Recent Developments on Serre’s Multiplicity Conjectures: Gabber’s Proof of the Nonnegativity

Conjecture

Paul C. Roberts

September 8, 1997

These notes are based on talks given at the Encuentro de Geometría Algebraica y Álgebra Conmutativa in Guanajuato in August 1997. They describe recent developments in the questions on intersection multiplicities, particularly Gabber’s recent proof of Serre’s conjecture that intersection multiplicities over regular local rings are non-negative. After an introductory section on Serre’s conjectures, we present an outline of this proof. In addition, we discuss related questions on Hilbert polynomials of bi-graded rings.

An outline of Gabber’s proof can be found in Berthelot [1], and a more complete exposition of the proof is given in Hochster [5]. Both of these articles had a strong influence on these notes.

1 The Serre multiplicity conjectures

In [7], Serre introduced a definition of intersection multiplicity for regular local rings and showed that it satisfied many of the properties which should hold for intersection multiplicities. The definition is as follows: let R be a regular local ring of dimension d, and let X = Spec(R). Let Y and Z be closed subschemes of X defined by ideals 𝔭 and 𝔮 such that Y ∩ Z consists only of the closed point of X, or, equivalently, that R/𝔭 ⊗ R/𝔮 is a module of finite length. (Despite the notation, it is not necessary that 𝔭 and 𝔮 be prime; however, they will usually be assumed to be prime in later sections of the paper.) Then the intersection multiplicity of Y and Z is defined to be

χ(Y, Z) = χ(R/𝔭, R/𝔮) = ∑_{i=0}^{d}(−1)^{i}length(Tor_{i}^{R}(R/𝔭, R/𝔮)).

More generally, if M and N are finitely generated R-modules such that M ⊗ N is a module of finite length, we define

χ(M, N) = ∑_{i=0}^{d}(−1)^{i}length(Tor_{i}^{R}(M, N)).

One of the motivations behind this definition is that it can be shown that Bézout’s theorem holds if multiplicities are defined in this way; that is, if Y and Z are closed subschemes of projective space meeting in a finite number of points, than the number of points of intersection counted with multiplicities is the product of the degrees of Y and Z.

On the other hand, there were certain properties which are not obviously satisfied and which were left as conjectures. In the form given by Serre [7], the conjectures are as follows: let R be a regular local ring, and let M and N be finitely generated R-modules such that M ⊗_{R} N has finite length. Then:

1. dim(M) + dim(N) ≤ dim(R).

2. (Non-negativity.) χ(M, N) ≥ 0.

3. χ(M, N) > 0 if and only if dim(M) + dim(N) = dim(R).

Another version of these conjectures is the following :

1. dim(M) + dim(N) ≤ dim(R).

2. (Vanishing.) If dim(M) + dim(N) < dim(R), χ(M, N) = 0.

3. (Positivity.) If dim(M) + dim(N) = dim(R), χ(M, N) > 0.

It is easy to see that the two sets of conjectures are equivalent. Serre proved the first statement in general, and he proved the others for regular rings containing a field by the method of reduction to the diagonal. We will discuss part of this method below. The question was left open for rings of mixed characteristic, and Serre also asked whether a proof existed which did not use reduction to the diagonal.

The vanishing conjecture was proven about ten years ago (Roberts [6], Gillet-Soulé [3]) using K-theoretic methods. The proof in [6] uses the theory of local Chern characters, while that in [3] uses the theory of Adams operations on Grothendieck groups of complexes.

The main topic of these notes is the recent proof of Gabber of the nonnegativity conjecture, in the course of which he also gives a new proof of the vanishing conjecture. In addition, we discuss some questions which arise when attempting to extend these ideas to prove the positivity conjecture. First, we recall a spectral sequence argument used by Serre in his proof and which was extended by Gabber to reduce these questions on modules over regular local rings to questions on locally free sheaves on projective space.

2 The Serre spectral sequence

The main theorem of this section relates the Euler characteristic of a Koszul complex on a module to the Samuel multiplicity of the module. Let A be a local ring, and let M be a finitely generated A-module of dimension at most k. Let 𝔞 be an ideal of A such that M/𝔞M has finite length. We recall that the associated Hilbert-Samuel polynomial P 𝔞_{M}(n) is defined to be the polynomial for which

P 𝔞_{M}(n) = length(M/𝔞^{n}M)

for large n. If the dimension of M is at most k, we define the Samuel multiplicity e_{k}(𝔞, M) to be k! times the coefficient of n^{k} in P 𝔞_{M}(n) (if the dimension of M is less than k, e_{k}(𝔞, M) will be zero).

Theorem 1 With notation as above, let x_{1}, …, x_{k} be a sequence of elements of A, and let 𝔞 be the ideal generated by x_{1}, …, x_{k}. Assume that M/𝔞M is a module of finite length. Let K_{•} be the Koszul complex on x_{1}, …, x_{k}, and let

χ(K_{•} ⊗ M) = ∑_{i=0}^{k}(−1)^{i}length(H_{i}(K_{•} ⊗ M)).

Then

χ(K_{•} ⊗ M) = e_{k}(𝔞, M).

We sketch the argument used to prove this theorem. The main idea is to examine the spectral sequence defined by the filtration on K_{•} induced by powers of 𝔞. For each n ≥ 0 and for each i we consider the quotient 𝔞^{n}K_{i}/𝔞^{n+1}K_{i}. For each r ≥ 0 we then take the subquotient E_{i,n}^{r} of this module defined by

E_{i,n}^{r} = \frac{\{k_{i} ∈ 𝔞^{n}K_{i}|d_{i}(k_{i}) ∈ 𝔞^{n+r}K_{i−1}\} + 𝔞^{n+1}K_{i}}{(\{d_{i+1}(k_{i+1})|k_{i+1} ∈ 𝔞^{n−r+1}K_{i+1}\} ∩ 𝔞^{n}K_{i}) + 𝔞^{n+1}K_{i}} .

The E_{i,n}^{r} define a spectral sequence (the usual spectral sequence associated to a filtered complex). While the precise definition is necessarily quite complicated, the idea is that E_{i,n}^{r} is the subquotient of 𝔞^{n}K_{i}/𝔞^{n+1}K_{i} consisting of elements whose boundaries lie r steps further down in the filtration modulo boundaries of elements which lie at most r−1 steps further up in the filtration. As r gets large, this subquotient approaches the submodule of elements whose boundaries are zero modulo the submodule consisting of all of the boundaries. In fact, it can be shown using the Artin-Rees lemma (see Serre [7]) that the spectral sequence does in fact converge to the 𝔞-adic filtration on the homology of K_{•}.

Part of the general theory of spectral sequences, which can be verified directly in this case from the above definition, is that the boundary map d_{i} on K_{•} induces a map d_{i,n}^{r} from E_{i,n}^{r} to E_{i−1,n+r}^{r} for each i, n and r, and that we have

E_{i,n}^{r+1} = Ker(d_{i,n}^{r})/Im(d_{i+1,n−r}^{r}).

Thus the modules at the r + 1 stage can be computed as the homology of those at the rth stage under maps induced by the boundary maps of K_{•}.

We next examine the complexes defined by E_{i,n}^{0} and E_{i,n}^{1}.

If we let r = 0 in the above definition of E_{i,n}^{r}, the condition that d_{i}(k_{i}) ∈ 𝔞^{n+r}K_{i−1} states that d_{i}(k_{i}) ∈ 𝔞^{n}K_{i−1}, which is always true since k_{i} is assumed to be in 𝔞^{n}K_{i} and d_{i} is a module homomorphism. Similarly, the condition that k_{i+1} ∈ 𝔞^{n+1}K_{i+1} implies that d_{i+1}(k_{i+1}) ∈ 𝔞^{n+1}K_{i}, so that when r = 0 the denominator in the above definition of E_{i,n}^{r} is just 𝔞^{n+1}K_{i}. Hence E_{i,n}^{0} is simply 𝔞^{n}K_{i}/𝔞^{n+1}K_{i}. Furthermore, since K_{•} is the Koszul complex on the generators of 𝔞, the maps d_{i} are all zero modulo 𝔞, and the maps induced by the boundary maps d_{i} on E_{i,n}^{0} are zero. It then follows that E_{i,n}^{1} is also equal to 𝔞^{n}K_{i}/𝔞^{n+1}K_{i}.

We next consider the maps d_{i,n}^{1} induced by d_{i} on E_{i,n}^{1}; we denote this map \overline{d}_{i}. Since K_{•} is the Koszul complex on x_{1}, …, x_{k}, the map d_{i} is defined by a matrix with ±x_{i} in certain positions and zeros in the remaining positions. Thus \overline{d}_{i} is defined by the same matrix in which x_{i} is considered as a map from 𝔞^{n}K_{i}/𝔞^{n+1}K_{i} to 𝔞^{n+1}K_{i−1}/𝔞^{n+2}K_{i−1} for each n. Let \overline{K}_{i} denote the associated graded module of K_{i} under the filtration by powers of 𝔞. Then \overline{d}_{i} defines a map of degree one from \overline{K}_{i} to \overline{K}_{i−1}, and the above description shows that the resulting complex is the Koszul complex on \overline{x}_{1}, …, \overline{x}_{k}, where \overline{x}_{i} denotes the image of x_{i} in the component of degree 1 of the associated graded ring gr𝔞(A).

Thus we have shown that if for each i we let

\overline{K}_{i} = ⊕_{n≥0}E_{i,n}^{1} = ⊕_{n≥0}𝔞^{n}K_{i}/𝔞^{n+1}K_{i},

the maps d_{i,n}^{1} induced by d_{i} define a complex \overline{K}_{•} which is the Koszul complex on \overline{x}_{1}, …, \overline{x}_{k} over gr𝔞(A).

Up to now we have considered the filtration on K_{•} without mentioning the module M. However, exactly the same argument holds for K_{•} ⊗ M, and we obtain a spectral sequence E_{i,n}^{r}(M) which converges to the homology of K_{•} ⊗ M and such that the modules E_{i,n}^{1}(M) form the Koszul complex induced by \overline{x}_{1}, …, \overline{x}_{k} on the associated graded module gr𝔞(M). This Koszul complex can also be expressed as \overline{K}_{•} ⊗gr𝔞(M). Since we are assuming that M/𝔞M has finite length, the homology of the Koszul complex induced by \overline{x}_{1}, …, \overline{x}_{k} on gr𝔞(M) also has finite length. Thus, since the r + 1 stage of the spectral sequence is obtained from the rth stage by taking homology, the Euler characteristic is preserved from each stage of the spectral sequence to the next. Hence, since the spectral sequence converges to the homology of K_{•} ⊗ M, we have

χ(K_{•} ⊗_{A} M) = χ (\overline{K}_{•} ⊗_{gr𝔞(A)} gr𝔞(M)) .

To complete the proof that this Euler characteristic is equal to k! times the Samuel multiplicity, we interpret the complex \overline{K}_{•} ⊗_{gr𝔞(A)} gr𝔞(M) as a complex of graded modules. Denote this complex \overline{K}_{•}^{M}. Each module \overline{K}_{i}^{M} has a Hilbert polynomial P_{i} such that

P_{i}(n) = ∑_{j=0}^{n−1} length(\overline{K}_{i}^{M})_{j},

where (\overline{K}_{i}^{M})_{j} denotes the component of \overline{K}_{i}^{M} of degree j. However, since \overline{K}_{•}^{M} is a Koszul complex on the associated graded module of M, we also have

P_{i}(n) = \binom{k}{i}P 𝔞_{M}(n − i)

for all i, where P 𝔞_{M} is the Hilbert polynomial of M. The shift by i in \overline{K}_{•}^{M} is necessary so that the boundary maps will be maps of graded modules of degree zero. By the additivity of Hilbert polynomials, ∑_{i=0}^{k}(−1)^{i}P_{i}(n) gives the Hilbert polynomial defined by the homology of \overline{K}_{•}^{M}, which is constant with value χ(\overline{K}_{•}^{M}). But a direct computation (we prove a more general version of this in a later section) shows that ∑_{i=0}^{k}(−1)^{i}\binom{k}{i}P 𝔞_{M}(n − i) is k! times the coefficient of n^{k} in P 𝔞_{M}(n − i), which proves the result.

The point of this computation is that it transforms questions about Euler characteristics into questions about Hilbert polynomials, which are often easier to deal with. We consider one particularly important case. Let R be a regular local ring, and let 𝔭 and 𝔮 be ideals of R such that R/𝔭 ⊗ R/𝔮 has finite length. Suppose 𝔮 is generated by a regular sequence x_{1}, …, x_{k}. Then dim(R/𝔮) = dim(R) − k, so that we have dim(R/𝔭) ≤ dim(R) − dim(R/𝔮) = k, and dim(R/𝔭) + dim(R/𝔮) = dim(R) if and only if dim(R/𝔭) = k. Since x_{1}, …, x_{k} is a regular sequence, the Koszul complex K_{•} on x_{1}, …, x_{k} is a free resolution of R/𝔮. Thus Tor_{i}^{R}(R/𝔮, R/𝔭) is the homology H_{i}(K_{•} ⊗ R/𝔭). Applying the above theorem with M = R/𝔭, we deduce that

χ(R/𝔮, R/𝔭) = e_{k}(𝔮, R/𝔭).

Since the Samuel multiplicity e_{k}(𝔮, R/𝔭) is always non-negative and is positive if and only if the dimension of R/𝔭 is equal to k, this proves the conjectures in this case.

Serre’s proof of the multiplicity conjectures in the equicharacteristic case proceeded by reducing to the case of a regular sequence by reduction to the diagonal. If R is a power series ring k[[X_{1}, …, X_{d}]] and M and N are R-modules with M ⊗_{R}N of finite length, he introduced a new set of variables Y_{1}, …, Y_{d} and considered N as a module over k[[Y_{1}, …, Y_{d}]]. He then defined a “complete” tensor product M ˆ⊗_{k}N over k as a module over the ring k[[X_{1}, …, X_{d}, Y_{1}, …, Y_{d}]] and showed that

Tor_{i}^{R}(M, N) ≅ Tor_{i}^{k[[X_{i},Y_{j}]]}(M ˆ⊗_{k}N, k[[X_{i}, Y_{j}]]/(X_{1} − Y_{1}, …, X_{d} − Y_{d})).

Since X_{1} − Y_{1}, …, X_{d} − Y_{d} form a regular sequence, this proves the result for power series rings, and the conjectures for general equicharacteristic rings can be reduced to this case by completion and the Cohen structure theorems.

3 Gabber’s reduction to regular embeddings

In this section we describe Gabber’s use of de Jong’s theorem on the existence of “regular alterations” to reduce the intersection conjectures to questions on regular embeddings in projective space over R.

As above, let R be a regular local ring and let 𝔭 and 𝔮 be prime ideals of R such that R/𝔭 ⊗ R/𝔮 has finite length. Let d be the dimension of R, let r be the dimension of R/𝔭 and let t be the dimension of R/𝔮.

The following theorem of de Jong ([2]) makes the reduction to a question on regular embeddings possible:

Theorem 2 Let A be a local integral domain which is a localization of a ring of finite type over a discrete valuation ring. Then there exists a projective map ϕ : X → Spec(A) such that

1. X is an integral regular scheme.

2. If K is the quotient field of A, then the extension k(X) of K is finite (we say that X is generically finite over Spec(A)).

For the proof of this theorem we refer to [2]. We show below that this reduces the questions on intersections over a regular local ring to corresponding questions on intersections on projective schemes where one of the schemes is regular. We note that the fact that ϕ is projective means that X is a closed subscheme of Proj(A[X_{0}, …, X_{n}]) for some n. In our application, we apply the theorem to A = R/𝔮. Suppose first that Spec(R/𝔮) is already regular, which means that R/𝔮 is a regular local ring. In that case, 𝔮 is generated by part of a regular system of parameters, so is in particular generated by a regular sequence. Hence the conjectures follow immediately from the results of the previous section on Koszul complexes.

We note that there is an extra assumption, that the ring be a localization of a ring of finite type over a discrete valuation ring (and there are also assumptions on the discrete valuation ring). However the general case can be reduced to this case (see Berthelot [1] or Hochster [5]), and we assume that our rings have this property.

Let X = Spec(R), Z = Spec(R/𝔮) and Y = Spec(R/𝔭). We denote a regular scheme which is projective and generically finite over Z (whose existence follows from de Jong’s theorem) by Z′. Then there exists an n such that Z′ is a closed subscheme of Proj(R/𝔮[X_{0}, …, X_{n}]) and hence also of Proj(R[X_{0}, …, X_{n}]). We let P denote Proj(R[X_{0}, …, X_{n}]) and let ϕ denote both the map from P to X and the induced map from Z′ to Z. Let I denote the graded ideal of R[X_{0}, …, X_{n}] which defines Z′. Let Y′ = ϕ^{−1}(Y) = Proj(R/𝔭[X_{0}, …, X_{n}]).

The generalization from rings to projective schemes involves a corresponding generalization from modules to sheaves. The sheaves we consider will be coherent (see for example Hartshorne [4] for the general theory of sheaves on projective schemes). We recall that a coherent sheaf M on W can be defined either by specifying its modules of sections over the open sets in an affine open cover or, alternatively, by taking the sheaf defined by a finitely generated graded A-module M. We will generally use the second definition, as it is usually more convenient in computing examples.

We let O_{P}, O_{Y′}, and O_{Z′} denote the structure sheaves of P, Y′, and Z′ respectively; they are defined by the graded rings R[X_{0}, …, X_{n}], R/𝔭[X_{0}, …, X_{n}], and R[X_{0}, …, X_{n}]/I. We will also sometimes denote R by O_{X} and similarly for O_{Y} and O_{Z}.

If M and N are sheaves on a projective scheme W defined by graded modules M and N, we define the sheaves T or_{i}^{O_{W}} (M, N) by taking a resolution of M (or N) by locally free sheaves F_{i}, and letting T or_{i}^{O_{W}} (M, N) be the ith homology of F_{•} ⊗ N. Usually we define F_{•} by defining a complex of graded modules F_{i} which define locally free sheaves and which give a resolution of M (or N). In the case where W = P, a bounded resolution can be constructed using direct sums of copies of O_{P} (n), so this process is quite easy to carry out. We also define the complex T or^{O_{W}} (M, N) to be the complex F_{•} ⊗ N. This complex is of course not well-defined as a complex, but it is well-defined up to quasi-isomorphism.

The last ingredient in the generalization to projective space is the pushdown of complexes from P to X by the map ϕ, which we denote ϕ_{∗}. In general this functor is the derived functor of the global section functor on sheaves, but in the case of projective space over R it is not difficult to give a direct definition using Cech cohomology. Let A = R[X_{0}, …, X_{n}], and let P = Proj(A) as above. Let C^{•} be the complex

0 → ∏ A_{X_{i}} → ∏ A_{X_{i}X_{j}} → … → A_{X_{0}X_{1}···X_{n}} → 0

where for any element Y ∈ A, A_{Y} denotes the localization of A obtained by inverting Y. If M_{•} is a bounded complex of coherent sheaves over P represented by a complex of graded modules M_{•}, we then define ϕ_{∗}(M_{•}) to be the graded part of degree zero of the complex C^{•} ⊗_{A} M_{•}. Then ϕ_{∗}(M_{•}) is a bounded complex of R-modules with finitely generated homology and is well-defined up to quasi-isomorphism.

Now suppose that M and N are coherent sheaves on P such that M⊗_{O_{P}} N has support which lies over the closed point of R, which we denote s. Then T or_{i}^{O_{P}} (M, N) has support lying over s for all i, so that the homology of ϕ_{∗}(T or_{i}^{O_{P}} (M, N)) is supported at the maximal ideal and thus has finite length. Hence we can define

χ(M, N) = ∑(−1)^{i}length (H^{i}(ϕ_{∗}(T or^{O_{P}} (M, N))) .

The first part of the reduction is to show that it suffices to show that the Euler characteristic χ(O_{Z′}, O_{Y′}) is non-negative and is zero if dim(R/𝔭) + dim(R/𝔮) < dim(R). The point is that the assumptions on ϕ imply that this new Euler characteristic is closely related to the Euler characteristic χ(R/𝔭, R/𝔮) defined earlier. Let G_{•} be a finite free resolution of R/𝔭 over R. Let F_{•} be a finite locally free resolution of O_{Z′} as above, and let F_{•} be defined by a complex F_{•} of graded modules. We then have the following “projection formula”:

ϕ_{∗}(F_{•}) ⊗ G_{•} ≅ ϕ_{∗}(F_{•} ⊗ ϕ^{∗}(G_{•})).

To prove this formula, we use the definition of ϕ_{∗} in terms of the complex C^{•} defined above. The associativity of the tensor product implies that we have isomorphisms of complexes:

(C^{•} ⊗_{A} F_{•}) ⊗_{R} G_{•} ≅ C^{•} ⊗_{A} (F_{•} ⊗_{R} G_{•}) ≅
≅ C^{•} ⊗_{A} (F_{•} ⊗_{A} (A ⊗_{R} G_{•})).

Since A⊗_{R}G_{•} defines a locally free resolution of O_{Y′}, it is clear that the complex C^{•} ⊗_{A} (F_{•} ⊗_{A} (A ⊗_{R} G_{•})) represents ϕ_{∗}(F_{•} ⊗ ϕ^{∗}(G_{•})). Since (C^{•} ⊗_{A} F_{•}) ⊗_{R} G_{•} represents ϕ_{∗}(F) ⊗ G_{•}, this proves the above isomorphism.

To complete the proof of the fact that it suffices to prove non-negativity and vanishing for χ(O_{Z′}, O_{Y′}), we use induction on the dimension of R/𝔮 together with the assumption that the map induced by ϕ from Z′ to Z is generically finite. Let F_{•} be a locally free resolution of O_{Z′} on P. If we localize at 𝔮, the generic finiteness of ϕ implies that the resulting map from Proj((A/I)𝔮) to Spec((R/𝔮)𝔮) is defined by a finite field extension of a given degree which we denote n. Thus ϕ_{∗}(F_{•}) localized at 𝔮 is isomorphic to ((R/𝔮)𝔮)^{n}, so the complex ϕ_{∗}(F_{•}) is isomorphic to the module (R/𝔮)^{n} up to a complex with homology of dimension strictly less than the dimension of R/𝔮.

By the projection formula, we have that

ϕ_{∗}(F_{•}) ⊗ G_{•} ≅ ϕ_{∗}(F_{•} ⊗ ϕ^{∗}(G_{•})),

where G_{•} is a free resolution of R/𝔭. Since F_{•} is a locally free resolution of O_{Z′} and ϕ^{∗}(G_{•}) is a locally free resolution of O_{Y′}, the complex ϕ_{∗}(F_{•} ⊗ ϕ^{∗}(G_{•})) is quasi-isomorphic to ϕ_{∗}(T or^{O_{P}} (O_{Z′}, O_{Y′})). Hence, taking Euler characteristics and using the above isomorphism, we have χ(ϕ_{∗}(F_{•}), R/𝔭) = χ(O_{Z′}, O_{Y′}). Applying the induction hypothesis, we have that χ(M, R/𝔭) is zero whenever the dimension of M is less than the dimension of R/𝔮. Thus, since ϕ_{∗}(O_{Z′}) is isomorphic to (R/𝔮)^{n} up to something dimension strictly less than the dimension of R/𝔮, we have that

χ(O_{Z′}, O_{Y′}) = χ(ϕ_{∗}(F_{•}), R/𝔭) = χ((R/𝔮)^{n}, R/𝔭) = n(χ(R/𝔮, R/𝔭)).

Thus the vanishing, non-negativity, and positivity of χ(O_{Z′}, O_{Y′}) are equivalent to the corresponding properties of χ(R/𝔮, R/𝔭).

Thus we have reduced the multiplicity conjectures to corresponding conjectures on Euler characteristics defined by subschemes Y′ and Z′ of projective space over R, where Z′ is regular and Y′ is the pullback of a subscheme of Spec(R). In particular, the ideal I defining Z′ is locally generated by a regular sequence, and this fact makes it possible to use the Serre spectral sequence to reduce to the case of associated graded rings.

Let gr_{I}(A) = A/I ⊕ I/I^{2} ⊕ … be the associated graded ring of I. Let B denote A/𝔭A, and let gr_{I}(B) denote the associated graded ring of the image of I in B. We note that both gr_{I}(A) and gr_{I}(B) are bigraded rings, with one grading induced by the grading on A and the other corresponding to powers of I. The ring gr_{I}(B) is also a bigraded module over gr_{I}(A). We make the convention that the i, j component of gr_{I}(A) is the component of I^{j}/I^{j+1} of degree i. We let E denote the scheme Proj(gr_{I}(A)), where gr_{I}(A) is considered to be a graded module by the grading in the first component (the grading induced from that on A). Then E can be defined locally as follows: if U is an affine open set in Z′ and O_{U} is the ring such that U = Spec(O_{U}), then the fiber of E over U is defined to be Spec(C), where C is the associated graded ring of O_{U} by the restriction of I to U. Since I is locally generated by a regular sequence, C is locally a polynomial ring over O_{U}. We note that O_{Z′} is a quotient of both A and gr_{I}(A). Let M denote the sheaf on E defined by the graded module gr_{I}(B).

We next show that the Serre spectral sequence implies that we have an equality:

χ_{E}(O_{Z′}, M)) = χ_{P} (O_{Z′}, O_{Y′}).

Let

0 → F_{k} → … → F_{1} → F_{0} → 0

be a locally free resolution of O_{Z′} over O_{P}. We apply the argument of section 2 to the filtration of F_{•} induced by the powers of I. Since I is locally generated by a regular sequence, the same argument goes through. However, there are two points which are different from the case of the Koszul complex. First of all, F_{•} will in general not be a minimal complex locally, so that it is locally a direct sum of a Koszul complex and a trivial (split exact) complex. However, in the local computation, the split exact part is eliminated in the step from E^{0} to E^{1}, so from that point the argument goes through as before. The second point is that in taking the homology at E^{1}, the homology is no longer of finite length, but only supported at the maximal ideal of R. However, it is still zero except for finitely many i and n and we can conclude that the Euler characteristic is the same using the additivity of ϕ_{∗} and the Euler characteristic on Spec(R). Thus the argument goes through, and we have the above equality

There is one more reduction, which reduces to the fibers over Spec(R/𝔪) = s. Let M be the sheaf on E associated graded ring gr_{I}(B) considered as a module over gr_{I}(A). Then, since for the original ideals 𝔭 and 𝔮 we had that R/𝔭 ⊗ R/𝔮 had finite length, M is annihilated by a power of the maximal ideal 𝔪 of R. Hence it has a finite filtration with quotients M_{i} which are annihilated by 𝔪. It then suffices to show that χ_{E}(O_{Z′}, M_{i}) is non-negative for each i. We can compute this Euler characteristic by taking a locally free resolution for O_{Z′} and tensoring with M_{i}, and, since M_{i} is annihilated by 𝔪, we can tensor first with R/𝔪. Let s denote the closed point of Spec(R) as above, let E_{s} = Proj(gr_{I}(A) ⊗_{R} k), where k = R/𝔪, and let Z′_{s} = Proj(A/I ⊗_{R} k). The above argument shows that for each i we have χ_{E}(O_{Z′}, M_{i}) = χ_{E_{s}}(O_{Z′_{s}}, M_{i}). Hence

χ_{E}(O_{Z′}, M) = ∑_{i} χ_{E}(O_{Z′}, M_{i}) = ∑_{i} χ_{E_{s}}(O_{Z′_{s}}, M_{i}).

We recall that the dimension of M is equal to dim(R/𝔭) + n (where A = Proj(R[X_{0}, …, X_{n}]). Thus to prove the vanishing and non-negativity conjectures it suffices to show that whenever M is a coherent sheaf on E_{s} and dim(M) + dim(Z′) ≤ dim(R) + n we have χ_{E_{s}}(O_{Z′}, M) ≥ 0, and that we have equality when dim(M) + dim(Z′) < dim(R) + n.

To prove the positivity conjecture it would of course suffice to show that if dim(M) + dim(Z′) = dim(R) + n, the Euler characteristic is positive. However, this is not true in generatl (we give an example below). However, assuming the non-negativity conjecture for a moment, we show that there is a simple criterion for positivity.

Proposition 1 Let notation be as above, and let M_{0} be the sehaf defined by gr_{I}(B)⊗_{R} k considered as a module over gr_{I}(A)⊗_{R} k. Assume that dim(R/𝔭)+ dim(R/𝔮) = dim(R). Then the positivity conjecture holds for the ideals 𝔭 and q if and only if χ_{E_{s}}(O_{Z′}, M_{0}) > 0.

Proof. Since M_{0} is a quotient of the sheaf M defined by gr_{I}(B) and Euler characteristics are non-negative, if χ_{E_{s}}(O_{Z′}, M_{0}) > 0, then χ_{E}(O_{Z′}, M) > 0. Conversely, suppose that χ_{E_{s}}(O_{Z′}, M_{0}) = 0. Since gr_{I}(B) is annihlated by a power of 𝔪, it has a filtration with quotients which are homomorphic images of direct sums of copies of gr_{I}(B) ⊗_{R} k. Again using non-negativity, we can deduce that if M_{i} is the sheaf defined by any of these quotients, then M_{i} is a quotient of a direct sum of copies of M_{0}, so we have χ_{E_{s}}(O_{Z′}, M_{i}) = 0. Thus the addivitiy of the Euler characteristic implies that χ_{E}(O_{Z′}, M) = 0.

4 Some examples

We give here three simple examples of the type of bigraded ring which might result from this construction. Each of these examples is obtained by taking a birational map from a regular scheme to Spec(R/𝔮), and the last two are simple resolutions of singularities.

We first summarize the construction up to this point. We began with a regular local ring R and prime ideals 𝔭 and 𝔮. We then took a regular subscheme Z′ of Proj(R[X_{0}, …, X_{n}]) which was generically finite over Spec(R/𝔮). The next step was to replace R[X_{0}, …, X_{n}] with the associated graded ring of I tensored with R/𝔪 = k. The sheaf O_{Y′} defined by B = R/𝔭[X_{0}, …, X_{n}] was then replaced with the sheaf M defined by gr_{I}(B), again tensored with k. The assumption of regularity implies that I/I^{2} is locally free over A/I; denote its rank r. Then the dimension of M is at most r, and it is equal to r if and only if we had dim(R/𝔭) + dim(R/𝔮) = dim(R). We note that the fiber Z′_{s} of Z′ over the maximal ideal of R has dimension at most dim(R/𝔮) − 1, but apart from that we do not know much about it. It is the projective scheme defined by the graded ring (A/I) ⊗_{R} k, which is the part of degree zero in the grading in the second component of the bigraded ring we are considering.

For the first example, let R have dimension four, let t, u, v, w be a regular system of parameters, and define the prime ideals 𝔭 and 𝔮 by letting 𝔭 = (t, u) and 𝔮 = (v, w). In this case, Spec(R/𝔮) is already regular, and we can simply take the projective scheme Proj(R[X]) = Spec(R).

For a slightly more complicated example, consider the subscheme of the projective space Proj(R[X, Y]) defined by the ideal I generated by Xu − Y t. Then Z′ is the blow up of R/𝔮 at the point defined by the maximal ideal, and the fiber over s is projective space of dimension 1. One could define similar examples in higher dimension.

For a third example, let R have dimension 2, and let I be generated by Xu − Y t, Zu − Xt, X^{2} − Y Z. The projective space P has dimension 2, and the fiber over the maximal ideal has codimension one in Proj(k[X, Y, Z]) and thus has dimension one. The sheaf defined by I/I^{2} has rank 2, but I is minimally generated by three elements.

In the above examples it was not really necessary to reduce to projective space since the original quotients R/𝔮 were regular. We next consider an example where the original scheme is not regular. Let 𝔪 be minimally generated by t, u, and let 𝔮 be the principal ideal generated by t^{2} − u^{3}. We can resolve the singularity by letting Z′ be defined by the ideal in R[X, Y] generated by t^{2} − u^{3}, uX − tY, X^{2} − uY^{2}. The fiber Z′_{s} in this case is Proj(k[X, Y]/(X^{2})).

Finally, we consider the case where 𝔮 is the determinantal ideal in R of dimension 4 generated by wu − t^{2}, wv − tu, and tv − u^{2}. In this case the resolution can be found by taking the ideal I in R[X, Y, Z, W] generated by the following elements:

Z^{2} − Y W, Y Z − XW, Y^{2} − XZ, uW − vZ, uZ − vY, uY − vX, u^{2} − tv, tW − vY,

tZ − vX, tY − uX, tu − wv, t^{2} − su, wW − vX, wZ − uX, wY − tX.

The fiber over the maximal ideal is a determinantal subvariety of dimension 1.

In a later section we will return to these examples and consider the question of computing the Euler characteristics χ_{E_{s}}(O_{Z′_{s}}, M) for sheaves M defined as above by certain prime ideals 𝔭 of R.

5 Hilbert polynomials of bigraded modules

In section 2 we showed how the Serre spectral sequence can be used to express the Euler characteristic defined by a Koszul complex in terms of the Samuel multiplicity. In this section we show that similar results hold in the present situation. We now let C denote the bigraded ring which we previously denoted gr_{I}(A)⊗_{R}k, where C_{i,j} consists of the elements of (I^{j}/I^{j+1})⊗k of degree i. Thus in our present notation, E_{s} = Proj(C), where the grading on C is that in the first coordinate. Let C_{0} denote the subring ⊕_{i}(C_{i,0}). Let r be the rank of I/I^{2}, and let M be a bigraded module defining a sheaf M on E_{s} of dimension at most r; we define the dimension of M to be the dimension of the associated sheaf. We consider the question of computing the Euler characteristic χ_{E_{s}}(O_{Z′_{s}}, M), which we also denote χ(C_{0}, M).

Let

0 → F_{k} → … → F_{1} → F_{0} → C_{0} → 0

be a complex of bigraded modules which defines a locally free resolution of C_{0} over C. For any finitely generated bigraded module N, we let P_{N}(m, n) by the Hilbert polynomial of N; more precisely, we define P_{N} to be the polynomial in two variables such that

P_{N}(m, n) = ∑_{i=0}^{n−1} length(N_{m,i})

for large m and n. The degree of P_{N} is equal to the dimension of N (that is, the dimension of the sheaf defined by N on E_{s}). Let M be a bigraded module of dimension at most r as above. Then M ⊗ F_{i} has dimension at most r, and we have that the alternating sum of P_{M⊗F_{i}} is constant with value equal to χ(C_{0}, M).

We will prove this in a special case below (and reduce to non-negativity conjecture to this special case in the next section). We first briefly consider the question of constructing a resolution F_{•} of C_{0}. One method is to take the E^{1} term of the Serre spectral sequence as defined in the previous section, starting from a locally free resolution of A/I over A. However, even though A/I has a nice resolution by sums of shifts of A, the resulting locally free sheaves in the resolution over the associated graded ring will not be so simple. An alternative approach is to take a global Kozsul complex

… Λ^{2}(I/I^{2}) ⊗ gr_{I}(A) → Λ^{1}(I/I^{2}) ⊗ gr_{I}(A) → gr_{I}(A) → A/I → 0.

The resolution over C can then be obtained in either of these constructions by tensoring with k. This resolution gives an expression for the Euler characteristic in terms of the Chern classes of I/I^{2}, but again it is not easy to see how to use this information to compute Euler characteristics.

For the remainder of this section we assume that I/I^{2} ⊗_{R} k is a sum of copies of O_{Z′_{s}}(−k_{i}) for various k_{i}, so that C is a polynomial ring C_{0}[T_{1}, …, T_{r}] over C_{0}, where T_{i} has degree (k_{i}, 1) in the bigrading on C. As mentioned above, the non-negativity conjecture will be reduced to this situation in the next section. In this case the resolution is the usual Koszul complex on T_{1}, …, T_{r}, and the Hilbert polynomial of M ⊗ F_{i} is a sum of Hilbert polynomials of M with shifts in the degrees. Furthermore, the Koszul complex on T_{1}, …, T_{r} is a tensor product of Koszul complexes on the individual T_{i}, and we can compute the Hilbert polynomial of the tensor product K_{•}(T_{1}, …, T_{r}) ⊗ M by tensoring by each factor K_{•}(T_{i}) in turn and keeping track of the result. As above, assume that the dimension of M is at most r, and let Q_{M}^{r}(m, n) be the component of P_{M}(m, n) of degree r. Let T_{i} have degree (k, 1), and consider the Hilbert polynomial obtained by tensoring with the complex

0 → C[(−k, −1)] →^{T_{i}} C → 0.

The Hilbert polynomial of the resulting complex K_{•}(T_{i}) ⊗ M will be given by the polynomial whose value at (m, n) is P_{M}(m, n) − P_{M}(m − k, n − 1). We compute this difference for a monomial m^{i}n^{j} and obtain

m^{i}n^{j} − (m − k)^{i}(n − 1)^{j} = m^{i}n^{j} − (m^{i} − ikm^{i−1} + …)(n^{j} − jn^{j−1} + …) =

m^{i}n^{j} − m^{i}n^{j} + ikm^{i−1}n^{j} + jm^{i}n^{j−1} + … = ikm^{i−1}n^{j} + jm^{i}n^{j−1} + …

where the remaining terms have lower degree. Since we are concerned with the component of highest degree, this suffices for our computation. We note that we can express this result by the formula

Q_{K_{•}(T_{i})⊗M}^{r−1} = \frac{∂Q^{r}}{∂n} + k \frac{∂Q^{r}}{∂m} .

Iterating this process, where we let T_{i} have degree (k_{i}, 1) for each i, we have

χ(C_{0}, M) = ∏_{i=1}^{r} (\frac{∂}{∂n} + k_{i} \frac{∂Q^{r}}{∂m} ) Q_{M}^{r}.

In this formula Q_{M}^{r} could be replaced with P_{M}.

Theorem 3 Let C = C_{0}[T_{1}, …, T_{r}], where T_{i} has degree (k_{i}, 1) as above, and let M be a bigraded C-module of dimension at most r.

1. If dim(M) < r, then χ(C_{0}, M) = 0.

2. If k_{i} ≥ 0 for all i, then χ(C_{0}, M) ≥ 0.

3. If k_{i} = 0 for all i, then χ(C_{0}, M) > 0 if and only if the coefficient of n^{r} in P_{M} is non-zero.

4. If k_{i} > 0 for all i and dim(M) = r, then χ(C_{0}, M) > 0.

Proof. If the dimension of M is less than r, its Hilbert polynomial has degree less that r, so the result of taking r partial derivatives is zero. Thus (i) holds.

We prove (ii) and (iv) by induction on r. By taking a filtration of M, we may assume that M is of the form (C/𝔭)[(i, j)], where 𝔭 is a bigraded prime ideal of C and [(i, j)] denotes a shift in degrees. Suppose some T_{i} is not in 𝔭. Then T_{i} is a non-zero divisor on M, and we can tensor with the Koszul complex on T_{i}, replacing M with M/T_{i}M and reducing r by one. Thus the result follows by induction. If all T_{i} are in 𝔭, then its Hilbert polynomial is constant with respect to n, so we have Q^{r}(m, n) = αm^{r} for some α ≥ 0. Hence the above formula states that

χ(C_{0}, M) = k_{1}k_{2} ⋯ k_{r}(r!)α.

If all the k_{i} are greater than or equal to zero, we thus have χ(C_{0}, M) ≥ 0. If all the k_{i} are greater than zero and M has dimension r, then α > 0 and χ(C_{0}, M) > 0. This proves (ii) and (iv).

If all the k_{i} are zero, then χ(C_{0}, M) is simply the rth derivative of P_{M}, so it is positive if and only if the coefficient of n^{r} is positive. On the other hand, this coefficient gives the length of the module ⊕_{i=1}^{n−1}M_{m,i} for sufficiently large n up to terms of lower degree in n, so it cannot be negative.

The graded ring obtained from the original situation will be of the form considered here when I is globally defined by a regular sequence, and the k_{i} will then be the degrees of the generators. We give an example to show that the condition the M has dimension r does not suffice for χ(C_{0}, M) to be positive. Let R have dimension 3 and let t, u, v be a regular system of parameters. Let 𝔮 be the ideal generated by v, and let I be the ideal of R[X, Y] generated by v and uX − tY. Then the fiber over the closed point is projective space of dimension one, C_{0} = k[X, Y], and C = C_{0}[T_{1}, T_{2}] with k_{1} = 0 and k_{2} = 1. Then if M = C/T_{1}, M has dimension 2 and χ(C_{0}, M) = 0.

Exercise: Prove (without using the Serre positivity conjecture) that the module M in the previous paragraph could not arise from a prime ideal 𝔭 such that R/𝔭 ⊗ R/𝔮 has finite length and dim(R/𝔭) + dim(R/𝔮) = dim(R).

6 Gabber’s proof of non-negativity

In this section we complete Gabber’s proof of the non-negativity of intersection multiplicities. We have seen in the last section that if gr_{I}(A) ⊗_{R} k is a polynomial ring over (A/I) ⊗_{R} k generated by elements of non-negative degree, then non-negativity follows. We show here that we can embed gr_{I}(A) ⊗_{R} k into a polynomial ring of this type. Let A_{0} denote A/I ⊗_{R} k. Actually, we show instead that we can embed the symmetric algebra Sym_{A_{0}}((I/I^{2}) ⊗_{R} k) into a polynomial ring by a locally flat map. Since I/I^{2} is locally free, the map from the symmetric algebra to the associated graded algebra defines an isomorphism of schemes, so this suffices to prove the result. Let S = Sym_{A_{0}}((I/I^{2}) ⊗_{R} k).

Let E_{s} denote Proj(gr_{I}(A)⊗_{R} k) = Proj(Sym_{A_{0}}((I/I^{2})⊗_{R} k)) as above. Let W = Proj(A_{0}[T_{1}, …, T_{r′}]) for T_{i} of degree (k_{i}, 1) for some integer r′. Suppose that f is a map from S into the polynomial ring A_{0}[T_{1}, …, T_{r′}] such that the map ϕ induced by f from W to E_{s} is flat of relative dimension r′ − r, where r is the rank of I/I^{2}. Then we have a commutative diagram

S ⟶^{ϕ} A_{0}[T_{1}, …, T_{r′}]
↘  ↓
A_{0}

which induces a commutative diagram of schemes

\begin{aligned}E_{s} & ⟵^{ϕ} & W \\ & ↖ & ↑ \\ & & Z′_{s} \end{aligned} .

Let M be a sheaf on E_{s} defined by a bigraded module M. Since we are assuming that ϕ is flat, we have an isomorphism

T or_{i}^{O_{Es}} (O_{Z′_{s}}, M) ≅ T or_{i}^{O_{W}} (O_{Z′_{s}}, ϕ^{∗}(M))

for all i. Thus we have an equality of Euler characteristics

χ_{E_{s}}(O_{Z′_{s}}, M) = χ_{W} (O_{Z′_{s}}, ϕ^{∗}(M)).

Thus if we can find such a map f such that all the k_{i} are non-negative, the conjecture will follow. We now show that such an embedding exists. Gabber’s proof uses the fact that the dual of I/I^{2} over s is generated by global sections; we define this map directly without dualizing. At this point we assume that R is ramified. Although this is an unusual assumption, it is possible to reduce to the ramified case by a finite flat extension of R, for example by adjoining a square root of p, where R has mixed characteristic p. Let t_{1}, …, t_{d} be a minimal set of generators of the maximal ideal 𝔪 of R. Since R is ramified, R/𝔪^{2} is isomorphic to a polynomial ring in t_{1}, …, t_{d} modulo the square of the

ideal generated by t_{1}, …, t_{d}. Thus for each i, the partial derivative \frac{∂}{∂t_{i}} defines

a map from R/𝔪^{2}[X_{0}, …, X_{n}] to R/𝔪[X_{0}, …, X_{n}]. By taking the composition with the map from I to R/𝔪^{2}[X_{0}, …, X_{n}] induced by the inclusion of I into A and with the map from R/𝔪[X_{0}, …, X_{n}] = A ⊗_{R} k to (A/I) ⊗_{R} k = A_{0}, we obtain a map from I to A_{0}. Since for all a in A and i in A the partial derivatives satisfy

\frac{∂(ai)}{∂t_{i}} = a\frac{∂i}{∂t_{i}} + i\frac{∂a}{∂t_{i}}

and A_{0} is annihilated by I, we can deduce that the map induced by \frac{∂}{∂t_{i}} vanishes

on I^{2} and defines a homomorphism of A/I-modules from I/I^{2} to A_{0}.

Similarly, for each i = 0, …, n we have a map induced by \frac{∂}{∂X_{i}} from I/I^{2}

to A_{0}[−1], where the shift in degree arises from the fact that these partial derivatives lower the degree by 1.

Putting these together, we have a map from I/I^{2} to A_{0}^{d} ⊕ A_{0}[−1]^{n+1}, which define a map f from S to A_{0}[T_{1}, …, T_{d}, S_{0}, …, S_{n}] where the T_{i} have degree 0 and the S_{i} have degree 1.

Theorem 4 The map ϕ is locally an inclusion of polynomial rings. In particular, it is locally flat of relative dimension d + n + 1 − r.

Proof. This is one of the main points of the proof, and it is the only place where the full strength of the assumption that Z′ is regular is used. It suffices to show that for every closed point p of Z′_{s}, the map from I/I^{2} to A_{0}^{d}⊕A_{0}[−1]^{n+1} defines a split inclusion locally at the point p. We assume that the residue field is algebraically closed (which we can do by a flat extension) and look at the maximal ideal 𝔪_{p} corresponding to p. The local ring at p in P, which we denote A_{p}, is isomorphic to R[u_{1}, …, u_{n}]_{𝔪_{p}}, where, after a change of coordinates, we may assume that u_{1}, …, u_{n} together with a set of generators of 𝔪_{R} generate 𝔪_{p}. Since Z′ is regular, I is generated locally by part of a regular system of parameters i_{1}, …, i_{r}. Furthermore, the quotient I/I^{2} is locally generated by the images of i_{1}, …, i_{r}. Since i_{1}, …, i_{r} form part of a regular system of paramteres, the images of their partial derivatives in (A_{p}/𝔪_{p})^{d} ⊕ (A_{p}/𝔪_{p})^{n+1} are linearly independent. Hence the map from (I/I^{2}) ⊗ k to A_{0}^{d} ⊕ A_{0}[−1]^{n+1} locally defines a split inclusion at p as was to be shown.

This completes the proof of the Serre nonnegativity conjecture. Since certain of the indeterminates in the polynomial ring used in the proof have degree zero, it does not show that the Euler characteristics must be positive. In fact, as we showed at the end of the previous section, the locally free sheaf defined by I/I^{2} is not itself positive enough to ensure positivity. Thus the positivity conjecture requires studying the sheaf M coming from the associated graded ring of I on R/𝔭[X_{0}, …, X_{n}].

We note that we can embed A[−1] into A^{n+1} by a locally split embedding which sends 1 to (X_{0}, …, X_{n}) and thus embed S into a polynomial ring D generated by d + (n + 1)^{2} elements all of which have degree zero. Thus one criterion for the positivity conjecture to hold is that if we take the quotient of D by the image in D of the kernel of the map from gr_{I}(A) to gr_{I}(R/𝔭[X_{0}, …, X_{n}]), then (under the usual assumptions) the coefficient of n^{d+(n+1)^{2}} in the Hilbert polynomial of this quotient is not zero.

We remark that the construction we have presented is quite computational in the sense that it is possible to compute the embedding ϕ explicitly in special cases. We give two simple examples. First, let R have dimension 2 with maximal ideal generated by t, u, let A = R[X, Y], and let I be generated by uX − tY. Then A_{0} = k[X, Y]. The map f to A_{0}[S_{1}, S_{2}, T_{0}, T_{1}] induced by the partial derivatives sends uX − tY to −Y S_{1} + XS_{2} + uT_{0} − tT_{1}, which, after dividing by 𝔪, is −Y S_{1} +XS_{2}. Let 𝔭 = (t, u). Then uX −tY is zero modulo 𝔭, so the kernel on the map of graded rings is generated by the image of uX −tY in I/I^{2}. Hence M is mapped to the sheaf associated to A_{0}[S_{1}, S_{2}, T_{0}, T_{1}]/(−Y S_{1} + XS_{2}). It can be verified that this quotient satisfies the condition on Hilbert polynomials; the positivity condition also follows from the fact that −Y S_{1} + XS_{2} has degree (1, 1).

Finally, we consider the example from section 3 in which I is generated by t^{2} − u^{3}, uX − tY, X^{2} − uY^{2}. Then I/I^{2} has rank 2. Taking derivatives, we see that the map ϕ (after dividing by 𝔪) satisfies ϕ(t^{2} − u^{3}) = 0, ϕ(uX − tY) = XS_{1} − T S_{2}, and ϕ(X^{2} − uY^{2}) = −Y^{2}S_{2} + 2XT_{0}. To compute the result of intersecting with Y′, where Y′ is generated by an ideal 𝔭, it suffices to compute the kernel of the map from the symmetric algebra on I/I^{2} to the associated graded ring of I on R/𝔭[X, Y] tensored with k, and then find the image of this kernel in A_{0}[S_{1}, S_{2}, T_{0}, T_{1}]. On the other hand, in this case Proj(A_{0}) = Proj(k[X, Y]/(X^{2})) has dimension zero, so that the locally free sheaf defined by (I/I^{2}) ⊗_{R} k is actually positive.

Similar examples can be computed from the other examples in section 3.

References

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