. In codimension one this
program was carried out by Foxby [7], using a construction of MacRae [15],
leading to proofs of some of the conjectures in low dimension.
However, Dutta, Höchster and McLaughlin [4] constructed an example which
showed that this generalization of Serre vanishing was false in dimension three.
They construct a module M of finite projective dimension and finite length over
the ring A = k[[X, Y,Z, W]]/(XY -ZW), which has dimension three, such that
if jV = A/(X,Z), then x(M,N) = - 1 . Thus the theory of local Chern characters,
even if it existed, could not have the hoped for vanishing properties. In the next
section we describe such a theory and then show how in spite of these setbacks,
it could solve some of the conjectures.
4. Localized Chern Characters
The Chern characters alluded to in the previous section were defined by Baum,
Fulton, and MacPherson [1]. There were two major problems in defining such a
theory for complexes over a local ring. The first was that there is no cohomology
theory. This problem was answered by defining Chern characters as operators on
the Chow group. The second problem was that there are no non-trivial vector
bundles over a local ring, so that straightforward definitions of Chern characters
Intersection Theory and the Homological Conjectures
365
tend to give zero. The solution here was to give a "localized" definition, with
values not in the Chow group of the entire ring, but in the Chow group of the
support of the complex. We give a very brief summary of the part of the theory
we need ; we refer to Fulton [8] for a more complete description.
Let V = Spec(>4), and let Y be a closed subset of V. For each integer i, let
Zi(Y) denote the rational group of cycles of dimension /'; that is, Z/(7) consists
of the free Q-module with generators all reduced and irreducible subschemes
of Y of dimension /, or, equivalently, all prime ideals P with dim(A/P) = /.
Then the y-th graded piece of the Chow group, denoted Aj(Y), is Z,(7) modulo
rational equivalence, where rational equivalence is defined by setting the divisor
of a rational function on a reduced and irreducible suscheme of dimension i + 1
to zero.
Let Fm be a bounded complex of free modules with support Z. Then there
are Chern characters chk(Fm) for each k ^ 0, which map A^(X) to Ai-k(X n Z).
The connection with Euler characteristics was given by the local RiemannRoch theorem (Fulton [8], Example 18.3.12). We state a very special case of this
theorem here. Let F. be a complex of free modules with homology of finite
length, or, equivalently, with support at the closed point of Spec(yl). Assume that
A is an integral domain, and let [A] denote the class of Spec(y4) in the highest
dimensional graded piece of the Chow group Ad(SpGc(A)). Then there is a class
T(A), with [A] as its highest component, which satisfies
X(Fm) =
^(äii(F.)(zi(A)).
feO
In this equation, the left hand side is an integer, and the right hand side is an
element of the Chow group of the support of F.. In this case, the support of F.
is a point p, so its Chow group reduces to Ao(p), which we identify with Q.
Furthermore, if A is regular (or, more generally, a complete intersection), then
T(A) = [A].
5. The Proofs of Serre's Vanishing Conjecture
and the Intersection Conjecture
Using the theory described in the last section, it is possible to prove Serre's
vanishing conjecture without regard to characteristic and to prove the PeskineSzpiro Intersection Theorem in mixed characteristic. We briefly outline these
proofs here and attempt to show how the theory of localized Chern characters is
used in their solution.
Let M and N be modules over a regular local ring A such that M ®AN has
finite length and dim(M) + dim(iV) < dim(>4). Let F. and G. be free resolutions
of M and N respectively. The problem is to show that #(F. ® G.) = 0, since the
homology of F. ® G. is precisely Tor(M,N). By the local Riemann-Roch formula
we have
x(F.®G.)=ch(F.®G.)([A\).
We must now use some properties of local Chern characters. First, they are
multiplicative, which means that we can write
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Paul C. Roberts
ch(F. ® G.)([A]) = £
chl(F.)(chi(G.))(M).
i+j=d
Let X and 7 denote the supports of M and N respectively; these are also
the supports of F. and G.. Now suppose that d — j > dim(7). Then the element
chj(G.)([A]), which is in the group Ad-j(Y), must be zero, since there can be no
subvarieties of 7 of dimension larger than the dimension of 7. Thus all terms
for which d—j> dim(7) are zero. Using the commutativity of Chern characters
(proven in Roberts [23]), one shows also that terms with d — i> dvax(X) are zero.
The hypothesis that dim(X) + dim(7) < d implies that all terms fall into one of
these two sets, and therefore the entire sum is zero, as was to be shown. .
This theorem also holds if A is a complete intersection. It was proven independently using Adams operations in X-theory by Gillet and Soulé [9,10]. It was
proven when the singular locus has dimension at most one in Roberts [23]. We
note that this argument gives a vanishing theorem for intersection multiplicities
defined by local Chern characters for modules of finite projective dimension in
general, and this approach appears to work better than the approach based on
Euler characteristics for singular points.
We next describe how localized Chern characters are used in the proof of the
New Intersection Theorem in mixed characteristic. Let A be a domain of mixed
characteristic of dimension d, and let
0 -> Fd-i ->...-> F0 -> 0
be a complex of free modules with homology of finite length which is not exact.
We wish to prove that such a complex cannot exist. Let F. denote F. ® A/pA,
where p is the characteristic of the residue field of A. Then F. is a complex of
free modules over A/pA.
The main idea of the proof is to compute chd-i([A/pA]) in two different
ways. First, using the properties of localized Chern characters and the fact that
F . comes from restriction to a divisor of a complex supported at one point, we
deduce that chd-i([A/pA]) = 0. Second, we use that F. is a non-trivial complex
of length d — 1 over a ring of positive characteristic of dimension d — 1. By means
of an asymptotic Euler characteristic introduced by Dutta [3] and put on a firm
foundation by Seibert [27], together with a local cohomology argument, we can
then show that it must be positive (details can be found in Roberts [24, 25]). This
contradiction proves the theorem.
6. Open Questions
We conclude with some remarks on three questions which are still open.
1. Serre's Positivity Conjecture. The method of local Chern characters provides a
geometric approach to the question of positivity, but as of yet it has not led to a
solution. A result of Tennison [30] states that if one computes x(A/P,A/Q), where
dim(A/P) + dim(A/Q) = dim(4), and if the subschemes defined by the prime
ideals P and Q are not tangent at the closed point of Spec(R) (this condition can
be stated precisely in terms of associated graded rings), then x(R/P,R/Q) is the
product of the multiplicities of the A/P and A/Q and is therefore positive. We
remark that an example of Roberts [26] shows that positivity is unlikely for two
Intersection Theory and the Homological Conjectures
367
modules of finite projective dimension in the non-regular case, even using Chern
characters rather than Euler characteristics, although this question is also open.
2. The Improved New Intersection Conjecture (Evans-Griffith [5, 6]). This conjecture states the following: Let F. = 0 -> Fk -» ... - • Fo -> 0 be a complex
of free modules with finite length homology except possibly for Ho(F.). Assume
that there is a minimal generator of Ho(Fm) which is annihilated by a power of
the maximal ideal of A. Then k > dim(A).
It seems reasonable to try to prove this theorem using the methods which
worked for the "unimproved" version, but this version appears to be more difficult.
This conjecture is equivalent to several others, such as Hochster's Monomial
Conjecture (see [13]), and is open in dimension three in mixed characteristic.
3. Another Conjecture on Bounded Complexes. The New Intersection Theorem
states that the minimum length of a bounded complex of free modules is d.
Suppose that F. has length exactly d. In this case, the question is whether
the cycles Z\ are integral over the boundaries B\ in degrees larger than zero.
This result would imply the Monomial Conjecture, it can be proven in positive
characteristic by the theory of tight closure of Höchster and Huneke [14], and it
has been proven by Rees [19] for the Koszul complex in any characteristic.
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