As a consequence, this provides an alternative proof of the classical fact that Schubert varieties in flag varieties are normal and have rational singularities

S 0019-2082

ON THE F -RATIONALITY AND COHOMOLOGICAL PROPERTIES OF MATRIX SCHUBERT VARIETIES

JEN-CHIEH HSIAO

Abstract. We characterize complete intersection matrix Schu- bert varieties, generalizing the classical result on one-sided lad- der determinantal varieties. We also give a new proof of the F -rationality of matrix Schubert varieties. Although it is known that such varieties are F -regular (hence F -rational) by the global F -regularity of Schubert varieties, our proof is of independent in- terest 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.

1. Introduction

Matrix Schubert varieties (MSVs) were introduced by W. Fulton in his theory of degeneracy loci of maps of flagged vector bundles [Ful92]. Such va- rieties 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 [GL00], [GM00], MSVs can be identified (up to product of an affine space) as the opposite big cells of the corresponding Schubert varieties. This observation in [Ful92] implies that the MSVs are normal and Cohen–Macaulay, since Schubert varieties are (see [Ram85]).

The Cohen–Macaulay property of MSVs was re-established by A. Knutson and E. Miller [KM05] using the Gr¨obner 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.

Received September 1, 2011; received in final form September 8, 2013.

The author was partially supported by NSF under Grant DMS 0901123.

2010 Mathematics Subject Classification.13C40, 14M15, 14M10, 05E40, 13A35.

1

2014 University of Illinoisc

Theorem 1.1 ([KM05, Theorem 2.4.3]). Let C be a local condition that holds for a variety X whenever it holds for the product of X with any vector space. Then C holds for every Schubert variety in every flag variety if and only ifC holds for all MSVs.

1.1. F -rationality of MSVs. In the same spirit, the first part of this paper is devoted to a new proof of F -rationality of MSVs. F -rationality is a notion that arises from the theory of tight closure introduced by M. Hochster and C. Huneke [Hun96] in positive characteristic. The results of [Smi97]

and [Har98] establish a connection between F -rationality and the notion of rational singularity in characteristic 0: A normal variety in characteristic 0 has at most rational singularities if and only if it is of F -rational type. Therefore, by Theorem1.1the F -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., [Bri05] and [BK05] 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: F - regularity and F -injectivity. The relation between these properties is

regular =⇒ F -regular =⇒ F -rational =⇒ F -injective.

We remark that MSVs are in fact F -regular by Theorem1.1 and the global F -regularity of Schubert varieties [LRPT06] (again, this relies on the Bott–

Samelson resolution).

Our proof of F -rationality of MSVs utilizes the results of Schubert determi- nantal ideals in [KM05] as well as the techniques developed in [CH97], where A. Conca and J. Herzog prove that arbitrary (possibly two-sided) ladder de- terminant varieties are F -rational. However, it is still unknown whether such varieties are F -regular.

One of the key ingredients in our proof is the following theorem.

Theorem 1.2 ([CH97, Theorem 1.2]). Let R be a complete local Cohen–

Macaulay ring and c be a nonzero-divisor of R such that R[¹_c] is F -rational and R/cR is F -injective. Then R is F -rational.

After recalling several known facts in the theory of tight closure (Sec- tion3), we will see that the most essential step is to find c such that Rw[¹_c] is F -rational and that the initial ideal in<(c + Iw) of c + Iw is Cohen–

Macaulay (where < is any antidiagonal term order, Rw and Iw is the co- ordinate ring and the defining ideal of the MSV associated to the partial permutation w as defined in Section 2). This goal is achieved by choosing c = xi0,w(io) where i0 is the smallest number such that {(p, q) | p > i0, q >

w(i0)} ∩ E>0(w)= ∅. See Section2for 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 determi- nantal varieties are obtained in [Con95], [Con96] and [GM00]. In one-sided cases, the characterization can be generalized as the following. Recall that there exists a characterization of smooth (respectively, Gorenstein) Schubert varieties [LS90] (respectively, [WY06]). 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 (re- spectively, 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 [WY06, Section 3.5] for more details.

The second goal of this paper is to characterize complete intersection MSVs.

We explain the characterization as the following. See Sections 2 and 5 for unexplained notation and more details.

Theorem (Theorem 5.2). The matrix Schubert variety Xw associated to a permutation w is a complete intersection if and only if, for any (p, q) in the diagram of w with rp,q(w) > 0, that is, (p, q)∈ D>0(w), w(p,q)is a permutation matrix in GLrp,q(w) such that Xw(p,q) is a complete intersection. Here, w(p,q)

is the connected (solid) square submatrix of size r_p,q(w) whose southeast corner lies at (p− 1, q − 1). In fact, in this case

xp,q| (p, q) ∈ D=0(w) 

det X_(p,q)| (p, q) ∈ D>0(w)

is a set of generators of Iw with cardinality|D(w)|, the codimension of Xwin M_n×n, where X_(p,q)is the connected (solid) square submatrix of size rp,q(w) + 1 whose southeast corner lies at (p, q).

Theorem 5.2 generalizes a result in [GS95] for one-sided ladder determi- nantal varieties. The proof of Theorem 5.2uses Nakayama’s lemma and the properties of Schubert determinantal ideals established in [KM05].

After this work is finished, A. Woo and H. Ulfarsson give a criterion of lo- cally complete intersection Schubert varieties. Theorem5.2may be recovered by their criterion (see [UW13, Corollary 6.3] and the comment after that).´ 1.3. This paper is organized as follows. We will recall some preliminary facts about matrix Schubert varieties as well as tight closure theory in Sec- tions2and3, respectively. The proof of F -rationality of MSVs is in Section4.

Section5 is devoted to the characterization of complete intersection MSVs.

2. Matrix Schubert varieties

We recall some fundamental facts about matrix Schubert varieties (see [Ful92], [KM05] and [MS05] for more information).

Denote M_l×m the space of l× m matrices over a field K. An l × m matrix w∈ Ml×m is called a partial permutation if all entries of w are equal to 0 except for at most one entry equal to 1 in each row and column. If l = m and w∈ GLl, then w is called a permutation. An element w in the permutation group Sn will be identified as a permutation matrix (also denoted by w) in GLn via

w_i,j=



1 if w(i) = j, 0 otherwise.

Let K[X] be the coordinate ring of M_l×mwhere X = (xi,j) is the generic l×m matrix of variables. For a matrix Z∈ Ml×m, denote Z_[p,q] the upper-left p× q submatrix of Z. Similarly, X_[p,q] denotes the upper-left p× q submatrix of X.

The rank of Z_[p,q] will be denoted by rank(Z_[p,q]) := rp,q(Z).

Given a partial permutation w∈ Ml×m, the matrix Schubert variety Xwis the subvariety

Xw=

Z∈ Ml×m| rp,q(Z)≤ rp,q(w) for all p, q

in M_l×m. The classical (one-sided ladder) determinantal varieties are special examples of MSVs.

It is known that MSVs are reduced and irreducible. Denote Rw= K[Xw] = K[X]/Iw

the coordinate ring of X_w. The defining ideal I_w of X_w (called Schubert determinantal ideal ) is generated by all minors of size r_p,q(w) + 1 in X_[p,q]. One can reduce the generating set of I_was the following. Consider the diagram of w

D(w) =

(i, j)∈ [1, l] × [1, m] : w(i) > j and w⁻¹(j) > i ,

that is, D(w) consists of elements that are neither due east nor due south of a nonzero entry of w. The essential set of w is defined to be

E(w) =

(p, q)∈ D(w) : (p + 1, q) /∈ D(w) and (p, q + 1) /∈ D(w) . One can check that (see [Ful92, Lemma 3.10])

Iw=

minors of size rp,q(w) + 1 in X[p,q]: (p, q)∈ D(w) (2.1)

=

minors of size rp,q(w) + 1 in X[p,q]: (p, q)∈ E(w) .

Also, the codimension of X_win M_l×mis the cardinality|D(w)| of D(w) which is actually the Coxeter length of w when w is a permutation. We often need to consider certain subsets of D(w) orE(w). For that, we will put the conditions as subscripts to indicate the constraints. For examples, D=0(w) ={(p, q) ∈ D(w)| rp,q(w) = 0} and D>0(w) ={(p, q) ∈ D(w) | rp,q(w) > 0}.

Questions on Xw for a partial permutation w∈ Ml×m is often reduced to the cases where w is a permutation. More precisely, extend w to the

permutationw∈ Sn, n = l + m via

 w(i) =

⎧⎪

⎨

⎪⎩

j if wi,j= 1 for some j,

min{[m + 1, n] \ { w(1), . . . ,w(i − 1)}} if w_i,j= 0 for all j, min{[1, n] \ { w(1), . . . ,w(i − 1)}} if i > l.

Then D(w) = D(w), E(w) = E( w), and the defining ideals I w and Iw_ share the same set of generators. Therefore,

(2.2) X_w∼= Xw× K^n2−lm.

The following substantial results due to A. Knutson and E. Miller is in- dispensable in the proofs of our main theorems. Recall that a term order on K[X] is called antidiagonal if the initial term of every minor of X is its antidiagonal term. We will fix an antidiagonal term order < and simply write in(I), in(f ) as the initial ideal of an ideal I and the leading term of an ele- ment f , respectively. We will call an antidiagonal term of a minor of size r an antidiagonal of size r.

Theorem 2.1 ([KM05]). Fix any antidiagonal term order. Then (1) The generators of Iw in (2.1) constitute a Gr¨obner basis, that is,

in(I_w) =

antidiagonals of size r_p,q(w) + 1 in X_[p,q]: (p, q)∈ E(w)

; (2) in(Iw) is a Cohen–Macaulay square-free monomial ideal.

3. F -rationality and F -injectivity

Recall that in the theory of tight closure, a Noetherian ring is F -rational if all its parameter ideals are tightly closed. There is a weaker notion called F -injectivity. A Noetherian ring R is F -injective if for any maximal ideal m of R the map on the local cohomology module H_mⁱ(R) induced by the Frobenius map is injective for all i. We collect some facts concerning F -rationality and F -injectivity. See [Hun96], or [BH93] for convenient resources.

Theorem 3.1. Let R be a Noetherian ring.

(1) R is F -rational if and only if R_m is F -rational for any maximal ideal m.

(2) If R is an F -rational ring that is a homorphic image of a Cohen–Macaulay ring, then R_S is F -rational for any multiplicative closed set S of R.

Theorem 3.2. Let (R, m) be a Noetherian local ring.

(1) R is F -injective if and only if R is F -injective.

(2) Suppose in addition R is excellent, then R is F -rational if and only if R is F -rational.

Theorem3.3. Let R be a positive graded K-algebra, where K is a field of positive characteristic. Let m be the unique maximal graded ideal of R.

(1) R is F -injective if and only if Rm is F -injective.

(2) R is F -rational if and only if R[T ] is F -rational for any variable T over R.

(3) Suppose in addition that K is perfect. Then R is F -rational if and only if Rm is F -rational.

4. Matrix Schubert varties are F -rational

Fix an antidiagonal term order. Denote Jw= in(Iw) the initial ideal of Iw. In this section, the ground field K is perfect and of positive characteristic. As mentioned in theIntroduction, consider c := x_i0,w(i0)where i0 is the smallest number such that{(p, q) | p > i0, q > w(i0)} ∩ E>0(w)= ∅. Note that such i0

exists exactly whenE>0(w)= ∅ (or equivalently Rw is not regular). We make this assumption (existence of c) for Lemmas 4.1, 4.2, 4.3 and set (p0, q0) = (i0, w(i0)). Note also that for this particular choice of i0,

(p, q)= (p0, q₀)| p ≤ p0, q≤ q0

⊆ 

(p, q)| p ≤ p0

∩ D(w) (4.1)

⊆ D=0(w).

In particular, the only nonzero entry in w_[p0,q0] is (p0, q0).

In the following, we use the notation [p1, . . . , pt| q1, . . . , qt] to denote the size t minor of the submatrix of X involving the rows of indices p1, . . . , ptand the columns of indices q1, . . . , qt.

Lemma4.1. Let Δ be any minor in X such that c| in Δ. Then Δ ∈ c+Jw

and hence so is Δ− in Δ.

Proof. Write Δ = [p_t, . . . , p₁, p₀, p^₁, . . . , p^_s| qs^, . . . , q₁^, q₀, q₁, . . . , q_t], so in Δ =

 _s



i=1

x_p i,q_i^



· c ·

 _t



j=1

x_pj,qj

 ,

where p^_s>· · · > p^1> p0> p1>· · · > ptand q_s^<· · · < q1^ < q0< q1<· · · < qt. Use induction on t. When t = 0, then Δ = [p0, p^₁, . . . , p^_s| qs^, . . . , q₁^, q0]. If s = 0, Δ = c = in Δ is obviously in c + Jw. Suppose s > 0, expanding Δ along the first row, we get

Δ = (−1)^sc

p^₁, . . . , p^_s| q_s^, . . . , q₁^ +

s i=1

(−1)ⁱ⁺¹x_p0,qi^

p^₁, . . . , p^_s| q^_s, . . . , q^_i, . . . , q₁^, q0

.

Since 0 = r_p0,q^₁(w) =· · · = rp0,q_s^(w) by (4.1), xp0,q^₁, . . . , xp0,q_s^ ⊆ Jw and hence Δ∈ c + Jw.

Suppose t > 0. For 1≤ i ≤ t, set Δi=

p_t−1, . . . , p1, p0, p^₁, . . . , p^_s| q^s, . . . , q₁^, q0, q1, . . . ,qi, . . . , qt

. Note that c| in Δi for 1≤ i ≤ t, since c is on the antidiagonal of Δi (the row deleted is above c and the column deleted is to the right of c). SoΔi| 1 ≤

i≤ t ⊆ c + Jw by the inductive hypothesis. Again, expanding Δ along the first row, we see that

Δ∈ Δi| 1 ≤ i ≤ t + xpt,q_j^ | j = 1, . . . , s + xpt,q0.

Once again [xpt,q^_j | j = 1, . . . , s + xpt,q0] ⊆ Jw, since 0 = rpt,q^₁(w) =· · · = r_pt,q

s(w) = r_pt,q0(w) by (4.1). Therefore, Δ∈ c + Jw as desired.  Lemma 4.2. in(c + Iw) =c + Jw.

Proof. The containment in(c + Iw)⊇ c + Jw is obvious. Conversely, let cf− g ∈ c + Iw for some f∈ K[X] and g ∈ Iw. If in(cf )= in(g), then in(cf− g) = in(cf) or in(−g). In either case, in(cf − g) ∈ c + Jw.

So we may assume that in(cf ) = in(g). Write g = m₁Δ₁+ m₂Δ₂+· · · + m_uΔ_uwhere the m_i’s are monomial elements in K[X] and the Δ_i’s are minors in the generating set of I_w. If in(cf− g) is a term in cf, then in(cf − g) ∈ c.

Also, if in(cf− g) = in(miΔi) for some i, then in(cf− g) = miin(Δ˙ i)∈ Jw. Therefore, we may assume that in(cf− g) is a term of mi0Δi0 for some i0

and that in(cf− g) is neither in(mi0Δ_i0) nor a term of cf . This implies that in(mi0Δi0) is a term of cf and hence c| in(mi0Δi0) = mi0in(Δ˙ i0). If c| mi0, then in(cf− g) ∈ c since it is a term of mi0Δi0. Otherwise, c| in(Δi0). Then by Lemma 4.1, Δi0− in(Δi0)∈ c + Jw. Therefore, mi0Δi0− in(mi0Δi0)∈

c + Jw. Now, since in(cf− g) is a term of mi0Δ_i0− in(mi0Δ_i0) and since

c + Jw is a monomial ideal, we conclude that in(cf− g) ∈ c + Jw.  Lemma 4.3. R_w/cR_w is F -injective.

Proof. By Theorem 2.1 in [CH97], it suffices to show that K[X]/ in(c+Iw) is Cohen–Macaulay and F -injective. By Lemma4.2, in(c + Iw) =c + Jw. Also, by Theorem2.1(2) Jwis a Cohen–Macaulay square-free monomial ideal.

Soc + Jw is also a square-free monomial ideal, and hence K[X]/(c + Jw) is F -injective by the discussion in the paragraph before corollary of [CH97]

involving Fedder’s criterion. The Cohen–Macaulayness of K[X]/(c + Jw) follows from that fact the c is a nonzero-divisor on K[X]/J_w.

To see this, first note that c = x_i0,w(i0)= x_p0,q0 ∈ J/ w. Suppose for some z∈ K[X] we have cz ∈ Jw. We will show that z∈ Jw. By Theorem 2.1, we may assume z is a monomial and cz = rD for some monomial r∈ K[X] and some antigonal D∈ Jw. If c| r, then z =^r_cD∈ Jw. Therefore, we may assume c r. Then c | D. We finish the proof by showing that ^D_c ∈ Jw.

Write

D =

 _s



i=1

x_p^_i,qi^



· c ·

 _t



j=1

xpj,qj

 ,

where p1^> p^₂>· · · > p^_s> p0> p1>· · · > pt and q₁^ < q^₂<· · · < q_s^ < q0<

q1<· · · < qt. Since c = xp0,q0∈ J/ w, either s > 0 or t > 0. Note also that D is

of size (s + t + 1), so rp^₁,qt(w)≤ s + t by Theorem2.1(1). On the other hand, as mentioned before the only nonzero entry in w_[p,q] is (p0, q0), so

{nonzero entries in w[p^₁,qt]}

⊇

(p₀, q₀)

{nonzero entries in w[p1,qt]} {nonzero entries in w[p^₁,q^_s]}.

Therefore, 1 + rp1,qt(w) + r_p^_1,q

s(w)≤ rp^₁,qt≤ s+t and hence either rp1,qt(w) <

t or r_p^_1,q

s(w) < s. We conclude that either s

i=1x_p^_i,q^_i∈ Jw ort

i=1xpi,qi∈

Jw, so ^D_c ∈ Jw. 

Theorem 4.4. R_w= K[X]/I_w is F -rational.

Proof. By (2.2) and Theorem 3.3(2), we may assume that w∈ Sn is a permutation. Use induction on n. If Rw is regular (this includes the cases n = 1, 2), then it is F -rational. Suppose n > 2 and Rwis not regular. Then the element c = xp0,q0 described as above exists. By Lemma 4.3, Rw/cRw is F - injective. Hence, R_w/c R_w is F -injective by Theorem3.2(1) and3.3(1). So by Theorems1.2,3.2(2) and3.3(3), it suffices to show that R_w[¹_c] is F -rational.

For (p, q) /∈ Γ = {(p, q) | p = p0 or q = q₀}, consider the change of variables x^_p,q= xp,q− c⁻¹xp,q0xp0,q.

Set X^= (x^_p,q| (p, q) /∈ Γ). Then K[X]

1 c



= K X^

x_p,q| (p, q) ∈ Γ1 c



in the field of fraction of K[X].

Let w^ be the permutation obtained by deleting the p₀th row and the q₀th column of w and let I_w be the corresponding Schubert determinantal ideal in K[X^]. Denote I = I_w· K[X][¹_c] the extended ideal of I_wand set

I^= Iw^+

xp,q| (p, q) ∈ Γ, p < p0or q < q0

· K X^

xp,q| (p, q) ∈ Γ1 c

 . We claim that

(4.2) I = I^.

It follows from (4.2) that K[X]

I_w

1 c



=K[X^] I_w

x_p,q| (p, q) ∈ Γ, p ≥ p0 and q≥ q0

1 c

 .

By inductive hypothesis, K[X^]/I_w is F -rational. So Theorem 3.1(2) and Theorem3.3(2) imply that R_w[¹_c] is F -rational. Therefore, it suffices to prove (4.2).

We prove (4.2) by showing that the generators of I belongs to I^ and conversely. First observe that

(a) For (p, q)∈ Γ satisfying p < p0 or q < q0, by (4.1) xp,q∈ Iw.

(b) Fix a (p, q) satisfying p < p0 or q < q0. Let Δ be an r-minor (r≥ 1) of X[p,q] that does not involve the p0th row and the q0th column. Denote Δ^the corresponding r-minor in X^_[p,q](replace xp,qby x^_p,q). Then direct computation shows

Δ− Δ^∈

x_p,q| (p, q) ∈ Γ, p < p0 or q < q₀ .

By (a), xp,q| (p, q) ∈ Γ, p < p0 or q < q0 ⊆ I, so we see that Δ − Δ^∈ I∩ I^.

(c) Let Δ be any r-minor in X that involves Γ but does not involve c. Then Δ = Δ^ where Δ^ is obtained from Δ by replacing xp,q((p, q) /∈ Γ) by x^_p,q. (d) Let Δ be any r-minor in X that does not involve Γ and let Δ^ the cor- responding r-minor in X^ (replace x_p,q by x^_p,q). Denote Δ and Δ^ the (r + 1)-minors obtained by adding the p0th row and q0th column to Δ and Δ^ , respectively. Then

cΔ = Δ^ and cΔ^= Δ.

Now, we are ready to prove (4.2), I = I^.

We first show that I⊆ I^. Fix (p, q)∈ D=0(w)∪ E>0(w). We need to show that the following set

x_p,q| (p, q) ∈ D=0(w)

∪

 

(p,q)∈E>0(w)

r_p,q(w) + 1

-minors in X_[p,q]

is contained in I^. Consider the following cases.

(i) (p, q)∈ D=0(w). We must have p < p₀ or q < q₀. (i.1) If (p, q)∈ Γ, then xp,q∈ I^ by construction.

(i.2) If (p, q) /∈ Γ, then (p, q) ∈ D=0(w^) and hence x^_p,q∈ Iw^⊆ I^. There- fore, xp,q= x^_p,q+ c⁻¹xp0,qxp,q0∈ I^, since either xp0,q or xp,q0 is in I^.

(ii) (p, q)∈ E>0(w), say r_p,q(w) = r. Let Δ be any (r + 1)-minor in X_[p,q]. In this case, p > p₀ by (4.1).

(ii.1) q < q₀. In this case, (p, q)∈ E=r(w^). If Δ involves the p₀th row, expanding along this row we see that Δ∈ xp0,q| q < q0 ⊆ I^. Oth- erwise, let Δ^ be the corresponding (r + 1)-minor in X^_[p,q] and we have Δ− Δ^∈ xp0,q| q < q0 ⊆ I^ by (b). But (p, q)∈ E=r(w^) im- plies Δ^∈ Iw^⊆ I^. So Δ∈ I^.

(ii.2) q > q0. In this case, (p, q)∈ E=r−1(w^).

(ii.2.1) If Δ involves c, then Δ = c(Δ\ [p0| q0])^ by (d), where Δ\ [p0| q0] is the r-minor obtained from Δ by deleting the row and the column involving c. But (p, q)∈ E=r−1(w^) implies (Δ\ [p0| q0])^∈ Iw^⊆ I^. So Δ∈ I^.

(ii.2.2) If Δ involves Γ but does not involve c, then by (c) Δ = Δ^ where Δ^is obtained from Δ by replacing xp,q((p, q) /∈ Γ) by

x^_p,q. Expanding Δ^along the row or column involving Γ, we see that Δ^∈ r-minors in X^_[p,q]. But (p, q) ∈ E=r−1(w^) impliesr-minors in X^_[p,q] ⊆ Iw^⊆ I^. So Δ = Δ^∈ I^. (ii.2.3) If Δ does not involve Γ, then cΔ = Δ^ by (d). Expand-

ing the (r + 2)-minor Δ^ along the row and the column involving Γ, we see that Δ^∈ r-minors in X^_[p,q]. Again,

r-minors in X^_[p,q] ⊆ I^ since (p, q)∈ E=r−1(w^). So Δ = c⁻¹Δ^∈ I^.

Conversely, we show that I^⊆ I. Fix (p, q) ∈ D=0(w^)∪ E>0(w^). Again, we show that the set

x^_p,q| (p, q) ∈ D=0

w^

∪

 

(p,q)∈E>0(w^)

r_p,q w^

+ 1

-minors in X^_[p,q]

is contained in I.

(i) (p, q)∈ D=0(w^).

(i.1) If p < p₀ or q < q₀, then (p, q)∈ D=0(w) and hence x_p,q∈ I. By (a), either xp0,q or xp,q0 is in I. Therefore,

x^_p,q= x_p,q− c⁻¹x_p0,qx_p,q0∈ I.

(i.2) If p < p0 and q > q0, then (p, q)∈ E=1(w). Hence, the 2-minor cx^_p,q= cx_p,q− xp0,qx_p,q0∈ Iw⊆ I. So x^p,q∈ I.

(ii) (p, q)∈ E>0(w^). In this case, p > p0 by (4.1). Suppose rp,q(w^) = r and let Δ^ be any (r + 1)-minor in X^_[p,q].

(ii.1) If q < q₀, then (p, q)∈ E=r(w). By (d), cΔ^ = Δ. Expanding Δ along the q ₀th column, we see Δ∈ (r + 1)-minors in X[p,q].

But (p, q)∈ E=r(w) implies(r + 1)-minors in X[p,q] ⊆ I. So Δ^= c⁻¹Δ∈ I.

(ii.2) If q > q0, then (p, q)∈ E=r+1(w). Again, cΔ^= Δ by (d). This time Δ ∈ Iw⊆ I since (p, q) ∈ E=r+1(w). Therefore, Δ^= c⁻¹Δ ∈ I. 

Example4.5. Consider w = 35142 in S₅. Use the same notation as in the proof of Theorem 4.4. We have c = x₁₃, I_w=x11, x₁₂, x₂₁, x₂₂, [12| 34], [34 | 12] and Iw^=x^21, x^₂₂, x^₂₄, [34| 12]^. Check that the following elements lie in x11, x12 and hence in I ∩ I^:

x21− x^₂₁, x22− x^₂₂, [34| 12] − [34 | 12]^, [12| 34] − c⁻¹x^₂₄. Therefore, we see that I = I^ and R_w[c⁻¹] = R_w[x₁₁, x₁₂, c⁻¹].

In the following diagram, the 1’s indicate the permutation and the dots indicate the elements in D>0(w).

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5. Complete intersection matrix Schubert varieties

We want to characterize the complete intersection MSVs. By (2.2), we may assume w∈ Sn. Denote w_(p,q) the r_p,q(w)× rp,q(w) submatrix of w involving the rows of indices p− rp,q(w), . . . , p− 1 and the columns of in- dices q− rp,q(w), . . . , q− 1. Define the submatrix X(p,q) of X similarily.

Furthermore, denote X_(p,q) the (r_p,q(w) + 1)× (rp,q(w) + 1) submatrix of X involving the rows of indices p− rp,q(w), . . . , p and the columns of in- dices q− rp,q(w), . . . , q. If r_p,q(w) = 0, X_(p,q)= det X_(p,q)= x_p,q. However, in order to make our proof more transparent, we will only use X_(p,q) for (p, q)∈ D>0(w).

Recall that the codimension of X_win M_n×nis|D(w)|. So Xwis a complete intersection if and only if I_w can be generated by|D(w)| many elements. We need the following lemma.

Lemma 5.1. Let w be such that X_w is a complete intersection. Then for any (p, q)∈ D>0(w) and any 1≤ i ≤ rp,q(w),

(p− i, q) /∈ D(w) and (p, q − i) /∈ D(w).

In particular,

D>0(w)⊆ E(w), that is, D>0(w) =E>0(w).

Proof. We only have to prove the first statement. The second statement follows by definition. Suppose the first statement does not hold, then by symmetry we may assume that there exist (p0, q0) in D>0(w) and some 1≤ i0≤ rp0,q0(w) so that (p0− i0, q0)∈ D(w) and (p0− j, q0) /∈ D(w) for all 1 ≤ j < i0.

Denote r := rp0,q0(w). Consider the (r + 1)-minor

Δ = [p0− r − 1, . . . , p0− i0, . . . , p0| q0− r, . . . , q0]

in the (r + 2)× (r + 1) submatrix XΔ of X involving the rows of indices p₀− r − 1, . . . , p0 and the columns of indices q₀− r, . . . , q0.

Consider the set G =

xp,q| (p, q) ∈ D=0(w) 

det X_(p,q)| (p, q) ∈ D>0(w) {Δ}.

Observe that the unions are disjoint and |G| = |D(w)| + 1. Denote m the maximal graded ideal of K[X]. We claim that

the image of G in I_w/mI_w form a (5.1)

K

= K[X]/m

-linearly independent set.

By Nakayama’s lemma, Iwis generated by at least |D(w)| + 1 elements. This contradicts the assumption that Xw is a complete intersection.

It remains to show the claim (5.1). Suppose

 

(p,q)∈D=0(w)

cp,qxp,q

 +

 

(p,q)∈D>0(w)

cp,qdet X_(p,q)

+ cΔΔ∈ mIw, where cp,q and cΔ are in K. Notice mIw is a homogeneous ideal whose gen- erators are of degree at least 2. This implies cp,q= 0 for all (p, q)∈ D=0(w) since otherwise we will have an element in mIw that has a nonzero degree 1 part. Therefore,

F :=

 

(p,q)∈D>0(w)

cp,qdet X_(p,q)

+ cΔΔ∈ mIw.

Fix any antidiagonal term order on K[X]. Then in(F ) is the antidiagonal term of one of the minors in

det X_(p,q)| (p, q) ∈ D>0(w) {Δ}.

So in(F ) is in the generating set of in(Iw) described in Theorem 2.1(1). On the other hand, F ∈ mIw implies that there exists an antidiagonal δ in the generating set of in(Iw) described in Theorem 2.1(1) such that δ is a factor of in(F ) but δ= in(F ). This means that δ ∈ in(Iw) is an antidiagonal in one of the submatrices of the matrices in

X_(p,q)| (p, q) ∈ D>0(w)

{XΔ},

and that δ is of size≤ rp,q(w) (if δ is in X_(p,q)) or of size≤ rp0,q0(w) (if δ is in XΔ). This is impossible in view of Theorem2.1(1). Therefore, we conclude that cp,q(w) = cΔ= 0 for all (p, q)∈ D(w) and the claim (5.1) is proved. 

Now we are ready for the following theorem.

Theorem 5.2. Let w∈ Sn. The following conditions are equivalent:

(1) Xw is a complete intersection,

(2) for any (p, q)∈ D>0(w), w_(p,q) is a permutation in S_rp,q(w) such that Xw(p,q) is a complete intersection.

In this case,

xp,q| (p, q) ∈ D=0(w) 

det X_(p,q)| (p, q) ∈ D>0(w) is a set of generators for Iw.

Proof. The conditions in (2) shows that for any (p, q)∈ D>0(w) the only nonzero entries of w[p,q]appear in w(p,q), so all size (rp,q(w) + 1) minors except det X_(p,q) belong to xp,q| (p, q) ∈ D=0(w). Therefore, the last statement follows immediately from (2.1) and the equivalence of (1) and (2).

We prove (2) implies (1). Let (p1, q1), . . . , (pt, qt) be all the elements in D>0(w) satisfying

(p, q)| p ≥ pi, q≥ qi

∩ D>0(w) =∅.

Denote ri= rpi,qi(w) and wi= w_(pi,qi)∈ Sri. By the assumptions in (2), the diagram D(wi) of wi is contained in D(w). Also, the ideal Iwi of Xwi is generated by |D(wi)| many elements, since Xwi is a complete intersection.

Note also that the conditions in (2) and the choice of (p_i, q_i) imply that D(w) can be decomposed as



(p, q) (p, q) ∈ D=0(w)

^t

i=1

D(wi)

 _t

i=1

D(wi)

 

(pi, qi)| 1 ≤ i ≤ t ,

where the unions are disjoint. Furthermore, by construction one can check using (2.1) that

Iw=

!

xp,q (p, q) ∈ D=0(w)

^t

i=1

D(wi)

"

+

t i=1

Iwi+

t i=1

det X_(pi,qi).

So Iw is generated by



(p, q)∈ D=0(w) (p, q) /∈

t i=1

D(w_i) +

t i=1

D(w_i)+ t =D(w)

many elements. Therefore, X_w is a complete intersection.

To prove (1) implies (2), let (p, q)∈ D>0(w) and use induction on r_p,q(w).

When r_p,q(w) = 1, we must have w_p−1,q−1= 1 since otherwise either (p− 1, q)∈ D=1(w) or (p, q− 1) ∈ D=1(w) which contradicts Lemma 5.1. Also, Xw(p,q) is the affine line, so we are done for rp,q(w) = 1.

Suppose rp,q(w) > 1 and denote r = rp,q(w). By Lemma5.1, (p− i, q) /∈ D(w) and (p, q − i) /∈ D(w), for i = 1, . . . , r.

This implies that rank(w_(p,q)) = r and w_(p,q)∈ Sr. Moreover, consider the diagram D(w_(p,q)). This is exactly the part of the diagram D(w) involving the rows of indices p− r, . . . , p − 1 and the columns of indices q − r, . . . , q − 1.

For any (p^, q^)∈ D>0(w_(p,q)), r_p,q^(w_(p,q)) = r_p,q^(w) < r. So by inductive hypothesis, w_(p,q)∈ Sr satisfies the conditions in (2). Therefore, by the im- plication of (2) ⇒ (1) we just proved, Xw(p,q) is a complete intersection as

desired. 

Example 5.3. Consider w∈ S6. Again, In the following diagrams the 1’s indicate the permutation and the dots indicate the elements in D>0(w).

(1) If w = 361452, then Xw is not a complete intersection since D>0(w) = {(2, 4), (2, 5), (4, 2), (5, 2)} but (2, 5) /∈ E>0(w).

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(2) If w = 352614, then D>0(w) ={(2, 4), (4, 4)} = E>0(w). But Xw is still not a complete intersection since w_(4,4)∈ S/ 2. So we see that the condition D>0(w) =E>0(w) is not sufficient for Xw to be a complete intersection.

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(3) If w = 462153, then Xw is a complete intersection, and Iw is generated by{xi,j| 1 ≤ i ≤ 2, 1 ≤ j ≤ 3} ∪ {x31, det X_(2,5), det X_(5,3)}.

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Acknowledgments. 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.

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Jen-Chieh Hsiao , Department of Mathematics, National Cheng Kung Univer- sity, No.1 University Road, Tainan 70101, Taiwan

E-mail address:jhsiao@mail.ncku.edu.tw