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

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1 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 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 HI i (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 Hm(R) i = 0 for i < d, and Hm(R) d 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 HI k (R) have only nitely many associated prime ideals. In [Si2] Singh constructed a counterexample with the hypersurface R = Z[x; y; z; u; v; w] (ux + vy + wz) : Singh proved that for each prime integer p, the element (ux) p + (vy) p + (wz) p p = p(xyz) p of the local cohomology module H 3 (R) is p-torsion; equivalently (x;y;z) H 3 (x;y;z)(r) contains an isomorphic copy of the abelian group Z=pZ for every prime integer p. I proved the following theorem: Theorem 1. [Chan] Let R = Z[x;y;z;u;v;w] (ux+vy+wz) and consider the local cohomology module H 3 (x;y;z)(r) with the standard Z-grading. Then each nitely generated abelian group embeds into a graded component H 3 (x;y;z) (R). Moreover, the result remains true if we assign to the ring a ne Z 4 -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 HI k (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 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 Ass R0 (H i R + (M) n ) is asymptotically stable as n! 1 (i.e. there is an n 0 2 Z such that Ass R0 (H i R + (M) n ) = Ass R0 (H i R + (M) n0 ) for all n n 0. They proved that if R 0 is local M is a nitely generated and graded R-module. In addition if the R-module H j R + (M) is nitely generated for all j < i then Ass R0 (H i R + (M) n ) is asymptotically stable for n! 1. My work shows explicitly that this may fail when R 0 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) = r p. A ring R is said to be F-injective if the Frobenius map induces an injective map on local cohomology F : H i m(r)! H i m(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 F- injectivity 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[x 1 ; : : : ; x m ; y 1 ; : : : ; y n ]=(f) be a normal N 2 -graded hypersurface where deg x i = (1; 0), deg y j = (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[x 1 ; : : : ; x n ] and G a nite group viewed as a subset of the symmetric group on n letters. Let f(x 1 ; : : : ; x n ) 2 R and g 2 G. We dene an action of G on R by g : f(x 1 ; : : : ; x n )! f(x g(1) ; : : : ; x g(n) ), and dene the ring of invariants as R G = ff(x 1 ; : : : ; x n ) 2 R such that f(x 1 ; : : : ; x n ) = f(x g(1) ; : : : ; x g(n) ) for all g 2 Gg:

3 3 When K = F q is a nite eld of characteristic p, the behavior of R G 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 R G need not even by Cohen-Macaulay. Others have attempted to characterize the behavior of R G 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 R G a direct summand of R as an R G module, and so R G 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 R G 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 = (y 1 ; : : : y n ) 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 cx q 2 (y q 1 ; : : : ; yq n) for all q = p e. 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[x 1 ; : : : ; x n ] 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 R H is F-rational so is R G. (2) If R H is F-regular so is R G. (3) If R H is F-pure so is R G. Consequently to prove that R G has one of the properties it is sucient to check the result for R H. When G is the alternating group on n letters Singh showed the following. Theorem 6. [Si1] Let R = K[x 1 ; : : : ; x n ] be a polynomial ring in n variables over a eld of characteristic p, an odd prime, and let the alternating group A n, act on R by permuting the variables. Then the invariant subring R An is F-regular if and only if the order of the group ja n j is relatively prime to p. Consider elementary symmetric X functions. X e 1 = x j ; e 2 = x j x k ; : : : ; 1jn 1j<kn e n = x 1 : : : x n Dene D = Q i>j (x i x j ). It is an exercise in a beginning graduate algebra class to show the ring of invariants of A 3 and S 3 are: R A3 = K[e 1 ; e 2 ; e 3 ; D] and R S3 = K[e 1 ; e 2 ; e 3 ]

4 4 One can show that R S3 is F-regular explicitly using the above representation, but by the above theorem R A3 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 R G is F-rational, but R H 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 R H is F-injective is R G F-injective? It has been an unresolved problem for some time to determine when the rings of invariants R G 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 x 3 + y 3 + z 3 which denes a smooth elliptic curve in E p for any characteristic p 6= 3. The dening ideal of E P 1 is I = (u 3 + v 3 + w 3 ; u 2 x + v 2 y + w 2 z; ux 2 + vy 2 + wz 2 ; x 3 + y 3 + 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 HI 4 (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), [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 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), [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, [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.