RESEARCH STATEMENT
Julian Chan
My research interests include algebra, algebraic geometry, and invariant theory. I have also done
some research in statistics, and will be receiving a masters of statistics in May of 2011. I list each
research area and will describe each research area below.
Local Cohomology
F-injectivity
Invariant theory
Tight Closure
Local Cohomology
Given an ideal I of a ring R the cohomology modules HIi (R) encode important information such
as the depth of R with respect to the ideal I . If R has dimension d then R is Cohen{Macaulay if
Hmi (R) = 0 for i < d, and Hmd (R) =
6 0. Cohomology is arguably one of the most important ideas
in all of mathematics. It is natural to study the structure of local cohomology. An object that
describes the structure is the associated primes of local cohomology modules. An associated prime
of local cohomology module is a prime ideal of R that is the annihilator of some element of the local
cohomology module. In certain situations we view the local cohomology module as a module over Z
and in this situation the associated primes are a subset of primes so that px = 0, for p a prime, and
x a nonzero element of the local cohomology module. Let R be a commutative ring and I an ideal
of R. In [Hu] Huneke asks if local cohomology modules HIk (R) have only nitely many associated
prime ideals. In [Si2] Singh constructed a counterexample with the hypersurface
[x; y; z; u; v; w] :
R = Z(ux
+ vy + wz )
Singh proved that for each prime integer p, the element
p
)p + (wz )p
p = (ux) +p((vy
xyz )p
of the local cohomology module H(3x;y;z)(R) is p-torsion; equivalently H(3x;y;z)(R) contains an isomorphic copy of the abelian group Z=pZ for every prime integer p. I proved the following theorem:
[x;y;z;u;v;w] and consider the local cohomology module H 3 (R) with
Theorem 1. [Chan] Let R = Z(ux
(x;y;z)
+vy+wz)
the standard Z-grading. Then each nitely generated abelian group embeds into a graded component
H(3x;y;z)(R).
Moreover, the result remains true if we assign to the ring a ne Z4 -grading. In addition, given
a graded component of the local cohomology module I determine the torsion and free components
of the group structure. In [Ly1], Lyubeznik conjectured that if R is a regular ring and I is an
ideal of R, then local cohomology modules HIk (R) have only nitely many associated prime ideals.
This conjecture has lead to several results. Huneke and Sharp, proved that if a regular local ring
R contains a eld of prime characteristic p, then the set of associated primes is a nite set [HS].
1
2
Lyubeznik proved that if R contains a eld of characteristic 0, and R is any regular K -algebra then
the set of associated primes is nite [Ly1]. Later Lyubeznik developed a characteristic free approach,
and proved if R is a regular ring containing a eld then the set of associated primes is nite [Ly2].
Broadmann and Hellus asked if AssR0 (HRi + (M )n ) is asymptotically stable as n ! 1 (i.e. there
is an n0 2 Z such that AssR0 (HRi + (M )n ) = AssR0 (HRi + (M )n0 ) for all n n0 . They proved that if
R0 is local M is a nitely generated and graded R-module. In addition if the R-module HRj + (M )
is nitely generated for all j < i then AssR0 (HRi + (M )n ) is asymptotically stable for n ! 1. My
work shows explicitly that this may fail when R0 is not a local ring and it can be shown in which
degrees of the graded components this fails. Similar work was done by Brodmann, Katzman, and
Sharp in [BKS].
F-injectivity
The study of F-injectivity dates back to Fedder and Watanabe [FW]. The action of Frobenius on
a ring is given by F : R ! R where F (r) = rp . A ring R is said to be F-injective if the Frobenius
map induces an injective map on local cohomology F : Hmi (R) ! Hmi (R) for all 0 i dim(R).
Fedder and Watanabe showed that for a Cohen{Macaulay ring R, R is F-rational if and only if R is
F-injective and F-unstable.
Diagonal subalgebras of multigraded rings are particularly interesting; in certain situations Kurano, Sato, Singh, and Watanabe showed they correspond to homogeneous coordinate rings of blow
ups of projective varieties. Working in this setting I proved the following result about the Finjectivity of diagonal subalgebras of multigraded rings.
Theorem 2. [Chan] Let K be a eld, let m; n be integers with m; n 2, and let
R = K [x1 ; : : : ; xm ; y1 ; : : : ; yn]=(f )
be a normal N2 -graded hypersurface where deg xi = (1; 0), deg yj = (0; 1), and deg f = (d; e) > (0; 0).
For positive integers g and h, consider the diagonal = (g; h)Z. Then the ring R is not F-injective
whenever d m + 1 or e n + 1.
The theorem below is a main consequence of the work I have done with the F-injectivity of multigraded rings. A problem of interest in many settings in mathematics is the property of deformation.
It is an open question if F-injectivity deforms i.e., let x 2 R be a nonzerodivisor of R, and R=(xR)
be F-injective then is R F-injective? In this setting I prove that diagonal subalgebras of multigraded
rings have the deformation property of F-injectivity with the following theorem.
Theorem 3. [Chan] Let S is a normal ring over a eld, and g 2 S is a nonzerodivisor of S such
that S=(g)S = R . If the ring R is F-injective then S is F-injective.
Invariant Theory.
Let R = K [x1 ; : : : ; xn ] and G a nite group viewed as a subset of the symmetric group on n
letters. Let f (x1 ; : : : ; xn ) 2 R and g 2 G. We dene an action of G on R by g : f (x1 ; : : : ; xn ) !
f (xg(1) ; : : : ; xg(n) ), and dene the ring of invariants as
RG = ff (x1 ; : : : ; xn ) 2 R such that f (x1 ; : : : ; xn ) = f (xg(1) ; : : : ; xg(n) ) for all g 2 Gg:
3
When K = Fq is a nite eld of characteristic p, the behavior of RG is wildly mysterious. M.-J
Bertin showed that that if G is a cyclic group acting on R by cyclically permuting the variables of
R, then RG need not even by Cohen-Macaulay. Others have attempted to characterize the behavior
of RG in this situation. Let S be a subring of R. We say that P : R ! S is a Reynolds operator
for (S; R) if P is a S -module homomorphism and the restriction of P to R is the identity map on S .
The Reynolds operator makes RG a direct summand of R as an RG module, and so RG is F-regular
in this case. Hochster and Eagon used the Reynolds operator to show that if G is a nite group and
the characteristic of K is coprime to the order of G, then RG is Cohen{Macaulay.
Related to Hochster and Eagon's results is the theory of tight closure which was invented by
Hochster and Huneke [HH].
Denition 4. [HH] Let I = (y1; : : : yn) be an ideal of a ring R of characteristic p > 0, and let R
denote the complement of the minimal primes of R. We say that x 2 I , the tight closure of I , if
there exists a c 2 R such that cxq 2 (y1q ; : : : ; ynq ) for all q = pe . If I = I then we say that I is
tightly closed.
A ring R is weakly F-regular if every ideal of R is tightly closed, and is F-regular if every localization is weakly F-regular. A ring R is said to be F-rational if every ideal generated by a system of
parameters is tightly closed.
I used the Reynolds operator to study Sylow p-groups and the tight closure of rings of invariants.
Theorem 5. [Chan] Let R = K [x1; : : : ; xn] be a polynomial ring in n variables over a eld of
characteristic p. Let G be a group acting on R, and H a Sylow p subgroup of G.
(1) If RH is F-rational so is RG .
(2) If RH is F-regular so is RG .
(3) If RH is F-pure so is RG .
Consequently to prove that RG has one of the properties it is sucient to check the result for
RH . When G is the alternating group on n letters Singh showed the following.
Theorem 6. [Si1] Let R = K [x1; : : : ; xn] be a polynomial ring in n variables over a eld of characteristic p, an odd prime, and let the alternating group An , act on R by permuting the variables.
Then the invariant subring RAn is F-regular if and only if the order of the group jAn j is relatively
prime to p.
Consider elementary symmetric functions.
e1 =
Q
X
1jn
xj ;
e2 =
X
1j<kn
xj xk ; : : : ;
en = x1 : : : xn
Dene D = i>j (xi xj ). It is an exercise in a beginning graduate algebra class to show the
ring of invariants of A3 and S3 are:
RA3 = K [e1; e2 ; e3 ; D] and RS3 = K [e1; e2 ; e3 ]
4
One can show that RS3 is F-regular explicitly using the above representation, but by the above
theorem RA3 is not F-regular. This computation shows that the converse of (1) to my theorem is
false.
Future Research
We have seen that Singh's results gives an example of a rings in which RG is F-rational, but
RH is not. An interesting project is to determine if the converse of (2) or (3) to theorem
7 also fails. Another interesting question is if RH is F-injective is RG F-injective? It has
been an unresolved problem for some time to determine when the rings of invariants RG
is F-rational or F-regular when working over a eld of characteristic p. The result I have
proven in my thesis is a rst step to understanding this type of behavior, and allows us to
consider the Sylow p-subgroups when trying to prove these properties.
Lyubeznik's conjecture in the case of polynomial rings over Z remains unresolved. Let A
and B be two N graded rings over a eld K . The Segre product of A and B is the ring
A#B =
M
n0
A n K B n :
This is a graded ring, and if A and B correspond to projective varieties this corresponds to
the homogeneous coordinate ring for the Serge embedding. Let R = Z[x; y; z; u; v; w] and
consider the cubic polynomial x3 + y3 + z 3 which denes a smooth elliptic curve in Ep for
any characteristic p 6= 3. The dening ideal of E P1 is
I = (u3 + v3 + w3 ; u2x + v2 y + w2 z; ux2 + vy2 + wz 2 ;
x3 + y3 + z 3 ; vz wy; wx uz; uy vx)
It is known that the cohomological dimension cd(R=pR; I ) varies with the characteristic
p. It is an open question if HI4 (R) has innitely many associated primes. If we can prove
this ring has innitely many associated primes we will answer Lyubeznik's conjecture in the
negative.
F-injectivity has several consequences both in algebra and algebraic geometry. It remains
unsolved if F-injectivity deforms in general, and I will be studying this in more detail.
Another interesting question about F-injectivity is the following: when is R=I \J F-injective?
One could ask if R=I and R=J are F-injective, does this imply R=I \ J is F-injective? Karl
Schwede proved a special case of this in [Sc]
References
[Be]
M.-J. Bertin, Anneaux d'invariants d'anneaux de polynomes, en caracteristique p, C. R. Acad. Sci. Paris
Ser. A-B 264 (1967), 653-656.
[BH] M. Broadmann and M. Hellus, Cohomological patterns of coherent sheaves over projective schemes, J.pure
and appl. Algebra. 172 (2001), 165{182.
[BKS] M. Broadmann, M. Katzman, and R. Sharp, Associated primes of graded components of local cohomology
modules. Trans. Amer. Math. 354 (2002), 4261{4283.
[Fe]
R. Fedder, F-purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), 461{480.
5
[FW]
R. Fedder, and K.Watanabe, A characterization of F-regularity in terms of F-purity, Commutative algebra
(Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ. 15 (1989), 227-245.
[HH] M. Hochster and C. Huneke, Tight closure, Invariant theory, and the Briancon-Skoda themorem. J. Amer.
Math. Soc. 3 (1990), 31{116.
[Hu] C. Huneke, Problems on local cohomology, in: Free resolutions in commutative algebra and algebraic geometry (Sundance, Utah, 1990), 93{108, Res. Notes Math. 2, Jones and Bartlett, Boston, MA, 1992.
[HS] C. Huneke and R. Sharp, Bass numbers of local cohomology modules, Trans. Amer, Math. Soc. 339 (1993),
765{779.
[KSSW] K. Kurano, E. Sato, A. K. Singh, and K. Watanabe, Multigraded rings, rational singularities, and diagonal
subalgebras, J. Algebra, to appear.
[Ly1] G. Lyubeznik, Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra), Invent. Math. 113 (1993), 41{55.
[Ly2] G. Lyubeznik, Finiteness properties of local cohomology modules: a characteristic-free approach, J.pure and
appl. Algebra. 151 (2000), 43{50.
[Sc]
K. Schwede, F-injective singularities are Du Bois, Amer J. Math. Soc. 131 (2009), 445{473.
[Si1] A. Singh, Failure of F-purity and F-regularity in certain rings of invariants,Illinois J. of Math 42 (1998),
441{448.
[Si2] A. Singh, p-torsion elements in local cohomology modules, Math. Res. Lett. 7 (2000), 165{176.
[SW] A. Singh, and U. Walther , On the arithmetic rank of certain Segre products,Commun Contemp Math. 390
(2005), 147{155.