5 Grassmannians, flag varieties, Schubert varieties
General references: [ GW1 ] Ch. 8.
Recall that for a field \(k\), \(\mathbb {P}^n(k)\) is identified with the set of lines (i.e., one-dimensional linear subspaces) in \(k^{n+1}\); in fact, we took that as a definition in Algebraic Geometry 1. For a description of \(\mathbb {P}^n(S)\) for a general scheme \(S\), one usually, following Grothendieck, prefers to talk about one-dimensional quotients rather than subspaces. Over a field one can switch back and forth between the two points of view by passing to dual vector spaces; this is even true over a general base scheme, since only locally free modules of finite rank are involved.
Jan. 22,
2024
The Grassmannian variety \(\operatorname{Grass}_{r, n}\) attached natural numbers \(0\le r\le n\) similarly, but more generally, parameterizes \(r\)-dimensional linear subspaces of the \(n\)-dimensional vector space \(k^n\). As for projective space, one can easily describe the functor represented by this scheme (and we will take this as our starting point); and also, the Grassmannian can be explicitly constructed by gluing copies of affine space in a suitable way.
For \(r=n-1\) (and dually, for \(r=1\)), this functor is represented by \(\mathbb {P}^{n-1}\).
Our first goal is to show that for any choice of \(r\le n\) this functor is representable by a scheme which admits an open cover by affine spaces. The open subsets of this cover admit a simple functorial description themselves (which of course is equivalent to the usual functorial description of affine space, but written a bit differently). This gives rise to “open subfunctors” of the Grassmann functor in the following sense.
A morphism \(F\to G\) of functors \({\rm (Sch)}{}^{\rm opp}\to {\rm (Sets)}\) is called representable, if for every scheme \(T\) (considered as a functor \({\rm (Sch)}{}^{\rm opp}\to {\rm (Sets)}\)) and every morphism \(T\to G\) the fiber product functor \(F\times _G T\) is representable by a scheme.
Given a morphism \(f\colon F\to G\) of functors \({\rm (Sch)}{}^{\rm opp}\to {\rm (Sets)}\), we call \(F\) an open subfunctor of \(G\), if the morphism \(f\) is representable by an open immersion, i.e., \(f\) is representable and for every scheme \(T\) and morphism \(T\to G\) the scheme morphism \(F\times _GT\to T\) is an open immersion.
Let \(F \colon {\rm (Sch)}{}^{\rm opp}\to {\rm (Sets)}\) be a functor. A family \((F_i\to F)_i\) of open subfunctors of \(F\) is called an open cover of the functor \(F\), if for every scheme \(T\) and morphism \(T\to G\) the open immersions \(F_i\times _FT\to T\) induce an open cover of \(T\).
With this terminology, “gluing of morphisms” via the Yoneda lemma becomes the following proposition.
\(F\) is a sheaf for the Zariski topology,
\(F\) has an open cover by representable subfunctors.
We construct a scheme representing the functor \(F\) by gluing the schemes representing the open subfunctors which cover \(F\). The gluing datum (isomorphisms between the intersections satisfying the cocycle conditions) is obtained from the open cover of the functor \(F\) via the Yoneda lemma. E.g., if \(U_i\) represents \(F_i\), then \(F_i\times _F F_j\) is representable by an open subscheme \(U_{ij}\) of \(U_i\) which will correspond to the intersection of \(U_i\) and \(U_j\) in the result of the gluing process. Let \(X\) denote the scheme obtained by gluing.
Since \(F\) is a sheaf for the Zariski topology, the maps \(U_i = F_i \to F\) can be glued to give a morphism \(X\to F\). For every scheme \(S\) this gives rise to a bijection \(X(S)\to F(S)\) (since this can be checked locally on \(S\) and hence reduces to a statement about every \(F_i\)), and thus is an isomorphism of functors, as desired.
Gluing of \({\mathscr O}_S\)-modules implies that the Grassmann functor satisfies Property (a):
Jan. 24,
2024
By definition, \(\operatorname{Grass}_{r, n}^I\) is a subfunctor of \(\operatorname{Grass}_{r,n}\) and we claim that it is even an open subfunctor. Indeed, consider a scheme \(S\) and a morphism \(S\to \operatorname{Grass}_{r, n}\) of functors. By the Yoneda lemma, it corresponds to a submodule \({\mathscr U}\subseteq {\mathscr O}^n_S\) with locally free quotient. Then the fiber product \(\operatorname{Grass}^I_{r,n}\times _{\operatorname{Grass}_{r, n}} S\) is representable by the open subscheme
of \(S\). Here, as usual, \({\mathscr U}(s) = {\mathscr U}\otimes _{{\mathscr O}_S}\kappa (s)\) denotes the fiber of \({\mathscr U}\) at \(s\).
It also follows from this description that the open subfunctors \(\operatorname{Grass}_{r,n}^I\) for all \(I\) give rise to an open cover of \(\operatorname{Grass}_{r, n}\) since this can be checked on field-valued points.
Write \(J:=\{ 1,\dots , n\} \setminus I = \{ j_1 {\lt} \cdots {\lt} j_{r} \} \). We view \(\mathbb {A}^{r(n-r)}\) as the space of \((n\times r)\)-matrices that have the unit matrix in rows \(j_1,\dots , j_r\), and arbitrary entries in the other \(n-r\) rows. For each such matrix (with entries in \(\Gamma (S, {\mathscr O})\) for some scheme \(S\)), the subspace of \({\mathscr O}_S^n\) generated by the columns of the matrix is an element of \(\operatorname{Grass}_{r, n}^I\), and this map defines an isomorphism of functors. In fact, it is clearly injective. To show surjectivity, take \({\mathscr U}\in \operatorname{Grass}^I(S)\) and consider the short exact sequence
The composition \({\mathscr O}_S^n/{\mathscr U}\cong {\mathscr O}_S^I\subset {\mathscr O}_S^n\) defines a splitting of this sequence, and this induces an isomorphism \({\mathscr O}_S^J\cong {\mathscr O}_S^n/{\mathscr O}_S^I\cong {\mathscr U}\). In particular \({\mathscr U}\) is free and \({\mathscr U}\oplus {\mathscr O}^I_S = {\mathscr O}_S^n\), so \({\mathscr U}\) has a basis which can be extended to a basis of \({\mathscr O}_S^n\) by adding the standard basis vectors of \({\mathscr O}_S^I\). This means precisely that it lies in the image of the above map.
See [ GW1 ] Lemma 8.13 for a different way of phrasing the proof.
Putting everything together, we obtain:
Let us show that the Grassmann scheme is projective. We will specify an explicit closed embedding \(\operatorname{Grass}_{r,n}\to \mathbb {P}^N_\mathbb {Z}\), where \(N=\binom {n}{r}-1\), the so-called Plücker embedding. Here we view \(\mathbb {P}^N\) as the space of lines in \(\bigwedge ^r {\mathscr O}^n\), i.e., we use the “dual point of view” on the functorial description of projective space.
We mostly skip the proof here. In terms of coordinates, the map maps a (free) \(r\)-dimensional subspace, with basis the columns of some \((n\times r)\)-matrix \(A\), say, to the vector with homogeneous coordinates all the \(r\)-minors (i.e., determinants of \((r\times r)\)-submatrices) of \(A\). It is easy to show that the morphism \(\operatorname{Grass}_{r, n}\to \mathbb {P}^N\) is injective (on the underlying sets). Given an element of \(\mathbb {P}^N(S)\) which we view as a locally free line bundle in \(\bigwedge ^r {\mathscr O}_S^n\) with locally free quotient, we construct a map \(\varphi _{\mathscr L}\colon {\mathscr O}_S^n\to \bigwedge ^{r+1}{\mathscr O}_S^n\) as follows. Locally on \(S\), \({\mathscr L}\) is free, say with basis the vector \(v\in \bigwedge ^r {\mathscr O}_S^n\). Then we map \(w\mapsto w\wedge v\). The condition that the kernel of \(\varphi _{\mathscr L}\) is a locally free submodule of \({\mathscr O}^n\) of rank \(r\) with locally free quotient is a closed condition on \(\mathbb {P}^N\). The image of the morphism from the Grassmannian is contained in the closed subscheme of \(\mathbb {P}^N\) defined in this way, and there the above morphism admits the inverse \({\mathscr L}\mapsto \operatorname{Ker}(\varphi _{\mathscr L})\).
We obtain the following corollary, which can also easily be shown directly via the valuative criterion for properness (Theorem 1.2).
Fix \(0\le r \le n\). The definition of the Grassmannian \({\rm Gr} = \operatorname{Grass}_{r, n}\) as representing a functor provides us with a universal object, i.e., an \({\mathscr O}_{\rm Gr}\)-submodule \({\mathscr U}\subseteq {\mathscr O}_{\rm Gr}^n\) of rank \(r\) with locally free quotient.
We obtain a short exact sequence
of locally free \({\mathscr O}_{\rm Gr}\)-modules. It is then straightforward to translate the proof of Proposition 2.46 (about the Euler sequence on projective space) to prove the following generalization.
From this we can easily compute the canonical bundle, i.e., the highest exterior power of the sheaf of differentials. This is also the dualizing sheaf of the Grassmannian (after base change to a field in order to get into the setting where we defined this).
The computation of \(\Omega _{{\rm Gr}/\mathbb {Z}}\) give us, denoting the highest exterior power of a locally free sheaf \({\mathscr E}\) by \(\det ({\mathscr E})\),
where in the second step we use the lemma below, and for the final step we have used that (as a result of the short exact sequence defining \({\mathscr Q}\)) \(\det \left({\mathscr Q}\right)^{\vee }\cong \det \left({\mathscr U}\right)\).
Locally \({\mathscr E}\) and \({\mathscr F}\) are free, with bases \((b_i)_i\) and \((c_j)_j\), say. Then \((b_i\otimes c_j)_{i,j}\) is a basis of \({\mathscr E}\otimes {\mathscr F}\), and we obtain a homomorphism \(\det ({\mathscr E}\otimes {\mathscr F})\to \det ({\mathscr E})^{\otimes f}\otimes _{{\mathscr O}_X}\det ({\mathscr F})^{\otimes e}\) by mapping
Since it maps a basis vector to a basis vector (source and target are free of rank \(1\)), it is an isomorphism. One checks that this homomorphism is independent of the choices of bases. Thus the local homomorphisms glue.
One can show that the Picard group of the Grassmannian (over a field, or more generally over any unique factorization domain) is isomorphic to \(\mathbb {Z}\), more precisely we have the following result.
Consider one of the open charts \(\operatorname{Grass}_{r, n}^I\) of \({\rm Gr}\). The key point is to show that its complement is irreducible of codimension \(1\), i.e., can be considered as a primitive Weil divisor. This follows from the analysis of “Schubert varieties” in the Grassmannian (which we omit here, but see below for “Schubert varieties in flag varieties”).
Then [ GW1 ] Proposition 11.42 and the fact that \(\operatorname{Pic}(\operatorname{Grass}_{r, n}^I) = 0\) (since \(\operatorname{Grass}_{r, n}^I\) is isomorphic to affine space) provide a surjection \(\mathbb {Z}\to \operatorname{Pic}(\operatorname{Grass}_{r, n})\). This maps \(1\) to the pullback of \({\mathscr O}(1)\) under the Plücker embedding. Since \(\operatorname{Grass}_{r, n}\) is projective of positive dimension, the Picard group cannot be finite; cf. [ GW1 ] Corollary 13.82 for one way to show this.
Recall that a (full) flag in a finite-dimensional vector space \(V\) over some field \(k\) is a chain \(0\subset F_1\subset \cdots \subset F_{\dim V-1} \subset V\) of subspaces with \(\dim F_i = i\) for all \(i\).
Jan. 29,
2024
The full flag functor \({\rm (Sch)}{}^{\rm opp}\to {\rm (Sets)}\),
\[ {\rm Flag}(S) = \left\{ (F_i)_i\in \prod _{i=1}^{n-1} \operatorname{Grass}_{i,n}(S);\ \forall i\colon F_i\subset F_{i+1} \right\} \]is representable by a closed subscheme of \(\prod _{i=1}^{n-1} \operatorname{Grass}_{i,n}(S)\) which we also denote by \({\rm Flag}\). The scheme \({\rm Flag}\) is projective over \(\mathbb {Z}\).
Similarly, given \(0{\lt}i_1{\lt}\cdots {\lt} i_r {\lt} n\), we have the partial flag variety \({\rm Flag}_{i_\bullet }\) of all partial flags \((F_\nu )_\nu \) with \(F_\nu \in \operatorname{Grass}_{i_\nu , n}\).
To show that the full flag functor is representable by a closed subscheme of the product of Grassmannians we need to show that the conditions \(F_i\subset F_{i+1}\) are “closed conditions”. By induction we reduce to the following
Claim. The subfunctor
is represented by a closed subscheme of the product \(\operatorname{Grass}_{r, n}\times \operatorname{Grass}_{r', n}\).
Proof of claim. We can check this locally on the product and therefore pass to a product \(\operatorname{Grass}_{r,n}^I\times \operatorname{Grass}_{r', n}^{I'}\) for suitable \(I\), \(I'\) (with notation as in Definition 5.7). Now given \((F, F') \in \operatorname{Grass}_{r,n}^I\times \operatorname{Grass}_{r', n}^{I'}\), \(F\) and \(F'\) are given by matrices \(M\), \(M'\) as in the proof of Lemma 5.8. The condition \(F\subseteq F'\) is then equivalent to the existence of a matrix \(A\in {\rm Mat}_{r'\times r}(\Gamma (S, {\mathscr O}_S))\) such that \(M = M'A\). But since \(M'\) has the unit matrix in rows \(J':=\{ 1,\dots , n\} \setminus J'\), there is at most one choice for \(A\), namely the submatrix \(M_{J', \bullet }\) of \(M\) in rows \(J'\). The condition \(F\subseteq F'\) thus translates to \(M_{I', \bullet } = M'_{I', \bullet }M_{J', \bullet }\). This is clearly a closed condition.
We have already proved the projectivity, so it remains to exhibit an open cover by affine spaces and compute the dimension. To simplify the notation, we give the proof for the full flag variety only.
Consider the free module \(\mathbb {Z}^n\) and denote by \(e_1,\dots , e_{n-1}\) its standard basis. For a permutation \(w\in S_n\) let \(G^w_\bullet \) be the flag defined as
For any scheme \(S\) by pullback we obtain a flag in \({\mathscr O}_S^n\) which we again denote by \(G^w_i\). We define
This is the intersection (in \(\prod \operatorname{Grass}_{i,n}\)) of \({\rm Flag}\) with a product of standard (up to permutation of the basis) open charts of Grassmannians and hence an open subfunctor of \({\rm Flag}\), represented by an open subscheme of \({\rm Flag}\).
It is now enough to show that each \({\rm Flag}^w\) is isomorphic to an affine space of dimension \(n(n-1)/2\) and that the \({\rm Flag}^w\) cover \({\rm Flag}\). This latter property can be checked on field-valued points where it becomes a simple linear algebra consideration.
To show that \({\rm Flag}^w\) is an affine space (of the correct dimension), by renumbering our basis we may assume that \(w = \operatorname{id}\). Denote by \(U^-\) the scheme of lower triangular matrices whose diagonal entries are all \(=1\). The scheme \(U^-\) is obviously isomorphic to \(\mathbb {A}^{n(n-1)/2}\). One then shows, similarly as in the proof of Lemma 5.8 that the map \(U^-\to {\rm Flag}^{\operatorname{id}}\) which maps a matrix \(A\in U^-(S)\) to the flag \((F_i)_i\) where \(F_i\) is the free \({\mathscr O}_S\)-submodule of \({\mathscr O}_S^n\) generated by the first \(i\) columns of \(A\), is an isomorphism.
(For general \(w\), one obtains an isomorphism \(U^-\to {\rm Flag}^w\) by mapping \(A\in U^-(S)\) to the flag \((F_i)_i\) where \(F_i\) is generated by the first \(i\) columns of \(wA\), where we view \(w\) as a permutation matrix.)
Let \(k\) be a field and denote by \({\rm Flag}\) the flag variety over \(k\), for some fixed \(n\). The group \(GL_n(k)\) of invertible \((n\times n)\)-matrices with entries in \(k\) acts transitively on \({\rm Flag}(k)\), and denoting by \(E_\bullet \) the standard flag in \(k^n\), the map \(g\mapsto g E_\bullet \) induces a bijection \(GL_n(k)/B(k)\overset {\sim }{\to }{\rm Flag}(k)\), where \(B(k)\subseteq GL_n(k)\) denotes the stabilizer of the standard flag (i.e., the subgroup of upper triangular matrices).
The action is given by a morphism \(GL_{n,k}\times _{\operatorname{Spec}k}{\rm Flag}\to {\rm Flag}\). One can make sense of the quotient \(GL_n/B\) in algebraic geometry, but (as usual) dealing with quotients is more complicated than in differential or complex geometry, for instance, because the Zariski topology is too coarse.
Fix a field \(k\) and a natural number \(n\ge 1\). We consider full flags \(F_\bullet \) in \(k^n\). The standard flag \(E\) is the flag with \(i\)-th entry the subspace \(\langle e_1,\dots ,e_i\rangle \) generated by the first \(i\) standard basis vectors. Let \(S_n\) denote the symmetric groups on \(n\) letters. In this context, it is natural to view \(S_n\) as the subgroup of permutation matrices of \(GL_n(k)\).
Recall that \(S_n\) is generated by the simple transpositions, i.e., by the transpositions \(s_i = (i, i+1)\) of two adjacent elements (\(i=1,\dots , n-1\)). We denote by \(\mathbb S\) the set of simple transpositions.
One shows that for \(w\in S_n\) and \(s\in \mathbb S\), \(\ell (ws) \ge \ell (w)-1\) (more precisely, one can show that \(\ell (ws)\in \{ \ell (w)-1, \ell (w)+1\} \). And moreover that for every \(w\in S_n\) with \(\ell (w) {\gt} 0\) there exists \(s\in \mathbb S\) with \(\ell (ws) {\lt} \ell (w)\). Both statements follow from elementary considerations about permutations.
(Gauß-Bruhat algorithm) Let \(k\) be a field, \(n\ge 1\), \(G = GL_n(k)\) the group of invertible matrices of size \(n\times n\) over \(k\). Let \(B\subset G\) be the subgroup of upper triangular matrices, and \(U\subset B\) the subgroup of unipotent upper triangular matrices (i.e., upper triangular matrices all of whose diagonal entries are equal to \(1\)). Let \(U^-\) be the group of unipotent lower triangular matrices in \(G\). (In this lemma, we consider all these groups as abstract groups, rather than \(k\)-schemes/\(k\)-varieties.)
We have \(G = \bigcup _w BwB\) (disjoint union of \(B\)-double cosets). This is called the Bruhat decomposition of \(GL_n(k)\).
Let \(w\in S_n\). Denote by \(U_w\) the following subgroup of the group of unipotent upper triangular matrices \(U\):
\[ U_w = U \cap wU^- w^{-1} \](where we view \(w\) inside \(GL_n\) as a permutation matrix). The map \(U_w\times B\overset {\sim }{\to }BwB\), \((u, b)\mapsto uwb\), is bijective.
The decomposition of \(GL_n/B = {\rm Flag}(k)\) as a disjoint union of \(B\)-orbits (for the \(B\)-action on \(GL_n/B\) by multiplication on the left, equivalently the action on \({\rm Flag}(k)\) induced by the standard action of \(B\) on \(k^n\)) is \(GL_n/B = \bigcup _w BwE\) (where \(E\), as usual, denotes the standard flag, which under the identification with \(GL_n/B\) is the residue class of the unit matrix). The bijections of Part (2) induce bijections \(BwE \leftrightarrow U_w\), and \(U_w\) is in bijection (as a set) with \(k^{\ell (w)}\). This decomposition is called the Bruhat decomposition of the space of flags \({\rm Flag}(k)\).
The key point is that every matrix \(g\in G\) can be written as \(g=uwb\) for \(u\in U\), \(b\in B\), \(w\) a permutations matrix. Equivalently, given \(g\), we can transform \(g\) to a permutation matrix by the following elementary row and column operations:
Add a multiple of a row of the matrix \(g\) to another row lying above the given one.
Add a multiple of a column of the matrix \(g\) to another column lying to the right of the given one.
Multiply a column with a non-zero scalar.
This can be checked fairly easily by “directly manipulating matrices”.
Looking carefully at the proof of the previous lemma, we obtain the following “scheme-theoretic version” of the Gauß-Bruhat algorithm.
Jan. 31,
2024
(Gauß-Bruhat algorithm, scheme version) Let \(k\) be a field, \(n\ge 1\), \(G = GL_{n,k}\) the \(k\)-group scheme of invertible matrices of size \(n\times n\). Let \(B\subset G\) be the subgroup scheme of upper triangular matrices, and \(U\subset B\) the subgroup scheme of unipotent upper triangular matrices (i.e., upper triangular matrices all of whose diagonal entries are equal to \(1\)). Similarly, let \(U^-\) be the scheme of unipotent lower triangular matrices.
Let \(w\in W\). The image of the morphism \(B\times B\to G\) that is given on \(R\)-valued points (\(R\) a \(k\)-algebra) by \((b, b')\mapsto bwb'\) is locally closed. We denote by \(BwB\) the reduced subscheme of \(G\) whose topological space is this locally closed subset.
We have \(GL_n = \bigcup _w BwB\) (disjoint union of subschemes), i.e., every point of the topological space \(GL_n\) lies in exactly one of the subschemes \(BwB\). This decomposition is called the Bruhat decomposition of the scheme \(GL_n\).
Let \(w\in S_n\). Denote by \(U_w\) the following subgroup scheme of \(U\):
\[ U_w = U \cap wU^- w^{-1}. \]The morphism \(U_w\times B\overset {\sim }{\to }BwB\) given on \(R\)-valued points by \((u, b)\mapsto uwb\), is an isomorphism of \(k\)-schemes.
The decomposition in (1) induces a decomposition of the flag variety \({\rm Flag}\) over \(k\) as a disjoint union of locally closed subschemes \(C(w)\), where \(C(w)\) is the reduced subscheme of \({\rm Flag}\) whose topological space is the image of \(BwB\) under the natural morphism \(G\to {\rm Flag}\).
The isomorphisms in Part (2) induce isomorphisms \(C(w)\cong U_w \cong \mathbb {A}^{\ell (w)}_k\).
The subscheme \(C(w)\) of \({\rm Flag}\) is called the Schubert cell attached to \(w\). This decomposition is called the Bruhat decomposition of \({\rm Flag}\).
We can make the partition of the flag variety into Schubert cells very explicit using the notion of relative position of flags in the following sense.
It follows from the Bruhat decomposition of \(GL_n(k)\) that for any two flags, there exists a unique element of \(S_n\) with the properties of the above definition. We obtain a map
Let \(F_\bullet \), \(F'_\bullet \) be flags. Then \(\operatorname{inv}(F_\bullet , F'_\bullet ) = \operatorname{id}\) if and only if \(F_\bullet =F'_\bullet \).
Let \(s\) be a simple transposition, i.e., \(s = (i, i+1)\) for some \(i\in \{ 1, \dots , n-1\} \). Then \(\inf (F_\bullet , F'_\bullet ) = s\) if and only if \(F_j = F'_j\) for all \(j\ne i\), and \(F_i\ne F'_i\).
It is clear that the relative position with the standard flag \(E_\bullet \) is constant on each Schubert cell. Thus basically by definition,
Let \(k\) be a field, and let \({\rm Flag}\) denote the flag variety over \(k\) for some fixed \(n\). The decomposition \({\rm Flag}(k) = \bigcup C(w)(k)\) is precisely the decomposition into \(B(k)\)-orbits for the natural action by \(B(k)\) (the group of upper triangular matrices) on \({\rm Flag}(k)\), namely the action induced by the standard action of \(B(k)\subseteq GL_n(k)\) on \(k^n\).
The theory of Schubert cells (and Schubert varieties as defined below, …) can be generalized to other “reductive groups” (including, for instance, (special) orthogonal groups, symplectic groups, unitary groups). See e.g. [ B ] , [ Mi ] .
Let \(k\) be a field. We denote by \({\rm Flag}\), \(C(w)\) the flag variety and Schubert cells over \(k\).
From the definition (and the projectivity of the flag variety) it is clear that \(X(w)\) is a projective \(k\)-scheme. In general, \(X(w)\) is not smooth and in fact Schubert varieties are a class of interesting varieties which are fairly explicit but have a “non-trivial” geometry.
One can show that each \(X(w)\) is a union of Schubert cells (cf. Remark 5.24). This defines a unique partial order on \(S_n\) which satisfies
the so-called Bruhat order. It follows immediately from the definition that \(v\le w\) implies \(\ell (v)\le \ell (w)\). The converse implication is not true. The Bruhat order can also be described in combinatorial terms, but the description is non-trivial because of the inherent complexity of the situation. The identity element is the unique minimal element and the unique element of maximal length, \(w_0 = \begin{pmatrix} 1 & 2 & \cdots & n \\ n & n-1 & \cdots & 1 \end{pmatrix}\), is the unique maximal element with respect to the Bruhat order. The Schubert cell \(C(w_0)\) is the unique open Schubert cell. It is equal to the open subscheme \({\rm Flag}^{w_0}\) defined in the proof of Proposition 5.17. Except for \(n\le 2\) the Bruhat order is not a total order.
We work over a fixed field \(k\) (i.e., \({\rm Flag}\) denotes the flag variety over \(k\), …).
If \(\operatorname{inv}(F_1, F_2) = w\), \(\operatorname{inv}(F_2, F_3) = s\), then \(\operatorname{inv}(F_1, F_3) = ws\).
Given \(F_1\), \(F_3\) with \(\operatorname{inv}(F_1, F_3) = ws\), there exists a unique \(F_2\) with \(\operatorname{inv}(F_1, F_2) = w\), \(\operatorname{inv}(F_2, F_3) = s\).
The proof is “just linear algebra” and more or less straightforward. Recall that \(\operatorname{inv}(F, F') = s_i\) is equivalent to saying that \(F_j = F'_j\) for all \(j\ne i\) and \(F_i\ne F'_i\).
From the lemma we obtain the following scheme-theoretic statement.
Let \(k\) be a field. Let \(w\in S_n\) and \(s\in \mathbb S\) such that \(\ell (ws) {\gt} \ell (w)\).
There is a (unique) morphism \(C(ws)\to C(w)\) which on \(k'\)-valued points (for any extension field \(k'\) of \(k\)) is given by mapping a flag \(F\in C(ws)(k)\) to the unique flag \(F'\) with \(\operatorname{inv}(E, F') = w\), \(\operatorname{inv}(F', F)=s\). (As usual, \(E\) denotes the standard flag.) Each fiber is isomorphic to the affine line \(\mathbb {A}^1\).
Let \(w = s_{i_1}\cdots s_{i_\ell }\) be a reduced expression, i.e., \(s_\nu \in \mathbb S\), \(\ell =\ell (w)\) (the minimal number of factors needed to express \(w\) as a product of elements of \(\mathbb S\)). Let \(D(i_\bullet ):=D(i_1,\dots , i_\ell )\) be the unique reduced closed subscheme of \(\prod _\nu {\rm Flag}^{\ell }\) such that for every \(k'/k\) we have
To see that this definition defines is a closed subset, recall that the condition \(\operatorname{inv}(F, F') \in \{ \operatorname{id}, s_i\} \) is equivalent to asking that \(F_j = F'_j\) for all \(j\ne i\).
Here we denote by \(F_0:=E\) the standard flag.
Similarly, we define
This defines an open subscheme of \(D(i_\bullet )\).
We denote by \(\pi :=\pi _{i_\bullet }\colon D(i_\bullet )\to {\rm Flag}\) the projection to the last factor, i.e., the map \((F_\nu )_\nu \mapsto F_\ell \).
The natural projections
\[ D(i_1, \dots , i_\ell ) \to D(i_1, \dots , i_{\ell -1}) \to \cdots \to D(i_1) \to \operatorname{Spec}(k) \]are Zariski-locally trivial \(\mathbb {P}^1_k\)-bundles (i.e., Zariski-locally on the target of each morphism the source is isomorphic to a product with \(\mathbb {P}^1_k\)).
The scheme \(D(i_1,\dots , i_\ell )\) is a smooth projective \(k\)-scheme
The restriction to \(\pi ^{-1}(C(w))\) induces an isomorphism \(\mathbb {A}^{\ell (w)}\cong D(i_\bullet )^\circ \overset {\sim }{\to }C(w)\).
The map \(\pi _{i_\bullet }\) has image \(X(w)\).
Part (1) is clear on \(k\)-valued points and with a bit more effort can be checked to be valid scheme-theoretically. Part (2) follows immediately from (1). Part (3) follows from Lemma 5.27.
Since \({\rm Flag}\) is proper, so is the closed subscheme \(D(i_\bullet )\subset \prod {\rm Flag}\). Hence the morphism \(D(i_\bullet )\to {\rm Flag}\) is proper and in particular has closed image. Since \(D(i_\ell )\) and \(X(w)\) are reduced, Part (4) follows from (3).
In particular, \(\pi \colon D(i_\bullet )\to X(w)\) is a “resolution of singularities” of \(X(w)\), i.e., a surjective proper birational map onto \(X(w)\) with smooth source. It is called the Demazure resolution (or the Bott-Samelson resolution).
Using the theory of “Frobenius splittings” (see [ Ra ] , [ BK ] ) one can show the following result.
\(\pi _*{\mathscr O}_{D(i_\bullet )} = {\mathscr O}_{X(w)}\),
\(R^q\pi _*{\mathscr O}_{D(i_\bullet )} = 0\) for all \(q {\gt} 0\),
\(R^q\pi _*\omega _{D(i_\bullet )} = 0\) for all \(q {\gt} 0\).
Problem 43. (See [ Ra ] .)