ULI WALTHER UNIVERSITY OF MINNESOTA

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1 ALGORITHMIC COMPUTATION OF LOCAL COHOMOLOGY MODULES AND THE COHOMOLOGICAL DIMENSION OF ALGEBRAIC VARIETIES arxiv:alg-geom/ v1 3 Oct 1997 ULI WALTHER UNIVERSITY OF MINNESOTA Abstract. In this paper we present algorithms that compute certain local cohomology modules associated to a ring of polynomials containing the rational numbers. In particular we are able to compute the local cohomological dimension of algebraic varieties in characteristic zero. 1. Introduction 1.1. Let R be a commutative Noetherian ring, I an ideal in R and M an R-module. The ith local cohomology functor with respect to I is the ith right derived functor of the functor HI 0 ( ) which sends M to the I-torsion 1 (0 : M I k ) of M and is denoted by HI i ( ). The cohomological dimension of I in R, denoted by cd(r, I), is the smallest integer c such that the local cohomology modules H q I (M) = 0 for all q > c and all R-modules M. Local cohomology was introduced by Grothendieck as an algebraic analog of (classical) relative cohomology. A brief introduction to local cohomology may be found in appendix 4 of [2] Knowledge of local cohomology modules provides interesting information, illustrated by the following three situations. Let I R and c = cd(r, I). Then I cannot be generated by fewer than c elements. In fact, no ideal J with the same radical as I will be generated by fewer than c elements. A second application is a family of results commonly known as Barth theorems which are a generalization of the classical Lefschetz theorem that states that if Y P n is a hypersurface then HdR i (Pn ) HdR i (Y ) is an isomorphism for i < dim Y 1 and injective for i = dim Y. For example, let Y P n be a closed subset and I R = k[x 0,..., x n ] the defining ideal of Y. Then HdR i (Pn ) HdR i (Y ) is an isomorphism for i depth OP n(o Y ) cd(r, I) (compare [13], 4.7 and [5], the theorem after III.7.6). 1

2 2 ULI WALTHER Finally, it is also a consequence of the work of Ogus and Hartshorne ([13], 2.2, 2.3 and [5], IV.3.1) that if I R = k[x 0,..., x n ] is the defining ideal of a complex smooth variety V P n then, for i < n codim V, dim C soc R Hm(H 0 n i (R)) equals dim C Hx(Ṽ i, C) where Hi x(ṽ, C) stands for the ith singular cohomology group of the affine cone Ṽ over V with support in the vertex x of Ṽ and with coefficients in C (soc R(M) denotes the socle (0 : M m) M for any R-module M) The cohomological dimension has been studied by many authors; see for example [4], [13], [6], [14], [3], [7] and [10]. Yet despite this extensive effort, the problem of finding an algorithm for the computation of cohomological dimension remained open. It is in fact enough to find an algorithm to decide whether or not the local cohomology module HI i(r) = 0 for given i, R, I. This is because Hq I (R) = 0 for all q > c implies cd(r, I) c. In [10] an algorithm for deciding whether or not HI i(r) = 0 has been given for all I R = K[x 1,..., x n ] where K is a field of positive characteristic. One of the main purposes of this work is to produce such an algorithm in the case where K is a field containing the rational numbers and R = K[x 1,..., x n ]. Since in such a situation the local cohomology modules HI i (R) have a natural structure of finitely generated D(R, k)-modules ([10]), D(R, k) being the ring of k-linear differential operators of R, explicit computations may be performed. Using this finiteness we employ the theory of Gröbner bases to develop algorithms that give a representation of HI i(r) and Hi m (Hj I (R)) for all triples i, j N, I R in terms of generators and relations over D(R, K) (where m = (x 1,..., x n )). This also leads to an algorithm for the computation of the invariants λ i,j (R/I) = dim K soc R (Hm i (Hn j I (R))) introduced in [10] The outline of the paper is as follows. The next section is devoted to a short survey of results on local cohomology and D-modules as they apply to our work, as well as their interrelationship. In section 3 we review the theory of Gröbner bases as it applies to A n and modules over the Weyl algebra. Most of that section should be standard and readers interested in proofs and more details are encouraged to look at the book by D. Eisenbud ([2], chapter 15) or the more fundamental article [8]. In section 4 we generalize some results due to B. Malgrange and M. Kashiwara on D-modules and their localizations. The purpose of sections 4 and 5 is to find a representation of R f N as a cyclic A n - module if N is a given holonomic D-module (for a definition and some properties of holonomic modules, see subsection 2.3 below). Many of the essential ideas in section 5 come from T. Oaku s work [12].

3 ALGORITHMS IN LOCAL COHOMOLOGY 3 In section 6 we describe our main results, namely algorithms that for arbitrary i, j, k, I determine the structure of HI k(r), Hi m (Hj I (R)) and find λ i,j (R/I). Some of these algorithms have been implemented in the programming language C and the theory is illustrated with examples. The final section is devoted to comments on implementations, effectivity and examples. It is a pleasure to thank my advisor Gennady Lyubeznik for suggesting the problem of algorithmic computation of cohomological dimension to me and pointing out that the theory of D-modules might be useful for its solution. 2. Preliminaries 2.1. Notation. Throughout we shall use the following notation: K will denote a field of characteristic zero, R = K[x 1,..., x n ] the ring of polynomials over K in n variables, A n = K x 1, 1,..., x n, n the Weyl algebra over K in n variables, or, equivalently, the ring of differential operators on R, m will denote the maximal ideal (x 1,..., x n ) of R and I will stand for the ideal (f 1,..., f r ) in R. We shall sometimes refer to elements of A n as polynomials although strictly speaking they are not. All tensor products in this work will be over R and all A n -modules will be left modules Local cohomology. It turns out that HI k (M) may be computed as follows. Let C (f i ) be the complex 0 R R fi 0. Then HI k (M) is the k-th cohomology group of the Čech complex defined by C (M; f 1,..., f r ) = M r 1 C (f i ). Unfortunately, explicit calculations are complicated by the fact that HI k (M) is rarely finitely generated as R-module. This difficulty disappears for HI k (R) if we enlarge the ring to A n, in essence because R f is finitely generated over A n for any f R D-modules. A good introduction to D-modules is the book by Björk, [1]. Let g R. Then the R-module R f has a structure as left A n -module: x i ( f ) = x if, g k g k i ( f ) = i(f)g k i (g)f. This may be thought of as a special g k g k+1 case of localizing an A n -module: if M is an A n -module and g R then R g R M becomes an A n -module via i ( f m) = g k i ( f ) m+ f g k g k i m. Localization of A n -modules lies at the heart of our arguments. Let N = A n /L be a nonzero finitely generated leftmodule over the Weyl algebra A n. Of particular interest are the holonomic modules which are those for which Ext j A n (M, A n ) = 0 unless j = n. Holonomic modules are always cyclic and of finite length over A n. Besides that,

4 4 ULI WALTHER if f R, s an indeterminate and g is some fixed generator of N, then there is a nonzero polynomial b(s) with rational coefficients and an operator P(s) A n [s] such that P(s)(f f s g) = b(s) f s g. The unique monic polynomial that divides all other polynomials satisfying an identity of this type is called the Bernstein polynomial of N and f and denoted by b L f (s). Any operator P(s) that satisfies P(s)fs+1 g = b L f (s)fs g we shall call a Bernstein operator and refer to the roots of b L f (s) as Bernstein roots of f on A n/l. Localizations of holonomic modules are holonomic (see [1], section 5.9) and in particular cyclic over A n, generated by f a g for sufficiently large a N (see also our proposition 4.3). So the complex C (N; f 1,..., f r ) consists of holonomic A n -modules whenever N is holonomic. As a special case we see that localizations of R are holonomic, and hence finite, over A n (since R = A n /( 1,..., n ) is holonomic). This facilitates the use of Gröbner bases as computational tool for maps between holonomic modules and their localizations The Čech complex. In [10] it is shown that local cohomology modules are not only D-modules but in fact holonomic: we know already that the modules in the Čech complex are holonomic, it suffices to show that the maps are A n -linear. All maps in the Čech complex are direct sums of localization maps. Suppose R f is generated by f s and R fg by (fg) t. We may replace s, t by their minimum u and then we see that the inclusion R f R fg is nothing but the map A n / ann(f u ) A n / ann((fg) u ) sending the coset of the operator P to the coset of the operator P g u. So C i (M; f 1,..., f r ) C i+1 (M; f 1,..., f r ) is an A n -linear map between holonomic modules. One can prove that kernels and cokernels of A n -linear maps between holonomic modules are holonomic. Holonomicity of HI k (R) follows. In the same way it can be seen that Hm i (Hj I (R)) is holonomic for i, j N (since H j I (R) is holonomic). 3. Gröbner bases over the Weyl algebra In this section we review some of the concepts and results related to the Buchberger algorithm in modules over Weyl algebras. It turns out that with a little care many of the important constructions from the theory of commutative Gröbner bases carry over to our case. For an introduction into non-commutative monomial orders and related topics, [8] is a good source. Let us agree that every time we write an element in A n, we write it as a sum of terms c αβ x α β in multi-index notation. That is, c αβ are

5 ALGORITHMS IN LOCAL COHOMOLOGY 5 scalars, x α = x α x αn n, β = β βn n and in every monomial we write first the powers of x and then the powers of the differentials. Further, if m = c αβ x α β, c αβ K, we will say that m has degree deg m = α + β and a polynomial p has degree equal to the largest degree of any monomial occuring in p. Recall that a monomial order < in A n is a total order on the monomials of A n, subject to m < m mm < m m for all nonzero monomials m, m, m. Since the product of two monomials in our notation is not likely to be a monomial (as i x i = x i i + 1) it is not obvious that such orderings exist at all. However, the commutator of any two monomials m 1, m 2 will be a polynomial of degree at most deg m 1 + deg m 2 2. That means that the degree of a polynomial is independent of the way it is represented. Thus we may for example introduce a monomial order on A n by taking any monomial order on à n = k[x 1,..., x n, 1,... n ] (the polynomial ring in 2n variables) and saying that m 1 > m 2 in A n if and only if m 1 > m 2 in Ãn. Let < be a monomial order on A n. Let G = d 1 A n γ i be the free A n -module on the symbols γ 1,..., γ d. We define a monomial order on G by m i γ i > m j γ j if either m i > m j in the order on A n, or m i = m j and i > j. The largest term mγ in an element g G will be denoted by in(f). Of fundamental importance is Algorithm 3.1 (Remainder). Let h and g = {g i } s 1 be elements of G. Set h 0 = h, σ 0 = 0, j = 0 and let ε i = ε(g i ) be symbols. Then Repeat If in(g i ) in(h j )set Until No in(g i ) in(h j ) {h j+1 := h j in(h j) g in(g i ) i, σ j+1 := σ j + in(h j) ε in(g i ) i, j := j + 1} The result is h a, called a remainder R(h, g) of h under division by g, and an expression σ r = s i=1 a iε i with a i A n. By Dickson s lemma ([8], 1.1) the algorithm terminates. It is worth mentioning that R(h, g) is not uniquely determined, it depends on which g i we pick amongst those whose initial term divides the initial term of h j. Note that if h a is zero, σ a tells us how to write h in terms of g. Such a σ a is called a standard expression for h with respect to {g 1,..., g s }. Definition 3.2. If in(g i ) and in(g j ) involve the same basis element of G, then we set s ij = m ji g i m ij g j and σ ij = m ji ε i m ij ε j where

6 6 ULI WALTHER m ij = lcm(in(g j),in(g i )) in(g i. Otherwise, σ ) ij and s ij are defined to be zero. s ij is the Schreyer-difference to g i and g j. Suppose R(s ij, g) is zero for all i, j. Then we call g a Gröbner basis for the module A n (g 1,..., g s ). The following proposition ([8], Lemma 3.8) indicates the usefulness of Gröbner bases. Proposition 3.3. Let g be a finite set of elements of G. Then g is a Gröbner basis if and only if h A n g implies i : in(g i ) in(h). Computation of Gröbner bases over the Weyl algebra works just as over polynomial rings: Algorithm 3.4 (Buchberger). Input: {g 1,..., g s } G. Output: a Gröbner basis for A n (g 1,..., g s ). Begin. Repeat If h = R(s ij, g) 0 Until all R(s ij, g) = 0. Return g. add hto g End Now we shall describe the construction of kernels of A n -linear maps using Gröbner bases. Again, this is similar to the commutative case and we first consider the case of a map between free A n -modules. Let E = r 1A n ε i, G = s 1A n γ j and φ : E G be a A n -linear map. Assume φ(ε i ) = g i. Suppose that in order to make g a Gröbner basis we have to add g 1,..., g s to g which satisfy g i = s k=1 a ikg k. We get an induced map s+s 1 A n ε i π φ s 1 A nε i r 1 A nγ j φ where π is the identity on ε i for i s and sends ε i+s into s k=1 a ikε k. Of course, φ = φπ. The kernel of φ is just the image of the kernel of φ under π. So in order to find kernels of maps between free modules one may assume that the generators of the source are mapped to a Gröbner basis and replace φ by φ. Recall from definition 3.2 that σ ij = m ji ε i m ij ε j or zero, depending on the leading terms of g i and g j.

7 ALGORITHMS IN LOCAL COHOMOLOGY 7 Proposition 3.5. Assume that {g 1,..., g s } is a Gröbner basis. Let s ij = d ijk g k be standard expressions for the Schreyer differences. Then {σ ij d ijk ε k } 1 i<j s generate the kernel of φ. The proof proceeds exactly as in the commutative case, see for example [2], section We explain now how to find a set of generators for the kernel of an arbitrary A n -linear map. Let E, G be as in subsection 3.1 and suppose A n (κ 1,..., κ k ) = K E, A n (λ 1,..., λ l ) = L G and φ : r 1A n ε i /K s 1A n γ j /L. It will be sufficient to consider the case K = 0 since we may lift φ to a free module surjecting onto E/K. Let as before φ(ε i ) = g i. A kernel element in E is a sum i a iε i, a i A n which if ε i is replaced by g i can be written in terms of the generators λ j of L. Let β = {β 1,..., β b } be such that g λ β is a Gröbner basis for A n (g, λ). We may assume that the β i are the results of applying algorithm 3.4 to g λ. Then from algorithm 3.1 we have expressions β i = j c ij g j + k c ik λ k. (3.1) Furthermore, by proposition 3.5, algorithm 3.4 returns a generating set σ for the syzygies on g λ β. Write σ i = j a ij ε gj + k a ikε λk + l a ilε βl (3.2) and eliminate the last sum using the relations (3.1) to obtain syzygies σ i = a ij ε gj + a ikε λk + ( a il c lv ε gv + ) c lwε λw. j k l v w (3.3) These will then form a generating set for the syzygies on β λ. Cutting off the λ-part of these syzygies we get a set of forms { a ij ε gj + ( )} a il c lv ε gv j l v which generate the kernel of the map E G/L The comments in this subsection will find their application in algorithm 6.2 which computes the structure of H i m(h j I (R)) as A n-module.

8 8 ULI WALTHER Let M 3 φ M 2 φ M 1 α M 3 α M β γ ψ 3 ρ β M 2 M 2 ψ M 1 γ be a commutative diagram of A n -modules. Note that the row cohomology of the column cohomology at N is given by ρ M 1 [ker(ψ ) β 1 (im ρ) + im(ψ)]/[β ker(φ ) + im(ψ)]. In order to compute this we need to be able to find: preimages of submodules, kernels of maps, intersections of submodules. It is apparent that the second and third calculation is a special case of the first: kernels are preimages of zero, intersections are images of preimages (if A r φ n A s n/m ψ A t n is given, then im(φ) im(ψ) = ψ(ψ 1 (im(φ))) ). So suppose in the situation φ : A r n /M As n /N, ψ : At n /P As n /N we want to find the preimage under ψ of the image of φ. We may reduce to the case where M and P are zero and then lift φ, ψ to maps into A s n. The elements x in ψ 1 (im φ) A t n are exactly the elements in ker(a t ψ n A s n/n A s n/(n + im φ)) and this kernel can be found according to the comments in D-modules after Kashiwara and Malgrange The purpose of this and the following section is as follows. Given f R, L A n an ideal such that A n /L is holonomic and L is f- saturated (i.e. f P L only if P L), we want to compute the structure of the module R f A n /L. It turns out that it is useful to know the ideal J L (f s ) which consists of the operators P(s) A n [s] that annihilate f s 1 R f [s]f s A n /L where the bar denotes cosets in A n /L. In order to find J L (f s ),we will consider the module M = R f [s] f s R A n /L over the ring A n+1 = A n t, t. It will turn out in 4.1 that one can easily compute the ideal J L n+1 (fs ) A n+1 consisting of all operators that kill f s 1. In section 5 we will then show how to compute J L (f s ) from J L n+1(f s ).

9 ALGORITHMS IN LOCAL COHOMOLOGY 9 The second basic fact in this section (4.3) shows how to compute the structure of R f A n /L as A n -module once J L (f s ) is known Consider A n+1 = A n t, t, the Weyl algebra in x 1,..., x n and the new variable t. B. Malgrange has defined an action of t and t on M = R f [s] f s R A n /L by t(g(x, s) f s P) = g(x, s + 1)f f s P and t (g(x, s) f s P) = s g(x, s 1) f fs P for P A n /L. A n acts on M as expected, the variables by multiplication on the left, the i by the product rule. One checks that this actually defines an structure of M as a left A n+1 -module and that t t acts as multiplication by s. We denote by Jn+1 L (fs ) the ideal in A n+1 that annihilates the element f s 1 in M. Then we have an induced morphism of A n+1 -modules A/Jn+1(f L s ) M sending P + Jn+1(f L s ) into P(f s 1). The operators t and t were introduced in [11]. The following lemma generalizes lemma 4.1 in [11] (as well as part of the proof given there) where the special case L = ( 1,..., n ), M = R is considered. Lemma 4.1. Suppose that L = A n (P 1,..., P r ) is f-saturated (i.e., if f P L, then P L). With the above definitions, J L n+1 (fs ) is the ideal generated by f t together with the images of the P j under the automorphism φ of A n+1 induced by x x, t t f. Proof. The automorphism sends i to i + f i t and t to t. So if we write P j = P j ( 1,..., n ), then φp j = P j ( 1 + f 1 t,..., n + f n t ). One checks that ( i + f i t )(f s Q) = f s i Q for all differential operators Q so that φ(p j ( 1,..., n ))(f s 1) = f s P j ( 1,..., n ) = 0. By definition, f (f s 1) = t (f s 1). So t f Jn+1 L (fs ) and φ(p j ) Jn+1(f L s ) for i = 1,..., r. Conversely let P(f s 1) = 0. We may assume, that P does not contain any t since we can eliminate t using f t. Now rewrite P in terms of t and the i +f i t. Say, P = c αβ t αxβ Q αβ ( 1 +f 1 t,..., n +f n t ), where the Q αβ are polynomials in n variables and c αβ K. Application to f s 1 results in t α(fs c αβ x β Q αβ ( 1,..., n )). Now α,β c αβ t α(fs x β Q αβ ( 1,..., n )) is a polynomial in s of degree, say, α with coefficients in R f f s A n /L. We show that the sum of terms that contain t α is in (P 1 ( 1,..., n ),..., P r ( 1,..., n )) as follows. In order for P(f s 1 to vanish, the coefficient to the highest s-power must vanish, which is to say c αβ ( 1/f) α f s x β Q αβ ( 1,..., n ) R f f s L β as R f is R-flat.

10 10 ULI WALTHER It follows, that β c αβx β Q αβ ( 1,..., n ) L (L is f-saturated!) and so by the first part, P β c αβ t α x β Q αβ ( 1 + f 1 t,..., n + f n t ) kills f s 1, but is of smaller degree in t than P was. The claim follows. Definition 4.2. Let J L (f s ) stand for the ideal in A n [s] = A n [ t t] that kills f s 1 R f [s]f s R A n /L. Note that J L (f s ) = J L n+1 (fs ) A n [ t t]. We will in the next section show how the lemma can be used to determine J L (f s ). Now we show why J L (f s ) is useful, generalizing [9], proposition 6.2. Recall that the Bernstein polynomial b L f (s) is defined to be the (nonzero) monic generator of the ideal of polynomials b(s) K[s] for which there exists an operator P(s) A n [s] such that P(s)(f s+1 1) = b(s)f s 1 ([1], chapter 1). Proposition 4.3. If a Z is such that no integer root of b L f (s) is smaller than a, then we have isomorphisms R f M = A n [s]/j L (f s ) s=a = An f a 1. (4.1) Proof. We mimick the proof given by Kashiwara, who proved the proposition for the case L = ( 1,..., n ), M = R. Let us first prove the second equality. Certainly J L (f s ) s=a kills f a 1. So we have to show that if P(f a 1) = 0 then P J L (f s ) + A n [s] (s a). To that end note that st acts as t(s 1) which means that t (A n [s]/j L (f s )) is a left A n [s]-module. Identify A n [s]/j L (f s ) with A n [s] (f s 1) and write Nf L = A n[s]/j L (f s ). By definition, b L f (s) is the minimal polynomial for which there is P(s) with b L f (s)(fs 1) = P(s)f s+1 = t P(s 1)(f s 1). So b L f (s) multiplies A n[s] (f s 1) into t A n [s](f s 1) and whenever the polynomial b(s) k[s] is relatively prime to b L f (s) its action on N f L/t N f L is injective. Since by hypothesis s a + j is not a divisor of b L f (s) for 0 < j N, (s a + j)n L f t N L f (s a + j)t N L f. (4.2) So (s a + m)nf L tm Nf L (s a + m)tn f L tm Nf L = t[(s a + m 1)Nf L tm 1 Nf L ] whenever m 1. We show now by induction on m that (s a + m)nf L tm Nf L (s a + m)t m Nf L for m 1. The claim is clear for m = 1 from equation 4.2. So let m > 1. The inductive hypothesis states that (s a + m 1)Nf L tm 1 Nf L (s a + m 1)tm 1 Nf L. The previous

11 ALGORITHMS IN LOCAL COHOMOLOGY 11 paragraph shows that (s a+m)nf L tm Nf L t[(s a+m 1)N f L t m 1 Nf L ]. Combining these two facts we get (s a + m)n L f tn t(s a + m 1)t m 1 N L f = (s a + m)t m N L f. Now if P(s) A n [s] is of degree m in the i and P(a)(f a 1) = 0, then P(s+m) f m +J L (f s ) (s a+m) Nf L because we can interprete P(s + m)(f s+m 1) as a polynomial in s + m with root a. But then P(s + m)(f s+m 1) = P(s + m)(f m f s 1) is in (s a + m)n L f tm N L f (s a + m)tm N L f, implying P(s + m)(f s+m 1) = (s a + m)q(s)(f s+m 1) for some Q(s) A n [s] (note that J L (f s ) kills f s 1). In other words, P(s) (s a)q(s m) J L (f s ). For the first isomorphism we have to show that A n (f a 1) = R f R M. It suffices to show that every term of the form f m f a Q is in the module generated by (f a 1) for all m Z. Furthermore, we may assume that Q is a monomial in the 1,..., n. Existence and definition of b L f (s) provides an operator P(s) with [b L f (s 1)] 1 P(s 1)(f s 1) = f 1 f s 1. As b L f (a m) 0 for all 0 < m N we have f m f a 1 A n (f a 1) for all m. Now let Q be a monomial in 1,..., n of degree j > 0 and assume that f m f a Q A n (f a 1) for all m and all operators Q of degree lower than j. Then Q = i Q for some 1 i n. By assumption on j, for some P we have P (f a 1) = f m f a Q. So f m f a Q = i P (f a 1) f i (a + m)f m 1 f a Q A n (f a 1). (4.3) The claim follows by induction. This completes the proof of the proposition. We remark that if any a Z satisfies the conditions of the proposition, then so does every integer smaller than a. 5. The algorithm of Oaku The purpose of this section is to review and generalize an algorithm due to Oaku. In [12], Oaku showed how to construct a generating set for J L (f s ) in the case where L = ( 1,..., n ). According to 4.2, J L (f s ) is the intersection of J L n+1 (fs ) with A n+1. Proposition 5.1 below is a simplified account of the results in [12], generalized to arbitrary L A n+1 : we shall demonstrate how one may calculate J A n [ t t]

12 12 ULI WALTHER whenever J A n+1 is any given ideal. The proof follows closely Oaku s argument. On A n+1 [y 1, y 2 ] define weights w(t) = w(y 1 ) = 1, w( t ) = w(y 2 ) = 1, w(x i ) = w( i ) = 0. Note that the Buchberger algorithm preserves homogeneity in the following sense: if a set of generators for an ideal is given and these generators are homogeneous with respect to the weights above, then any new generator for the ideal constructed with the classical Buchberger algorithm will also be homogeneous. (This is a consequence of the facts that the y i commute with all other variables and that t t = t t + 1 is homogeneous of weight zero.) Proposition 5.1. Let J = A n+1 (Q 1,..., Q r ) and let y 1, y 2 be two new variables. Let I be the left ideal in A n+1 [y 1 ] generated by the y 1 - homogenizations (Q i ) h of the Q i, relative to the weight w above, and let Ĩ = A n+1[y 1, y 2 ] (I, 1 y 1 y 2 ). Let G be a Gröbner basis for Ĩ under a monomial order that eliminates y 1, y 2. For each P G set P = t w(p) P if w(p) < 0 and P = w(p) t P if w(p) > 0 and let G = {P : P G}. Then G 0 = G A n [ t t] generates J A n [ t t]. Proof. Note first that G consists of w-homogeneous operators and so w(p) is well defined for P G. Suppose P G 0. Hence P Ĩ. So P = Q 1 (1 y 1 y 2 )+ a i (Q i ) h where the a i are in A n+1 [y 1, y 2 ]. Since P A n [ t t], the substitution y i = 1 shows that P = a i (1, 1) (Q i ) h (1, 1) = a i (1, 1) Q i J. Now assume that P J A n [ t t]. So P is w-homogeneous of weight 0. Also, P J and J is contained in I(1), the ideal of operators Q(1) where Q(y 1 ) I. By lemma 5.2 below (taken from [12]), y1 ap I for some a N. Therefore P = (1 (y 1 y 2 ) a )P + (y 1 y 2 ) a P Ĩ. Let G = {P 1,..., P b, P b+1,..., P c } and assume that P i A n+1 if and only if i b. Buchberger algorithm gives a standard expression P = a i P i with all in(a i P i ) in(p). That implies that a b+i is zero for positive i and a i does not contain y 1, y 2 for any i. Since P, P i are w-homogeneous, the same is true for all a i, from Buchberger algorithm. In fact, w(p) = w(a i ) + w(p i ) for all i. As w(p) = 0 (and t, t are the only variables with nonzero weight that may appear in a i ) we find a i = a i t w(pi) or a i = a i w(p i) t, depending on whether w(p i ) is negative or positive. It follows that P = b 1 a ip i = b 1 a i P i A n[ t t] G 0. Lemma 5.2. Let I be a w-homogeneous ideal in A n+1 [y 1 ] with respect to the weights introduced before the proposition and I(1) defined as in the proof of the proposition. Assume P is a w-homogeneous operator. Then P I(1) implies y a 1P I for some a.

13 ALGORITHMS IN LOCAL COHOMOLOGY 13 Proof. Note first that y 1 1 will not lead to cancellation of terms in any homogeneous operator as w(y 1 ) 0. If P I(1), P = Q i (1), with all Q i w-homogeneous in I. Then the y 1 -homogenization of Q i (1) will be a divisor of Q i and the quotient will be some power of y 1, say y η i 1. Homogenization of the equation P = Q i (1) results in y η 1 P = Q i (1) h so that y η+max(η i) 1 P = y max(η i) η i 1 Q i I. Corollary 5.3. If J L (f s ) is known or it is known that L is f-saturated, then the Bernstein polynomial b L f (s) of R f R A n /L can be found from (b L f (s)) = A n[s] (J L (f s ), f) k[s]. Moreover, suppose b L f (s) = sd + b d 1 s d b 0 and define B = max i { b i 1/(d i) }. In order to find the smallest integer root of b L f (s), one only needs to check the integers between 2B and 2B. If in particular L = ( 1,..., n ), it suffices to check the integers between b d 1 and -1. Proof. If L is f-saturated, propositions 4.1 and 5.1 enable us to find J L (f s ). The first part follows then easily from the definition of b L f (s): as (b L f (s) P f L f)(fs 1) = 0 it is clear that b L f (s) is in k[s] and in A n [s](j L (f s ), f). Using an elimination order on A n [s], b L f (s) will be (up to a scalar factor) the unique element in the reduced Gröbner basis for J L (f s ) + (f) that contains no x i nor i. Now suppose that s = 2Bρ where B is as defined above and ρ > 1. Assume also that s is a root of b L f (s). We find (2Bρ) d = s d d 1 d 1 = b i s i B d i s i (5.1) 0 0 d 1 = B d (2ρ) i B d ((2ρ) d 1), (5.2) 0 using ρ 1. By contradiction, s is not a root. The final claim is a consequence of Kashiwara s work [9] where it is proved that if L = ( 1,..., n ) then all roots of b L f (s) are negative and hence b n 1 is a lower bound for each single root. So we have Algorithm 5.4. Input: f R, L A n such that A n /L is holonomic and f-torsionfree. Output: Generators for J L (f s ). Begin

14 14 ULI WALTHER End. For each generator Q i of L compute the image φ(q i ) under x i x i, t t f, i i + f i t, t t. Add t f to the list. Homogenize all φ(q i ) with respect to the new variable y 1 relative to the weight w introduced before 5.1. Compute a Gröbner basis for the ideal (φ(q 1 ) h,..., φ(q r ) h, 1 y 1 y 2, t y 1 f) in A n+1 [y 1, y 2 ] using an order that eliminates y 1, y 2. Select the operators {P j } b 1 in that basis which do not contain any y i. For each P j, 1 j b, if w(p j ) > 0 replace P j by P j = w(p j) t P j. Otherwise replace P j by P j = t w(pj) P j. Return the new operators {P j }b 1. For purposes of reference we also list algorithms that compute the Bernstein polynomial to a holonomic module and the localization of a holonomic module. The algorithms 5.5 and 5.4 appear already in [12] in the special case L = ( 1,..., n ), A n /L = R. Algorithm 5.5. Input: f R, L A n such that A n /L is holonomic and f-torsionfree. Output: The Bernstein polynomial b L f (s). Begin End. Determine J L (f s ) following algorithm 5.4. Find a reduced Gröbner basis for the ideal J L (f s )+A n [s] f using an elimination order for x and. Pick the unique element in that basis contained in k[s] and return it. Algorithm 5.6. Input: f R, L A n such that A n /L is holonomic and f-torsionfree. Output: An ideal J such that R f A n /L = A n /J. Begin End. Determine J L (f s ) following algorithm 5.4. Find the Bernstein polynomial b L f (s) using algorithm 5.5. Find the smallest integer root a of b L f (s) using corollary 5.3. Replace s by a in all generators for J L (f s ) and return these generators.

15 ALGORITHMS IN LOCAL COHOMOLOGY Local cohomology as A n -module In this section we will combine the results from previous sections to obtain algorithms that compute for given i, j, k N, I R the local cohomology modules H k I (R), Hi m(h j I (R)) and the invariants λ i,j(r/i) associated to I HI k (R). Here we will describe an algorithm that takes in a finite set of polynomials f = {f 1,..., f r } R and returns a presentation of HI k(r) where I = (f 1,..., f r ). Consider the Čech complex associated to f 1,..., f r in R, 0 R r 1 R f i 1 i<j r R fi f j R f1... f r 0. (6.1) Its kth cohomology group is the local cohomology module HI k (R). The map C k = R fi1... f ik R fj1... f jk+1 = C k+1 1 i 1 < <i k r 1 j 1 < <j k+1 r (6.2) is the sum of maps R fi1... f ik R fj1... f jk+1 (6.3) which are either zero (if {i 1,..., i k } {j 1,..., j k+1 }) or send 1 to 1 1, up to sign. Let = A 1 n ( 1,..., n ) so that A n / = R and identify R fi1... f ik with A n /J((f i1... f ik ) s ) s=a and R fj1... f jk+1 with A n /J((f j1... f jk+1 ) s ) s=b where J(f s ) = J (f s ) and a, b are sufficiently small integers. By proposition 4.3 we may assume that a = b 0. Then the map (6.2) is in the nonzero case multiplication from the right by (f l ) a where l = {j 1,..., j k+1 } \ {i 1,..., i k }, again up to sign. It follows that the matrix representing the map C k C k+1 in terms of A n -modules is very easy to write down once the annihilator ideals and Bernstein polynomials for all k- and k + 1-fold products of the f i are known: the entries are 0 or ±f a l where f l is the new factor. Let Θ r k be the set of k-element subsets of 1,..., r and for θ Θr k write F θ for the product i Θ f r i. We have demonstrated the correctness and k finiteness of the following algorithm. Algorithm 6.1. Input: f 1,..., f r R; k N. Output: HI k (R) in terms of generators and relations as finitely generated A n -module. Begin

16 16 ULI WALTHER Compute the annihilator ideal J (Fθ s ) and the Bernstein polynomial b F θ (s) for all k 1-, k- and k + 1-fold products of f1 s,..., fs r as in 5.4 and 5.5 (so θ runs through Θ r k 1 Θr k Θr k+1 ). Compute the smallest integer root for each b F θ (s), let a be the minimum and replace s by a in all the annihilator ideals. Compute the two matrices M k 1, M k representing the A n -linear maps C k 1 C k and C k C k+1. Compute the kernel G of the map A n /J((F θ ) s M ) k s=a A n /J((F θ ) s ) s=a θ Θ r k θ Θ r k+1 End. as in 3.2. Compute a Gröbner basis G 0 for the module im(m k 1 ) + θ Θ r k J((F θ ) s ) s=a. Compute the remainders of all elements of G with respect to G 0. Return these remainders and G 0. The nonzero elements of G generate the quotient G/G 0 = H k I (R) so that HI k (R) = 0 if and only if all returned remainders are zero Hm i (Hj I (R)). As a second application of Gröbner basis computations over A n we show now how to compute Hm i (Hj I (R)). Note that we cannot apply lemma 4.1 to A n /L = H j I (R) since Hj I (R) may well contain some torsion. As in the previous sections, C j (f 1,..., f r ) denotes the jth module in the Čech complex to R and {f 1,..., f r }. Let C be the double complex with C i,j = C i (x 1,..., x n ) R C j (f 1,..., f r ), the vertical maps φ induced by the identity on the first factor and the usual Čech maps on the second, whereas the horizontal maps ξ are induced by the Čech maps on the first factor and the identity on the second. Since C i (x 1,..., x n ) is R-projective, the column homology of C at (i, j) is C i (x 1,..., x n ) R H j I (R) and the induced horizontal maps in the jth row are simply the maps in a Čech resolution of Hj I (R). It follows that the row cohomology of the column cohomology ot (i 0, j 0 ) is Hm(H i j I (R)). Now note that C i,j is a direct sum of modules R g where g = x α1... x αi f β1... f βj. So the whole double complex can be rewritten

17 ALGORITHMS IN LOCAL COHOMOLOGY 17 in terms of A n -modules and A n -linear maps using 5.6: i 1,j+1ξi 1,j+1 C C i,j+1 ξi,j+1 C i+1,j+1 C i 1,j φ i 1,j ξ i 1,j φ i 1,j 1 C i,j φ i,j φ i,j 1 ξ i,j C i+1,j φ i+1,j φ i+1,j 1 i 1,j 1ξi 1,j 1 C C i,j 1 ξi,j 1 C i+1,j 1 Using the comments in subsection 3.3, we may then compute H i m(h j I (R)). More concisely, we have the following Algorithm 6.2. Input: f 1,..., f r R; i 0, j 0 N. Output: H i 0 m (Hj 0 I (R)) in terms of generators and relations as finitely generated A n -module. Begin. For i = i 0 1, i 0, i and j = j 0 1, j 0, j compute the annihilators J ((F θ X θ ) s ) and Bernstein polynomials b F (s) θ X θ of F θ X θ where θ Θ r i, θ Θ n j and X θ denotes in analogy to F θ the product α θ x α. Let a be the minimum integer root of the product of all these Bernstein polynomials and replace s by a in all the annihilators computed in the previous step. Compute the matrices to the A n -linear maps C i,j C i,j+1 and C i,j C i+1,j, again for i = i 0 1, i 0, i 0 +1 and j = j 0 1, j 0, j Compute Gröbner bases for the modules G = ker(φ i 0j 0 ) (ξ i 0j 0 ) 1 (im(φ i 0+1,j 0 1 )) + im(φ i 0,j 0 1 ) and G 0 = ξ i 0 1,j 0 (ker(φ i 0 1,j 0 )) + im(φ i 0,j 0 1 ). Compute the remainders of all elements of G with respect to G 0 and return these remainders together with G 0. End. Again, the elements of G will be generators for Hm i (Hj I (R)) and the elements of G 0 generate the relations that are not syzygies λ i,n j (R/I). In [10] it has been shown that Hm i (Hj I (R)) is an injective m-torsion R-module of finite socle dimension λ i,n j (which depends only on i, j and R/I) and so isomorphic to (E R (K)) λ i,n j where E R (K) is the injective hull of K over R. We are now in a position that allows computation of these invariants of R/I. For, let Hm(H i j I (R)) be generated by g 1,..., g l A d n modulo the relations h 1,..., h e A d n. Let H be the module generated by the h i. We know that (A n g 1 + H)/H is

18 18 ULI WALTHER m-torsion and so it is possible (with trial and error) to find a multiple of g 1, say mg 1 with m a monomial in R, such that (A n mg 1 +H)/H is nonzero but x i mg 1 H for all 1 i n. Then the element mg 1 +H/H has annihilator equal to m and hence generates an A n -module isomorphic to A n /A n m = E R (k). The injection A n mg 1 + H/H A n (g 1,..., g l )+H/H splits as map of R-modules because of injectivity and so the cokernel A n (g 1,..., g l )/A n (mg 1, h 1,..., h e ) is isomorphic to (E R (K)) λ i,n j 1. Reduction of the g i with respect to a Gröbner basis of the new relation module and repetition of the previous will lead to the determination of λ i,n j. 7. Implementation and examples The algorithms described above have been implemented as C-scripts and tested on some examples The algorithm of Oaku 5.4 has been implemented in two different ways, the first time by Oaku himself using the package Kan (see [15]) which is a postscript language for computations in the Weyl algebra and in polynomial rings. The second implementation is written by the current author and part of a program that deals exclusively with computations around local cohomology ([16]). [16] is theoretically able to compute H i I (R) for arbitrary i, R = Q[x 1,..., x n ], I R in the above described terms of generators and relations, using algorithm 6.1. It is expected that in the near future also algorithms for computation of H i m (Hj I (R)) and λ i,j(r) will be implemented, but see the comments in 7.2 below. Example 7.1. Let I be the ideal in R = K[x 1,(..., x 6 ] that is ) generated by the 2 2 minors f, g, h of the matrix 2 x 3 x1 x. It is x 4 x 5 x 6 known that then HI i (R) 0 is possible only for i = 2, 3 and nonvanishing is guaranteed for i = 2. It is however rather complicated to make any statement about HI 3 (R). Using the program [16], one finds that HI 3(R) is isomorphic to a cyclic A 6-module generated by 1 A 6 subject to relations x 1 =... = x 6 = 0. It follows that HI 3(R) = E R (K) and λ 0,3 (R/I) = Computation of Gröbner bases in many variables is in general a time- and space consuming enterprise. Already in (commutative) polynomial rings the worst case performance for the number of elements in reduced Gröbner bases is doubly exponentially. In the (relatively small) example above R is of dimension 6, so that the intermediate ring

19 ALGORITHMS IN LOCAL COHOMOLOGY 19 A n+1 [y 1, y 2 ] contains 16 variables. In view of these facts the following idea has proved useful. The general context in which lemma 4.1 and proposition 4.3 were stated allows successive localization of R fg in the following way. First one computes R f according to algorithm 5.6 as quotient of A n by a holonomic ideal L(f). Then R fg may be computed using 5.6 again since R fg = Rg A n /L(f). (Note that all localizations of R are automatically f-torsion free for f R as R is a domain.) This process may be iterated for products with any finite number of factors. Note also that the exponents for the various factors might be different. This requires some care as the following situation illustrates. Assume first that -1 is the smallest integer root of the Bernstein polynomial of f and g (both in R) with respect to the holonomic module R. Assume further that R fg = An f 2 g 1 A n (fg) 1. Then R f R fg can be written as A n / ann(f 1 ) A n / ann((fg) 1 ) sending P A n to P f g. Suppose on the other hand that we are interested in HI 2 (R) where I = (f, g, h) and we know that R f = A n f 2 A n f 1, R g = A n g 2 and R fg = A n f 1 g 2. (In fact, the degree 2 part of the Čech complex of example 7.1 consists of such localizations.) It is tempting to write the embedding R f R fg with the use of a Bernstein operator (if P f (s)f s+1 = b f (s)f s then take s = 2) but as f 1 is not a generator for R f, b f ( 2) will be zero. In other words, we must write R fg as A n / ann((fg) 2 ) and then send P ann(f 2 ) to P g 2. The two examples indicate how to write the Čech complex in terms of generators and relations over A n while making sure that the maps C k C k+1 can be made explicit using the f i : the exponents used in C i have to be at least as big as those in C i 1 (for the same f i ). Remark 7.2. We suspect that for all holonomic f g-torsionfree modules M = A n /L we have (with R g M = A n /L ): min{x Q : b L f (s) = 0} min{x Q : b L f (s) = 0}. In other words, localizing R g M at f is not more complicated than localizing M at f. The advantage of localizing R fg as (R f ) g is threefold. For once, it allows the exponents of the various factors to be distinct which is useful for the subsequent cohomology computation: it helps keeping degrees of the maps small. (So for example R x (x 2 +y 2 ) can be written as A n x 1 (x 2 +y 2 ) 2 instead of A n (x 3 +xy 2 ) 2 ). On the other hand, computing C i uses already available information from C i 1. Finally, since the computation of Gröbner bases is is doubly exponentially it

20 20 ULI WALTHER seems to be advantageous to break a big problem (localization at a product) into many easy problems (successive localization). An extreme case of this behaviour is our example 7.1: if we compute R fgh as ((R f ) g ) h, the calculation uses approximately 2.5 kb and lasts 32 seconds on a Sun workstation using [16]. If one tries to localize R at the product of the three generators at once, [16] crashes after about 30 hours having used up the entire available memory (1.2 GB). References [1] J.-E. Björk. Rings of Differential Operators. North Holland, [2] D. Eisenbud. Commutative Algebra with a View toward Algebraic Geometry. Springer Verlag, [3] G. Faltings. Über lokale Kohomologiegruppen hoher Ordnung. Math. Ann., 255:45 56, [4] R. Hartshorne. Cohomological Dimension of Algebraic Varieties. Ann. of Math., 88: , [5] R. Hartshorne. On the de Rham Cohomology of Algebraic Varieties. Publ. Math. Inst. Hautes Sci., 45:5 99, [6] R. Hartshorne and R. Speiser. Local cohomological dimension in characteristic p. Ann. Math.(2), 105:45 79, [7] C. Huneke and G. Lyubeznik. On the vanishing of local cohomology. Invent. Math., 102:73 93, [8] A. Kandri-Rody and V. Weispfenning. Non-commutative Gröbner bases in algebras of solvable type. J. Symb. Comp., 9:1 26, [9] M. Kashiwara. B-functions and holonomic systems, rationality of B-functions. Invent. Math., 38:33 53, [10] G. Lyubeznik. Finiteness Properties of Local Cohomology Modules: an Application of D-modules to Commutative Algebra. Invent. Math., 113:41 55, [11] B. Malgrange. Le polynôme de Bernstein d une singularité isolée. Lecture Notes in Mathematics, Springer Verlag, 459:98 119, [12] T. Oaku. An algorithm for computing B-functions. Duke Math. Journal, 87, [13] A. Ogus. Local cohomological dimension of algebraic varieties. Ann. of Mathematics, 98: , [14] C. Peskine and L. Szpiro. Dimension projective finie et cohomologie locale. Inst. Hautes Études Sci. Publ. Math., 42:47 119, [15] N. Takayama. Kan, a computer package for symbolic computations in Weyl algebras [16] U. Walther. Locol, a computer program for symbolic computations in local cohomology. walther@math.umn.edu. School of Mathematics, University of Minnesota, Minneapolis, MN address: walther@math.umn.edu