GEOMETRIC METHODS IN COMMUTATIVE ALGEBRA

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1 GEOMETRIC METHODS IN COMMUTATIVE ALGEBRA STEVEN DALE CUTKOSKY 1. Local Cohomology Suppose that R is a Noetherian ring, and M is an R-module. Suppose that I is an ideal in R, with generators I = (f 1,..., f n ). Consider the modified Cech complex Where C 0 = R and C : 0 C 0 C 1 C d 0 C t = The Local Cohomology of M is 1 i 1 <i 2 < <i t n R fi1 f i2 f it. H i I(M) = H i (M R C ). H 0 I (M) = {f M I k f = 0 for some k 0} = Γ I (M), the set of elements of M which have support in I. HI i (M) does not depend on the choice of generators of I. Further, H i I(M) = H i I (M). Γ I ( ) is a left exact functor, and the cohomology modules are its right derived functors. If 0 A B C 0 is a short exact sequence of R-modules, then there is a long exact sequence 0 H 0 I (A) H 0 I (B) H 0 I (C) H 1 I (A) Sheaf Cohomology. Let X = spec(r). Let M = M be the quasi coherent sheaf on X associated to M. M is determined by the stalks ( M)p = M p for all p X. We have that for f R, the sections over the open set D(f) = X \ V (f) are Γ(D(f), M) = M f. From the sheaf axioms, and since n i=1 D(f i) = U, we have that Γ(U, M), where U = X \ V (I), is computed as the kernel of d 0 : n Γ(D(f i ), M) d 0 Γ(D(f j ) D(f k ), M). j<k This map is just i=1 Consider the Cech complex n i=1 M fi d 0 M fj f k. j<k F : F 0 F 1 F n 1, 1

2 where F t = 1 i 1 <i 2 < <i t+1 n The sheaf cohomology of M on U = X \ V (I) is We have that the sections of M over U = X \ V (I). R fi1 f i2 f it+1. H i (U, M) = H i (M R F ). H 0 (U, M) = Γ(U, M), H i (U, M) depends only on U, so it is independent of the choice of generators of I, even up to the radical of I. Γ(U, M) is a left exact functor on modules, and the cohomology modules are its right derived functors. If 0 A B C 0 is a short exact sequence of R-modules, then there is a long exact sequence 0 H 0 (U, Ã) H0 (U, B) H 0 (U, C) H 1 (U, Ã) Comparison of Local Cohomology and Sheaf Cohomology. The modified Cech complex C (used to compute local cohomology) is obtained from the Cech complex F by shifting the Cech complex one to the right, and setting C 0 = R. From this we see that there is an exact sequence 0 H 0 I (M) M H 0 (U, M) H 1 I (M) 0 and isomorphisms H i (U, M) = H i+1 I (M) for i 1, where U = X \ V (I). We have the interpretation of H 0 (U, M) = Hom R (I n, M) as an ideal transform. A particularly important case of this is when R = k[x 0,..., x n ], a polynomial ring over a field with the standard grading, and m = (x 0,..., x n ). Suppose that M is a graded module over R. Then the local and sheaf cohomology modules H i (Spec(R) \ {m}, M) and Hm(R) i are graded, and the maps of the previous slide are graded. From the natural surjection of the affine cone Spec(R) \ {m} onto the projective space Proj(R) = P n k, we obtain graded isomorphisms H i (Spec(R) \ {m}, M) = j Z H i (P n k, M(j)). In the first cohomology module, M is the sheaf associated to M on Spec(R). In the second cohomology module, M(j) is the sheaf associated to M(j) on P n (M(j) d = M j+d ). We thus have an exact sequence of graded R-modules 0 H 0 m(m) M j Z H 0 (P n, M(j)) H 1 m(m) 0 and isomorphisms j Z H i (P n, M(j)) = H i+1 m (M) for i 1. 2

3 We have the interpretation as an ideal transform. j Z H 0 (P n, M(j)) = Hom R (m n, M) 1.3. Regularity. We continue to study the graded polynomial ring R = k[x 0,..., x n ], and assume that M is a finitely generated graded R-module. { a i sup{j H i (M) = m (M) j 0} if Hm(M) i 0 otherwise The regularity of M is defined to be reg(m) = max{a i (M) + i}. i Interpreting R as the coordinate ring of P n = Proj(R), and considering the sheaf M on P n associate to M, we can define the regularity of M to be Thus reg( M) = max{m H i (P n, M(m i 1)) 0 for some i 1} = max i 2 {a i (M) + i}. reg( M) reg(m) 1.4. Geometric Consequences of Regularity of Sheaves of Modules. The classical interpretations are for sections of line bundles on a projective variety X. Theorem 1.1. (Geometric Regularity Theorem) (Mumford [29]) Suppose that F is a coherent sheaf on P n = Proj(R), and m Z satisfies H i (P n, F(m i)) = H i (F(m i)) = 0 for all i 1. Then a) H 0 (F(k)) is spanned by H 0 (F(k 1)) H 0 (O(1)) if k > m. b) H i (F(k)) = 0 whenever i > 0, k + i m. c) H 0 (F(d)) d Z is minimally generated as an R-module in degrees m Interpretations of Regularity of Modules. Let F : 0 F j F 1 F 0 M 0 be a minimal free resolution of M as a graded R-module. Let b j be the maximum degree of the generators of F j. Then reg(m) = max{b j j j 0}. In fact, we have (Eisenbud and Goto [14]) that reg(m) = max{b j j j 0} = max{n j such that Tor R j (k, M) n+j 0} = max{n j such that Hm(M) j n j 0}. The equality of the first and second of these conditions follows since Tor R j (k, M) = H j (F R/m), and as F is minimal, we have that the maps of the complex F R/m are 3

4 all zero. To obtain the equality of the first and third conditions, we take the cohomology of the dual of F, to compute Ext j R (M, R), and then apply graded local duality. We will now give another proof, which brings out the role of sheaf cohomology, of the most interesting part of these equivalent conditions, namely, that for a finitely generated graded R-module M, reg(m) m implies b j m + j for all j. Here b j is the maximal degree of a generator of F j in the minimal free resolution of M (our proof is from Bayer and Mumford [1]). max{a i (M) + i} = reg(m) m implies a i (M) m i for i 2 which implies that c) of the geometric regularity theorem holds, so d Z H0 ( M(d)) is minimally generated in degree m. a 0 (M) m, a 1 (M) m 1 implies M d H 0 ( M(d)) is an isomorphism if d > m and a surjection if d = m. Thus M is minimally generated in degree m. We thus have a surjection with kernel K 1, 0 K 1 F 0 = α R( d α ) M 0 with d α m for all α. Sheafify to get 0 K 1 α O P n( d α ) M 0. Consider the long exact cohomology sequence: α H0 (O P n(m d α )) H 0 ( M(m)) H 1 ( K 1 (m)) α H1 (O P n(m d α )) H 1 ( M(m)) H 2 ( K 1 (m)) H i (O P n(j)) = 0 for i 1 and j 1 i. By b) of the the geometric regularity theorem, H i ( M(j)) = 0 for i 1 and j m i. Thus H i ( K 1 (m + 1 i)) = 0 for i 1. Induction Assumption: Assume we have constructed F k F 0 M 0 such that the minimal generators of F i have degree m + i for 0 i k and where H i ( K k (m + k i)) = 0 for i 1, K k = Kernel(F k F k 1 ). Then we will construct such a sequence of length k + 1. By c) of the geometric regularity theorem, we have a surjection with kernel K k+1, 0 K k+1 F k+1 = α R( d α ) K k 0 where d α m + k for all α. 4

5 Sheafify, and compute the long exact cohomology sequence, to get H i ( K k+1 (m + k + 1 i)) = 0 for i 1. We define reg i (M) = max{n Tor R i (k, M) n 0} i. Then reg(m) = max{reg i (M) i 0}. We have that reg 0 (M) is the maximum degree of a homogeneous generator of M. 2. Regularity of Powers of Ideals We outline the proof of, Herzog and Trung [12] showing that reg(i n ) is a linear polynomial for large n. Let F 1,..., F s be homogeneous generators of I R = k[x 0,..., x n ], with deg(f i ) = d i. The map y i F i induces a surjection of bigraded R-algebras S = R[y 1,..., y s ] R(I) = m 0 where we have bideg(x i ) = (1, 0) for 0 i n, and bideg(y j ) = (d j, 1) for 1 j s. We have Tor R i (k, I m ) a = Tor S i (S/mS, R(I)) (a,m) Theorem 2.1. Let E be a finitely generated bigraded module over k[y 1,..., y s ]. Then the function ρ E (m) = max{a E (a,m) 0} is a linear polynomial for m 0. Since Tor S i (S/mS, R(I)) is a finitely generated bigraded S/mS module, we have that Theorem 2.2. (, Herzog and Trung [12]) For all i 0, the function reg i (I n ) is a linear polynomial for n 0. Theorem 2.3. (, Herzog and Trung [12] and Kodiyalam [25]) reg(i n ) is a linear polynomial for n 0. In the expression for n 0, we have that reg(i n ) = an + b a = reg(in ) = d(in ) = ρ(i) n n where d(i n ) = reg 0 (I n ) is the maximal degree of a homogeneous generator of I n, and (Kodiyalam [25]) I m ρ(i) = min{max{d(j) J is a graded reduction of I}. Theorem 2.4. (Trung, Wang [37]) Suppose that R is standard graded over a commutative Noetherian ring with unity, I is a graded ideal of R and M is a finitely generated graded R-module. Then there exists a constant e such that for n 0, where reg(i n M) = ρ M (I)n + e ρ M (I) = min{d(j) J is a M-reduction of I}. 5

6 J is an M-reduction of I if I n+1 M = JI n M for some n 0. Theorem 2.5. (Eisenbud, Harris [15]) If I is R + primary (R = k[x 1,..., x m ]), and generated in a single degree, then the constant term of reg(i n ) (for n 0) is the maximum of the regularity of the fibers of the morphism defined by a minimal set of generators. Theorem 2.6. (Tai Hà [19], Chardin [4]) The constant term of the regularity reg(i n ), for I homogeneous in R = k[x 1,..., x m ] with generators all of the same degree, can be computed as the maximum of regularities of the localization of the structure sheaf of the graph of a rational map of P m determined by I above points in the projection of the graph onto its second factor (the image of the rational map) Comparison of reg i (I m ), a i (I m ) and reg(i m ). We continue to study the graded polynomial ring R = k[x 0,..., x n ], and assume that I is a homogeneous ideal. Recall that { a i (I m sup{j H i ) = m (I m ) j 0} if Hm(I i m ) 0 otherwise, and the regularity is reg i (I m ) = max{n Tor R i (k, I m ) n 0} i, reg(i m ) = max{a i (I m ) + i} = max{reg i (I m ) i 0}. i We have shown that all of the functions reg i (I m ) are eventually linear polynomials, so reg(i m ) is eventually a linear polynomial. 3. Behavior of a i (I m ) Theorem 3.1. ( [6]) There is a homogeneous height two prime ideal I in k[x 0, x 1, x 2, x 3 ] of a nonsingular space curve, such that a 2 (I m ) = m(9 + 2) σ(m) for m > 0, where x is the greatest integer in a real number x and { 0 if m = q2n for some n N σ(m) = 1 otherwise where q n is defined recursively by q 0 = 1, q 1 = 2, q n = 2q n 1 + q 2n 2, computed from the convergents pn q n of the continued fraction expansion of 2. The m such that σ(m) = 0 are very sparse, as q 2n 3 n. We also compute that a 3 (I m ) = m(9 2) τ(m) where 0 τ(m) constant is a bounded function, and Since m reg(i m ) m a 4 (I m ) = 4. Z +, we have that a 1 (I m ) = reg(i m ) = linear function for m 0. 6

7 3.1. Numerical Equivalence. A good introduction to this subject can be found in [27]. Let k be an algebraically closed field, and X be a nonsingular projective variety over k. Numerical equivalence: Div(X) = divisors on X := formal sums of codimension 1 subvarieties of X. D 1 D 2 if (D 1 C) = (D 2 C) for all curves C on X. N(X) = (Div(X)/ ) Z R, a finite dimensional R-vector space. A divisor D on X is ample if H 0 (O X (md)) gives a projective embedding of X for some m 0. Theorem 3.2. A divisor D is ample if and only if (D d V ) > 0 for all d-dimensional irreducible subvarieties V of X. (taking V = X this condition is (D dim X ) > 0). Theorem 3.3. A(X) = ample cone = convex cone in N(X) generated by ample divisors. Nef(X) = nef cone = convex cone generated by numerically effective divisors ((D C) 0 for all curves C on X.) NE(X) = convex cone generated by effective divisors (h 0 (O X (nd)) > 0 for some n > 0.) A(X) A(X) = Nef(X) NE(X) Here T denotes closure of T in the euclidean topology. Suppose that S is a nonsingular projective surface. Then N(S) has an intersection form q(d) = (D 2 ) for D a divisor on S. K3 surface: a nonsingular projective surface with H 1 (O S ) = 0 and such that K S is trivial. From theory of K3 surfaces it follows that there exists a K3 S such that N(S) = R 3 and q(d) = 4x 2 4y 2 4z 2 for D = (x, y, z) R 3. Lemma 3.4. Suppose that C is an integral curve on S. Then (C 2 ) 0. Proof. Suppose otherwise. Then (C 2 ) = 2, since S is a K3. But 4 divides q(c) = (C 2 ), a contradiction. Corollary 3.5. NE(S) = A(S), and NE(S) = { (x, y, z) 4x 2 4y 2 4z 2 0, x 0 }. 7

8 Let H = (1, 0, 0) (that is, let H be a divisor whose class is (1, 0, 0)). H 0 (O S (H)) gives an embedding of S as a quartic surface in P 3. Choose (a, b, c) Z 3 such that a > 0, a 2 b 2 c 2 > 0 and b 2 + c 2 Q. (a, b, c) is in the interior of NE(S), which is equal to A(S). There exists a nonsingular curve C on S such that C = (a, b, c). Let Suppose that m, r N. λ 2 = a + b 2 + c 2 and λ 1 = a b 2 + c 2. mh rc NE(S) and is ample if rλ 2 < m, mh rc, rc mh NE(S) if rλ 1 < m < rλ 2, rc mh NE(S) and is ample if m < rλ 1. Choose C = (a, b, c) so that 7 < λ 1 < λ 2 and λ 2 λ 1 > 2. Then by Riemann-Roch, χ(mh rc) = h 0 (mh rc) h 1 (mh rc) + h 2 (mh rc) = 1 2 (mh rc) Theorem 3.6. h 1 (mh rc) = { 0 if rλ2 < m 1 2 (mh rc)2 2 if rλ 1 < m < rλ 2. h 2 (mh rc) = 0 if rλ 1 < m. Let H be a linear hyperplane on P 3 such that H S = H. Let I C = ĨC, where I C is the homogeneous ideal of C in the coordinate ring R of P 3. Let π : X P 3 be the blow up of C. Let E = π (C), the exceptional surface, S be the strict transform of S on X. S = S and E S = C. For m, r N and i 0, In particular, I (r) C = (Ir C) sat = H i (P 3, I r C(m)) = H i (X, O X (mh re)). m 0 H 0 (P 3, I r C(m)) = m 0 0 O X ( S) O X O S 0 S 4H E. Tensor with O X ((m + 4)H (r + 1)E) to get H 0 (X, O X (mh re)). 0 O X (mh re) O X ((m + 4)H (r + 1)E) O S ((m + 4)H (r + 1)C) 0. h i (O X (mh)) = 0 for i > 0 and m 0, our calculation of cohomology on S and induction gives: { h 1 0 if m > rλ2 (mh re) = h 1 (mh re) if m = rλ 2 or m = rλ 2 1 h 2 (mh re) = 0 if m > λ 1 r h 3 (mh re) = 0 if m > 4r. By our calculation of cohomology on S, we have for r, t N, { h 1 (IC(t r 1)) = h 1 0 if t rλ σ(r) ((t 1)H re) = 0 if t = rλ 2 + σ(r), 8

9 where with We obtain for r N, σ(r) = { 0 if h 1 ( rλ 2 H rc) = 0 1 if h 1 ( rλ 2 H rc) 0. a 2 (I r C) = reg(i (r) C ) = reg((ir C) sat ) = rλ σ(r) a 2 (IC r ) r r reg((ic r = )sat ) r r In this example, we have shown that the function reg((i n C) sat ) = reg(i (n) C ) = λ 2 Q. has irrational behavior asymptotically. This is perhaps not so surprising, as its symbolic algebra n 0 is not a finitely generated R-algebra. An example of an ideal of a union of generic points in the plane whose symbolic algebras is not finitely generated was found and used by Nagata [30] to give his counterexample to Hilbert s 14th problem. Roberts [33] interpreted this example to give an example of a prime ideal of a space curve. Even for rational monomial curves this algebra may not be finitely generated, by an example of Goto, Nishida and Watanabe [17] Regularity of Coherent Sheaves. Suppose that X is a projective variety, over a field k, and H is a very ample divisor on X. Suppose that J O X is an ideal sheaf. Let π : B(I) X be the blow up of I, with exceptional divisor F. The Seshadri constant of J is defined to be I (n) C s H (J ) = inf{s R π (sh) F is a very ample R-divisor on B(J ).} The regularity of J is defined to be reg H (J ) = max{m H i (X, H O X ((m i 1)H)) 0. Theorem 3.7. (, Ein and Lazarsfeld [8]) Suppose that I O X is an ideal sheaf. Then For an ideal sheaf J, reg H (I m ) d H (I m ) = m m m m = s H(I). d H (J ) = least integer d such that J (dh) is globally generated. If H is a linear hyperplane on P n, and I = Ĩ, we get the statement that the it reg((i m ) sat ) = m m m d((i m ) sat ). m exists, where d((i m ) sat ) is the maximal degree of a generator of (I m ) sat. 9

10 4. An example of an Irrational Seshadri Constant The ideal I of a nonsingular curve in P 3 contained in a quadric which we considered earlier gives an example ( [6]). reg s H (Ĩ) = H (Ĩm ) reg((i m ) sat ) = = m m m m We do have something like linear growth of regularity reg H in the example. Recall that the example is of the homogeneous height two prime ideal I in k[x 0, x 1, x 2, x 3 ] of a nonsingular projective space curve, such that reg H (Ĩn ) = reg((i n ) sat ) = max{a i (I n ) + i 2 i 4} = m(9 + 2) σ(m) for m > 0, where x is the greatest integer in a real number x and { 0 if m = q2n for some n N σ(m) = 1 otherwise where q n is defined recursively by q 0 = 1, q 1 = 2, q n = 2q n 1 + q 2n 2. Theorem 4.1. (Wenbo Niu [32]) Suppose that I = Ĩ is an ideal sheaf on Pn. Then there is a bounded function τ(m), with 0 τ(m) constant, such that for all m > 0. reg H (I m ) = reg((i n ) sat ) = s H m + τ(m). 5. Fat Points in Weighted Projective Space Let k be an algebraically closed field, ands = k[x, y, z] be a polynomial ring, with the grading wt(x) = a, wt(y) = b, wt(z) = c, where a, b, c are pairwise relatively prime positive integers. Let P = P(a, b, c), a weighted projective plane (as a set, P consists of the weighted homogeneous prime ideals of S, other than (x, y, z)). Let P 1,..., P r P(a, b, c) be a set of distinct nonsingular closed points, and e i be positive integers. Let I Pi be the weighted homogeneous ideal of P i in S. Let the ideal of a fat point. We have that 5.1. Rational Monomial Primes. I = r i=0i e i P i I (n) = (I n ) sat. I = P (a, b, c), where P is the kernel of the graded k-algebra homomorphism k[x, y, z] k[t] defined by x t a, y t b, z t c is an example of such an ideal; it is a (nonsingular) point in P(a, b, c). 10

11 5.2. General Points in a Projective Plane. Suppose that P 1,..., P r are independent generic points in the ordinary projective plane P 2. Then is an example of such an ideal. I = I P1 I Pr Nagata s conjecture: Suppose that r 10. Then if d rm. [I m P 1 I m P r ] d = 0 Nagata proved this if r is a square [30] (from which his counterexample to Hilbert s 14th problem follows). If the graded K-algebra m 0 I(m) is a finitely generated K-algebra, then reg(i m ) must be a quasi polynomial for m 0. A quasi polynomial is a polynomial in m with coefficients which are periodic functions in m. In general, m 0 I(m) is not a finitely generated K-algebra. Some examples where this algebra are not finitely generated are given by Nagata s Theorem, which implies that it is not finitely generated when r 16 is a perfect square, and I is the intersection of the ideals of r general points in the plane. Theorem 5.1. (Goto, Nishida, Watanabe [17]) There are examples of monomial primes P (a, b, c) such that the symbolic algebra P (a, b, c) (n) is not finitely generated. n 0 Let s(i) = s OP (1)(I) where I = Ĩ is the sheafification on P of I. Then reg(i (m) ) = ms(i) + σ I (m) where 0 σ I (m) constant is a bounded function. Theorem 5.2. (, Kurano [13]) Let I = r i=0 Ie i p i, and u = r i=1 e2 i. Then s(i) abcu. 1. If s(i) > abcu, then s(i) is a rational number. 2. If s(i) > abcu and k has characteristic zero or is the algebraic closure of a finite field, then σ I (m) is eventually periodic. Nagata s conjecture is equivalent to the statement that s(i) = r if r 10 e i = 1 for 1 i r and P 1,..., P r are independent generic points in ordinary projective space P 2. This is the case of equality in the inequality s(i) abcu of the previous theorem. The assumption on the field K is necessary for the periodicity statement in the previous theorem. 11

12 Theorem 5.3. (, Herzog and Trung [12]) There exists a field k which is of positive characteristic and is transcendental over the prime field, and an ideal of a fat point in ordinary projective space P 2, such that s(i) = 29 5 > abcu = 29 but σ I (m) is not eventually periodic. 6. Local Cohomology of Powers of Ideals Over a Local Ring We now consider a local ring (R, m), an ideal I R, and consider the local cohomology H i m(i n ). Theorem 6.1. (Grothendieck) [18] Suppose that (R, m) is a quotient of a regular local ring, and M is a finitely generated R module. Then TFAE: 1. The length λ(h i m(m)) < for i n 2. depth(m p ) > n dim R/p for p Spec(R \ {m}). Suppose that (R, m) is a local domain of dimension d 2, and I R is an ideal. Then for all n 0. The saturation of I n is λ(h 1 m(i n )) < (I n ) sat = I n : R m = i=1i n : R m i. Saturation removes the m-primary component of I n Some Interpretations of H 1 m. Let L be the quotient field of R. Then and H 1 m(i n ) = I n : L m /I n H 0 m(r/i n ) = I n : R m /I n = (I n ) sat /I n. If depth(r) 2, then these modules are all equal, so that H 1 m(i n ) = (I n ) sat /I n Example 1. Suppose that (R, m) is a d-dimensional local domain of depth 2 and I is an m-primary ideal. Then λ(hm(i 1 n )) = λ((i n ) sat /I n ) = λ(r/i n ) = e I(R) n d + d! is a polynomial for n >> 0 (the Hilbert Samuel polynomial). We have 6.3. Example 2. λ(h m(i 1 n )) n n d = e I(R) Q. d! Theorem 6.2. (, Hà, Srinivasan, Theodorescu [9]) There exists a homogeneous prime ideal I in the polynomial ring R = k[x 1, x 2, x 3, x 4 ] such that λ(h m(i 1 n )) λ((i n ) sat /I n ) n n 4 = n n 4 Q. 12

13 6.4. Outline of the Construction. Let I be the homogeneous ideal in P 3 of the nonsingular curve on a quartic surface found in our example giving strange a 2 (I r ). Since reg(i n ) is a linear polynomial for large n, there exists a constant e such that H 1 m(i n ) k = 0 for k > ne. We compute from the short exact sequences: 0 (I n ) k (I n ) sat k = H 0 (P 3, Ĩn (k)) H 1 m(i n ) k 0 ne ne λ(hm(i 1 n )) = dim(i n ) k + h 0 (P 3, I n (k)). k=0 The first sum is a polynomial of degree 4 in n for n 0. By Riemann-Roch, k=0 h 0 (P 3, I n (k)) = χ(i n (k)) + h 1 (I n (k)) h 2 (I n (k)) + h 3 (I n (k)) where χ(i n (k)) is a polynomial in k and n. We get from our computations of cohomology in the earlier example that the it λ(h m(i 1 n )) λ((i n ) sat /I n ) n n 4 = n n 4 exists, but is irrational. Suppose that (R, m) is a local ring, and I R is an ideal. Question 6.3. Does exist? λ(h m(i 1 n )) n n We will show: Yes if R is a domain of dimension d 2 which is essentially of finite type over a field. From now on, we will assume that R is essentially of finite type over a field of dimension d 2. Theorem 6.4. ( [7]) Suppose that E is a rank e submodule of a finitely generated R- module F. Let Then the it exists. R[E] = E k be the algebra generated by E in sym(f ) = F k. λ(h m(e 1 k )) k k d+e 1 If depth(r) 2 or rank(f ) < d + e then the epsilon multiplicity ε(e) = k (d + e 1)! k d+e 1 λ(h 0 m(f k /E k )) exists as a it. Epsilon multiplicity is defined as a sup by Ulrich and Validashti in [38]. 13

14 The above Theorem is proven in the case when E = I is a homogeneous ideal and R is a standard graded normal K-algebra in our paper with Hà, Srinivasan and Theodorescu [9]. The theorem is proven with the additional assumptions that R is regular, E = I is an ideal in F = R, and the singular locus of Spec(R/I) is m in our paper with Herzog and Srinivasan [11]. Kleiman has proven Theorem 1 in the case that E is a direct summand of F locally at every nonmaximal prime of R [24]. The theorem is proven for E of low analytic deviation in our paper with Herzog and Srinivasan [11], for the case of ideals, and by Ulrich and Validashti [38] for the case of modules; in the case of low analytic deviation, the it is always zero. A generalization of this problem to the case of saturations with respect to non m-primary ideals is investigated by Herzog, Puthenpurakal and Verma [21]; they show that an appropriate it exists for monomial ideals. An algorithm for computing this it in some cases is given by Nishida [31]. We give the proof in the case of an ideal I R, when R has depth 2. Let I = (f 1,..., f t ), and let Z be the blow up of I. Z = B(I) = Proj( n 0 In ) P t 1 R h X = Spec(R) I k O Z = O Z (k), k 0. By Serre s fundamental theorem, there exists k such that for k k, (1) H 0 (Z, I k O Z ) = I k and h (I k O Z ) = I k. Let f : Y = B(mI) X be the blow up of mi. Define line bundles on Y by Y = B(mI) g Z h X L = g O Z (1) = IO Y, M = mo Y Proposition 6.5. There exist positive integers k 0 and τ such that 1) For k k 0, n Z and p X \ {m}, Γ(Y, M n L k ) p = (I k ) p. 2) For k k 0, I k : R m = Γ(Y, M kτ L k ). Proof of 1): M n L k (Y \ f 1 (m)) = I k O Y (Y \ f 1 (m)). Now 1) follows from (1). Proof of 2): For all q Y, k, n 0 implies ( I k : m n [mo Y ] n I k) = q (M n L k ) q. Thus (2) I k : m n q Y (M n L k ) q Γ(Y, M n L k ). 14

15 1) and depth(r) 2 implies that for k k 0, (3) Γ(Y, M n L k ) Γ(X \ f 1 (m), I n O X ) Γ(X \ f 1 (m), O X ) = R. Katz and McAdam [23] (also [35]): There exists τ such that (4) I k : m = I k : m kτ for all k 0. (2), (3) and (4) imply I k : m Γ(Y, M kτ L k ) R for k k 0. 2) now follows from 1) since I k : m is the largest ideal J of R with the property that J p = I k p for all p X \ {m}. Let R(I) = n 0 In. (5) 0 Γ(Y, L k )/I k I k : m /I k I k : m /Γ(Y, L k ) 0 1) implies Γ(Y, L k )/I k is supported on m for k k 0. L = h O Z (1) is the pullback of an ample line bundle on Z = Proj(R(I)), so k 0 Γ(Y, Lk ) is a finitely generated R(I)-module (Serre s theorem). 1) implies there exists an r such that for all k k 0. Now and R/m r is an Artin local ring imply m r ( Γ(Y, L k )/I k) = 0 dim R(I)/mR(I) dim R(I) 1 dim R d λ(γ(y, L k )/I k ) is a polynomial of degree d 1 for k 0. By (5), we are reduced to computing λ ( I k : m /Γ(Y, L k ) ) k k d. Taking global sections of ) 0 L k M kτ L k M kτ L k (O Y /m kτ O Y We obtain for k k 0, 0 0 I k : m /Γ(Y, L k ) Γ(Y, M kτ L k ( O Y /m kτ O Y ) ) H 1 (Y, L k ) Since L is the pullback of an ample line bundle on Z, we have that h 1 (Y, L k ) = polynomial in k of degree d 1 for large k. We have reduced to showing λ ( Γ(Y, M kτ L k ( O Y /m kτ )) O Y k k d exists. For simplicity, assume there exists a field K R such that R/m = K. 15

16 There exist projective varieties X and Y which are closures of X, Y and a commutative diagram Y Y f f X X such that L has an extension to a line bundle C on Y which is generated by global sections and is big (global sections of large multiples of C give a birational map to its image). This follows since L itself has these properties on Y. Set M = mo Y, B = C M τ. Tensor with B k to get 0 M kτ O Y O Y /m kτ O Y = OY /m kτ O Y 0 ) 0 C k B k M kτ (O Y /m kτ O Y 0 for k 0. 0 H 0 (Y, C k ) H 0 (Y, B k ) H 0 (Y, M kτ L k ( O Y /M kτ ) O Y ) H 1 (Y, C k ) C generated by global sections and is big implies and So h 1 (Y, C k ) k k d = 0 h 0 (Y, C k ) k k d Q. λ ( I k : m /I k) k k d h 0 (Y, B k ) = k k d exists since B is big on a projective variety. This last statement is by Fujita Approximation (Fujita [16], S. Takagi [36], Lazarsfeld and Mustata [28]). We construct examples of Rees algebras 7. Failure of Tameness A = R(I) = n 0 I n, associated to an ideal I in a local ring R, which is an algebra over a field K, such that the function (6) j dim K (H i A + (A) j ) is an interesting function for j 0. In our examples, this dimension will be finite for all j. Suppose that A 0 is a Noetherian local ring, A = j 0 A j is a standard graded ring and set A + := j>0 A j. Let M be a finitely generated graded A-module and F := M be the sheafification of M on Y = Proj(A). We then have graded A-module isomorphisms H i+1 A + (M) = n Z H i (Y, F(n)) for i 1, and a similar expression for i = 0 and 1. 16

17 By Serre vanishing, H i A + (M) j = 0 for all i and j 0. However, the asymptotic behaviour of H i A + (M) j for j 0 is much more mysterious. In the case when A 0 = K is a field, the function j dim K (H i A + (A) j ) is in fact a polynomial for large enough j. The proof is a consequence of graded local duality, or follows from Serre duality on a projective variety. For more general A 0, H A+ (M) j are finitely generated A 0 modules, but need not have finite length. The following problem was proposed by Brodmann and Hellus [3]. Tameness problem: Are the local cohomology modules H i A + (M) tame? That is, is it true that either {H i A + (M) j 0, j 0} or {H i A+(M) j = 0, j 0}? The problem has a positive solution for A 0 of small dimension (some of the references are Brodmann [2], Brodmann and Hellus [3], Lim [26], Rotthaus and Sega [34]). Theorem 7.1. (Brodmann and Hellus [3]) If dim A 0 2, then M is tame Examples of Failure of Tameness. Theorem 7.2. (, Herzog [10] and Chardin,, Herzog and Srinivasan [5]) Suppose that K is an algebraically closed field. Then there exists a normal, standard graded K algebra T 0 with dim(t 0 ) = 3 and a graded ideal A T 0 such that the Rees algebra T = T 0 [At] of A is normal, and for j > 0, { dim K (HB(T 2 2 if j is even, ) j ) = 0 if j is odd. where B is the graded ideal AtT of T. A more exotic example of failure of tameness can be constructed in characteristic p > 0. Theorem 7.3. (Chardin,, Herzog and Srinivasan [CCHS]) Suppose that p is a prime number such that p 2( mod )3 and p 11. Then there exists a normal standard graded K-algebra T 0 over a field K of characteristic p with dim(t 0 ) = 4, and a graded ideal A T 0 such that the Rees algebra T = T 0 [At] of A is normal, and for j > 0, 1 if j 0( mod )(p + 1), dim K (HQ(T 2 ) j ) = 1 if j = p t for some odd t 0, 0 otherwise, where B is the graded ideal AtT of T. We have p t 1( mod )(p + 1) for all odd t 0. Theorem 7.4. (Chardin,, Herzog and Srinivasan [5]) Suppose that K is an algebraically closed field. Then there exists a normal, standard graded K-algebra T 0 with dim(t 0 ) = 4 and a graded ideal A T 0 such that the Rees algebra T = R 0 [At] of A is normal, and for j > 0, { dim K (HB(T 3 6j if j is even, ) j ) = 0 if j is odd, where B is the graded ideal AtT of T. 17

18 Theorem 7.5. (Chardin,, Herzog and Srinivasan [5]) Suppose that K is an algebraically closed field. Then there exists a normal standard graded K algebra T 0 with dim(t 0 ) = 3, and a graded ideal A T 0 such that the Rees algebra T = T 0 [At] of A is normal, and for j > 0, dim K (HB 2 (T ) j) = 162 ( j 2 ([ j ] ) [ ] ([ ] ) ( [ ] )) 1 j j j and dim K (HB 2 (T ) j) j j 3 = 54 2 where B is the graded ideal AtT of T. In all of these examples, T 0 is generalized Cohen Macaulay, but is not Cohen Macaulay. This follows since in all of these examples, H 2 P 0 (R 0 ) 0 = H 1 (X, O X ) Some Questions Question 8.1. Suppose that I is a homogeneous ideal in a polynomial ring S. Does a i (I n ) n n exist for all i? Does exist? reg(i (n) ) n n (Yes if the singular locus of S/I has dimension 1 Herzog, LT Hoa, NV Trung [20]). Question 8.2. Does exist? reg(in(i n )) n n Question 8.3. Suppose that I is generated in a single degree. Explain (geometrically) the constant term b in the linear polynomial reg(i n ) = an + b for n 0 There are partial answers by Eisenbud and Harris [15], Ha [19] and Chardin [4]. Suppose that S is a polynomial ring and M is a graded S-module of dimension d. Then the Hilbert polynomial d ( ) x + d i P M (x) = ( 1) i e i (M) d i for x 0. i=0 Question 8.4. Suppose that I, J are homogeneous ideals in a polynomial ring S = k[x 1,..., x n ]. Define I k (J) = I k : J. Let d be the it dimension of I k (J)/I k. Does exist? e 0(I k (J)/I k )/k n d k 18

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