On the -rationality and cohomological properties of matrix Schubert varieties
We characterize complete intersection matrix Schubert varieties, generalizing the classical result on one-sided ladder determinantal varieties. We also give a new proof of the -rationality of matrix Schubert varieties. Although it is known that such varieties are -regular (hence -rational) by the global -regularity of Schubert varieties, our proof is of independent interest since it does not require the Bott–Samelson resolution of Schubert varieties. As a consequence, this provides an alternative proof of the classical fact that Schubert varieties in flag varieties are normal and have rational singularities.
2010 Mathematics Subject Classification:13C40, 14M15, 14M10, 05E40, 13A35
Matrix Schubert varieties (MSVs) were introduced by W. Fulton in his theory of degeneracy loci of maps of flagged vector bundles . Such varieties are reduced and irreducible. Classical (one-sided ladder) determinantal varieties are special examples of MSVs (they are so-called vexillary MSVs). Just like one-sided ladder determinantal varieties , , MSVs can be identified (up to product of an affine space) as the opposite big cells of the corresponding Schubert varieties. This observation in  implies that the MSVs are normal and Cohen–Macaulay, since Schubert varieties are (see ).
The Cohen–Macaulay property of MSVs was re-established by A. Knutson and E. Miller  using the Gröbner basis theory, pipe dreams, and their theory of subword complexes. Interestingly, this gives a new proof of the Cohen–Macaulayness of Schubert varieties by the following principle.
Theorem 1.1 ( Theorem 2.4.3).
Let be a local condition that holds for a variety whenever it holds for the product of with any vector space. Then holds for every Schubert variety in every flag variety if and only if holds for all MSVs.
1.1. -rationality of MSVs
In the same spirit, the first part of this paper is devoted to a new proof of -rationality of MSVs. -rationality is a notion that arises from the theory of tight closure introduced by M. Hochster and C. Huneke  in positive characteristic. The results of  and  establish a connection between -rationality and the notion of rational singularity in characteristic : A normal variety in characteristic has at most rational singularities if and only if it is of -rational type. Therefore, by Theorem 1.1 the -rationality of matrix Schubert varieties is equivalent to the classical fact that Schubert varieties are normal and have at most rational singularities (see e.g.  and  for the classical proofs of the later statement using the Bott–Samelson resolution).
Two other notions in tight closure theory will also be used later: -regularity and -injectivity. The relation between these properties is
Our proof of -rationality of MSVs utilizes the results of Schubert determinantal ideals in  as well as the techniques developed in , where A. Conca and J. Herzog prove that arbitrary (possibly two-sided) ladder determinant varieties are -rational. However, it is still unknown whether such varieties are -regular.
One of the key ingredients in our proof is the following
Theorem 1.2 ( Theorem 1.2).
Let be a complete local Cohen–Macaulay ring and be a non zero-divisor of such that is -rational and is -injective. Then is -rational.
After recalling several known facts in the theory of tight closure (section 4.4), we will see that the most essential step is to find such that is -rational and that the initial ideal of is Cohen–Macaulay (where is any antidiagonal term order, and is the coordinate ring and the defining ideal of the MSV associated to the partial permutation as defined in section 2). This goal is achieved by choosing where is the smallest number such that See section 2 for unexplained notation.
1.2. Complete intersection MSVs
Since MSVs are Cohen–Macaulay, it is then natural to ask when such varieties are smooth, complete intersection, or Gorenstein. Classically, characterizations of Gorenstein ladder determinantal varieties are obtained in , , and . In one-sided cases, the characterization can be generalized as the following. Recall that there exists a characterization of smooth (respectively, Gorenstein) Schubert varieties  (respectively, ). Since the singular (respectively, non-Gorenstein) locus of a Schubert variety is closed and invariant under the Borel subgroup action, the opposite big cell must be contained in the singular (respectively, non-Gorenstein) locus. Hence, a Schubert variety is smooth (respectively, Gorenstein) if and only if its corresponding MSV is so. Therefore, one can deduce a criterion of smooth (respectively, Gorenstein) MSVs by the corresponding result for Schubert varieties. See [20, section 3.5] for more details.
Theorem (Theorem 5.2).
The matrix Schubert variety associated to a permutation is a complete intersection if and only if, for any in the diagram of with , i.e. , is a permutation matrix in such that is a complete intersection. Here, is the connected (solid) square submatrix of size whose southeast corner lies at . In fact, in this case
is a set of generators of with cardinality , the codimension of in , where is the connected (solid) square submatrix of size whose southeast corner lies at .
Theorem 5.2 generalizes a result in  for one-sided ladder determinantal varieties. The proof of Theorem 5.2 uses Nakayama’s lemma and the properties of Schubert determinantal ideals established in .
This paper is organized as follows. We will recall some preliminary facts about matrix Schubert varieties as well as tight closure theory in section 2 and 4.4, respectively. The proof of -rationality of MSVs is in section 4. Section 5 is devoted to the characterization of complete intersection MSVs.
The author would like to thank A. Knutson, E. Miller, K. Smith, U. Walther, and A. Woo for their comments and suggestions about this work. Special thanks go to the referee for carefully reading this paper and many useful suggestions on the presentation of the results
2. Matrix Schubert varieties
Denote the space of matrices over a field . An matrix is called a partial permutation if all entries of are equal to except for at most one entry equal to in each row and column. If and , then is called a permutation. An element in the permutation group will be identified as a permutation matrix (also denoted by ) in via
Let be the coordinate ring of where is the generic matrix of variables. For a matrix , denote the upper-left submatrix of . Similarly, denotes the upper-left submatrix of . The rank of will be denoted by .
Given a partial permutation , the matrix Schubert variety is the subvariety
in . The classical (one-sided ladder) determinantal varieties are special examples of MSVs.
It is known that MSVs are reduced and irreducible. Denote
the coordinate ring of . The defining ideal of (called Schubert determinantal ideal) is generated by all minors of size in . One can reduce the generating set of as the following. Consider the diagram of
i.e. consists of elements that are neither due east nor due south of a nonzero entry of . The essential set of is defined to be
One can check that (see [7, Lemma 3.10])
Also, the codimension of in is the cardinality of which is actually the Coxeter length of when is a permutation. We often need to consider certain subsets of or . For that, we will put the conditions as subscripts to indicate the constraints. For examples, and .
Questions on for a partial permutation is often reduced to the cases where is a permutation. More precisely, extend to the permutation via
Then , , and the defining ideals and share the same set of generators. Therefore,
The following substantial results due to A. Knutson and E. Miller is indispensable in the proofs of our main theorems. Recall that a term order on is called antidiagonal if the initial term of every minor of is its antidiagonal term. We will fix an antidiagonal term order and simply write , as the initial ideal of an ideal and the leading term of an element , respectively. We will call an antidiagonal term of a minor of size an antidiagonal of size .
3. -rationality and -injectivity
Recall that in the theory of tight closure, a Noetherian ring is -rational if all its parameter ideals are tightly closed. There is a weaker notion called -injectivity. A Noetherian ring is -injective if for any maximal ideal of the map on the local cohomology module induced by the Frobenius map is injective for all . We collect some facts concerning -rationality and -injectivity. See , or  for convenient resources.
Let be a Noetherian ring.
is -rational if and only if is -rational for any maximal ideal .
If is an -rational ring that is a homorphic image of a Cohen–Macaulay ring, then is -rational for any multiplicative closed set of .
Let be a Noetherian local ring.
is -injective if and only if is -injective.
Suppose in addition is excellent, then is -rational if and only if is -rational.
Let be a positive graded -algebra, where is a field of positive characteristic. Let be the unique maximal graded ideal of .
is -injective if and only if is -injective.
is -rational if and only if is -rational for any variable over .
Suppose in addition that is perfect. Then is -rational if and only if is -rational.
4. Matrix Schubert varties are -rational
Fix an antidiagonal term order. Denote the initial ideal of . In this section, the ground field is perfect and of positive characteristic. As mentioned in the introduction, consider where is the smallest number such that Note that such exists exactly when (or equivalently is not regular). We make this assumption (existence of ) for Lemmas 4.1, 4.2, 4.3 and set . Note also that for this particular choice of ,
In particular, the only nonzero entry in is .
In the following, we use the notation to denote the size minor of the submatrix of involving the rows of indices and the columns of indices .
Let be any minor in such that . Then and hence so is .
Write , so
where and .
Use induction on . When , then . If , is obviously in . Suppose , Expanding along the first row, we get
Since by (4.1), and hence .
Suppose . For , set
Note that for , since is on the antidiagonal of (the row deleted is above and the column deleted is to the right of ). So by the inductive hypothesis. Again, expanding along the first row, we see that
Once again , since by (4.1). Therefore, as desired. ∎
The containment is obvious. Conversely, let for some and . If , then or . In either case,
So we may assume that . Write where the ’s are monomial elements in and the ’s are minors in the generating set of . If is a term in , then . Also, if for some , then . Therefore, we may assume that is a term of for some and that is neither nor a term of . This implies that is a term of and hence . If , then since it is a term of . Otherwise, . Then by Lemma 4.1, . Therefore, . Now, since is a term of and since is a monomial ideal, we conclude that . ∎
By Theorem 2.1 in , it suffices to show that is Cohen–Macaulay and -injective. By Lemma 4.2, Also, by Theorem 2.1(2) is a Cohen–Macaulay square-free monomial ideal. So is also a square-free monomial ideal, and hence is -injective by the discussion in the paragraph before Corollary of  involving Fedder’s criterion. The Cohen–Macaulayness of follows from that fact the is a non zero-divisor on .
To see this, first note that . Suppose for some we have . We will show that . By Theorem 2.1, we may assume is a monomial and for some monomial and some antigonal . If , then . Therefore, we may assume . Then . We finish the proof by showing that .
where and . Since , either or . Note also that is of size , so by Theorem 2.1(1). On the other hand, as mentioned before the only nonzero entry in is , so
Therefore, and hence either or . We conclude that either or , so . ∎
By (2.2) and Theorem 3.3(2), we may assume that is a permutation. Use induction on . If is regular (this includes the cases ), then it is -rational. Suppose and is not regular. Then the element described as above exists. By Lemma 4.3, is -injective. Hence, is -injective by Theorem 3.2(1) and 3.3(1). So by Theorem 1.2, 3.2(2), and 3.3(3), it suffices to show that is -rational.
For , consider the change of variables
Set . Then
in the field of fraction of .
Let be the permutation obtained by deleting the th row and the th column of and let be the corresponding Schubert determinantal ideal in . Denote the extended ideal of and set
We claim that
It follows from (4.2) that
We prove (4.2) by showing that the generators of belongs to and conversely. First observe that
For satisfying or , by (4.1) .
Fix a satisfying or . Let be an -minor () of that does not involve the th row and the th column. Denote the corresponding -minor in (replace by ). Then direct computation shows
By (a), , so we see that .
Let be any -minor in that involves but does not involve . Then where is obtained from by replacing by .
Let be any -minor in that does not involve and let the corresponding -minor in (replace by ). Denote and the -minors obtained by adding the th row and th column to and respectively. Then
Now, we are ready to prove (4.2), .
We first show that . Fix . We need to show that the following set
is contained in . Consider the following cases.
. We must have or .
If , then by construction.
If , then and hence . Therefore, , since either or is in .
, say . Let be any -minor in . In this case, by (4.1).
. In this case, . If involves the th row, expanding along this row we see that . Otherwise, let be the corresponding -minor in and we have by (b). But implies . So .
. In this case, .
If involves , then by (d), where is the -minor obtained from by deleting the row and the column involving . But implies . So .
If involves but does not involve , then by (c) where is obtained from by replacing by . Expanding along the row or column involving , we see that . But implies . So .
If does not involve , then by (d). Expanding the -minor along the row and the column involving , we see that . Again, since . So .
Conversely, we show that . Fix . Again, we show that the set
is contained in .
If or , then and hence . By (a), either or is in . Therefore,
If and , then . Hence, the -minor . So .
. In this case, by (4.1). Suppose and let be any -minor in .
If , then . By (d), . Expanding along the th column, we see . But implies . So .
If , then . Again, by (d). This time since . Therefore, .
Consider in . Use the same notation as in the proof of Theorem 4.4. We have , and . Check that the following elements lie in and hence in :
Therefore, we see that and .
In the following diagram, the ’s indicate the permutation and the dots indicate the elements in .
5. Complete intersection matrix Schubert varieties
We want to characterize the complete intersection MSVs. By (2.2), we may assume . Denote the