Cohomological dimension and arithmetical rank of some determinantal ideals
Davide Bolognini, Alessio Caminata, Antonio Macchia, Maral Mostafazadehfard
aa r X i v : . [ m a t h . A C ] M a r COHOMOLOGICAL DIMENSION AND ARITHMETICAL RANKOF SOME DETERMINANTAL IDEALS
DAVIDE BOLOGNINI, ALESSIO CAMINATA, ANTONIO MACCHIA, MARAL MOSTAFAZADEHFARDAbstract. Let M be a (2 × n ) non-generic matrix of linear forms in a polynomial ring. For large classes of suchmatrices, we compute the cohomological dimension (cd) and the arithmetical rank (ara) of the ideal I ( M ) generatedby the -minors of M . Over an algebraically closed fi eld, any (2 × n ) -matrix of linear forms can be written in theKronecker-Weierstrass normal form, as a concatenation of scroll, Jordan and nilpotent blocks. B ă descu and Vallacomputed ara( I ( M )) when M is a concatenation of scroll blocks. In this case we compute cd( I ( M )) and extendthese results to concatenations of Jordan blocks. Eventually we compute ara( I ( M )) and cd( I ( M )) in an interestingmixed case, when M contains both Jordan and scroll blocks. In all cases we show that ara( I ( M )) is less than thearithmetical rank of the determinantal ideal of a generic matrix. Mathematics Subject Classification (2010):
Keywords: ideals of minors, cohomological dimension, arithmetical rank.IntroductionDeterminantal ideals are a classical topic in Commutative Algebra and have been extensively studied becauseof their connections with other fields, such as Algebraic Geometry, Combinatorics, Invariant Theory and Repre-sentation Theory (see e.g. [6]). In this paper we focus on the ideals I ( M ) generated by the -minors of a (2 × n ) non-generic matrix M in a polynomial ring R over a field K . In particular, we compute the cohomologicaldimension ( cd ) and the arithmetical rank ( ara ) for large classes of such matrices.We recall that the cohomological dimension of an ideal I of a Noetherian ring R is cd R ( I ) = max { i ∈ Z : H iI ( R ) = 0 } , where H iI ( R ) denotes the i -th local cohomology module of R with support in I , and the arithmetical rank of I isthe smallest integer s for which there exist s elements of R , a , . . . , a s , such that √ I = p ( a , . . . , a s ) . If thereis no ambiguity, we will write simply cd( I ) and omit the subscript R . In general, the following inequalities hold(see, e.g., [14, Proposition 9.2]): ht( I ) ≤ cd( I ) ≤ ara( I ) , where ht is the height of the ideal. If ht( I ) = ara( I ) , then I is called a set-theoretic complete intersection . Inparticular, if I is a squarefree monomial ideal in the polynomial ring R = K [ x , . . . , x n ] , then(1) ht( I ) ≤ pd R ( R/I ) = cd( I ) ≤ ara( I ) ≤ µ ( I ) , where µ ( I ) denotes the minimum number of generators of I and the equality between the projective dimension(pd) and the cohomological dimension was proved by Lyubeznik in [16, Theorem 1]. The third author was supported by Università degli Studi di Bari.The fourth author was partially supported by IMPA (Instituto Nacional de Matemática Pura e Aplicada), Rio de Janeiro, Brazil. For a generic (2 × n ) -matrix X , Bruns and Schwänzl have shown in [5] that ara( I ( X )) = 2 n − and it isindependent of the field. On the other hand, the cohomological dimension has a different behavior: cd( I ( X )) = ( ht( I ( X )) = n − if char( K ) = p > I ( X )) = 2 n − if char( K ) = 0 . Motivated by [17, Question 8.1], we investigate the following special case.
Question 1.
Let M = ( x ij ) be a (2 × n ) non-generic matrix of linear forms and consider the ideal I ( M ) in thepolynomial ring R = K [ x ij ] generated by the -minors of M . If X is a (2 × n ) -generic matrix, is it true that I ( M ) can be generated up to radical by less than ara( I ( X )) = 2 n − elements, i.e. ara( I ( M )) < ara( I ( X )) ?In order to study non-generic matrices, we first introduce the Kronecker-Weierstrass normal form of a matrix:a (2 × n ) -matrix M , whose entries are linear forms, can be written, by means of an invertible transformation,as a concatenation of blocks. Each block can be a nilpotent, a scroll or a Jordan matrix (see Section 2). Firstwe treat all the possible mixed cases of (2 × -matrices in Remark 2.4. In Example 2.5 we compute cd andara when M consists of exactly one block. In all three cases I ( M ) is a set-theoretic complete intersectionwith ara( I ( M )) = n − . In the rest of the paper we deal with matrices consisting of at least blocks andwith n ≥ columns. In Proposition 2.6 we show that, if X is a matrix of linear forms and we add a nilpotentblock N n with length n + 1 defining a new matrix M = ( X | N n ) , then cd( I ( M )) = cd( I ( X )) + n and ara( I ( M )) ≤ ara( I ( X ))+ n . This implies that, if we have a matrix X for which cd( I ( X )) = ara( I ( X )) , thenthe concatenation of an arbitrary number of nilpotent blocks to X preserves the equality between cohomologicaldimension and arithmetical rank.In all the cases examined throughout the paper, we noticed a behavior similar to the generic case: the upperbound for the arithmetical rank is independent of the field, while the cohomological dimension is equal to theheight of the ideal in positive characteristic and to the arithmetical rank in characteristic zero.In Section 3 we analyze concatenations of scroll blocks. Bădescu and Valla, in [1], computed the arithmeticalrank of the ideal I ( M ) , showing that it is independent of the fi eld. On the other hand, using some tools fromAlgebraic Geometry, we prove that the cohomological dimension equals the height of the ideal if char( K ) = p > , while it is equal to the arithmetical rank if char( K ) = 0 (see Theorem 3.2).In Section 4 we consider concatenations of Jordan blocks when char( K ) = 0 . We show that also in thissituation cd( I ( M )) = ara( I ( M )) .Finally, in Section 5, we study an interesting mixed case. We start with a (2 × n ) -matrix M with zerosin di ff erent rows and columns, and we transform it in the Kronecker-Weierstrass form. In this way M can bewritten as a concatenation of two Jordan blocks of length with di ff erent eigenvalues and n − scroll blocksof length . The ideal I ( M ) is generated by both monomials and binomials. First we fi nd an upper bound forthe arithmetical rank independent of the fi eld, showing that ara( I ( M )) ≤ n − . In the proof of Theorem5.4 we combine the classical result by Bruns and Schwänzl (Theorem 1.2) and a well-known technique due toSchmitt and Vogel (Lemma 1.1). To reduce the number of generators up to radical, we sum some of them in asuitable way and use Plücker relations to prove the claim. Concerning the cohomological dimension, for smallvalues of n , the ideal I ( M ) is a set-theoretic complete intersection. For n ≥ , in Theorem 5.5 we prove that cd( I ( M )) = ht( I ( M )) if char( K ) = p > , while cd( I ( M )) = ara( I ( M )) if char( K ) = 0 . For the last fact,we prove a stronger result, showing also the vanishing of all local cohomology modules with indices betweenthe height and n − , if char( K ) = 0 . For all the classes of (2 × n ) -matrices considered in Sections 3, 4 and 5, except for small values of n , we alwaysprove that I ( M ) can be generated with less than n − polynomials up to radical. Hence we give a positiveanswer to Question 1. 1. P reliminaries In this section we recall some results that will be useful in the rest of the paper.A well-known technique that provides an upper bound for the arithmetical rank of an arbitrary ideal is due toSchmitt and Vogel.
Lemma 1.1. (Schmitt,
Vogel [23, Lemma p. 249] ) Let R be a ring, P be a finite subset of elements of R and P , . . . , P r subsets of P such that ( i ) S rℓ =0 P ℓ = P , ( ii ) P has exactly one element, ( iii ) if p and p ′′ are different elements of P ℓ , with ≤ ℓ ≤ r , there is an integer ℓ ′ , with ≤ ℓ ′ < ℓ , and anelement p ′ ∈ P ℓ ′ such that pp ′′ ∈ ( p ′ ) .We set q ℓ = X p ∈ P ℓ p e ( p ) , where e ( p ) ≥ are arbitrary integers. We will write ( P ) for the ideal of R generated by theelements of P . Then p ( P ) = p ( q , . . . , q r ) . In [4] and [5], Bruns and Schwänzl computed the cohomological dimension and the arithmetical rank ofdeterminantal ideals of generic matrices. Let X be an ( m × n ) -matrix of indeterminates and I t ( X ) be the idealgenerated by the t -minors of X . Theorem 1.2. (Bruns, Schwänzl, [5, Theorem 2] ) Let X be an ( m × n ) -matrix of indeterminates over a ring R .Then ara( I t ( X )) = mn − t + 1 . In [4, Corollary 2.2], Bruns proved that ara( I t ( X )) ≤ mn − t − over any commutative ring, by de fi ning aposet attached to the matrix X . We recall here this construction. We denote by [ a , . . . , a t | b , . . . , b t ] the minorof X with row indices a , . . . , a t and column indices b , . . . , b t . On the set ∆( X ) of all minors of X we de fi ne apartial order given by(2) [ a , . . . , a u | b , . . . , b u ] ≤ [ c , . . . , c v | d , . . . , d v ] ⇐⇒ u ≥ v, a i ≤ c i and b i ≤ d i , i = 1 , . . . , v. The polynomials that generate I t ( X ) up to radical have the form(3) p j = X ξ ∈ ∆( X ) , rk( ξ )= j ξ e ( ξ ) , for j = 1 , . . . , rk(∆( X )) , where rk(∆( X )) denotes the rank of the poset, e ( ξ ) = m deg ξ and m is the least common multiple of the degreesof the elements ξ ∈ ∆( X ) .In particular, we are interested in the case t = m = 2 , for which p j = ⌊ j +12 ⌋− − δ j X k =0 [ k + 1 + δ j , j − k + 1 − δ j ] for j = 1 , . . . , n − , where δ j = ( j − n + 1) ⌊ jn ⌋ . Here and in what follows, when we deal with -minors, weuse the notation [ a, b ] instead of [ a, b | , . Example 1.3.
We give an explicit example of the construction of the poset and of the polynomial generators upto radical for the ideal I ( X ) , where X = x x x x x x x x x x . The poset ∆( X ) is b b b bb b bb b b [1 ,
2] [1 ,
3] [1 ,
4] [1 , ,
3] [2 ,
4] [2 , ,
4] [3 ,
5] [4 , F igure I ( X ) is generated by the following polynomials up to radical: p = [1 ,
2] = x x − x x ,p = [1 ,
3] = x x − x x ,p = [1 ,
4] + [2 ,
3] = x x − x x + x x − x x ,p = [1 ,
5] + [2 ,
4] = x x − x x + x x − x x ,p = [2 ,
5] + [3 ,
4] = x x − x x + x x − x x ,p = [3 ,
5] = x x − x x ,p = [4 ,
5] = x x − x x . While the arithmetical rank of I t ( X ) is independent of the ring, the cohomological dimension has a di ff erentbehavior. In fact, if R is a polynomial ring on a fi eld of characteristic , then cd( I t ( X )) = ara( I t ( X )) = mn − t + 1 (see [5, Corollary p. 440]). On the other hand, if R is a polynomial ring on a fi eld of primecharacteristic p > , then cd( I t ( X )) = ht( I t ( X )) = ( m − t + 1)( n − t + 1) by [19, Proposition 4.1, p. 110], since I t ( X ) is a perfect ideal in light of [12].In Sections 3, 4 and 5, we will see that a similar result occurs also for some classes of non-generic matrices.The following Lemma will be employed more than once in the rest of the paper. Even if it was proved in [22,Lemma 1.19 p. 258], we give a more explicit proof for the sake of completeness. Lemma 1.4.
Let R be a Noetherian commutative ring and I be an ideal of R . Consider a set of variables y , . . . , y k and the polynomial ring S = R [ y , . . . , y k ] . Then cd S ( I + ( y , . . . , y k )) = cd R ( I ) + k. Proof.
We proceed by induction on k ≥ . It su ffi ces to prove the statement for k = 1 . For simplicity, let y = y .Consider the following long exact sequence · · · → H cI ( S ) ϕ → ( H cI ( S )) y → H c +1 I +( y ) ( S ) → H c +1 I ( S ) → · · · . Since S is a free R -module, it follows that H c +1 I ( S ) = 0 . Then H c +1 I +( y ) ( S ) is the cokernel of the map ϕ , andhence it is isomorphic to H cI ( S y /S ) , which is nonzero since S y /S is a free R -module.Thus cd S ( I + ( y )) ≥ c + 1 and, on the other hand, the inequality cd S ( I + ( y )) ≤ cd S ( I ) + 1 is clear. Noticethat cd S ( I ) = c by virtue of the invariance of local cohomology with respect to the change of basis. (cid:3)
2. K ronecker- W eierstrass decomposition Let K be an algebraically closed fi eld and R be a polynomial ring over K . We require K to be algebraicallyclosed in order to transform the matrix into the Kronecker-Weierstrass form, but we can drop this assumption ifthe matrix is already in that form.We consider a (2 × n ) -matrix M , whose entries are linear forms of R . From the Kronecker-Weierstrasstheory of matrix pencils, there exist two invertible matrices C and C ′ such that the matrix X = CM C ′ is aconcatenation of blocks,(4) X = (cid:0) N n | · · · | N n c | J λ ,m | · · · | J λ d ,m d | B ℓ | · · · | B ℓ g (cid:1) , where the blocks are matrices of the form N n i = x i, x i, · · · x i,n i x i, · · · x i,n i − x i,n i ,J λ j ,m j = y j, y j, · · · y j,m j λ j y j, y j, + λ j y j, · · · y j,m j − + λ j y j,m j ,B ℓ p = z p, z p, · · · z p,ℓ p − z p,ℓ p − z p, z p, · · · z p,ℓ p − z p,ℓ p . Here, x = { x i,h } , y = { y j,h } , z = { z p,h } are independent linear forms of R , c, d, g ≥ , n i , m j , ℓ p arepositive integers, and λ j ∈ K . We call N n i nilpotent block of length n i + 1 , J λ j ,m j Jordan block of length m j andeigenvalue λ j and B ℓ p scroll block of length ℓ p , respectively. The number of scroll and nilpotent blocks g and c ,together with the lengths ℓ p and n i of each of these blocks, are invariants for M , while the eigenvalues λ j of theJordan blocks and the length m j of each of them are not invariant. We call the matrix X a Kronecker-Weierstrassnormal form of M . Since the matrices C and C ′ are invertible, the determinantal ideals de fi ned by X and M coincide. For a detailed discussion of Kronecker-Weierstrass theory we refer to [8, Chapter 12]. Remark 2.1.
We point out that the blocks of length are the following: N = , J λ, = y λy and B = z z . In particular, a (2 × n ) -matrix with generic entries is a concatenation of exactly n scroll blocks of the form B . Example 2.2.
Consider the following matrix of linear forms over the polynomial ring K [ x , . . . , x ] x + x x x + x x x + x x − x x x − x + x − x + x x − x − x + x + x . Subtracting the second column from the fi fth and the fourth from the sixth, we get x + x x x + x x x − x x x − x + x − x + x − x x . Subtracting the second column from the third and the fi fth from the fi rst, we get x x x x x x − x + x − x + x − x x . Then adding the fi rst row to the second one we obtain the canonical form x x x x x x x + x x x x , which is a concatenation of a Jordan block J , of length and eigenvalue , a scroll block B of length and anilpotent block N of length .When the matrix is in the Kronecker-Weierstrass form, a result due to Nasrollah Nejad and Zaare-Nahandiallows us to easily compute the height of the ideal of -minors. Since we will use it several times, we state it herefor ease of reference. Proposition 2.3. (Nasrollah Nejad, Zaare-Nahandi, [18, Proposition 2.2] ) Let X be a matrix in the Kronecker-Weierstrass form (4) . Then the height of I ( X ) in K [ x , y , z ] is given by the following formulas.(1) If X consists of exactly c ≥ nilpotent blocks, then ht (cid:0) I ( X ) (cid:1) = c X i =1 n i . (2) If X consists of c ≥ nilpotent blocks and g ≥ scroll blocks, then ht (cid:0) I ( X ) (cid:1) = c X i =1 n i + g X p =1 ℓ p − . (3) If X consists of c ≥ nilpotent blocks, g ≥ scroll blocks and d ≥ Jordan blocks, then ht (cid:0) I ( X ) (cid:1) = c X i =1 n i + g X p =1 ℓ p + d X j =1 m j − γ, where γ is the maximum number of Jordan blocks with the same eigenvalue. We are interested in computing the cohomological dimension and the arithmetical rank of I ( X ) for somespecial Kronecker-Weierstrass decompositions. We begin with some easy cases.If X is (2 × -matrix, then the ideal I = I ( X ) is principal. Hence cd( I ) = ara( I ) = 1 , provided that I is not the zero ideal. The fi rst non trivial case occurs for matrices of size × . In [13, Corollary 6.5], Huneke,Katz, and Marley proved that, if A is a commutative Noetherian ring containing the fi eld of rational numbers,with dim( A ) ≤ , and I = I ( M ) is the ideal generated by the -minors of a (2 × -matrix M with entries in A , then H I ( A ) = 0 . In the following remark we show that, under these assumptions, the arithmetical rank isstrictly less than whenever M is a matrix of linear forms. Remark 2.4.
Let
A, M and I be as the above. Suppose that M is in the Kronecker-Weierstrass form. If M contains at least one nilpotent block, the result is clear. If M consists of only scroll blocks, the arithmetical rankhas been settled in [1] and the cohomological dimension is explicitly studied in Section 3. On the other hand, thecase of a concatenation of Jordan blocks is studied in Section 4. It remains to consider the concatenation of scrolland Jordan blocks. The matrix M with a scroll block of length and a Jordan block of length is a special case of[24, Theorem 2.1]. Suppose now that M consists of two Jordan blocks of length and one scroll block of length . If the Jordan blocks have the same eigenvalue, then M can be transformed into a matrix with two zeros on thesame row, hence I ( M ) is a squarefree monomial ideal generated by monomials and the arithmetical rank is .This is also the case if M consists of a scroll block of length and a Jordan block of length . Otherwise, if theJordan blocks have di ff erent eigenvalues, M can be transformed into a matrix with one zero and the arithmeticalrank is in light of [2, Example 2]. This is also the case if M has two scroll blocks of length and a Jordan blockof length . Thus we completely settle the case of (2 × -matrices of linear forms.This is the starting point of our investigation about the cohomological dimension and the arithmetical rank ofdeterminantal ideals of (2 × n ) -matrices of linear forms. Example 2.5.
Let X be a (2 × ( n + 1)) -matrix in the Kronecker-Weierstrass form and assume that X consistsof exactly one block.i) If X = B n +1 is a scroll block, where B n +1 = z z · · · z n − z n z z · · · z n z n +1 , then I ( X ) is the de fi ning ideal of a rational normal curve of degree n in P n . In [21], Robbiano and Vallaproved that I ( X ) is set-theoretic complete intersection with ht( I ( X )) = cd( I ( X )) = ara( I ( X )) = n . In particular p I ( X ) = p ( F , . . . , F n ) , where F i ( z , . . . , z n +1 ) = i X α =0 ( − α (cid:18) iα (cid:19) z i − αi +1 z α z αi , i = 1 , . . . , n. ii) If X = N n is a nilpotent block of length n + 1 , where(5) N n = x x · · · x n x · · · x n − x n , it easy to check that p I ( X ) = ( x , . . . , x n ) . Then ht( I ( X )) = cd( I ( X )) = ara( I ( X )) = n . Inparticular, I ( X ) is set-theoretic complete intersection. iii) If X = J λ,n +1 is a Jordan block of eigenvalue λ and length n + 1 , then, by subtracting λ times the fi rstrow from the second one, we transform the matrix into the following: y y · · · y n y n +1 y · · · y n − y n . It is now easy to see that p I ( X ) = ( y , . . . , y n ) . Then I ( X ) is set-theoretic complete intersectionwith ht( I ( X )) = cd( I ( X )) = ara( I ( X )) = n .Remark 2.4 and Example 2.5 describe completely the situation where the number of blocks is or the numberof columns is n = 3 , respectively. So for the rest of the paper we may assume, if necessary, that the number ofblocks is at least and n ≥ .As it appears in Example 2.5, the ideal of minors of nilpotent blocks correspond to linear subspaces. These arecomplete intersections. Precisely we have the following result. Proposition 2.6.
Let X = ( l i ) be a matrix of linear forms, where l i ∈ R = K [ y , . . . , y m ] . Let J = I ( X ) , N n bea nilpotent block of length n + 1 as in (5) and S = R [ x , . . . , x n ] . Consider the matrix M = ( X | N n ) given by theconcatenation of X and N n , then: cd S (cid:0) I ( M ) (cid:1) = cd R ( J ) + n and ara (cid:0) I ( M ) (cid:1) ≤ ara( J ) + n. Proof.
Set r = ara( J ) . Then √ J = ( p , . . . , p r ) , for some polynomials p i ∈ R . We de fi ne n = ( x , . . . , x n ) ,then p I ( N n ) = n by Example 2.5 ii). We consider the ideals J , n and I ( M ) in the ring S and we prove that(6) p I ( M ) = q √ J + n . We have I ( N n ) ⊆ I ( M ) and J ⊆ I ( M ) , hence J + I ( N n ) ⊆ I ( M ) . It follows that q √ J + n = q √ J + p I ( N n ) = p J + I ( N n ) ⊆ p I ( M ) , where the second equality holds in general for every pair of ideals in a polynomial ring. For the other inclusion,consider a -minor q of M . If q involves two columns of X or two columns of N n , then clearly q ∈ J or q ∈ n respectively. Otherwise q = l i x α − l j x β or q = − l i x or q = l i x n . In any case it is clear that q ∈ n . This showsthat I ( M ) ⊂ √ J + n , which implies p I ( M ) ⊂ p √ J + n .From (6) and Lemma 1.4 we get cd S (cid:0) I ( M ) (cid:1) = cd S (cid:0)p I ( M ) (cid:1) = cd S (cid:18)q √ J + n (cid:19) = cd S (cid:0) √ J + n (cid:1) = cd R (cid:0) √ J (cid:1) + n = cd R ( J ) + n. Moreover the equality (6) implies ara (cid:0) I ( M ) (cid:1) ≤ ara( J ) + n . (cid:3) We close this Section by providing explicitly an upper bound for the arithmetical rank that was implicit in[2]. Let n, k be positive integers and f , . . . , f k be polynomials in R = K [ x , . . . , x n ] . We recall that a syzygy of ( f , · · · , f k ) is a vector [ s , · · · , s k ] ∈ R k such that P ki =1 s i f i = 0 . Lemma 2.7.
Let k ≥ be an integer and I = ( f , . . . , f k ) be a homogeneous ideal in R = K [ x , . . . , x n ] .Assume that there exist a positive integer r and a syzygy [ g , . . . , g k − ] ∈ R k − of ( f , . . . , f k − ) such that f rk ∈ ( g , . . . , g k − ) . Then ara( I ) ≤ k − . Proof.
Since f rk ∈ ( g , . . . , g k − ) , there exist h , . . . , h k − ∈ R such that f rk = h g + · · · + h k − g k − . Let q i = f k h i + f i for ≤ i ≤ k − . We claim that √ I = p ( q , . . . , q k − ) . Clearly p ( q , . . . , q k − ) ⊂ √ I ,since ( q , . . . , q k − ) ⊂ I . For the other inclusion, let g ∈ √ I . Then there exist r , . . . , r k ∈ R such that g s = r f + · · · + r k f k for some positive integer s . Then g s = k − X i =1 r i q i − f k k − X i =1 r i h i − r k ! . We claim that f r +1 k ∈ ( q , . . . , q k − ) . In fact, k − X i =1 g i q i = k − X i =1 g i ( f k h i + f i ) = f k k − X i =1 g i h i ! + k − X i =1 g i f i = f r +1 k , where the last equality holds since [ g , . . . , g k − ] is a syzygy of ( f , . . . , f k − ) . Then g s ( r +1) = r X j =0 ( − j (cid:18) r + 1 j (cid:19) k − X i =1 r i q i ! r +1 − j k − X i =1 r i h i − r k ! j f jk + ( − r +1 k − X i =1 r i h i − r k ! r +1 f r +1 k ∈ ( q , . . . , q k − ) . Hence g ∈ p ( q , . . . , q k − ) , as desired. (cid:3) Up to fi nding a syzygy with the required properties, we are able to decrease by one the number of generatorsof I up to radical. We give a simple application of Lemma 2.7. Example 2.8.
Let M = x x x x x x x and I = I ( M ) in the polynomial ring R = K [ x , . . . , x ] , where K is a fi eld of characteristic . We prove that ara( I ) = 4 . By [17, Remark 5.2], we have cd R ( I ) = 4 . Then ara( I ) ≥ . To prove the claim it su ffi ces to fi nd polynomials that generate I up to radical. Recall that [ i, j ] denotes the minor corresponding to the i -th and j -th columns of M . Then I = ([1 , , [1 , , [2 , , [1 , , [2 , , [3 , . Notice that [ x , − x ] is a syzygy for ([1 , , [1 , and [2 ,
3] = x x − x x ∈ ( x , − x ) . Following the proof ofLemma 2.7, de fi ne q = − x [2 ,
3] + [1 , and q = − x [2 ,
3] + [1 , . Then √ I = p ( q , q , [1 , , [2 , , [3 , . By the Plücker relations (see (15)) [1 , , − [2 , ,
3] + [3 , ,
2] = 0 we have that [[2 , , − [1 , , [1 , is a syzygy for ([1 , , [2 , , [3 , . Notice that q = − x [2 , − ( − [1 , ∈ ([2 , , − [1 , , [1 , . Again, following the proof of Lemma 2.7, we de fi ne p = − x q + [1 , , p = − q + [2 , , p = [3 , . Then √ I = √ q , p , p , p , and hence ara( I ) ≤ .
3. S croll blocks
In this section we assume that the Kronecker-Weierstrass decomposition of our matrix contains only scrollblocks. We fi x an algebraically closed fi eld K and some integers d ≥ and n , n , . . . , n d > . We consider thematrix(7) M = ( B n | · · · | B n d ) = x , x , . . . x ,n − . . . x d, x d, . . . x d,n d − x , x , . . . x ,n . . . x d, x d, . . . x d,n d , where x i,j are algebraically independent variables over K . We also denote by N = P di =1 n i + d − the numberof variables minus and by I n ,...,n d = I ( M ) the homogeneous ideal generated by the -minors of the matrix M in the polynomial ring R = K [ x i,j ] .The projective variety R n ,...,n d = Proj( R/I n ,...,n d ) ⊂ P NK associated to I n ,...,n d has dimension d and iscalled d -dimensional rational normal scroll . These varieties have been widely studied and many properties areknown. In the following Proposition we collect a few facts that will be used later on. For a proof and a survey onrational normal scrolls the reader may consult [20, Chapter 2]. Proposition 3.1.
Let d ≥ , n , . . . , n d > be integers and let I n ,...,n d , R and R n ,...,n d be as above. Then(1) R n ,...,n d is irreducible, i.e. I n ,...,n d is a prime ideal,(2) R/I n ,...,n d is a Cohen-Macaulay ring of dimension d + 1 ,(3) Pic( R n ,...,n d ) ∼ = Z ⊕ Z , where Pic( R n ,...,n d ) is the Picard group of R n ,...,n d . In their paper [1], B ă descu and Valla proved that ara( I n ,...,n d ) = N − . They exhibit N − polynomialswhich generate the rational normal scroll set-theoretically and they use Grothendieck-Lefschetz theory to showthat ara( I n ,...,n d ) ≥ N − . In particular, it turns out that R n ,...,n d is a set-theoretic complete intersection ifand only if d = 2 and, in this case, ht( I n ,n ) = cd R ( I n ,n ) = ara( I n ,n ) = n + n − .The goal of this section is to compute the cohomological dimension of I n ,...,n d . We are going to prove thefollowing result. Theorem 3.2.
Let K be an algebraically closed field, d ≥ and n , . . . , n d > integers, and I n ,...,n d = I ( M ) bethe ideal generated by the -minors of the matrix (7) in the polynomial ring R = K [ x i,j ] in N + 1 variables. Then cd R ( I n ,...,n d ) = ht( I n ,...,n d ) = N − d = d X i =1 n i − if char( K ) = p > I n ,...,n d ) = N − d X i =1 n i + d − if char( K ) = 0 . The proof of this theorem will use geometric tools. In fact, we will study the variety R n ,...,n d rather than theideal I n ,...,n d . We recall some Algebraic Geometry facts. When not explicitly stated, we refer to [11] and [3,Chapter 20] for proofs and further details.Let S = L n ∈ N S n be a positively graded ring where S is a fi eld and let m = L n> S n its homogeneousmaximal ideal. We consider a fi nitely generated graded S -module N and the associated coherent sheaf F = e N on X = Proj( S ) . The Serre-Grothendieck Correspondence states that there are isomorphisms of S -modules between the sheaf cohomology modules and the local cohomology modules:(8) H i ( X, F ( n )) ∼ = H i +1 m ( N ) n , for all i > and n ∈ Z .The cohomological dimension of X is de fi ned as cd( X ) = min { n ∈ N : H i ( X, F ) = 0 for every i > n and F coherent sheaf over X } . If S is a fi eld and I is a homogeneous non-nilpotent ideal, then by a result of Hartshorne [9] we have(9) cd S ( I ) − S ) \ Proj(
S/I )) . Thus, in order to bound cd S ( I ) , we can fi nd bounds on cd( X \ Y ) , where Y = Proj( S/I ) .When the base fi eld S is the fi eld of complex numbers C , we have a strong connection between the van-ishing of the sheaf cohomology modules H i ( X \ Y, − ) and the singular cohomology groups H i sing ( X an , C ) and H i sing ( Y an , C ) . Here X an and Y an denote X and Y regarded as topological spaces with the euclidean topologyand are called analyti fi cation of X and Y . Theorem 3.3. (Hartshorne [10, Theorem 7.4, p. 148] ) Let X be a complete scheme of dimension N over C , Y bea closed subscheme, and assume that X \ Y is non-singular. Let r be an integer. Then cd( X \ Y ) < r implies thatthe natural maps H i sing ( X an , C ) −→ H i sing ( Y an , C ) are isomorphisms for i < N − r , and injective for i = N − r . The assumption S = C is not restrictive. In fact, the following Remark shows that we may assume it in manycases. Remark 3.4.
Let K be a fi eld of characteristic , R K = K [ x , . . . , x n ] the polynomial ring in n variables over K and I an ideal of R K . Since R K is Noetherian, I is fi nitely generated, say I = ( f , . . . , f m ) . The coe ffi cientsof the polynomials f i are elements of a fi nite extension of Q , say L . We denote by R L = L [ x , . . . , x n ] thecorresponding polynomial ring. Notice that L is a sub fi eld of K and a sub fi eld of C . We consider the ideal I L = I ∩ R L , then I = I L R K by construction. Set R C = C [ x , . . . , x n ] and I C = I L R C . We claim that cd R K ( I ) = cd R C ( I C ) . Let i and j be integers, we look at the j -th graded piece of the local cohomology modules with support in I : H iI ( R K ) j = H iI L R K ( R L ⊗ L K ) j = H iI L ( R L ) j ⊗ L K. Since the fi eld extension L ⊂ K is faithfully fl at, we have that H iI ( R K ) j = 0 if and only if H iI L ( R L ) j = 0 . Inparticular, cd R K ( I ) = cd R L ( I L ) . The same argument applied to the ideals I L and I C and to the faithfully fl at fi eld extension L ⊂ C , yields cd R L ( I L ) = cd R C ( I C ) , which proves the claim.We are now ready to prove Theorem 3.2. Proof of Theorem 3.2.
For ease of notation, we set I = I n ,...,n d and Y = R n ,...,n d .If K is a fi eld of positive characteristic p , then the statement follows from [19, Proposition 4.1, p. 110].Now let char( K ) = 0 . In light of Remark 3.4, we may assume K = C . We know that cd R ( I ) ≤ ara( I ) and ara( I ) = N − , so we need to prove that cd R ( I ) ≥ N − . We consider the exponential sequence of sheaves over Y an , the analyti fi cation of Y :(10) → Z → O Y an → O ∗ Y an → , where Z denotes the constant sheaf and the map O Y an → O ∗ Y an is given by f exp(2 πif ) . The sequence (10)induces a long exact sequence of sheaf cohomology modules, in particular we have(11) · · · → H ( Y an , O Y an ) → H ( Y an , O ∗ Y an ) → H ( Y an , Z ) → H ( Y an , O Y an ) → · · · . By de fi nition H ( Y an , O ∗ Y an ) = Pic( Y ) and, since Z is a constant sheaf, it follows that H ( Y an , Z ) = H sing ( Y an , Z ) .An application of the GAGA principle and (8) yield H ( Y an , O Y an ) = H ( Y, O Y ) = H m ( R/I ) , where m is thehomogeneous maximal ideal of R . Since R/I is a Cohen-Macaulay ring of dimension d + 1 ≥ we have that H m ( R/I ) = 0 , therefore (11) yields the group injection(12) Pic( Y ) ֒ → H sing ( Y an , Z ) . Now we assume that cd R ( I ) < N − and proceed by contradiction. From (9) it follows that cd( P N \ Y ) = cd R ( I ) − < N − − N − . Theorem 3.3 with r = N − yields H i sing ( P N an , C ) ∼ = H i sing ( Y an , C ) for i < , which implies dim C H i sing ( P N an , C ) = dim C H i sing ( Y an , C ) . By the Universal Coe ffi cients Theorem, this is equiva-lent to rank Z H i sing ( P N an , Z ) = rank Z H i sing ( Y an , Z ) . It is well known that H i sing ( P N an , Z ) = ( Z if i even , ≤ i ≤ N otherwise . In particular, rank Z H i sing ( Y an , Z ) ≤ . On the other hand, Pic( Y ) = Z , which contradicts (12). (cid:3) From Theorem 3.2 and Proposition 2.6 we immediately deduce
Corollary 3.5.
Let K be an algebraically closed field of characteristic , let R be a polynomial ring over K and M be a (2 × n ) -matrix of linear forms over R . If the Kronecker-Weierstrass decomposition of M is ( B n | · · · | B n d | N m | · · · | N m c ) for some integers d ≥ , c ≥ , n , . . . , n d > and m , . . . , m c ≥ , then cd R (cid:0) I ( M ) (cid:1) = ara (cid:0) I ( M ) (cid:1) = d X i =1 n i + c X j =1 m j + d − .
4. J ordan blocks
Let K be a fi eld of characteristic zero, d ≥ and α i ≥ for i = 1 , . . . , d . We consider the following (2 × n ) -matrix M consisting of α i Jordan blocks with eigenvalue λ i for i = 1 , . . . , d , such that α i ≥ α j if j > i :(13) M = (cid:16) J λ ,m (cid:12)(cid:12) J λ ,m (cid:12)(cid:12) · · · (cid:12)(cid:12) J α λ ,m α (cid:12)(cid:12) J λ ,m (cid:12)(cid:12) J λ ,m (cid:12)(cid:12) · · · (cid:12)(cid:12) J α λ ,m α (cid:12)(cid:12) · · · (cid:12)(cid:12) J λ d ,m d (cid:12)(cid:12) J λ d ,m d (cid:12)(cid:12) · · · (cid:12)(cid:12) J α d λ d ,m dαd (cid:17) . Here we use the following notation for the Jordan blocks, for j = 1 , . . . , d and i = 1 , . . . , α j : J iλ j ,m ji = y ij, y ij, · · · y ij,m ji λ j y ij, y ij, + λ j y ij, · · · y ij,m ji − + λ j y ij,m ji , where m ji is the length of the block.Consider the ideal I ( M ) in the polynomial ring R = K [ y ij,h : 1 ≤ j ≤ d, ≤ i ≤ α j , ≤ h ≤ m ji ] . Let α = P di =1 α i be the number of blocks in M and N = P ≤ j ≤ d ≤ i ≤ α j m ji be the number of variables in R .The following Theorem shows that, even though the height of I ( M ) depends on the maximum number ofblocks with the same eigenvalue, the cohomological dimension equals the arithmetical rank of I ( M ) and theyare independent on how many blocks have the same eigenvalue. Theorem 4.1.
Let K be a field of characteristic zero and M be a matrix of the form (13) . Then cd( I ( M )) = ara( I ( M )) = ( N − α if d = 1 N − if d > . Proof.
First we observe that(14) p I ( M ) = J + L M , where J is the ideal generated by all the N − α variables y ij,h , for every j = 1 , . . . , d , i = 1 , . . . , α j and h = 1 , . . . , m ji − . To describe the ideal L M fi rst we simplify the notation: we denote the last variable y ij,m ji of each block by y ij . Then L M is the squarefree monomial ideal generated by the quadratics monomialsof the form y ij y ℓk , for j = k , ≤ j, k ≤ d , ≤ i ≤ α j and ≤ ℓ ≤ α k . Notice that L M is an idealin the ring S = K [ y ij : 1 ≤ j ≤ d, ≤ i ≤ α j ] . The equality (14) holds because if we consider a minorinvolving at most one of the last columns of the blocks, then it is a multiple of some y ij,h ∈ J ; otherwise ifthe minor involves the last columns of two blocks, then it is a multiple of some monomial y ij y ℓk ∈ L M . Thisimplies that I ( M ) ⊂ J + L M , hence p I ( M ) ⊂ J + L M , since J + L M is a radical ideal. Vice versa, fi rst we show that J ⊂ p I ( M ) . We fi x a block J iλ j ,m ji and we prove that y ij,h ∈ p I ( M ) by induction on h ≥ . For h = 1 , (cid:16) y ij, (cid:17) = y ij, ( y ij, + λ j y ij, ) − λ j y ij, y ij, ∈ I ( M ) since it is the minor corresponding tothe fi rst two columns of the block J iλ j ,m ji . Suppose that h > and y ij,k ∈ p I ( M ) for ≤ k < h . Then (cid:16) y ij,h (cid:17) = (cid:16) y ij,h (cid:17) − y ij,h − y ij,h +1 + y ij,h − y ij,h +1 ∈ p I ( M ) , since (cid:16) y ij,h (cid:17) − y ij,h − y ij,h +1 ∈ I ( M ) is the minorcorresponding to the columns h and h + 1 and y ij,h − y ij,h +1 ∈ p I ( M ) by induction hypothesis. Now we prove that L M ⊂ p I ( M ) . Notice that ( λ k − λ j ) y ij y ℓk = ( λ k − λ j ) y ij,m ji y ℓk,m kℓ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ij,m ji y ℓk,m kℓ y ij,m ji − + λ j y ij,m ji y ℓk,m kℓ − + λ k y ℓk,m kℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:0) y ij,m ji y ℓk,m kℓ − − y ij,m ji − y ℓk,m kℓ (cid:1) ∈ p I ( M ) , since y ℓk,m kℓ − , y ij,m ji − ∈ J ⊂ p I ( M ) . This yields the equality (14).If d = 1 , all the blocks have the same eigenvalue λ . Hence L M = (0) and p I ( M ) = J . This implies that cd( I ( M )) = ara( I ( M )) = N − α .Let d ≥ . Notice that L M is the edge ideal of a complete d -partite graph K α ,α ,...,α d . By [15, Theorem 4.2.6],we have cd( L M ) = pd S ( S/L M ) = α − . Then cd( I ( M )) = cd( J ) + cd( L M ) = N − α + α − N − byProposition 2.6.Now we show that ara( I ( M )) ≤ N − . In light of Example 2.5 iii), ara (cid:0) I (cid:0) J iλ j ,m ji (cid:1)(cid:1) = m ji − and I (cid:0) J iλ j ,m ji (cid:1) is generated by the variables y ij, , y ij, , . . . , y ij,m ji − up to radical.Since J is generated by N − α variables, in order to prove the claim, it su ffi ces to show that L M is generated by α − polynomials up to radical. We construct the following matrix with P di =2 α i rows and P d − i =1 α i columns: Q = y y d y y d · · · y α y d y y d · · · y α y d · · · y d − y d · · · y α d − d − y d y y d y y d · · · y α y d y y d · · · y α y d · · · y d − y d · · · y α d − d − y d ... ... . . . ... ... . . . ... ... ... . . . ... y y α d d y y α d d · · · y α y α d d y y α d d · · · y α y α d d · · · y d − y α d d · · · y α d − d − y α d d y y d − y y d − · · · y α y d − y y d − ... y α y d − ... ... . . . ... ... . . . ... y y α y y α · · · y α y α y y α · · · y α y α y y y y · · · y α y ... ... . . . ... y y α y y α · · · y α y α , The fi rst block of Q is obtained by multiplying the variables y i by y hj for ≤ j ≤ d and ≤ h ≤ α j ; the secondblock is obtained by multiplying the variables y i by y hj for ≤ j ≤ d and ≤ h ≤ α j and so on.Let T to be the set of all the entries of Q , that are the generators of L M . For every ℓ = 1 , . . . , α − , wede fi ne T ℓ as the set of all the monomials of the ℓ -th antidiagonal of Q and q ℓ as the sum of these monomials.In particular, T = { y y d } and T = S α − ℓ =1 T ℓ . To show the last equality, we count the number of nonzeroantidiagonals of Q . Every element on the fi rst row is contained in exactly one T ℓ , hence we have P d − i =1 α i sets.Moreover, every nonzero element in the last column is contained in exactly one T ℓ , thus we have α d sets. In totalwe have P di =1 α i − sets, since the element y α d − d − y d has been counted twice. All the other antidiagonals of Q are zero because the elements of the form y α j j y α h h belong to the ( α − -th antidiagonal. This shows that the fi rsttwo conditions of Lemma 1.1 are ful fi lled.As for the third condition, if we pick two monomials on the ℓ -th antidiagonal of Q , they have the form y i j y i j and y h k y h k . We may assume that either j < k or ( j = k and i < h ). Hence their product y i j y i j · y h k y h k is a multiple of y i j y h k that belongs to the m -th antidiagonal, for some ≤ m < ℓ (this element is placed in theintersection of the column containing y i j y i j and the row containing y h k y h k ). From Lemma 1.1 it follows that L M = √ L M = p ( q , . . . , q α − ) and thus ara( L M ) ≤ α − . Therefore ara( I ( M )) ≤ ara( J ) + ara( L M ) ≤ N − α + α − N − . (cid:3) From Theorem 4.1 and Proposition 2.6 we deduce
Corollary 4.2.
Let K be a field of characteristic , let R be a polynomial ring over K and M ′ be a (2 × n ) -matrixof linear forms over R . Suppose that the Kronecker-Weierstrass decomposition of M ′ is ( M | N m | · · · | N m c ) for some integers d ≥ , c ≥ , α , . . . , α d ≥ and m , . . . , m c ≥ , and where M is the matrix (13) . Then cd (cid:0) I ( M ′ ) (cid:1) = ara (cid:0) I ( M ′ ) (cid:1) = ( N − α + P ck =1 m k if d = 1 N − P ck =1 m k if d > . (2 × n ) -matrices with a zero diagonal In Sections 3 and 4 we analyzed the cases of concatenations of scroll blocks or Jordan blocks. In this Section westudy a mixed case, in which there are both scroll and Jordan blocks. Precisely, let n ≥ , R = K [ x , . . . , x n − ] and J n = I ( A n ) be the ideal generated by the -minors of the matrix A n = x x · · · x n − x n − x n x n +1 x n +2 · · · x n − . Remark 5.1.
We add the fi rst row of A n to the second one and we apply the following linear change of variables y i = x i + x n + i for every i = 1 , . . . , n − . We get the matrix A ′ n = x x · · · x n − x n − x n y y · · · y n − x n − which is a Kronecker-Weierstrass form of A n . In particular, A ′ n = ( J , | B | · · · | B | J , ) is a concatenation ofa Jordan block of length and eigenvalue , n − scroll blocks of length and a Jordan block of length andeigenvalue . From Proposition 2.3 it follows that ht( J n ) = n − . Notation 5.2.
We label the columns of A n with the indices from to n − . Recall that [ i, j ] denotes the -minor x i x n + j − x j x n + i corresponding to the columns i and j . Remark 5.3.
We recall that, if M is a (2 × n ) -matrix of indeterminates and we label the columns with indicesfrom to n − , then the Plücker relations are the following: for every h ∈ { , . . . , n − } and for every ≤ j < j < j ≤ n − ,(15) [ h, j ][ j , j ] − [ h, j ][ j , j ] + [ h, j ][ j , j ] = 0 . As in the case of generic matrices, we fi nd an upper bound for the arithmetical rank of J n , independent of the fi eld. Theorem 5.4.
Let A n the matrix above with entries in a commutative ring R . For every n ≥ , ara( J n ) ≤ n − . Proof.
For n ≥ , the ideal J n contains both monomials and minors and it can be written in the form J n = J ′ n + J ′′ n , where J ′ n = ( x x n , x x n , . . . , x n − x n , x n − x n +1 , . . . , x n − x n − ) ,J ′′ n = ( x i x n + j − x n + i x j : 1 ≤ i < j ≤ n − . In particular, the ideal J ′′ n is the ideal of -minors of the submatrix C n of A n , obtained by removing the fi rstand the last column from A n . We prove that ara( J n ) ≤ n − . To do this we will de fi ne n − polynomialscontaining all the monomial generators of J n and n − − n − polynomials containing all thebinomial generators of J n . In total we get n − polynomials that generate J n up to radical. Then we will reducethese polynomials to n − by summing in a suitable way some of the polynomials in the fi rst group to some ofthe polynomials in the second group.First we de fi ne the following polynomials containing all the monomial generators of J n : q = x n − x n ,q = x x n + x n − x n +1 ,q = x x n + x n − x n +2 , ... q n − = x n − x n + x n − x n − . From Lemma 1.1, it follows that J ′ n = p ( q , . . . , q n − ) . On the other hand, by applying Theorem 1.2 weget ara( J ′′ n ) = 2( n − − n − , where J ′′ n = p ( p , . . . , p n − ) and p i is the sum of the minorscorresponding to rank i elements in the poset ∆( C n ) (see (2) and (3)).For n ≥ , we prove that J n = √ K n , where K n = ( p , . . . , p n − , q + p n − , q + p n − , . . . , q n − + p n − , q n − , q n − ) . In other words, we consider the lowest n − levels of the poset ∆( C n ) and the corresponding polynomials p , . . . , p n − will also be generators of J n up to radical. Then each of the remaining n − polynomials p n − i will be summed to q i for i = 1 , . . . , n − . Finally we consider q n − and q n − .Let e J n = e J ′ n + e J ′′ n , where e J ′ n = ( q , . . . , q n − ) and e J ′′ n = ( p , . . . , p n − ) . Notice that p J ′ n = q e J ′ n and p J ′′ n = q e J ′′ n . Then p J n = p J ′ n + J ′′ n = qp J ′ n + p J ′′ n = rq e J ′ n + q e J ′′ n = q e J ′ n + e J ′′ n = q e J n , where the second and the fourth equality are true for any pair of ideals. Hence it su ffi ces to prove that q e J n = √ K n . Of course K n ⊂ e J n , thus √ K n ⊂ q e J n . Conversely, we show that the generators of e J n belong to √ K n . We know that p , . . . , p n − , q n − , q n − ∈ K n .We need to prove that(16) q , . . . , q n − ∈ p K n . It will follow that p n − , . . . , p n − ∈ √ K n , thus e J n ⊂ √ K n .With respect to the Notation 5.2, the polynomials q i and p n − i can be written in the form q = − [0 , n − , q i = − [0 , i − − [ i − , n − for i = 2 , . . . , n − ,p n − i = ⌊ n − i ⌋− i X k =0 [ i + k, n − − k ] for i = 1 , . . . , n − . We know that q n − , q n − ∈ K n . Let i ∈ { , . . . , n − } and suppose that q j ∈ √ K n for every j ∈ { i +1 , . . . , n − } . We prove that q i ∈ √ K n . Notice that q i = q i ( q i + p n − i ) − q i p n − i Since q i + p n − i ∈ K n , it is enough to show that − q i p n − i ∈ √ K n . By using the Notation 5.2, this elementcan be rewritten in the form(17) − q i p n − i = ⌊ n − i ⌋− i X k =0 (cid:16) [0 , i −
1] + [ i − , n − (cid:17) [ i + k, n − − k ] . Let k ∈ { , . . . , ⌊ n − i ⌋ − i } , then the k -th summand of (17) is (cid:16) [0 , i −
1] + [ i − , n − (cid:17) [ i + k, n − − k ]= [0 , i − i + k, n − − k ] + [ i − , n − i + k, n − − k ]= [0 , i + k ][ i − , n − − k ] − [0 , n − − k ][ i − , i + k ]+[ i + k, n − i − , n − − k ] − [ n − − k, n − i − , i + k ]= [ i − , n − − k ] (cid:16) [0 , i + k ] + [ i + k, n − (cid:17) − [ i − , i + k ] (cid:16) [0 , n − − k ] + [ n − − k, n − (cid:17) = − [ i − , n − − k ] q i + k +1 + [ i − , i + k ] q n − k − , where the second equality follows from the Plücker relations (15) with respect to the indices h = 0 , j = i − , j = i + k, j = n − − k for the fi rst summand and h = n − , j = i − , j = i + k, j = n − − k for the second summand. Hence − q i p n − i = ⌊ n − i ⌋− i X k =0 (cid:16) − [ i − , n − − k ] q i + k +1 + [ i − , i + k ] q n − k − (cid:17) , where q i + k +1 , q n − k − ∈ √ K n since i + k + 1 and n − k − are both greater than i and less than or equal to n − . Thus q i ∈ √ K n and therefore q i , p n − i ∈ √ K n .It remains to prove that q ∈ √ K n . Notice that q = q ( q + p n − ) − q p n − Since q + p n − ∈ K n , it is enough to show that − q p n − ∈ √ K n . By using the Notation 5.2, this elementcan be rewritten in the form(18) − q p n − = ⌊ n − ⌋− X k =0 [0 , n − k, n − − k ] . Let k ∈ { , . . . , ⌊ n − ⌋ − } , then the k -th summand of (18) is [0 , n − k, n − − k ] = [0 , k ][ n − − k, n − − [0 , n − − k ][1 + k, n − , k ][ n − − k, n − − [0 , n − − k ][1 + k, n − k, n − n − − k, n − − [1 + k, n − n − − k, n − n − − k, n − (cid:16) [0 , k ] + [1 + k, n − (cid:17) − [1 + k, n − (cid:16) [0 , n − − k ] + [ n − − k, n − (cid:17) = − [ n − − k, n − q k + [1 + k, n − q n − k − , where the fi rst equality follows from the Plücker relations (15) with respect to the indices h = 0 , j = 1 + k, j = n − − k, j = n − . Hence − q p n − = ⌊ n − ⌋− X k =0 (cid:16) − [ n − − k, n − q k + [1 + k, n − q n − k − (cid:17) , where q k , q n − k − ∈ √ K n since k and n − k − are both greater than and less than or equal to n − .Thus q ∈ √ K n and therefore q , p n − ∈ √ K n . (cid:3) Now we compute the cohomological dimension of J n . Again, as for the generic matrices, it depends on thecharacteristic of the fi eld. Theorem 5.5.
Let n ≥ , R = K [ x , . . . , x n − ] and J n = I ( A n ) be the ideal generated by the -minors of A n .Then i) ht( J ) = cd( J ) = ara( J ) = 1 and ht( J ) = cd( J ) = ara( J ) = 2 ,ii) for n ≥ , cd( J n ) = ( ht( J n ) = n − if char( K ) = p > J n ) = 2 n − if char( K ) = 0 . First we consider the case char( K ) = 0 . Under this assumption, not only we prove that cd( J n ) = 2 n − , butwe also show the vanishing of all local cohomology modules with indices between ht( J n ) and n − . Theorem 5.6.
Let K be a field of characteristic and R = K [ x , . . . , x n − ] . For every n ≥ , H iJ n ( R ) = 0 ⇐⇒ i = n − or i = 2 n − . In particular, cd R ( J n ) = 2 n − . Notation 5.7. We fi x S = K [ x, y, x , . . . , x n − ] and A = S/ ( x ) , then R = A/ ( y ) = S/ ( x, y ) . We considerthe generic (2 × n ) -matrix M n over SM n = x x · · · x n − x n − x n x n +1 · · · x n − y and I = I ( M n ) the ideal of -minors of M n . Notice that, if x = y = 0 , the ideal I coincides with the ideal J n .In other words, J n = IR .The basic idea is to reduce the vanishing of H iIR ( R ) to the vanishing of H iI ( S ) by using the multiplicationmaps by x and by y . The modules H iI ( S ) are well-understood thanks to the following results due to Witt andLyubeznik, Singh and Walther. Theorem 5.8. (Witt, [25, Theorem 1.1] ) Let S and I be as above. Then H iI ( S ) = 0 ⇐⇒ i = n − or i = 2 n − . Theorem 5.9. (Lyubeznik, Singh, Walther, [17, Theorem 1.2] ) Let S and I be as above, and let m = ( x, y, x , . . . , x n − ) the homogeneous maximal ideal of S . Then we have an isomorphism of S -modules H n − I ( S ) ∼ = H n m ( S ) . Proof of Theorem 5.6.
It is clear that H n − J n ( R ) = 0 and H iJ n ( R ) = 0 for i < n − , since ht( J n ) = n − . ByTheorem 5.4, we have also H iJ n ( R ) = 0 for i > n − . For n = 4 , one has that n − J ) , then H J ( R ) = 0 .Now let n ≥ and let S , A and I be as in Notation 5.7.We consider the map S · x −→ S , it induces a long exact sequence of local cohomology modules:(19) · · · → H jI ( S ) → H jIA ( A ) → H j +1 I ( S ) · x −→ H j +1 I ( S ) → · · · . For j = 2 n − we get H n − IA ( A ) = ker (cid:0) H n − I ( S ) · x −→ H n − I ( S ) (cid:1) , since H n − I ( S ) = 0 by Theorem 5.8. On the other hand, by using the Č ech complex, it is easy to see that H n − m A ( A ) = ker (cid:0) H n m ( S ) · x −→ H n m ( S ) (cid:1) . Then the isomorphism of Theorem 5.9 yields H n − IA ( A ) ∼ = H n − m A ( A ) , the latter being non-zero since m A is the homogeneous maximal ideal of A . Moreover, by Theorem 5.8, if n − < j < n − , then H jI ( S ) = H j +1 I ( S ) = 0 . Therefore H jIA ( A ) = 0 by virtue of (19).Now we consider the multiplication map A · y −→ A and the corresponding long exact sequence:(20) · · · → H iIA ( A ) → H iIR ( R ) → H i +1 IA ( A ) · y −→ H i +1 IA ( A ) → · · · . For i = 2 n − , from Theorem 5.9 it follows that H n − IR ( R ) = ker (cid:0) H n − IA ( A ) · y −→ H n − IA ( A ) (cid:1) ∼ = ker (cid:0) H n − m A ( A ) · y −→ H n − m A ( A ) (cid:1) = H n − m R ( R ) , since H n − IA ( A ) = 0 and H n − m A ( A ) = 0 . Then m R is the homogeneous maximal ideal of R , hence H n − m R ( R ) =0 . This implies H n − J n ( R ) = 0 , since J n = IR .It remains to prove that H iJ n ( R ) = 0 for n − < i < n − . For such i , we have H iIA ( A ) = H i +1 IA ( A ) = 0 ,as shown above. Then (20) yields H iIR ( R ) = 0 , as required. (cid:3) Proof of Theorem 5.5.
For n = 2 , the ideal J is principal, thus cd( J ) = ara( J ) = 1 . For n = 3 , we have cd( J ) = ara( J ) = 2 , as computed in Example 2.4. Let n ≥ . If char( K ) = 0 , the claim follows from Theorem5.6. If char( K ) = p > , the claim follows from [19, Proposition 4.1, p. 110], since J n is a perfect ideal. In fact, ht( J n ) = grade( J n ) = n − . Moreover, by [7, Theorem 2, p. 201], pd R ( R/I ) = n − n − . (cid:3) A cknowledgments The authors wish to thank Aldo Conca, Srikanth Iyengar and, in particular, Anurag Singh, together withthe organizers of Pragmatic 2014 (Catania, Italy), for the beautiful school, the interesting lectures, many usefulsuggestions and insights. The authors are also in debt with Matteo Varbaro and Margherita Barile for helpfuldiscussions and hints. R eferences [1] L. B ădescu , G. V alla , Grothendieck-Lefschetz theory, set-theoretic complete intersections and rational normal scrolls , J.Algebra (2010), 1636–1655.[2] M. B arile , On ideals generated by monomials and one binomial , Algebra Coll. (2007), 4, 631–638.[3] M. P. B rodmann , R. Y. S harp , Local Cohomology. An algebraic introduction with geometric applications , Cambridgestudies in Advanced Mathematics (1998), Cambridge University Press.[4] W. B runs , Addition to the theory of algebras with straightening law , Commutative Algebra (Berkeley, CA, 1987), 111—138, Springer, New York, 1989.[5] W. B runs , R. S chwänzl , The number of equations defining a determinantal variety , Bull. London Math. Soc. (1990),439—445.[6] W. B runs , U. V etter , Determinantal rings , Lecture Notes in Mathematics (2nd ed.), Vol. (1988), Springer, Heidel-berg.[7] J. A. E agon , D. G. N orthcott , Ideals defined by matrices and a certain complex associated with them , Proc. R. Soc. Lond.A (1962), 188–204.[8] F. R. G antmacher , The theory of matrices,
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