GEOMETRIC METHODS in COMMUTATIVE ALGEBRA 1. Local

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 = (f1, . . . , fn). Consider the modiﬁed Cech complex C∗ : 0 → C0 → C1 → · · · → Cd → 0 Where C0 = R and M Ct = R . fi1 fi2 ···fit 1≤i1

0 k HI (M) = {f ∈ M | I f = 0 for some k ≥ 0} = ΓI (M), the set of elements of M which have support in I.

i HI (M) does not depend on the choice of generators of I. Further, Hi(M) = H√i (M). I I

ΓI (∗) 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 0 0 1 0 → HI (A) → HI (B) → HI (C) → HI (A) → ... 1.1. 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 = Mp 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) = Mf . n From the sheaf axioms, and since ∪i=1D(fi) = U, we have that Γ(U, M), where U = X \ V (I), is computed as the kernel of d0: n M d0 M Γ(D(fi), M) → Γ(D(fj) ∩ D(fk), M). i=1 j

Hi(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 → H0(U, A˜) → H0(U, B˜) → H0(U, C˜) → H1(U, A˜) → ... 1.2. 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 C0 = R. From this we see that there is an exact sequence 0 0 ˜ 1 0 → HI (M) → M → H (U, M) → HI (M) → 0 and isomorphisms i ˜ ∼ i+1 H (U, M) = HI (M) for i ≥ 1, where U = X \ V (I). We have the interpretation of 0 ∼ n H (U, M˜ ) = lim HomR(I ,M) → as an “ideal transform”. A particularly important case of this is when R = k[x0, . . . , xn], a polynomial ring over a field with the standard grading, and m = (x0, . . . , xn). Suppose that M is a graded module over R. Then the local and sheaf cohomology modules Hi(Spec(R) \{m}, M˜ ) i and Hm(R) are graded, and the maps of the previous slide are graded. From the natural n surjection of the affine cone Spec(R) \{m} onto the projective space Proj(R) = Pk , we obtain graded isomorphisms i ˜ ∼ M i n ˜ H (Spec(R) \{m}, M) = H (Pk , M(j)). j∈Z In the first cohomology module, M˜ is the sheaf associated to M on Spec(R). In the second ˜ n cohomology module, M(j) is the sheaf associated to M(j) on P (M(j)d = Mj+d). We thus have an exact sequence of graded R-modules 0 M 0 n ˜ 1 0 → Hm(M) → M → H (P , M(j)) → Hm(M) → 0 j∈Z and isomorphisms M i n ˜ ∼ i+1 H (P , M(j)) = Hm (M) for i ≥ 1. j∈Z 2 We have the interpretation M 0 n ∼ n H ( , M˜ (j)) = lim HomR(m ,M) P → j∈Z as an ideal transform.

1.3. Regularity. We continue to study the graded polynomial ring R = k[x0, . . . , xn], and assume that M is a finitely generated graded R-module. sup{j | Hi (M) 6= 0} if Hi (M) 6= 0 ai(M) = m j m −∞ otherwise The regularity of M is defined to be reg(M) = max{ai(M) + i}. i n Interpreting R as the coordinate ring of P = Proj(R), and considering the sheaf M˜ on n P associate to M, we can define the regularity of M˜ to be i n reg(M˜ ) = max{m | H (P , M˜ (m − i − 1)) 6= 0 for some i ≥ 1} i = maxi≥2{a (M) + i}. Thus 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 n i n i coherent sheaf on P = Proj(R), and m ∈ Z satisﬁes H (P , F(m−i)) = H (F(m−i)) = 0 for all i ≥ 1. Then a) H0(F(k)) is spanned by H0(F(k − 1)) ⊗ H0(O(1)) if k > m. b) Hi(F(k)) = 0 whenever i > 0, k + i ≥ m. c) M H0(F(d)) d∈Z is minimally generated as an R-module in degrees ≤ m. 1.5. Interpretations of Regularity of Modules. Let

F∗ : 0 → · · · → Fj → · · · → F1 → F0 → M → 0 be a minimal free resolution of M as a graded R-module. Let bj be the maximum degree of the generators of Fj. Then

reg(M) = max{bj − j | j ≥ 0}. In fact, we have (Eisenbud and Goto [14]) that

reg(M) = max{bj − j | j ≥ 0} R = max{n | ∃j such that Torj (k, M)n+j 6= 0} j = max{n | ∃j such that Hm(M)n−j 6= 0}. R The equality of the first and second of these conditions follows since Torj (k, M) = Hj(F∗ ⊗ R/m), and as F∗ is minimal, we have that the maps of the complex F∗ ⊗ R/m are 3 all zero. To obtain the equality of the first and third conditions, we take the cohomology j of the dual of F∗, to compute ExtR(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 bj ≤ m + j for all j.

Here bj is the maximal degree of a generator of Fj in the minimal free resolution of M (our proof is from Bayer and Mumford [1]). max{ai(M) + i} = reg(M) ≤ m implies ai(M) ≤ m − i for i ≥ 2 which implies that c) of the geometric regularity theorem holds, so L H0(M˜ (d)) is minimally generated in d∈Z degree ≤ m.

0 1 0 a (M) ≤ m, a (M) ≤ m − 1 implies Md → H (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 K1, M 0 → K1 → F0 = R(−dα) → M → 0 α with dα ≤ m for all α. Sheaﬁfy to get M ˜ n ˜ 0 → K1 → OP (−dα) → M → 0. α Consider the long exact cohomology sequence:

0 0 1 L n ˜ ˜ α H (OP (m − dα)) → H (M(m)) → H (K1(m)) → 1 1 2 L n ˜ ˜ α H (OP (m − dα)) → H (M(m)) → H (K1(m)) → · · · i n H (OP (j)) = 0 for i ≥ 1 and j ≥ 1 − i. By b) of the the geometric regularity theorem, Hi(M˜ (j)) = 0 for i ≥ 1 and j ≥ m − i. Thus i H (K˜1(m + 1 − i)) = 0 for i ≥ 1.

Induction Assumption: Assume we have constructed

Fk → · · · → F0 → M → 0 such that the minimal generators of Fi have degree ≤ m + i for 0 ≤ i ≤ k and i H (K˜k(m + k − i)) = 0 for i ≥ 1, where Kk = Kernel(Fk → Fk−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 Kk+1, M 0 → Kk+1 → Fk+1 = R(−dα) → Kk → 0 α where dα ≤ m + k for all α. 4 Sheaﬁfy, and compute the long exact cohomology sequence, to get i H (K˜k+1(m + k + 1 − i)) = 0 for i ≥ 1. We deﬁne R regi(M) = max{n | Tori (k, M)n 6= 0} − i. Then reg(M) = max{regi(M) | i ≥ 0}. We have that reg0(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(In) is a linear polynomial for large n. Let F1,...,Fs be homogeneous generators of I ⊂ R = k[x0, . . . , xn], with deg(Fi) = di. The map yi 7→ Fi induces a surjection of bigraded R-algebras M m S = R[y1, . . . , ys] → R(I) = I m≥0 where we have bideg(xi) = (1, 0) for 0 ≤ i ≤ n, and bideg(yj) = (dj, 1) for 1 ≤ j ≤ s. We have R m ∼ S Tori (k, I )a = Tori (S/mS, R(I))(a,m)

Theorem 2.1. Let E be a ﬁnitely generated bigraded module over k[y1, . . . , ys]. Then the function ρE(m) = max{a | E(a,m) 6= 0} is a linear polynomial for m 0. S Since Tori (S/mS, R(I)) is a ﬁnitely generated bigraded S/mS module, we have that

n Theorem 2.2. (–, Herzog and Trung [12]) For all i ≥ 0, the function regi(I ) is a linear polynomial for n 0. Theorem 2.3. (–, Herzog and Trung [12] and Kodiyalam [25]) reg(In) is a linear polynomial for n 0. In the expression reg(In) = an + b for n 0, we have that reg(In) d(In) a = lim = lim = ρ(I) n n n n n where d(I ) = reg0(I ) is the maximal degree of a homogeneous generator of I , and (Kodiyalam [25]) 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, n reg(I M) = ρM (I)n + e where ρM (I) = min{d(J) | J is a M-reduction of I}. 5 J is an M-reduction of I if In+1M = JInM for some n ≥ 0.

Theorem 2.5. (Eisenbud, Harris [15]) If I is R+ primary (R = k[x1, . . . , xm]), and generated in a single degree, then the constant term of reg(In) (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(In), for I homogeneous in R = k[x1, . . . , xm] with generators all of the same degree, can be computed as the maximum of regularities of the localization of the structure sheaf of the m graph of a rational map of P determined by I above points in the projection of the graph onto its second factor (the image of the rational map).

m i m m 2.1. Comparison of regi(I ), a (I ) and reg(I ). We continue to study the graded polynomial ring R = k[x0, . . . , xn], and assume that I is a homogeneous ideal. Recall that sup{j | Hi (Im) 6= 0} if Hi (Im) 6= 0 ai(Im) = m j m −∞ otherwise,

m R m regi(I ) = max{n | Tori (k, I )n 6= 0} − i, and the regularity is m i m m reg(I ) = max{a (I ) + i} = max{regi(I ) | i ≥ 0}. i m We have shown that all of the functions regi(I ) are eventually linear polynomials, so reg(Im) is eventually a linear polynomial.

3. Behavior of ai(Im)

Theorem 3.1. (– [6]) There is a homogeneous height two prime ideal I in k[x0, x1, x2, x3] of a nonsingular space curve, such that √ a2(Im) = bm(9 + 2)c + 1 + σ(m) for m > 0, where bxc is the greatest integer in a real number x and 0 if m = q for some n ∈ σ(m) = 2n N 1 otherwise where qn is deﬁned recursively by

q0 = 1, q1 = 2, qn = 2qn−1 + q2n−2, √ computed from the convergents pn of the continued fraction expansion of 2. qn n The m such that σ(m) = 0 are very sparse, as q2n ≥ 3 . We also compute that √ a3(Im) = bm(9 − 2)c − τ(m) where 0 ≤ τ(m) ≤ constant is a bounded function, and a4(Im) = −4. √ reg(Im) Since 9 + 2 ≤ limm→∞ m ∈ Z+, we have that a1(Im) = reg(Im) = linear function for m 0. 6 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. Div(X) = divisors on X := formal sums of codimension 1 subvarieties of X. Numerical equivalence:

D1 ≡ D2 if (D1 · C) = (D2 · C) for all curves C on X.

N(X) = (Div(X)/ ≡) ⊗Z R, a ﬁnite dimensional R-vector space. 0 A divisor D on X is ample if H (OX (mD)) gives a projective embedding of X for some m 0.

Theorem 3.2. A divisor D is ample if and only if (Dd · V ) > 0 for all d-dimensional irreducible subvarieties V of X.

(taking V = X this condition is (Ddim X ) > 0).

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 0 (h (OX (nD)) > 0 for some n > 0.) Theorem 3.3. 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) = (D2) for D a divisor on S.

1 K3 surface: a nonsingular projective surface with H (OS) = 0 and such that KS is trivial.

∼ 3 From theory of K3 surfaces it follows that there exists a K3 S such that N(S) = R and q(D) = 4x2 − 4y2 − 4z2 3 for D = (x, y, z) ∈ R . Lemma 3.4. Suppose that C is an integral curve on S. Then (C2) ≥ 0. Proof. Suppose otherwise. Then (C2) = −2, since S is a K3. But 4 divides q(C) = (C2), a contradiction. Corollary 3.5. NE(S) = A(S), and NE(S) = (x, y, z) | 4x2 − 4y2 − 4z2 ≥ 0, x ≥ 0 . 7 0 Let H = (1, 0, 0) (that is, let H be a divisor whose class is (1, 0, 0)). H (OS(H)) gives 3 3 an embedding of S as√ a quartic surface in P . Choose (a, b, c) ∈ Z such that a > 0, 2 2 2 a − b − c > 0 and b2 + c2 6∈ 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

p 2 2 p 2 2 λ2 = a + b + c and λ1 = a − b + c . Suppose that m, r ∈ N.

mH − rC ∈ NE(S) and is ample if rλ2 < m,

mH − rC, rC − mH 6∈ NE(S) if rλ1 < m < rλ2,

rC − mH ∈ NE(S) and is ample if m < rλ1. Choose C = (a, b, c) so that 7 < λ1 < λ2 and λ2 − λ1 > 2. Then by Riemann-Roch, χ(mH − rC) = h0(mH − rC) − h1(mH − rC) + h2(mH − rC) 1 2 = 2 (mH − rC) + 2.

Theorem 3.6. 1 0 if rλ2 < m h (mH − rC) = 1 2 − 2 (mH − rC) − 2 if rλ1 < m < rλ2.

2 h (mH − rC) = 0 if rλ1 < m. 3 ˜ Let H be a linear hyperplane on P such that H · S = H. Let IC = IC , where IC is the 3 homogeneous ideal of C in the coordinate ring R of P . 3 ∗ Let π : X → P 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, i 3 r ∼ i H (P , IC (m)) = H (X, OX (mH − rE)). In particular, (r) r sat ∼ M 0 3 r M 0 IC = (IC ) = H (P , IC (m)) = H (X, OX (mH − rE)). m≥0 m≥0

0 → OX (−S) → OX → OS → 0 S ∼ 4H − E. Tensor with OX ((m + 4)H − (r + 1)E) to get

0 → OX (mH − rE) → OX ((m + 4)H − (r + 1)E) → OS((m + 4)H − (r + 1)C) → 0. i h (OX (mH)) = 0 for i > 0 and m ≥ 0, our calculation of cohomology on S and induction gives: 1 0 if m > rλ2 h (mH − rE) = 1 h (mH − rE) if m = brλ2c or m = brλ2c − 1 2 h (mH − rE) = 0 if m > λ1r h3(mH − rE) = 0 if m > 4r. By our calculation of cohomology on S, we have for r, t ∈ N, 1 r 1 0 if t ≥ brλ2c + 1 + σ(r) h (IC (t − 1)) = h ((t − 1)H − rE) = 6= 0 if t = brλ2c + σ(r), 8 where 0 if h1(brλ − 2cH − rC) = 0 σ(r) = 1 if h1(brλ − 2cH − rC) 6= 0. We obtain for r ∈ N, 2 r (r) r sat a (IC ) = reg(IC ) = reg((IC ) ) = brλ2c + 1 + σ(r) with 2 r r sat a (IC ) reg((IC ) ) lim = lim = λ2 6∈ Q. r→∞ r r→∞ r In this example, we have shown that the function

n sat (n) reg((IC ) ) = reg(IC ) has irrational behavior asymptotically. This is perhaps not so surprising, as its symbolic algebra M (n) IC 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].

3.2. 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 ⊂ OX 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 ∗ sH (J ) = inf{s ∈ R | π (sH) − F is a very ample R-divisor on B(J ).} The regularity of J is defined to be i regH (J ) = max{m | H (X, H ⊗ OX ((m − i − 1)H)) 6= 0.

Theorem 3.7. (–, Ein and Lazarsfeld [8]) Suppose that I ⊂ OX is an ideal sheaf. Then m m regH (I ) dH (I ) lim = lim = sH (I). m→∞ m m→∞ m For an ideal sheaf J,

dH (J ) = least integer d such that J (dH) is globally generated.

n If H is a linear hyperplane on P , and I = I˜, we get the statement that the limit reg((Im)sat) d((Im)sat) lim = lim . m→∞ m m→∞ m exists, where d((Im)sat) is the maximal degree of a generator of (Im)sat. 9 4. An example of an Irrational Seshadri Constant 3 The ideal I of a nonsingular curve in P contained in a quadric which we considered earlier gives an example (– [6]). ˜m m sat √ regH (I ) reg((I ) ) sH (I˜) = lim = lim = 9 + 2. m→∞ m m→∞ m

We do have something like linear growth of regularity regH in the example. Recall that the example is of the homogeneous height two prime ideal I in k[x0, x1, x2, x3] of a nonsingular projective space curve, such that ˜n n sat i n regH (I ) = reg((I )√ ) = max{a (I ) + i | 2 ≤ i ≤ 4} = bm(9 + 2)c + 1 + σ(m) for m > 0, where bxc is the greatest integer in a real number x and 0 if m = q for some n ∈ σ(m) = 2n N 1 otherwise where qn is deﬁned recursively by

q0 = 1, q1 = 2, qn = 2qn−1 + q2n−2.

n Theorem 4.1. (Wenbo Niu [32]) Suppose that I = I˜ is an ideal sheaf on P . Then there is a bounded function τ(m), with 0 ≤ τ(m) ≤ constant, such that m n sat regH (I ) = reg((I ) ) = bsH mc + τ(m). for all m > 0.

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 P1,...,Pr ∈ P(a, b, c) be a set of distinct nonsingular closed points, and ei be positive integers. Let IPi be the weighted homogeneous ideal of Pi in S. Let I = ∩r Iei i=0 Pi the ideal of a “fat point”. We have that I(n) = (In)sat.

5.1. Rational Monomial Primes. 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 7→ ta, y 7→ tb, z 7→ tc is an example of such an ideal; it is a (nonsingular) point in P(a, b, c). 10 5.2. General Points in a Projective Plane. Suppose that P1,...,Pr are independent 2 generic points in the ordinary projective plane P . Then

I = IP1 ∩ · · · ∩ IPr is an example of such an ideal.

Nagata’s conjecture: Suppose that r ≥ 10. Then

m m [IP1 ∩ · · · IPr ]d = 0 √ if d ≤ rm.

Nagata proved this if r is a square [30] (from which his counterexample to Hilbert’s 14th problem follows). L (m) m If the graded K-algebra m≥0 I is a finitely generated K-algebra, then reg(I ) must be a quasi polynomial for m 0. A quasi polynomial is a polynomial in m with L (m) coefficients which are periodic functions in m. In general, m≥0 I 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 M P (a, b, c)(n) n≥0 is not ﬁnitely generated.

Let s(I) = s (I) OP(1) where I = I˜ is the sheaﬁﬁcation on P of I. Then (m) reg(I ) = bms(I)c + σI (m) where 0 ≤ σI (m) ≤ constant is a bounded function.

r ei Pr 2 Theorem 5.2. (–, Kurano [13]) Let I = ∩i=0Ipi , and u = i=1 ei . 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 σI (m) is eventually periodic. √ Nagata’s conjecture is equivalent to the statement that s(I) = r if r ≥ 10 ei = 1 for 2 1 ≤ i ≤ r and P1,...,Pr are independent generic points√ in ordinary projective space P . 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 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 2 ordinary projective space P , such that 29 √ √ s(I) = > abcu = 29 5 but σI (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 i n Hm(I ). 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: i 1. The length λ(Hm(M)) < ∞ for i ≤ n 2. depth(Mp) > 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 1 n λ(Hm(I )) < ∞ for all n ≥ 0.

The saturation of In is n sat n ∞ ∞ n i (I ) = I :R m = ∪i=1I :R m . Saturation removes the m-primary component of In.

1 6.1. Some Interpretations of Hm. Let L be the quotient field of R. Then 1 n ∼ n ∞ n Hm(I ) = I :L m /I and 0 n ∼ n ∞ n n sat n Hm(R/I ) = I :R m /I = (I ) /I . If depth(R) ≥ 2, then these modules are all equal, so that 1 n ∼ n sat n Hm(I ) = (I ) /I . 6.2. Example 1. Suppose that (R, m) is a d-dimensional local domain of depth ≥ 2 and I is an m-primary ideal. Then e (R) λ(H1 (In)) = λ((In)sat/In) = λ(R/In) = I nd + ··· m d! is a polynomial for n >> 0 (the Hilbert Samuel polynomial). We have 1 n λ(Hm(I )) eI (R) lim = ∈ Q. n→∞ nd d! 6.3. Example 2. Theorem 6.2. (–, Hà,Srinivasan, Theodorescu [9]) There exists a homogeneous prime ideal I in the polynomial ring R = k[x1, x2, x3, x4] such that 1 n n sat n λ(Hm(I )) λ((I ) /I ) lim = lim 6∈ Q. n→∞ n4 n→∞ n4 12 3 6.4. Outline of the Construction. Let I be the homogeneous ideal in P of the nonsingular curve on a quartic surface found in our example giving strange a2(Ir). Since reg(In) is a linear polynomial for large n, there exists a constant e such that 1 n Hm(I )k = 0 for k > ne. We compute from the short exact sequences: n n sat 0 3 ñ 1 n 0 → (I )k → (I )k = H (P , I (k)) → Hm(I )k → 0

ne ne 1 n X n X 0 3 n λ(Hm(I )) = − dim(I )k + h (P , I (k)). k=0 k=0 The ﬁrst sum is a polynomial of degree 4 in n for n 0. By Riemann-Roch, 0 3 n n 1 n 2 n 3 n h (P , I (k)) = χ(I (k)) + h (I (k)) − h (I (k)) + h (I (k)) where χ(In(k)) is a polynomial in k and n. We get from our computations of cohomology in the earlier example that the limit λ(H1 (In)) λ((In)sat/In) lim m = lim n→∞ n4 n→∞ n4 exists, but is irrational. Suppose that (R, m) is a local ring, and I ⊂ R is an ideal. Question 6.3. Does λ(H1 (In)) lim m n→∞ n exist?

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 M M R[E] = Ek be the algebra generated by E in sym(F ) = F k. Then the limit λ(H1 (Ek)) lim m k→∞ kd+e−1 exists.

If depth(R) ≥ 2 or rank(F ) < d + e then the epsilon multiplicity

(d + e − 1)! 0 k k ε(E) = lim λ(Hm(F /E )) k→∞ kd+e−1 exists as a limit. Epsilon multiplicity is deﬁned as a limsup by Ulrich and Validashti in [38]. 13 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 H`a,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 limit 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 limit exists for monomial ideals. An algorithm for computing this limit 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 = (f1, . . . , ft), and let Z be the blow up of I. L n t−1 Z = B(I) = Proj( n≥0 I ) ⊂ PR h ↓ X = Spec(R)

k 0 I OZ = OZ (k), k ≥ 0. By Serre’s “fundamental theorem”, there exists k such that for k ≥ k0,

0 k k k ˜k (1) H (Z,I OZ ) = I and h∗(I OZ ) = I . Let f : Y = B(mI) → X be the blow up of mI. g Y = B(mI) → Z →h X Deﬁne line bundles on Y by ∗ L = g OZ (1) = IOY ,

M = mOY

Proposition 6.5. There exist positive integers k0 and τ such that 1) For k ≥ k0, n ∈ Z and p ∈ X \{m}, n k k Γ(Y, M ⊗ L )p = (I )p.

2) For k ≥ k0, k ∞ −kτ k I :R m = Γ(Y, M ⊗ L ).

Proof of 1): n k −1 ∼ k −1 M ⊗ L |(Y \ f (m)) = I OY |(Y \ f (m)). Now 1) follows from (1). Proof of 2): For all q ∈ Y , k, n ≥ 0 implies

k n −n k −n k I : m ⊂ [mOY ] I = (M ⊗ L )q. q Thus k n −n k −n k (2) I : m ⊂ ∩q∈Y (M ⊗ L )q ⊂ Γ(Y, M ⊗ L ). 14 1) and depth(R) ≥ 2 implies that for k ≥ k0, −n k −1 n −1 (3) Γ(Y, M ⊗ L ) ⊂ Γ(X \ f (m),I OX ) ⊂ Γ(X \ f (m), OX ) = R. Katz and McAdam [23] (also [35]): There exists τ such that (4) Ik : m∞ = Ik : mkτ for all k ≥ 0.

(2), (3) and (4) imply Ik : m∞ ⊂ Γ(Y, M−kτ ⊗ Lk) ⊂ R k ∞ for k ≥ k0. 2) now follows from 1) since I : m is the largest ideal J of R with the k property that Jp = Ip for all p ∈ X \{m}. L n Let R(I) = n≥0 I . (5) 0 → Γ(Y, Lk)/Ik → Ik : m∞/Ik → Ik : m∞/Γ(Y, Lk) → 0 k k 1) implies Γ(Y, L )/I is supported on m for k ≥ k0. ∗ L k L = h OZ (1) is the pullback of an ample line bundle on Z = Proj(R(I)), so k≥0 Γ(Y, L ) is a ﬁnitely generated R(I)-module (Serre’s theorem). 1) implies there exists an r such that mr Γ(Y, Lk)/Ik = 0 for all k ≥ k0. Now dim R(I)/mR(I) ≤ dim R(I) − 1 ≤ dim R ≤ d and R/mr is an Artin local ring imply λ(Γ(Y, Lk)/Ik) is a polynomial of degree ≤ d − 1 for k 0. By (5), we are reduced to computing λ Ik : m∞/Γ(Y, Lk) lim . k→∞ kd Taking global sections of

k −kτ k −kτ k kτ 0 → L → M ⊗ L → M ⊗ L ⊗ OY /m OY → 0

We obtain for k ≥ k0, k ∞ k −kτ k kτ 0 → I : m /Γ(Y, L ) → Γ(Y, M ⊗ L ⊗ OY /m OY ) → H1(Y, Lk) Since L is the pullback of an ample line bundle on Z, we have that h1(Y, Lk) = polynomial in k of degree ≤ d − 1 for large k. We have reduced to showing −kτ k kτ λ Γ(Y, M ⊗ L ⊗ OY /m OY lim k→∞ kd exists. For simplicity, assume there exists a ﬁeld K ⊂ R such that R/m =∼ K. 15 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 . −τ Set M = mOY , B = C ⊗ M . kτ kτ ∼ kτ 0 → M → OY → OY /m OY = OY /m OY → 0 Tensor with Bk to get k k −kτ kτ 0 → C → B → M ⊗ OY /m OY → 0 for k ≥ 0. 0 → H0(Y, Ck) → H0(Y, Bk) → 0 −kτ k kτ 1 k H (Y, M ⊗ L ⊗ OY /M OY ) → H (Y, C ) C generated by global sections and is big implies h1(Y, Ck) lim = 0 k→∞ kd and h0(Y, Ck) lim ∈ Q. k→∞ kd So λ Ik : m∞/Ik h0(Y, Bk) lim = lim k→∞ kd k→∞ kd exists since B is big on a projective variety. This last statement is by “Fujita Approxima- tion” (Fujita [16], S. Takagi [36], Lazarsfeld and Mustata [28]).

7. Failure of Tameness We construct examples of Rees algebras M A = R(I) = In, n≥0 associated to an ideal I in a local ring R, which is an algebra over a field K, such that the function i (6) j 7→ dimK (HA+ (A)−j) is an interesting function for j 0. In our examples, this dimension will be finite for all j. L Suppose that A0 is a Noetherian local ring, A = j≥0 Aj is a standard graded ring and L ˜ set A+ := j>0 Aj. 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 M Hi+1(M) =∼ Hi(Y, F(n)) A+ n∈Z for i ≥ 1, and a similar expression for i = 0 and 1. 16 By Serre vanishing, Hi (M) = 0 for all i and j 0. However, the asymptotic A+ j behaviour of Hi (M) for j 0 is much more mysterious. A+ −j In the case when A0 = K is a field, the function i j 7→ dimK (HA+ (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 A0, HA+ (M)−j are ﬁnitely generated A0 modules, but need not have ﬁnite length. The following problem was proposed by Brodmann and Hellus [3].

Tameness problem: Are the local cohomology modules Hi (M) tame? That is, is it A+ true that either i i {HA+ (M)j 6= 0, ∀j 0} or {HA+(M)j = 0, ∀j 0}?

The problem has a positive solution for A0 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 A0 ≤ 2, then M is tame. 7.1. 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 T0 with dim(T0) = 3 and a graded ideal A ⊂ T0 such that the Rees algebra T = T0[At] of A is normal, and for j > 0, 2 if j is even, dim (H2 (T ) ) = K B −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 T0 over a ﬁeld K of characteristic p with dim(T0) = 4, and a graded ideal A ⊂ T0 such that the Rees algebra T = T0[At] of A is normal, and for j > 0,

 1 if j ≡ 0( mod )(p + 1), 2  t dimK (HQ(T )−j) = 1 if j = p for some odd t ≥ 0,  0 otherwise, where B is the graded ideal AtT of T . We have pt ≡ −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 T0 with dim(T0) = 4 and a graded ideal A ⊂ T0 such that the Rees algebra T = R0[At] of A is normal, and for j > 0, 6j if j is even, dim (H3 (T ) ) = K B −j 0 if j is odd, where B is the graded ideal AtT of T . 17 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 T0 with dim(T0) = 3, and a graded ideal A ⊂ T0 such that the Rees algebra T = T0[At] of A is normal, and for j > 0, 2 dimK (H (T )−j) B h i h i h i h i = 162 j2 √j + 1 − 1 √j √j + 1 2 √j + 1 2 2 3 2 2 2 and dim (H2 (T ) ) √ lim K B −j = 54 2 j→∞ j3 where B is the graded ideal AtT of T .

In all of these examples, T0 is generalized Cohen Macaulay, but is not Cohen Macaulay. This follows since in all of these examples, 2 ∼ 1 HP0 (R0)0 = H (X, OX ) 6= 0. 8. Some Questions Question 8.1. Suppose that I is a homogeneous ideal in a polynomial ring S. Does ai(In) lim n→∞ n exist for all i? Does reg(I(n)) lim n→∞ n exist? (Yes if the singular locus of S/I has dimension ≤ 1 Herzog, LT Hoa, NV Trung [20]). Question 8.2. Does reg(in(In)) lim n→∞ n exist? Question 8.3. Suppose that I is generated in a single degree. Explain (geometrically) the constant term b in the linear polynomial reg(In) = 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 x + d − i P (x) = (−1)ie (M) M i d − i i=0 for x 0. Question 8.4. Suppose that I,J are homogeneous ideals in a polynomial ring S = k ∞ k k[x1, . . . , xn]. Define Ik(J) = I : J . Let d be the limit dimension of Ik(J)/I . Does k n−d lim e0(Ik(J)/I )/k k→∞ exist? 18 The limit exists if I,J are monomial ideals (Herzog, Puthenpurakal, Verma [21]). Question 8.5. Suppose that R is a Noetherian local ring of dimension d, M a finitely k generated R-module, I an ideal of R. Are the Hilbert functions ei(M/I M) polynomial functions for large k? In the case when R = S is a graded ring of dimension n, I ⊂ S is a homogeneous ideal, k the Hilbert functions ei(M/I M) are polynomial functions of degree ≤ n − d − i for i 0 (Hoang and Trung [22], Herzog, Puthenpurakal, Verma [21]). A complexity in the case of L n a general (nonhomogeneous) ideal I is that the algebra n≥0 in(I ) may not be finitely generated ([11]).

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