A FORMULA FOR THE -core OF AN IDEAL

A FORMULA FOR THE -core OF AN IDEAL LOUIZA FOULI, JANET C. VASSILEV, AND ADELA N. VRACIU Abstract. Expanding on the work of Fouli and Vassilev [FV], we determine a formula for the -core for ideals in two different settings: (1) those contained in the tight closure of a parameter ideal in a Cohen Macaulay local ring with perfect infinite residue field and test ideal with depth at least two and (2) those contained in the tight closure of the Frobenius power of an ideal generated by star independent elements in a normal domain with perfect infinite residue field and an m-primary test ideal. 1. Introduction In a recent paper [FV], Fouli and Vassilev introduced the cl-core of an ideal for a closure operation cl. Like the core of an ideal, the cl-core of an ideal is the intersection of all the cl-reductions of an ideal, where J is a cl-reduction of an ideal I if J is a reduction of I and J cl = I cl. Of particular interest for rings of characteristic p, is the -core, where denotes tight closure. The tight closure of an ideal is contained in the integral closure, but is generally much closer to the ideal than the integral closure. Note that a -reduction of an ideal I is also a reduction. Hence, core(i) -core(i). Recall that core(i) sits deep within I; -core(i) will lie within I, but will be generally much closer to I. In [FV], Fouli and Vassilev determined that the -core and the core of an ideal, I, agree if the spread of I is equal to the -spread of I. In this paper, we determine a formula for the -core of an ideal as in the papers of Corso, Polini and Ulrich [CPU2] and [PU]. Our formula is similar to one found in [CPU2] for the core. We show when (R, m) is a Cohen Macaulay local ring with perfect infinite residue field with test ideal of depth at least two, then if J is generated by a system of parameters and I is contained in the tight closure of J, then - core(i) = (J : I)J. We also show that when (R, m) is a normal domain with perfect infinite residue field and m-primary test ideal, then if J is an ideal generated by -independent elements and I is an ideal contained in the tight closure of J, then -core(i [q 0] ) = (I [q 0] : J [q 0] )J [q 0] for q 0 a sufficiently large power of p. 1991 Mathematics Subject Classification. 13A30, 13A35, 13B22. Key words and phrases. tight closure, reduction, core, *-independent, spread. 1

2 LOUIZA FOULI, JANET C. VASSILEV, AND ADELA N. VRACIU 2. Tight Closure Preliminaries Since the -core of an ideal I is based on reductions stemming from the tight closure of I, we will review some tight closure concepts to help clarify these notions for the reader. Definition 2.1. Let (R, m) be a Noetherian local ring of prime characteristic p > 0. We denote positive powers of p by q and the set of elements of R which are not contained in the union of minimal primes by R o. Then (a) For any ideal I R, I [q] is the ideal generated by the qth powers of elements in I. (b) We say an element x R is in the tight closure, I, of I if there exists a c R o, such that cx q I [q] for all large q. Finding the tight closure of an ideal would be hard without test elements and test ideals. A test element is an element c R which is not in any minimal prime and ci I for all I R. Note that c (I : I ). The ideal τ = (I : I ) is called the test ideal of R. A result that we will use repeatedly is the following due to Aberbach: I R Proposition 2.2. ([Ab, Proposition 2.4]) Let (R, m) be an excellent, analytically irreducible local ring of characteristic p, let I be an ideal, and let f R. Assume that f I. Then there exists q 0 = p e 0 such that for all q q 0 we have I [q] : f q m [q/q 0]. In a Noetherian local ring of characteristic p Vraciu [Vr1] defined the special tight closure, I sp, to be the elements x R such that x (mi [q 0] ) for some q 0. Huneke and Vraciu show in [Vr1, Proposition 4.2] that I sp I = mi if I is generated by -independent elements. Note that the minimal -reductions of I are generated by -independent elements. Epstein showed in [Ep, Lemma 3.4] that I sp = J sp for all -reductions of I. 3. When the -core is equal to J(J : I) for some minimal -reduction J The following observation is an immediate consequence of the definition of the test ideal: Observation 3.1. Let R be a ring with test ideal τ. Then τi core(i) for every ideal I R. Proof. Let J I be an arbitrary -reduction. Since I J, it follows that τ J : I, and thus τi J. We will use the following characterization of minimal -reductions from [Vr2]: Theorem 3.2. [Thm. 1.13, [Vr2]] Let (R, m) be a local excellent normal ring of positive characteristic, with perfect residue field, and let K I be ideals, such that I R

A FORMULA FOR THE -core OF AN IDEAL 3 K is generated by l elements, where l is the -spread of I. Assume that I is tightly closed. The following are equivalent: a. K is a minimal -reduction of I b. I sp K mi c. I = I sp + K. Lemma 3.3. Let (R, m) be a local excellent normal ring of positive characteristic, with perfect residue field. Let J be a -independent ideal, with f 1,..., f n a minimal system of generators, and let I = J + (u 1,...,u s ), with u 1,...,u s J sp. Let K I. Then K is a minimal -reduction of I if and only if K = (f 1 +v 1, f 2 +v 2,...,f n +v n ), with v 1,...,v n (u 1,..., u s ). Proof. We check that for every K = (f 1 + v 1,...,f n + v n ) as in the statement, condition (b) in Theorem 3.2 holds. Indeed, if c 1 (f 1 + v 1 ) +... + c n (f n + v n ) I sp, then we must have c 1 f 1 +...+c n f n I sp J mj, where the last inclusion follows from Prop. 1.10 in [Vr2], noting that J is a minimal -reduction of I. Conversely, let K I be a minimal -reduction. Since J is a minimal -reduction of I, we know that the -spread of I is n, the minimal number of generators of J. Thus, K must also be minimally generated by n elements, and we can write K = (g 1 + v 1,..., g n + v n ), with g i (f 1,...,f n ), and v i (u 1,...,u s ) for all i = 1,..., n. We claim that we must have (g 1,...,g n ) = (f 1,...,g n ). If not, we can choose a minimal system of generators f 1,...,f n for J such that (g 1,...,g n ) mj+(f 2,...,f n ). Since K is a minimal -reduction of I, we must have f 1 K, which implies f 1 (f 2,...,f n ) + (u 1,..., u s )). Since u 1,...,u s J sp, this implies that in fact we must have f 1 (f 2,...,f n ) (to see this, write c 2 f q 1 = ca 2f q 2 +... + ca nf q cb 1 u q 1 +...+cb s u q s = a 1f q 1 + (ca 2 + a 2)f q 2 +...+ (ca n + a n)f q n with a 1,...,a n m q/q 0 with q 0 fixed, and apply Proposition 2.2). This contradicts the fact that f 1,...,f n are -independent. Theorem 3.4. Let (R, m) be a normal domain of characteristic p with perfect residue field and J = (f 1,...,f n ) I = J + (u 1,...,u s ), with u 1,...,u s J sp. (a) Assume that (f 1,...,f i 1, f i+1,...,f n ) : f i J : I, and u j (J : u j ) J(J : I) for all j = 1,...,s. Then core(i) J(J : I). (b) If u j (J : I) mj(j : I) for all j = 1,..., s, then core(i) J(J : I). Proof. (a) Let A 1 f 1 +...+A n f n core(i). Fix i {1,...,n} and j {1,..., s}. We wish to show that A i u j J. Since j is arbitrary, it will follow that A i J : I. Let J = (f 1,...,f i 1, f i +u j, f i+1,...,f n ) be another minimal -reduction of I. We must have A 1 f 1 +...+A n f n = B 1 f 1 +...+B i 1 f i 1 +B i (f i +u j )+...+B n f n for some B 1,..., B n R. Note we must have B i u j u j (J : u j ) J(J : I) by assumption, and thus we can write B i u j = C 1 f 1 +... + C n f n with C 1,...,C n J : I. The conclusion now follows by noting that we have A i B i C i (f 1,...,f i 1, f i+1,...,f n ) : f i J : I. n +

4 LOUIZA FOULI, JANET C. VASSILEV, AND ADELA N. VRACIU (b) Let A 1,...,A n J : I, and let J be another minimal -reduction of I. Then by Lemma 3.3 we can assume that J = (f 1 + v 1,..., f n + v n ), where v 1,..., v n (u 1,...,u s ). We wish to find B 1,..., B n J : I such that (1) A 1 f 1 +... + A n f n = B 1 (f 1 + v 1 ) +... + B n (f n + v n ). Write J : I = (g 1,...,g l ), and A i = Σ l j=1 α ijg j, B i = Σ l j=1 y ijg j, where y ij are unknowns. By assumption, we can write v i g k = Σ n ν=1 Σl j=1 γ ikjνg j f ν, with γ ikjν m for all i = 1,...,n and all k = 1,..., l. A sufficient condition for equation 1 to hold is that the coefficients of f ν on each side of the equation are the same for all ν = 1,...,n, which leads to the system of linear equations (2) Σ l j=1 y νjg j + Σ l j=1 Σn i=1 Σl k=1 y ikγ ikjν g j = Σ l j=1 α νjg j for all ν = 1,...,n. A further sufficient condition is obtained by identifying the coefficients of each g j on the two sides of the equations in 2. The following system of linear equations is thus obtained: (3) (ν, j) : y νj + Σ n i=1σ l k=1y ik γ ikjν = α νj for all ν = 1,...,n, j = 1,...,l. This is a system of nl equations with nl unknowns. Note that each y νj appears with a unit coefficient in equation (ν, j) and with coefficient in m in all the other equations, and therefore the matrix of coefficients can be row-reduced to the identity matrix, showing that there exists a (unique) solution for the system. Remark 3.5. When Theorem 3.4 holds, -core(i) is obtained as a finite intersection of minimal *-reductions, namely J = (f 1,...,f n ), and those of the form J i,j := (f 1,..., f i 1, f i + u j, f i+1...f n ), for all i {1,..., s} and j {1,..., s}. Proof. The proof of part a. of Theorem 3.4 shows that J i, jj i,j J(J : I), while part b. of Theorem 3.4 shows that J(J : I) core(i). The conclusion follows, since -core(i) J i,j J i,j by the definition of the -core. Theorem 3.6. Let (R, m) be Cohen Macaulay normal domain of characteristic p with perfect residue field and test ideal with depth at least 2. Let x 1,...,x d be part of a system of parameters, with x 1, x 2 τ. Let J = (x t 1,...,x t d ) I J, with t 0. Then core(i) = J(J : I). Proof. Let I = J + (u 1,...,u s ). We need to see that all the hypotheses listed in (a) and (b) in Theorem 3.4 hold, with K = J : I for (b). Check that the hypotheses of (a) hold: (x t 1,...,x t i 1, x t i+1,...,x t d) : x t i = (x t 1,...,x t i 1, x t i+1,...,x t d) J J : I. For the second hypothesis in (a), note that J J : τ J : (x 1, x 2 ) = (x t 1,...,xt d, (x 1x 2 ) t 1 ).

A FORMULA FOR THE -core OF AN IDEAL 5 Thus, J sp mj + (x 1 x 2 ) t 1, and we may assume without loss of generality that we can write u j = (x 1 x 2 ) t 1 u j. We have u j (J : u j ) = (x 1 x 2 ) t 1 (u j(x 1, x 2, x t 3,...,x t d) : u j) (x 1, x 2 )J (J : I)J. The last inclusion follows from the fact that (x 1, x 2 ) τ J : I. Now we check that the hypothesis in (b) holds, when we take K = J : I. As above, we can write u j = (x 1 x 2 ) t 1 u j, and we have J : I (x 1, x 2, x t 3,...,x t d ) : (u 1,...,u s), and thus u j (J : I) (x 1 x 2 ) t 1 (x 1, x 2, x t 3,...,x t d) (x t 1 1, x t 1 2 )J mj(j : I), where the last inclusion follows from the fact that (x t 1 1, x t 1 2 ) m(x 1, x 2 ) m(j : I) when we take t 3. The following observation can be obtained as a corollary of Theorem3.6 and Theorem 4.8 in[fv], using Theorem 2.6 in [CPU2], but we choose to give an elementary direct proof instead. Observation 3.7. Let (R, m) be a Cohen-Macaulay ring with test ideal τ, and x 1,...,x d part of a system of parameters with x 1, x 2 τ. If J = (x t 1,...,xt d ) I J with t 2, then r J (I) = 1. Proof. As seen in the proof of Theorem 3.6, we can write I = J+(x 1 x 2 ) t 1 (u 1,...,u s ). Thus, it suffices to see that (x 1 x 2 ) t 1 u i (x 1x 2 ) t 1 u j J2 for all i, j {1,...,s}. This is immediate, since (x 1 x 2 ) 2t 2 (x t 1 xt 2 ) J2. Corollary 3.8. Let (R, m) be a Gorenstein local ring of characteristic p with perfect residue field and test ideal with depth at least 2. Suppose x 1,...,x d form a system of parameters for R. For any t >> 0 and (x t 1,...,xt d ) I (xt 1,...,xt d ), gr I R is Cohen Macaulay. Proof. Notice that with the above conditions I is an m-primary ideal and satisfies G l, where l is the analytic spread. Also the condition depth R/I j d g j + 1 for all 1 j l g + 1, where g = ht(i) is immediately satisfied. Also, by Observation 3.7 r(i) = 1. Therefore, by [JU, Theorem 3.1] we have gr I (R) is Cohen-Macaulay. Theorem 3.9. Let (R, m) be a normal domain of characteristic p with perfect residue field. If the test ideal is m, then -core(i) = J(J : I) for any minimal -reduction J of I. Proof. Since τ J : I, we have J : I = m or R. Note that J : I = R is equivalent to I = J, which means that J is the only minimal -reduction of I, and thus -core(i) = J = J(J : I) holds in this case. Hence, we may assume that J : I = m. We have (J : I)J core(i) by Observation 3.1. In order to verify the other inclusion, we will check that the assumptions in part a. of Theorem 3.4 hold. The fact that (f 1,..., ˆf i,..., f n ) : f i m

6 LOUIZA FOULI, JANET C. VASSILEV, AND ADELA N. VRACIU is immediate, since f 1,...,f n is a minimal system of generators of J. It remains to check that u(j : I) mj, for u J sp. Assuming the contrary, it would follow that f i (u, f 1,..., ˆf i,...,f n ). Let c R o and q 0 = p e 0 be such that cu q m q/q 0 J [q]. It follows that cf q i (m q/q 0 f q i, fq 1,..., ˆf q i,..., fq n ), and thus f i (f 1,..., ˆf i,...,f n ) from Proposition 2.2. This contradicts the assumption that f 1,...,f n are -independent. Lemma 3.10. Assume that the test ideal τ is m-primary. Let J = (f 1,...,f n ), and I = (f 1,..., f n, u 1,...,u s ) with u 1,...,u s J sp. We can choose q 0 sufficiently large so that the following hold: (1) (f q 0 1,...,f q 0 i 1, fq 0 i+1,...,fq 0 n ) : f q 0 i τ; for all i = 1,...,n, and (2) u q 0 k (J[q 0] : u q 0 k ) mτj[q 0] for all k = 1,...,s Proof. We know (1) from 2.2. We prove (2). Fix j sufficiently large so that τ j : τ mτ (this is possible by the Artin-Rees Lemma, since τ j : τ τ j : d for a fixed non-zerodivisor d τ, and Fd τ j Fd dτ j c F τ j c, where c is a constant), and fix q 0 sufficiently large so that (f q 0 1,...,f q 0 i 1, fq 0 i+1,..., fq 0 n ) : fq 0 i τ j for all i = 1,..., n. Since u k J sp, we can choose q 0 0 so that u q 0 k (τj J [q0] ) (τ j J [q0] ) : τ. Let g J q 0 : u q 0 k, and write guq 0 k = c 1f q 0 1 +...+c n f q 0 n. Multiplying by t τ yields tguq 0 k = (tc 1 )f q 0 1 +... + (tc n)f q 0 n. But on the other hand we have tuq 0 k = d 1f q 0 1 +... + d nf q 0 with d 1,...,d n τ j. It follows that tc i gd i (f q 0 1,...,fq 0 i 1, fq ) i+1,...,fq 0 n ) : fq 0 and therefore c i τ j : τ mτ. n, i τ j, Theorem 3.11. We assume that (R, m) is a local normal Noetherian ring of positive characteristic p with perfect residue field with m-primary test ideal τ. Let J I be such that J is a minimal -reduction of I. We have core(i [q] ) = J [q] (J [q] : I [q] ) for all q = p e 0. Proof. Let I = J + (u 1,...,u s ). We need to prove -core(i [q] ) = J [q] (J [q] : I [q] ) by showing the hypotheses of Theorem 3.4 hold when I, J are replaced by I [q], J [q] respectively. First note that by Lemma 3.10 (1) (f q 1,..., ˆf q i,...,fq n ) : fq i τ (J : I) showing the first hypothesis of part (a) in Theorem 3.4 is satisfied. Now by Lemma 3.10 (2) u q j (J[q] : u q j ) mτj[q] (J [q] : I [q] )J [q] for all j = 1,..., s showing that Theorem 3.4 (a) holds. Now we verify that Theorem 3.4 (b) holds. This follows from Lemma 3.10 (2), by noting that τ J [q] : I [q]. 4. Examples The first example was an inspiration for Theorem 3.6 although the ring is not normal.

A FORMULA FOR THE -core OF AN IDEAL 7 Example 4.1. Let R = k[[x, y, z]]/(xyz) and I = (x, yz). By [Va], τ = (xy, xz, yz) and I = (x+ayz) for any 0 a k. Note, that (x+ayz)(x+ayz) : I = (x 2, y 2 z 2 ) = a k (x + ayz) = -core(i). The next example illustrates the formula for Theorem 3.6. Example 4.2. Let R = k[[x, y, z]]/(x 4 + y 4 + z 4 ) with k a field of characteristic p > 4. The test ideal is m 2. Thus for any ideal I, m 2 I I. Let J = (y, z) and I = J = (x 2, y, z). By Theorem 3.6, -core(i) = J(J : I) = (x 2 y, x 2 z, y 2, yz, z 2 ). The following example shows that our formula does not work for all ideals. Example 4.3. Let R = k[[x, y, z, w]]/(x 2 y 3 z 7, xw) and I = (x, y, z). Note that J = (y, z) is both a reduction and a -reduction of I and I = J. Similar to Example 4.10 in [PU], core(i) = I 2 is a finite intersection of minimal reductions. By [FV][Theorem 4.5], any ideal generated by l (I) general elements is a minimal - reduction of I. Thus -core(i) core(i), and since the reverse containment is always true, we have -core(i) = core(i) = I 2. Note, that J(J : I) = I 2 + (yw, zw). Hence, -core(i) J(J : I). We are going to use the following result of Brenner: Theorem 4.4. ([Br, Example 5.2]) If K is an algebraically closed field, and R = K[x, y, z] (x k + y k + z k ), where k is not divisible by the characteristic of K, and f = x d 1, g = y d 2, h = z d 3 are such that d i k for i {1, 2, 3}, and d 1 + d + 2 + d 3 2k, then R k (f, g, h). The following examples seem to suggest that the inclusion J(J : I) core(i) might hold much more generally than under the assumptions in (b) of Theorem 3.4. Note however that we have not been able to calculate -core(i) in these examples; we have only approximated it by intersecting a large number of -reductions. Example 4.5. Let K be an algebrically closed field of characteristic p 5, and R = K[x, y, z] (x 5 + y 5 + z 5 ), J = (x4, y 4, z 2 ), I = J + (x 3 y 2 ) By Theorem 4.4, we have I J, and thus J is a minimal -reduction of I. Let u = x 3 y 2. Note that (x 4, z 2 ) : y 4 = (y, z 2, x 4 ) J : I = (x, y 2, z 2 ), thus the first assumption in a. of Theorem 3.4 fails. However, since y y 4 J(J : I), the proof of part a. in Theorem 3.4 can still be used to show core(i) J(J : I). The assumption of part b. in Theorem 3.4 does not hold: u(j : I) (J : f). Experimental calculations with Macaulay 2 seem to suggest that we nevertheless still have J(J : I) core(i) (we have calculated the intersection of a large

8 LOUIZA FOULI, JANET C. VASSILEV, AND ADELA N. VRACIU number of minimal -reductions of I, and have seen that J(J : I) is contained in this intersection). Example 4.6. Let K be an algebraically closed field of characteristic p 2, 3, and let k[x, y, z] R = (x 6 + y 6 + z 6 ), J = (x5, y 5, z 2 ), I = J + (x 4 y 2 ), u = x 4 y 2. Both assumptions in part a., and the assumption in part b. of Theorem 3.4 fail. However, we have u(j : u) J[(J : u) + K 1 + K 2 + K 3 ], where K 1 = (x 5, y 5 ) : z 2, K 2 = (x 5, z ) : y 5, K 3 = (y 5, z 2 ) : x 5. The proof of part a. of Theorem 3.4 can be modified to show that in this case we have core(i) J[(J : u) + K 1 + K 2 + K 3 ] Experimental calculations with Macaulay 2 seem to suggest that J(J : I) core(i) holds, but the reverse containment fails (it seems that y 7 core(i) \J(J : I)). Example 4.7. R as above, J = (x 5, y 4, z 3 ), u = x 4 y 2. The assumptions in part a. of Theorem 3.4 hold, but the assumption in part b. fails. Thus, we have core(i) J(J : I). Experimental evidence seems to suggest that equality holds. References [Ab] I. Aberbach, Extensions of weakly and strongly F-rational rings by flat maps, J. Algebra, 241 (2001), 799 807. [Br] Brenner, H., Computing tight closure in dimension two [CPU1] Corso, A., Polini, C., Ulrich, B., The structure of the core of ideals, Math. Ann. 321 (2001), no. 1, 89 105. [CPU2] Corso, A., Polini, C., Ulrich, B., Core and residual intersections of ideals, Trans. Amer. Math. Soc. 354 (2002), no. 7, 2579 2594. [Ep] Epstein, N., A tight closure analogue of analytic spread, Math. Proc. Camb. Phil. Soc., 139, (2005), 371-383. [EV] Epstein, N., Vraciu, A., A Length Characterization of *-spread, Osaka J. Math., 45 (2008), no. 2, 445 456. [FPU] Fouli, L., Polini, C., Ulrich, B., Annihilators of Graded Components of the Canonical Module and the Core of Standard Graded Algebras, preprint, to appear in Trans. A.M.S. [FV] Fouli, L., Vassilev, J., The cl-core of an ideal, preprint. [HH] Hochster, M., Huneke, C., Tight closure, invariant theory, and the Brianon-Skoda theorem, J. Amer. Math. Soc. (1990), no. 1, 31 116. 3 [Hu] Huneke, C., Tight Closure and Its Applications, CBMS Lect. Notes Math. 88, American Math. Soc., Providence 1996. [HS1] Huneke, C., Swanson, I., Cores of ideals in 2-dimensional regular local rings, Michigan Math. J., 42 (1995), 193 208. [HS2] Huneke, C., Swanson, I., Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, 336, Cambridge University Press, Cambridge, 2006. [HV] Huneke, C., Vraciu, A., Special tight closure, Nagoya Math. J., 170 (2003), 175 183. [JU] Johnson, M., Ulrich, B., Artin-Nagata properties and Cohen-Macaulay associated graded rings, Compositio Math. 103 (1996), no. 1, 7 29.

A FORMULA FOR THE -core OF AN IDEAL 9 [NR] Northcott, D.G., Rees, D., Reductions of ideals in local rings, Proc. Camb. Phil. Soc., 50 (1954), 145-158. [PU] Polini, C., Ulrich, B., A formula for the core of an ideal, Math. Ann. 331 (2005), no. 3, 487 503. [RS] Rees, D., Sally, J., General elements and joint reductions, Michigan Math. J., 35 (1988), no. 2, 241 254. [Va] Vassilev, J., Test Ideals in quotients of F-finite regular local rings, Trans. A.M.S., 350 (1998), 4041-4051. [Vr1] Vraciu, A., -independence and special tight closure J. Algebra 249 (2002), no. 2, 544 565. [Vr2] Vraciu, A., Chains and families of tightly closed ideals, Bull. London Math. Soc. 38 (2006), no. 2, 201 208. The University of Texas, Austin, TX E-mail address: lfouli@math.utexas.edu University of New Mexico, Albuquerque, NM E-mail address: jvassil@math.unm.edu University of South Carolina, Columbia, SC E-mail address: vraciu@math.sc.edu