Research Statement
Sean Sather-Wagstaff
Note. All rings are commutative and Noetherian with identity. All modules are finitely
generated and unital.
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Historical Context
Assume that (R, ) is a local ring with modules M and N such that M has finite projective
dimension and M ⊗R N has finite length. Under these assumptions, Serre [11] defined the
intersection multiplicity of M and N as
X
dim(R)
χ(M, N ) =
(−1)i length(TorR
i (M, N )).
i=0
If the ring R is regular, he proved that
dim(M ) + dim(N ) ≤ dim(R)
and conjectured the following.
(Vanishing) If dim(M ) + dim(N ) < dim(R), then χ(M, N ) = 0.
(Nonnegativity) χ(M, N ) ≥ 0.
(Positivity) If dim(M ) + dim(N ) = dim(R), then χ(M, N ) > 0.
Serre was able to verify each statement in the case when R is unramified. Since χ(M, N )
has many of the characteristics we desire from an intersection multiplicity (for example,
Bézout’s Theorem holds), it was not unreasonable to suppose that these further properties
are satisfied for an arbitrary regular local ring. The results were left unproved for ramified
rings.
The vanishing conjecture was proven about fourteen years ago [4, 9] using K-theoretic
methods. The proof in [4] uses the theory of Adams operations on Grothendieck groups
of complexes, while that in [9] uses the theory of local Chern characters. Gabber proved
the nonnegativity conjecture very recently1 using a theorem of de Jong [2]. The positivity
conjecture remains open in the ramified case. Using methods introduced by Gabber, K.
Kurano and P. Roberts have proved the following.
Theorem 1.1. ([6] Theorem 3.2) Assume that (R, ) is a regular local ring which either
contains a field or is ramified. Also, assume that and are prime ideals in R such that
+ is -primary and ht + ht = dim R. If χ(R/ , R/ ) > 0 then
(n)
∩ ⊆
n+1
1I
for all n ≥ 0
(1)
have no direct reference to Gabber’s work as it remains unpublished. However, detailed treatments of
Gabber’s work on this problem can be found in [1, 5, 10].
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As a result, they conjectured that (1) should hold in all regular local rings. More specifically,
Conjecture 1.2. Assume that (R, ) is a regular local ring and that and are prime
ideals in R such that + is -primary and ht + ht = dim R. Then (n) ∩ ⊆ n+1 ,
for all n ≥ 0.
Note. There are straightforward examples showing that none of the assumptions we place
on the local ring R and the prime ideals and may be relaxed.
As positivity is known when R contains a field, this conjecture holds for equicharacteristic
regular local rings. Kurano and Roberts asked whether there was a proof in this case which
does not use positivity.
We study Conjecture 1.2 because, if we can verify this conjecture, then the tools used may
be applicable to the positivity conjecture.
Definition 1.3. Assume that (R, ) is a Noetherian local ring. Given a finitely generated
R-module M 6= 0 and a proper ideal of R such that M/ M has finite length, the Hilbert
polynomial of on M , denoted H[ , M ](n), is the polynomial in n of degree r = dim(M )
with rational coefficients such that for n 0
H[ , M ](n) = length(M/
n+1
M ).
If er is the leading coefficient of H[ , M ](n), then the multiplicity of
integer e( , M ) = r!er . We denote e( , M ) by e(M ).
on M is the positive
If R is regular with prime ideal and 0 6= f ∈ then it is straightforward to verify that
(e+1)
e = e(R/(f )) if and only if f ∈ (e)
. It follows that Conjecture 1.2 may be rephrased
as the following.
Conjecture 1.20 Assume that (R, ) is a regular local ring and that and are prime
ideals in R such that + is -primary. If there exists 0 6= f ∈ ∩ such that e(R /(f )) =
e(R/(f )), then dim(R/ ) + dim(R/ ) ≤ dim(R) − 1.
Conjecture 1.20 motivates the following generalization.
Conjecture 1.4. Assume that (R, ) is a universally catenary local ring with module M
such that √
R/Ann(M ) is equidimensional. Also, assume that and are prime ideals in R
such that
+ = and e(M ) = e(M ). Then dim(R/ ) + dim(R/ ) ≤ dim(M ).
Primarily, my research focuses on verifying Conjectures 1.2 and 1.4.
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Secondary Context
As noted above, Serre proved Conjecture 1.4 in the case where R is regular. Many attempts
have been made to generalize Serre’s result to the nonregular situation. It can be shown
easily that, if one drops the assumption of regularity on the ring, then one must assume
that the objects under investigation have additional properties which would be automatic if
the ring were regular. In our conjectures and results, we assume that the ring satisfies the
property e(R ) = e(R). When the ring is regular, this is automatic because the localization
R is also regular so that e(R) = 1 and e(R ) = 1. In a famous conjecture, Peskine and
Szpiro [8] focus on the finiteness of projective dimensions over regular local rings.
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Conjecture 2.1. (Peskine and Szpiro [8]) Assume that (R, ) is a local
√ ring and and
are prime ideals in R such that has finite projective dimension and
+ = . Then
dim(R/ ) + dim(R/ ) ≤ dim(R).
In addition to verifying the conjectures listed above, I am interested in finding connections,
if any exist, between Conjectures 1.4 and 2.1.
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Properties of Multiplicities
In Conjecture 1.4 the assumption that R is universally catenary may seem a bit odd. The
reason for this assumption is as follows.
The condition e(M ) = e(M ) is quite strong. For instance, if M = R and R/ is regular,
then e(R ) = e(R) if and only if has its analytic spread equal to its height. In order
for the multiplicity condition to give us information about the module, we must restrict
our attention to a class of rings and modules where the multiplicity is well-behaved. In
particular, we focus on the rings and modules where e(M ) ≤ e(M ) for all prime ideals of
R. Lech [7] asked what conditions guarantee that this multiplicity inequality holds. It can
be shown that, if R0 = R/Ann(M ) is quasi-unmixed and Nagata, then R and M satisfy this
condition. It is my belief that the assumption that R0 is Nagata is unnecessary. It can be
shown easily that R0 must be equidimensional. It can also be shown, although not so easily,
that R0 must be catenary. Currently, I am investigating whether R 0 must be universally
catenary in order for this multiplicity inequality to hold.
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Past Research
My initial research centered on Conjecture 1.2. I found a proof of this conjecture for regular local rings containing a field which does not use positivity, thus answering one of the
questions of Kurano and Roberts.
Subsequently, I considered general, “well-behaved” situations and proved the conjecture in
the following cases.
1. dim(R) ≤ 3.
2. The ideal
is generated by a regular sequence. (Here the ideal
need not be prime.)
3. The factor ring R/ is regular.
4. The sum
+ =
.
Conjecture 1.4 was suggested more recently and since then it has been my main research
focus. I have established this conjecture, most notably, in the case where R is excellent and
R/Ann(M ) contains a field.
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Future Research
Primarily, I am interested in continuing my research on the conjectures discussed above. As
the recent work of Gabber, Kurano and Roberts shows, current researchers in the field of
commutative algebra and algebraic geometry are very interested in the questions raised by
Serre in his work on intersection multiplicities. Progress toward a solution to the questions
posed by Kurano and Roberts could yield new methods to apply to the positivity conjecture.
Furthermore, there is a current interest in the properties of symbolic powers of ideals (cf. [3]
for example) and these methods may improve our understanding in this area.
My plan of attack for the future is threefold.
1. Investigate the conditions under which the inequality e(M ) ≤ e(M ) holds for all prime
ideals .
2. Investigate the Conjecture 1.4 in the mixed-characteristic or nonexcellent situation.
3. Investigate the relationship between Conjectures 1.4 and 2.1 and other homological
conjectures in commutative algebra (the existence of big Cohen-Macaulay modules, the
Direct Summand Conjecture, and so on). We have already seen that Conjecture 1.4 is
related to the Positivity Conjecture, so this is a reasonable line of inquiry. Since many
of the classical conjectures are known in the unramified cases, this would certainly
give us new results.
References
[1] P. Berthelot. Altérations de variétés algébriques [d’après A. J. de Jong]. Séminaire
Bourbaki, (815), 1996.
[2] A. de Jong. Smoothness, stability, and alterations. Publ. Math. IHES 83, pages 51–93,
1996.
[3] D. Eisenbud and B. Mazur. Evolutions, symbolic squares, and Fitting ideals. J. Reine
Angew. Math., 488:189–201, 1997.
[4] H. Gillet and C. Soulé. K-théorie et nullité des multiplicitiés d’intersection. C. R.
Acad. Sci. Paris Sér. I Math. no. 3, pages 71–74, 1985.
[5] M. Hochster. Nonnegativity of intersection multiplicities in ramified regular local rings
following Gabber/De Jong/Berthelot. unpublished notes.
[6] K. Kurano and P.C. Roberts. The positivity of intersection multiplicities and symbolic
powers of prime ideals. to appear in Compositio Math.
[7] C. Lech. Inequalities related to certain couples of local rings. Acta Math., 112:69–89,
1964.
[8] C. Peskine and L. Szpiro. Dimension projective finie et cohomologie locale. applications
à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck. Inst.
Hautes Études Sci. Publ. Math., (42):47–119, 1973.
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[9] P. Roberts. The vanishing of intersection multiplicities of perfect complexes. Bull.
Amer. Math. Soc. (N.S.), 13(2):127–130, 1985.
[10] P. Roberts. Recent developments on Serre’s multiplicity conjectures: Gabber’s proof
of the nonnegativity conjecture. Enseign. Math. (2), 44:305–324, 1998.
[11] J.-P. Serre. Algèbre Locale - Multiplicités, Troisième édition. Springer-Verlag, 1975.
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