aa r X i v : . [ m a t h . A C ] J a n KNUTSON IDEALS OF GENERIC MATRICES
LISA SECCIA
Abstract.
In this paper we show that determinantal ideals of generic matrices areKnutson ideals. This fact leads to a useful result about Gr¨obner bases of certain sums ofdeterminantal ideals. More specifically, given I = I + . . . + I k a sum of ideals of minorson adjacent columns or rows, we prove that the union of the Gr¨obner bases of the I j ’sis a Gr¨obner basis of I . Introduction
Let K be a field of any characteristic. Fix f ∈ S = K [ x , . . . , x n ] a polynomial suchthat its leading term in ≺ ( f ) is a squarefree monomial for some term order ≺ . We candefine many more ideals starting from the principal ideal ( f ) and taking associated primes,intersections and sums. Thereby, if K has characteristic p , we obtain a family of idealswhich are compatibly split with respect to Tr( f p − • ) (see [Kn] for more details).Geometrically this means that we start from the hypersurface defined by f and we con-struct a family of new subvarieties { Y i } i by taking irreducible components, intersectionsand unions. Definition 1 (Knutson ideals) . Let f ∈ S = K [ x , . . . , x n ] be a polynomial such that itsleading term in ≺ ( f ) is a squarefree monomial for some term order ≺ . Define C f to bethe smallest set of ideals satisfying the following conditions:1. ( f ) ∈ C f ;2. If I ∈ C f then I : J ∈ C f for every ideal J ⊆ S ;3. If I and J are in C f then also I + J and I ∩ J must be in C f .This class of ideals has some interesting properties which were first proved by Knutsonin the case K = Z /p Z and then generalized to fields of any characteristic in [Se]:i) Every I ∈ C f has a squarefree initial ideal, so every Knutson ideal is radical.ii) If two Knutson ideals are different their initial ideals are different. So C f is finite.iii) The union of the Gr¨obner bases of Knutson ideals associated to f is a Gr¨obnerbasis of their sum. Remark . Actually, assuming that every ideal of C f is radical, the second condition inDefinition 1 can be replaced by the following:2 ′ . If I ∈ C f then P ∈ C f for every P ∈
Min( I ).In this paper we will continue the study undertaken in [Se] about Knutson ideals.So far, it has been proved that determinanatal ideals of Hankel matrices are Knutsonideals for a suitable choice of f ( [Se, Theorems 3.1,3.2]). As a consequence of theseresults, one can derive an alternative proof (see [Se, Corollary 3.3]) of the F -purity of ankel determinantal rings, a result recently proved in [CMSV].In this paper we are going to show that also determinantal ideals of generic matricesare Knutson ideals (see Theorem 2.1). In particular, they define F -split rings. This wasalready known since the 1990’s from a result by Hochster and Huneke ([HH]).As a corollary we obtain an interesting result about Gr¨obner bases of certain sums ofdeterminantal ideals. More specifically, given I = I + . . . + I k a sum of ideals of minorson adjacent columns or rows, we will prove that the union of the Gr¨obner bases of the I j ’s is a Gr¨obner basis of I (see Corollary 2.4). Example 1.
Let X = ( x ij ) be the generic square matrix of size 6 and consider the ideal J = I ( X [1 , ) + I ( X [1 , )in the polynomial ring S = K [ X ]. Then J is the ideal generated by the 3-minors of thefollowing highlighted ladder X = x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x . From Corollary 2.4, we get that set of 3-minors that generate J is a Gr¨obner basis of J with respect to any diagonal term order. Actually, this result was already known forladder determinantal ideals (see [Na, Corollary 3.4]). Nonetheless, Corollary 2.4 can beapplied to more general sums of ideals. Consider for instance the ideal J = I ( X [12] ) + I ( X [12] ) + I ( X [56] ) + I ( X [56] )that is the ideal generated by the 2-minors inside the below coloured region of XX = x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x . In this case, J is not a ladder determinantal ideal but we can use Corollary 2.4 to provethat the 2-minors that generate J form a Gr¨obner basis for the ideal J with respect toany diagonal term order. In fact, J is a sum of ideals of the form I t ( X [ a,b ] ) or I t ( X [ c,d ] )which are Knutson ideals from Theorem 2.1. Then a Gr¨obner basis for J is given by theunion of their Gr¨obner bases.Furthermore, we can also consider sums of ideals of minors of different sizes, such as J = I ( X [2 , ) + I ( X [2 , ) . n this case, J is generated by the 2-minors of the blue rectangular submatrix and the3-minors of the red rectangular submatrix illustrated below X = x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x . Again from Corollary 2.4, being I ( X [2 , ) and I ( X [2 , ) Knutson ideals, the union of theirGr¨obner bases is a Gr¨obner basis for J . So, a Gr¨obner basis of J is given by the 2-minorsof X [2 , and the 3-minors of X [2 , .Unlike in the case of Hankel matrices, a characterization of all the ideals belonging tothe family C f has not been found yet. A first step towards this result would be to under-stand the primary decompostion of certain sums belonging to the family. Some knownresults (see [HS], [MR]) suggest us what these primary decompositions might be and com-puter experiments seem to confirm this guess. Finding this characterization could lead tointeresting properties on the Gr¨obner bases of determinantal-like ideals and it would alsoanswer to the question asked by F.Mohammadi and J. Rhau in [MR]. Acknowledgments . The author is grateful to her advisor Matteo Varbaro for hisvaluable comments and precious advice.2.
Knutson ideals and determinantal ideals of generic matrices
Let m, n be two positive integers with m < n , we will denote by X mn the generic matrixof size m × n with entries x ij , that is X mn = x x x . . . x n x x x . . . x n x x x . . . x n ... ... ... . . . ... x m x m x m . . . x mn . Moreover, for any 1 ≤ i < j ≤ n and 1 ≤ k < l ≤ m , we denote by X [ k,l ][ i,j ] the submatrixof X mn with column indices i, i + 1 , . . . , j and row indices k, k + 1 , . . . , l . In the case[ k, l ] = [1 , m ], we omit the superscript and we simply write X [ i,j ] .Given a generic matrix X mn and an integer t ≤ min( m, n ), we will denote by I t ( X ) thedeterminantal ideal in S = K [ X ] = K [ x i,j | ≤ i ≤ m, ≤ j ≤ n ] generated by all the t -minors of X .We are going to prove that determinantal ideals of a generic matrix are Knutson idealsfor a suitable choice of the polynomial f . heorem 2.1. Let X = X mn be the generic matrix of size m × n with entries x ij and m < n . Consider the polynomial f = m − Y k =0 (cid:16) det X [ m − k,m ][1 ,k +1] · det X [1 ,k +1][ n − k,n ] (cid:17) · n − m +1 Y k =1 (cid:0) det X [ k,m + k − (cid:1) in S = K [ x ij | ≤ i ≤ m, ≤ j ≤ n ] . Then I t ( X ) ∈ C f for t = 1 , . . . , m .Furthermore, I t ( X [ a,b ] ) (respectively, I t ( X [ a,b ] ) ) are Knutson ideals associated to f for t = 1 , . . . , m and ≤ a < b ≤ n (respectively, ≤ a < b ≤ m ) with b − a + 1 ≥ t . A first step towards the proof of Theorem 2.1 is showing that all the ideals generatedby the t -minors on t adjacent columns are in C f . This fact is formally stated in the lemmabelow. Lemma 2.2.
Let X = X m × n be the generic square matrix of size m × n with entries x ij and let f to be as in Theorem 2.1. If we fix t ≤ m , then: I t ( X [ i,t + i − ) ∈ C f ∀ i = 1 , . . . , n − t + 1 . Proof.
It is known (e.g. see [Co] and [BC]) that every determinanatal ideal of a genericmatrix X = X m × n is prime and its height is given by the following formula:(1) ht( I t ( X )) = ( n − t + 1)( m − t + 1) . Thereforeht( I t ( X [ i,t + i − )) = ( t + i − − i + 1 − t + 1)( m − t + 1) = m − t + 1 . We have three possibilities for i . : m − t + 1 ≤ i ≤ n − m + 1. Then I t ( X [ i,t + i − ) ⊇ (cid:16) det X [ i,i + m − , det X [ i − ,i + m − , . . . , det X [ i − m + t,i + t − (cid:17) . : i ≤ m − tI t ( X [ i,t + i − ) ⊇ (cid:16) det X [1 ,m ] , det X [2 ,m +1] , . . . , det X [ i,i + m − , det X [ m − t − i +2 ,m ][1 ,t + i +1] , det X [ m − t − i +1 ,m ][1 ,t + i ] . . . , det X [2 ,m ][1 ,m − (cid:17) . : i ≥ n − m + 2 I t ( X [ i,t + i − ) ⊇ (cid:16) det X [ n − m +1 ,n ] , det X [ n − m,n − , . . . , det X [ t + i − m,t + i − , det X [1 ,n − i +1][ i,n ] , det X [1 ,n − i +2][ i − ,n ] . . . , det X [1 ,m − n − m +2 ,n ] (cid:17) . Define H to be the right hand side ideal for each of the previous cases. Note that theinitial ideal of H is given by some of the diagonals of the matrix X . Since these monomialsare coprime, this ideal is a complete intersection andht( H ) = m − t + 1in each of the above mentioned cases. So I t ( X [ i,t + i − ) is minimal over H .By Definition 1, ( f ) : J ∈ C f for every ideal J ⊆ S . Taking J to be the principal idealgenerated by the product of some of the factors of f , we have that all the principal ideals enerated by one of the factors of f are Knutson ideal associated to f . Being H a sum ofthese ideals, H ∈ C f .In conclusion, we get that I t ( X [ i,t + i − ) is a minimal prime over an ideal of C f . So it is in C f . (cid:3) Using Lemma 2.2, we can then prove Theorem 2.1.
Proof.
Fix t ∈ { , . . . , m } . We want to prove that I t ( X ) ∈ C f . By lemma 2.2, we knowthat I t ( X [1 ,t ] ) , I t ( X [2 ,t +1] ) ∈ C f and so their sum.We claim that that the minimal prime decomposition of the sum is given by I t ( X [1 ,t ] ) + I t ( X [2 ,t +1] ) = I t ( X [1 ,t +1] ) ∩ I t − ( X [2 ,t ] ) . To simplify the notation, we set I := I t ( X [1 ,t ] ) , I := I t ( X [2 ,t +1] ) , P = I t ( X [1 ,t +1] ) and P = I t − ( X [2 ,t ] ). We want to prove that the minimal prime decomposition is given by: I + I = P ∩ P . We already know that I + I ⊆ P ∩ P . Passing to the correspondent algebraic varieties,we get the reverse inclusion V ( I + I ) ⊇ V ( P ∩ P )If we prove that V ( I + I ) ⊆ V ( P ∩ P ), then V ( I + I ) = V ( P ∩ P )and this is equivalent to say that √ I + I = √ P ∩ P . Since I + I ∈ C f , it is radicaland P ∩ P is radical because P and P are both radical ideals, then I + I = P ∩ P and we are done.For this aim, let X ∈ V ( I + I ) = V ( I ) ∩ V ( I ). This means that X [1 ,t ] and X [2 ,t +1] haverank less or equal than t −
1. Now we consider two cases:
Case 1.
Suppose that X [2 ,t ] has rank less or equal than t −
2. This implies that all the( t − × ( t − X . So X ∈ V ( P ). Case 2.
Suppose that X [2 ,t ] has full rank, namely t −
1. Then it generates a vectorspace V of dimension t −
1. But by assumption, X [1 ,t ] and X [2 ,t +1] have rank less or equalthan t −
1, so they also generate the vector space V . Consequently, X [1 ,t +1] generates thevector space V and this means that all the t × t - minors of our matrix X vanish on X .Therefore we have proved that X ∈ V ( P ).This proves the claim and shows that I t ( X [1 ,t +1] ) ∈ C f , being a minimal prime over aKnutson ideal.In the same way, simply shifting the submatrices, we get that I t ( X [ k,t + k ] ) ∈ C f for every k = 1 , . . . , n − t .In particular I t ( X [2 ,t +2] ) ∈ C f ; therefore the sum I t ( X [1 ,t +1] ) + I t ( X [2 ,t +2] ) belongs to C f .Using a similar argument to that used to prove the claim, it can be shown that theprimary decomposition of the latter sum is given by I t ( X [1 ,t +1] ) + I t ( X [2 ,t +2] ) = I t ( X [1 ,t +2] ) ∩ I t − ( X [2 ,t +1] ) . Therefore I t ( X [1 ,t +2] ) is a Knutson ideal associated to f . gain, shifting the submatrices, the same argument shows that I t ( X [ k,t + k +1] ) ∈ C f forevery k = 1 , . . . , n − t − I t ( X [ a,b ] ) ∈ C f for every 1 ≤ a < b ≤ n such that b − a + 1 ≥ t . In particular, I t ( X [1 ,n − ) , I t ( X [2 ,n ] ) ∈ C f . Hence their sum belongs to C f .Again, one can show that the primary decomposition of the sum is given by I t ( X [1 ,n − ) + I t ( X [2 ,n ] ) = I t ( X [1 ,n ] ) ∩ I t − ( X [2 ,n − ) . This shows that I t ( X [1 ,n ] ) ∈ C f and we are done.Notice that an identical proof shows that I t ( X [ a,b ] ) ∈ C f for every 0 ≤ a < b ≤ m with b − a + 1 ≥ t . (cid:3) As an immediate consequence of the previous theorem, we get an alternative proof of F -purity of determinantal ideals of generic matrices. Corollary 2.3.
Assume that K is a field of characteristic p and let X be a generic matrixof size m × n . Then S/I t ( X ) is F-pure.Proof. We may assume that K is a perfect field of positive characteristic. In fact, we canalways reduce to this case by tensoring with the algebraic closure of K and the F -purityproperty descends to the non-perfect case. Using Lemma 4 in [Kn], we know that the ideal( f ) is compatibly split with respect to the Frobenius splitting defined by Tr( f p − • ) (where f is taken to be as in the previous theorems). Thus all the ideals belonging to C f arecompatibly split with respect to the same splitting, in particular I t ( X ). This implies thatsuch Frobenius splitting of S provides a Frobenius splitting of S/I t ( X ). Being S/I t ( X ) F -split, it must be also F -pure. (cid:3) Furthermore, we obtain an interesting and useful result about Gr¨obner bases of certainsums of determinantal ideals.
Corollary 2.4.
Let X be a generic matrix of size m × n and let I be a sum of ideals, say I = I + I + . . . + I k , where each I i is of the form either I t i ( X [ a i ,b i ] ) or I t i ( X [ a i ,b i ] ) . Then G I = G I ∪ G I ∪ . . . ∪ G I k where G J denotes a Gr¨obner basis of the ideal J .Furthermore, if K has positive characteristic, I is also F -pure.Proof. By Theorem 2.1, we know that I t i ( X [ a i ,b i ] ) and I t i ( X [ a i ,b i ] ) are Knutson ideals.From property ( iii ) of Knutson ideals, we get the thesis. (cid:3) References [BC] W.Bruns, A.Conca,
Algebra of minors , Journal of Algebra 246, 311–330, 2001[BH] W.Bruns, J. Herzog,
Cohen-Macaulay rings , Cambridge University Press, 1993.[CMSV] A.Conca, M. Mostafazadehfard, A. K. Singh, M.Varbaro,
Hankel determinantal rings have ra-tional singularities , Adv. Math. 335 (2018), 111–129.[Co] A. Conca,
Straightening law and powers of determinantal ideals of Hankel matrices , Adv. Math. 138(1998), no. 2, 263–292.[CV] A.Conca, M.Varbaro,
Squarefree Grobner degeneration , arXiv:1805.11923, 2018.[HH] M.Hochster, C.Huneke,
Tight closure of parameter ideals and splitting in module-finite extensions ,Journal of Algebraic Geometry 3 (4), p. 599-670, 1994. HS] S.Hosten, S.Sullivant
Ideals of adjacent minors , J.Algebra vol.277,no.2, 2004.[Kn] A.Knutson,
Frobenius splitting, point-counting, and degeneration , arXiv:0911.4941, 2009.[MR] F.Mohammadi, J.Rauh,
Prime splitting of determinantal ideals , Communications in Algebra, 46:5,p. 2278-2296, 2018.[Na] H.Narasimhan,
The irreducibility of ladder determinantal varieties , J. Algebra 102, p. 162-185,(1986).[Se] L.Seccia,
Knutson ideals and determinantal ideals of Hankel matrices , arXiv:2003.14232 , 2020.
Universit`a di Genova, Dipartimento di Matematica. Via Dodecaneso 35, 16146 Genova,Italy
Email address : [email protected]@dima.unige.it