A GENERALIZED GAETA’S THEOREM ELISA GORLA 7 0 0 Abstract: We generalize Gaeta’s Theorem to the family of determinantal schemes. 2 In other words, we show that the schemes defined by minors of a fixed size of a matrix n a with polynomial entries belong to the same G-biliaison class of a complete intersection J whenever they have maximal possible codimension, given the size of the matrix and of 6 the minors that define them. 1 ] G Introduction A . In this paper we study the G-biliaison class of a family of schemes, whose saturated h t ideals are generated by minors of matrices with polynomial entries. Other families of a m schemes defined by minors have been studied in the same context. The results obtained in this paper are a natural extension of some of the results proven in [19], [14] and [11]. In [ [19]Kleppe, Migliore, Mir´o-Roig,Nagel,andPetersonprovedthatstandarddeterminantal 1 v schemes are glicci, i.e. that they belong to the G-liaison class of a complete intersection. 6 Wereferto[21]forthedefinitionofstandardandgooddeterminantalschemes. Hartshorne 5 pointed out in [14] that the double G-links produced in [19] can indeed be regarded as 4 1 G-biliaisons. Hence, standard determinantal schemes belong to the G-biliaison class of 0 a complete intersection. In [11] we defined symmetric determinantal schemes as schemes 7 whosesaturatedidealisgeneratedbytheminorsofsize t×tofanm×m symmetric matrix 0 / with polynomial entries, and whose codimension is maximal for the given t and m. In the h t same paper we proved that these schemes belong to the G-biliaison class of a complete a m intersection. We recently proved in [10] that mixed ladder determinantal varieties belong : to the G-biliaison class of a linear variety, therefore they are glicci. Ladder determinantal v varieties are defined by the ideal of t × t minors of a ladder of inderterminates. We i X call them mixed ladder determinantal varieties, since we allow minors of different sizes r in different regions of the ladder. The results in this paper provide us with yet another a family of arithmetically Cohen-Macaulay schemes, for which we can produce explicit G- biliaisons that terminate with a complete intersection. The question that one would hope to answer is whether every arithmetically Cohen-Macaulay scheme is glicci. Considerable progress have been made by several authors in showing that special families of schemes are glicci (see e.g. [3], [4], [19], [22], [13], [5], [6], and [18]). In this paper, we study a family of schemes that correspond to ideals of minors of fixed size of some matrix with polynomial entries. We call them determinantal schemes (see Definition 1.3). In Section 1 we establish the setup, and some preliminary results Theauthorwaspartiallysupportedby the SwissNationalScienceFoundation. Partofthe researchin this paper was done while the author was a guest at the Max Planck Institut fu¨r Mathematik in Bonn. The author would like to thank the Max Planck Institute for its support and hospitality. 1 2 ELISA GORLA aboutdeterminantal schemes. InRemarks1.7andLemma 1.13, we characterize thedeter- minantal schemes which are complete intersections or arithmetically Gorenstein schemes. In Theorem 1.16 and Proposition 1.19 we relate the property of being locally complete intersection outside a subscheme to the height of the ideal of minors of size one less. Section 2 contains results about heights of ideals of minors. It contains material that will be used to obtain the linkage results, but it can be read independently from the rest of the article. In this section we consider an m × n matrix M, such that the ideal I (M) t has maximal height (m− t+ 1)(n−t+ 1). In Proposition 2.2 we show that deleting a column of M we obtain a matrix O whose ideal of t×t minors I (O) has maximal height t (m−t+1)(n−t). In Theorem 2.4, we show that if we apply generic invertible row oper- ations to O and then delete a row, we obtain a matrix N whose ideal of (t−1)×(t−1) minors has maximal height (m−t+1)(n−t+1). Under the same assumptions, we show that if we apply generic invertible row operations to M and then delete one entry, we obtain a ladder L whose ideal of t×t minors has maximal height (m−t+1)(n−t+1)−1 (see Corollary 2.9). The consequence which is relevant in terms of the liaison result is that starting from a determinatal scheme X we can produce schemes X′ and Y such that X′ is determinantal and both X and X′ are generalized divisors on Y (see Theorem 2.11). Section 3 contains the G-biliaison results. The main result of the paper is Theorem 3.1, where we show that any determinantal scheme can be obtained from a linear variety by a finitesequence ofascending elementary G-biliaisons. Inparticular, determinantal schemes are glicci (Corollary 3.2). As a consequence of a result of Huneke and Ulrich, we obtain that determinantal schemes are in general not licci (see Corollary 3.4). 1. Determinantal schemes LetX beaschemeinPr = Pr ,whereK isanalgebraicallyclosedfield. LetI bethe K X saturatedhomogeneousidealassociatedtoX inthepolynomialringR = K[x ,x ,...,x ]. 0 1 r For an ideal I ⊆ R, we denote by H0(I) the saturation of I with respect to the maximal ∗ ideal m = (x ,x ,...,x ) ⊆ R. 0 1 r Let IX ⊆ OPr be the ideal sheaf of X. Let Y be a scheme that contains X. We denote by I the ideal sheaf of X restricted to Y, i.e. the quotient sheaf I /I . For X|Y X Y i ≥ 0, we let H∗i(Pr,I) = ⊕t∈ZHi(Pr,I(t)) denote the i-th cohomology module of the sheaf I on Pr. We simply write Hi(I) when there is no ambiguity on the ambient space ∗ Pr. Notation 1.1. Let I ⊆ R be a homogeneous ideal. We let µ(I) denote the cardinality of a set of minimal generators of I. In this paper we deal with homogeneous ideals in the polynomial ring R. Definition 1.2. Let M be a matrix with entries in R of size m×n, where m ≤ n. We say that M is t-homogeneous if the minors of M of size s×s are homogeneous polynomials for all s ≤ t. We say that M is homogeneous if its minors of any size are homogeneous. We always consider t-homogeneous matrices. We study a family of schemes whose homogeneous saturated ideal I (M) is generated by the t×t minors of a t-homogeneous t A GENERALIZED GAETA’S THEOREM 3 matrix M. We regard matrices up to invertible transformations, since they do not change the ideal I (M). We always assume that the transformations that we consider preserve t the t-homogeneity of the matrix. Definition 1.3. Let X ⊂ Pr be a scheme. We say that X is determinantal if: (1) there exists a t-homogeneous matrix M of size m×n with entries in R, such that the saturated ideal of X is generated by the minors of size t×t of M, I = I (M), X t and (2) X has codimension (m−t+1)(n−t+1). Remark 1.4. The ideal I (M) generated by the minors of size t×t of an m×n matrix t M has ht I (M) ≤ (m−t+1)(n−t+1). t This is a classical result of Eagon and Northcott. For a proof see Theorem 2.1 in [2]. Therefore the schemes of Definition 1.3 have maximal codimension for fixed m,n,t. The matrix M defines a morphism of free R-modules ϕ : Rn −→ Rm. Invertible row and column operations on M correspond to changes of basis in the domain and codomain of ϕ. The scheme X is the locus where rk ϕ ≤ t−1. So it only depends on the map ϕ and not on the matrix M chosen to represent it. In some cases, we will be interested in ideals that are generated by a subset of the minors of M. Notation 1.5. Let M = (F ) be an m×n matrix with entries in the poly- ij 1≤i≤m,1≤j≤n nomial ring R. Fix a choice of row indexes 1 ≤ i ≤ i ≤ ... ≤ i ≤ m and of column 1 2 t indexes 1 ≤ j ≤ j ≤ ... ≤ j ≤ n. We denote by M the determinant of the 1 2 t i1,...,it;j1,...,jt submatrix of M consisting of the rows i ,...,i and of the columns j ,...,j . 1 t 1 t Remark 1.6. Let L be the subladder of M consisting of all the entries except for F . mn The ideal I (L) = (M | i 6= m or j 6= n) ⊆ I (M) t i1,...,it;j1,...,jt t t t has height ht I (L) ≤ (m−t+1)(n−t+1)−1. t This is a special case of Corollary 4.7 of [15]. The family of determinantal schemes contains well-studied families of schemes, such as complete intersections and standard determinantal schemes. Remarks 1.7. (i) Standard determinantal schemes are a subfamily of determinantal schemes. In fact, a determinantal scheme is standard determinantal whenever t = m ≤ n, that is whenever its saturated ideal is generated by the maximal minors of M. (ii) Complete intersections are a subfamily of determinantal schemes, since they are a subfamily of standard determinantal schemes. They coincide with the determinantal schemes that have t = 1 or t = m = n (see also Lemma 1.13). 4 ELISA GORLA (iii) The Cohen-Macaulay type of a determinantal scheme as of Definition 1.3 is t−1 n−i t−1 (cid:0)m−i(cid:1) Yi=1 t−1 (cid:0) (cid:1) (see [2]). In particular, a determinantal scheme is arithmetically Gorenstein if and only if m = n. Glicciness of arithmetically Gorenstein schemes is established in [6]. In [20] it is shown that the determinantal arithmetically Gorenstein schemes with t+1 = m = n are glicci. Theorem 3.1 will imply that an arithmetically Gorenstein determinantal scheme belongs to the G-biliaison class of a complete intersection. The ideal of minors of size t×t of a generic matrix is an example of a determinantal scheme in Pr for r = mn−1 and for each t ≤ m. Example 1.8. For any fixed 1 ≤ m ≤ n, and for any choice of t with 1 ≤ t ≤ m, let r = mn−1. Let X ⊂ Pr be the determinantal scheme whose saturated ideal is generated by the minors of size t×t of the matrix of indeterminates x x ··· x 1,1 1,2 1,n x x ··· x  2,1 2,2 2,n  I = I . . . . X t . . . . . .    x x ··· x   m,1 m,2 m,n  X has codim(X) = depth(I ) = (m−t+1)(n−t+1) (see Theorem 2.5 of [2]). Then X X is arithmetically Cohen-Macaulay and determinantal. In [10] we proved that X belongs to the G-biliaison class of a complete intersection. Remark 1.9. Complete intersections are standard determinantal, hence determinantal (as observed in part (ii) of Remarks 1.7). Notice that the family of determinantal schemes strictly contains the family of standard determinantal schemes. For example, the schemes ofExample 1.8aredeterminantal, but not standarddeterminantal for 2 ≤ t ≤ m−1. This can be checked e.g. by comparing the number of minimal generators for the saturated ideals of determinantal and standard determinantal schemes. We now establish some properties of determinantal schemes that will be needed in the sequel. We use the notation of Definition 1.3. We start with a result due to Hochster and Eagon (see [16]). We state only a special case of their theorem, that is sufficient for our purposes. Theorem 1.10. (Hochster, Eagon) Determinantal schemes are arithmetically Cohen- Macaulay. In the sequel, we will also need the following theorem proven by Herzog and Trung. InCorollary4.10of[15]theyestablishCohen-Macaulaynessofladderdeterminantalideals, but we state their result only for the family of ideals that we are interested in. Theorem 1.11. (Herzog, Trung) Let U = (x ) be a matrix of indeterminates of size ij m×n, and let V be the subladder consisting of the all entries of U except for x . Then mn I (V) = (U | i 6= m or j 6= n) t i1,...,it;j1,...,jt t t A GENERALIZED GAETA’S THEOREM 5 is a Cohen-Macaulay ideal of height ht I (V) = (m−t+1)(n−t+1)−1. t We recall that if a scheme defined by the t×t minors of a matrix of indeterminates is a complete intersection, then it is generated by the entries of the matrix or by its determinant (in the case of a square matrix). We are now going to prove the analogous result for a t-homogeneous matrix M whose entries are arbitrary polynomials. We also prove a similar result for a subset of the t×t minors of M. We start by proving an easy numerical lemma. Lemma 1.12. Let m,n,t be positive integers satisfying 2 ≤ t ≤ m − 1, m ≤ n. The following inequality holds: (mn−t2)(m−1)·...·(m−t+2)(n−1)·...·(n−t+2) > (t!)2. Proof. Since t ≤ m−1 ≤ n−1, (m−1)·...·(m−t+2)(n−1)·...·(n−t+2) ≥ [(t!)/2]2. Therefore it suffices to show that mn−t2 > 4. But mn−t2 ≥ m2 −(m−1)2 = 2m−1 > 4 since m ≥ t+1 ≥ 3. (cid:3) The following lemma is analogous to Lemma 1.16 of [11]. Lemma 1.13. Let M be a t-homogeneous matrix of size m×n with entries in R or in R for some prime P. Let L be the subladder consisting of the all entries of M except P for F . mn (i) If M has no invertible entries and I (M) is a complete intersection of codimension t (m−t+1)(n−t+1), then t = 1 or t = m = n. (ii) If L has no invertible entries and I (L) is a complete intersection of codimension t (m−t+1)(n−t+1)−1, then t = 1 or t = m = n−1. Proof. (i) The minors of the t×t submatrices of M are a minimal system of generators of I (M). If I (M) is a complete intersection, then t t m n µ(I (M)) = = ht I (M) = (m−t+1)(n−t+1). t t (cid:18)t(cid:19)(cid:18)t(cid:19) Computations yield [m·...·(m−t+2)][n·...·(n−t+2)] = [t·...·2][t·...·2]. Both sides of the equality contain the same number of terms, and t−i ≤ m−i ≤ n−i for all i = 0,...,t−2. So the equality holds if and only if t = 1 or t = m = n. (ii) For a generic matrix M = (x ), the minors of the t × t submatrices that do ij not involve the entry x are a minimal system of generators of I (L). This follows mn t 6 ELISA GORLA e.g. from the observation that they are linearly independent. By Theorem 3.5 in [2], if we substitute F for x in a minimal system of generators of I (L), we obtain a minimal ij ij t systemofgeneratorsforI (L)inthecaseM = (F )andht I (L) = (m−t+1)(n−t+1)−1. t ij t In particular, the cardinality of a minimal generating system for I (L) is in both cases t m n m−1 n−1 µ(I (L)) = − . t (cid:18) t(cid:19)(cid:18)t(cid:19) (cid:18) t−1(cid:19)(cid:18)t−1(cid:19) If I (L) is a complete intersection, then t m n m−1 n−1 (1) ht I (L) = − = (m−t+1)(n−t+1)−1. t (cid:18) t(cid:19)(cid:18)t(cid:19) (cid:18)t−1 (cid:19)(cid:18)t−1(cid:19) It follows that (mn−t2)(m−1)·...·(m−t+1)(n−1)·...·(n−t+1) = (t!)2[(m−t+1)(n−t+1)−1] By Lemma 1.12 we have that if t 6= 1,m, then the left hand side of the equality is greater than (t!)2(m−t+1)(n−t+1). This is a contradiction, so t = 1 or t = m. Moreover, if t = m then (1) simplifies to n n−1 − = n−m (cid:18)m(cid:19) (cid:18)m−1(cid:19) or equivalently to n−1 (n−1)·...·(n−m) = = n−m. (cid:18) m (cid:19) m! Therefore m = 1 or m = n−1. Hence either t = 1 and I (L) is generated by the entries t of L, or t = m = n−1 and I (L) corresponds to a hypersurface (whose equation is the t determinant of the first m columns of M). (cid:3) Definition 1.14. Let X ⊂ Pr be a scheme. We say that X is generically complete intersection if it is locally complete intersection at all its components. That is, if the localization (I ) is generated by an R -regular sequence for every P minimal associated X P P prime of I . X We say that X is locally complete intersection outside a subscheme of codimension d in Pr if the localization (I ) is generated by an R -regular sequence for every P ⊇ I X P P X prime of ht P ≤ d−1. We say that X is generically Gorenstein, abbreviated G , if it is locally Gorenstein 0 at all its components. That is, if the localization (I ) is a Gorenstein ideal for every P X P minimal associated prime of I . X Remark 1.15. The locusof pointsat which a scheme failsto belocallycomplete intersec- tion is closed. Therefore, a scheme of codimension c in Pr is locally complete intersection outside a subscheme of codimension c + 1 in Pr if and only if it is generically complete intersection. Both of these assumption imply that the scheme is generically Gorenstein. We now prove two results that relate the height of the ideal of (t−1)-minors of M with local properties of the scheme defined by the vanishing of the t-minors of M or L. The notation is as in Definition 1.3. A GENERALIZED GAETA’S THEOREM 7 Theorem 1.16. Let X be a determinantal scheme with defining matrix M, I = I (M). X t Let c = (m−t+1)(n−t+1) be the codimension of X. Assume that X is not a complete intersection, i.e. t 6= 1 and t,m,n are not all equal. Let d ≥ c +1 be an integer. Then the following are equivalent: (1) X is locally complete intersection outside of a subscheme of codimension d in Pr. (2) ht I (M) ≥ d. t−1 Proof. (1) =⇒ (2): let P ⊇ I (M) be a prime ideal of height c ≤ ht P ≤ d−1. In order t to prove (2), it suffices to show that P 6⊇ I (M). Let M denote the localization of M t−1 P at P. The matrix M can be reduced after invertible row and column operations to the P form I 0 M = s , P (cid:20) 0 B (cid:21) where I is an identity matrix of size s ×s, 0 represents a matrix of zeroes, and B is a s matrixofsize(m−s)×(n−s) thathasnoinvertible entries. Byassumption, I (M) ⊆ R t P P is a complete intersection ideal. Since I (M ) = I (B) and B has no invertible entries, t P t−s it follows by Lemma 1.13 that either t−s = 1, or t−s = m−s = n−s. If the latter holds, then t = m = n and X is a hypersurface. Then t−s = 1 and I (M ) = R , so t−1 P P P 6⊇ I (M). t−1 (2) =⇒ (1): let P ⊇ I (M) be a prime of height c ≤ ht P ≤ d − 1. The thesis t is proven if we show that I (M) is locally generated by a regular sequence at P. Since t ht P < ht I (M), then P 6⊇ I (M), and the localization M of M at P can be t−1 t−1 P reduced, after invertible row and column operations, to the form I 0 M = t−1 , P (cid:20) 0 B (cid:21) where I is an identity matrix of size (t−1)×(t−1), 0 represents a matrix of zeroes, t−1 and B is a matrix of size (m−t+1)×(n−t+1). Since PR ⊇ I (M ) = I (B), we have P t P 1 µ(I (M) ) ≤ (m−t+1)(n−t+1) = c = ht I (M) . t P t P Then I (M) is locally generated by a regular sequence at P. (cid:3) t Remark 1.17. AssumethatX isnotacompleteintersection. Ford = c+1,theconclusion of Theorem 1.16 can be restated as: X is generically complete intersection if and only if ht I (M) > ht I (M). t−1 t The implication (2) =⇒ (1) of Theorem 1.16 clearly holds true without the assump- tion that X is not a complete intersection. The next example shows that the assumption that X is not a complete intersection is necessary for the implication (1) =⇒ (2). 8 ELISA GORLA Example 1.18. Let F ∈ R be a homogeneous form and consider the t×t matrix F 0 ... ... 0  0 F 0 ... 0  M = ... ... ... ... ... .    ... ... ... 0     0 ... ... 0 F    Let X ⊆ Pr be the scheme with I = I (M) = (Ft). Then X is a hypersurface, hence X t a complete intersection, therefore locally complete intersection outside any subscheme. However the ideal I (M) = (Ft−1) defines a hypersurface in Pr, hence ht I (M) = 1. t−1 t−1 The following proposition gives a sufficient condition for the scheme defined by I (L) t to be generically complete intersection. Proposition 1.19. Let M = (F ) be a t-homogeneous matrix of size m × n. Let L be ij the subladder of M consisting of all the entries except for F . Let N be the submatrix mn obtained from M bydeleting the last row and column, and let I (N) be the ideal generated t−1 by the minors of size (t − 1)× (t − 1) of N. Let Y be the scheme corresponding to the ideal I (L). Assume that ht I (L) = c−1 = (m−t+1)(n−t+1)−1 and ht I (N) = c. t t t−1 Then Y is generically complete intersection. Proof. Let P be a minimal associated prime of I = I (L), then P 6⊇ I (N). Denote Y t t−1 by L ,N the localizations of L,N at P. Then N ⊆ L contains an invertible minor P P P P of size t−1. We can assume without loss of generality that the minor involves the first t −1 rows and columns. After invertible row and column operations (that involve only the first t−1 rows and columns) we have I 0 L = t−1 , P (cid:20) 0 B (cid:21) where B is the localization at P of the ladder obtained by removing the entry in the lower right corner from the submatrix of M consisting of the last m−t+1 rows and n−t+1 columns. We have µ((I ) ) = µ(I (B)) ≤ (m−t+1)(n−t+1)−1 = ht (I ) . Y P 1 Y P Then I is locally generated by a regular sequence at P, i.e. Y is generically complete Y (cid:3) intersection. Remark 1.20. By Proposition 1.19, the condition that I (N) = c implies that Y t−1 contains a determinantal subscheme X′ of codimension 1, whose defining ideal is IX′ = I (N). Notice that whenever this is the case, Y is generically complete intersection, t−1 hence it is G . Under this assumption we have a concept of generalized divisor on Y (see 0 [14] about generalized divisors). Then X′ is a generalized divisor on Y. Proposition 1.19 proves that the existence of such a subscheme X′ of codimension 1 guarantees that Y is locally a complete intersection. Notice the analogy with standard determinantal ([19]) and symmetric determinantal schemes ([11]). A GENERALIZED GAETA’S THEOREM 9 2. Heights of ideals of minors In this section we study the schemes associated to the matrix obtained from M by deleting a column, or a column and a generalized row. We assume that the ideal I (M) t has maximal height according to Remark 1.4. This section can be read independently from the rest of the paper. As before, let M be a t-homogeneous matrix of size m×n with entries in R. Assume thatI (M)definesadeterminantalschemeX ⊂ Pr ofcodimensionc = (m−t+1)(n−t+1). t We assume that m,n,t are not all equal. In fact, if m = n = t then X is a hypersurface and all the results about the heights are easily verified. Definition 2.1. Fix a matrix O of size m×(n−1). Following [21], we call generalized row any row of the matrix obtained from O by applying generic invertible row operations. By deleting a generalized row of O we mean that we first apply generic invertible row operations to O, and then we delete a row. We start by deleting a column of M and look at the scheme defined by the t × t minors of the remaining columns. Proposition 2.2. Let X ⊂ Pr be a determinantal scheme with associated matrix M, I = I (M). Let O be the matrix obtained from M by deleting a column. Then I (O) is the X t t saturated ideal of a determinantal scheme Z of codimension (m−t+1)(n−t). Moreover, Z is locally complete intersection outside a subscheme of codimension (m−t+1)(n−t+1) in Pr. Proof. From the Lemma following Theorem 2 in [1] ht I (M)/I (O) ≤ m−t+1. t t Hence ht I (O) ≥ (m−t+1)(n−t+1)−(m−t+1) = (m−t+1)(n−t), so equality t holds. Then I (O) is the saturated ideal of a determinantal scheme Z of codimension t (m−t+1)(n−t). Since ht I (O) ≥ ht I (M) = (m−t+1)(n−t+1), by Theorem 1.16 t−1 t Z is locally complete intersection outside a subscheme of codimension (m−t+1)(n−t+1) in Pr. (cid:3) Notation 2.3. We let ϕ : F −→ G be the morphism of free R-modules associated to the matrix O, F = Rn−1, G = Rm. Our goal is to prove that if we delete a generalized row of O, the minors of size t − 1 of the remaining rows define a determinantal scheme of the same codimension as X. By the upper-semicontinuity principle, it suffices to show that one can apply chosen invertible row and column operations to O, then delete a row, and obtain a matrix whose t−1 minors define a determinantal scheme. Theorem 2.4. Let O be as in Proposition 2.2. Deleting a generalized row of O, one obtains a matrix N with ht I (N) = (m−t+1)(n−t+1). t−1 10 ELISA GORLA Proof. If t = m ≤ n then I (O) defines a good determinantal scheme, and the result was m proven by Kreuzer, Migliore, Nagel, and Peterson in [21]. Assume then that t < m ≤ n, and consider the exact sequence associated to the morphism ϕ 0 −→ B −→ F −ϕ→ G −→ Coker ϕ −→ 0. Deleting a row of O corresponds to a commutative diagram with exact rows and columns 0 ↓ 0 0 R ↓ ↓ ↓ (2) 0 −→ B −→ F −ϕ→ G −→ Coker ϕ −→ 0 ↓ k ↓ ↓ 0 −→ B′ −→ F −ϕ→′ G′ −→ Coker ϕ′ −→ 0 ↓ ↓ ↓ 0 0 0 where ϕ′ is the morphism associated to the submatrix obtained from O after deleting a row (possibly after applying invertible row operations). We first consider the case when m < n. Since I (M) defines a standard determi- m nantal scheme and O is obtained from M by deleting a column, then I (O) defines a good m determinantal scheme (see Chapter 3 of [19]). By Proposition 3.2 in [21], we have that Coker ϕ is an ideal of positive height in R/I (O). Then there is a minimal generator of m Coker ϕ as an R-module that is non zero-divisor modulo I (O). Call it f. Denote by s m the multiplication map by f: s (3) 0 −→ R/I (O) −→ Coker ϕ −→ Coker s −→ 0. m Since I (O)+(f) ⊆ Ann (Coker s), Coker s is supported ona subscheme ofcodimension m R at least ht I (O)+1. We have a commutative diagram with exact rows and columns m 0 0 ↓ ↓ R −→ R/I (O) −→ 0 m ↓ ↓ F −ϕ→ G −→ Coker ϕ −→ 0 ↓ ↓ G′ −β→ Coker s −→ 0 ↓ ↓ 0 0 Let π denote the morphism G −→ G′ in the diagram above, and define ϕ′ = π◦ϕ. Using the snake lemma, one can check that F −ϕ→′ G′ −β→ Coker s −→ 0 is exact. Therefore Coker ϕ′ = Coker s, and by taking kernels of ϕ and ϕ′ we produce a diagram as (2).