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5 Grassmannians, flag varieties, Schubert varieties

General references: [ GW1 ] Ch. 8.

(5.1) The Grassmannian variety

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.

Definition 5.1
Let \(0\le r\le n\). We define the Grassmann functor by
\[ \operatorname{Grass}_{r,n}(S) = \{ {\mathscr U}\subseteq {\mathscr O}_S^n;\ {\mathscr O}_S^n/{\mathscr U}\ \text{is a locally free ${\mathscr O}_S$-module of rank}\ n-r\} . \]
We turn this into a functor as follows. For a morphism \(f\colon S'\to S\) the map \(\operatorname{Grass}_{r,n}(S)\to \operatorname{Grass}_{r,n}(S')\) is given by pullback, i.e., \({\mathscr U}\mapsto f^*{\mathscr U}\). Note that the natural map \(f^*{\mathscr U}\to f^*{\mathscr O}_S^n = {\mathscr O}_{S'}^n\) is in fact injective (since \({\mathscr O}^n/{\mathscr U}\) is locally free, cf. Lemma 4.6) and that \({\mathscr O}_{S'}^n/f^*{\mathscr U}\cong f^*({\mathscr O}_S^n/{\mathscr U})\) is locally free of rank \(n-r\). The same lemma implies that \({\mathscr U}\) itself is locally free (of rank \(r\)).

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.

Definition 5.2
  1. 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.

  2. 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.

  3. 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 \(T\).

Definition 5.3
A functor \(F\colon {\rm (Sch)}{}^{\rm opp}\to {\rm (Sets)}\) is a sheaf for the Zariski topology, if for every scheme \(S\) and open cover \(S= \bigcup _i U_i\) the “sheaf sequence”
\[ F(S)\rightarrow \prod _i F(U_i)\rightrightarrows \prod _{i,j} F(U_i\cap U_j) \]
is exact (i.e., is an equalizer in the category of sets).

With this terminology, “gluing of morphisms” via the Yoneda lemma becomes the following proposition.

Proposition 5.4
Every representable functor \(F\colon {\rm (Sch)}{}^{\rm opp}\to {\rm (Sets)}\) is a sheaf for the Zariski topology.

Proposition 5.5
( [ GW1 ]  Theorem 8.9) Let \(F\colon {\rm (Sch)}{}^{\rm opp}\to {\rm (Sets)}\) be a functor that has the following properties:
  1. \(F\) is a sheaf for the Zariski topology,

  2. \(F\) has an open cover by representable subfunctors.

Then \(F\) is representable.

Sketch of proof

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):

Lemma 5.6
The Grassmann functor \(\operatorname{Grass}_{r,n}\) is a sheaf for the Zariski topology.

Definition 5.7
Let \(0\le r\le n\). For \(I\subseteq \{ 1, \dots , n\} \) of cardinality \(\# I = n-r\) let \({\mathscr O}_S^I\subseteq {\mathscr O}_S^n\) denote the free \({\mathscr O}_S\)-module generated by the standard basis vectors \(e_i\), \(i\in I\). We define
\begin{align*} \operatorname{Grass}_{r, n}^I(S) = \{ & {\mathscr U}\in \operatorname{Grass}_{r,n}(S);\\ & \text{the composition}\ {\mathscr O}_S^I\hookrightarrow {\mathscr O}_S^n\to {\mathscr O}^n_S/{\mathscr U}\ \text{is an isomorphism}\} . \end{align*}

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

\[ \{ s\in S;\ \kappa (s)^I\to \kappa (s)^n\to {\mathscr O}_S^n/{\mathscr U}(s) \} \]

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.

Lemma 5.8
For \(r, n, I\) as above, the functor \(\operatorname{Grass}_{r, n}^I\) is representable by \(\mathbb {A}^{r(n-r)}\).

Proof

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

\[ 0\to {\mathscr U}\to {\mathscr O}_S^n \to {\mathscr O}_S^n/{\mathscr U}\to 0. \]

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:

Theorem 5.9
The Grassmann functor \(\operatorname{Grass}_{r,n}\) is representable by a scheme of finite type over \(\mathbb {Z}\). It admits an open cover by copies of \(\mathbb {A}^{r(n-r)}\), in particular it is smooth over \(\mathbb {Z}\).

(5.2) The Plücker embedding

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.

Theorem 5.10
( [ GW1 ] Section (8.10)) Let \(0\le r\le n\) and \(N=\binom {n}{r}-1\). The morphism \(\operatorname{Grass}_{r,n}\to \mathbb {P}^N\) of functors given by
\[ \operatorname{Grass}_{r,n}(S)\to \mathbb {P}^N(S), \quad {\mathscr U}\mapsto \left(\bigwedge ^r\nolimits {\mathscr U}\subseteq \bigwedge ^r\nolimits {\mathscr O}_S^n\right) \]
gives rise to a closed immersion of schemes.

Remarks on the proof

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).

Corollary 5.11
Let \(0\le r\le n\) and \(N=\binom {n}{r}-1\). The Grassmann scheme \(\operatorname{Grass}_{r, n}\) is proper over \(\operatorname{Spec}(\mathbb {Z})\).

(5.3) Canonical bundle and Picard group of the Grassmannian

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

\[ 0\to {\mathscr U}\to {\mathscr O}_{\rm Gr}^n\to {\mathscr Q}\to 0 \]

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.

Proposition 5.12
( [ GW2 ] Theorem 17.46) We have an isomorphism
\[ \Omega _{{\rm Gr}/\mathbb {Z}}\cong \mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_{\rm Gr}}({\mathscr Q}, {\mathscr U}). \]

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).

Corollary 5.13
With notation as above, let \(\omega _{\rm Gr} := \bigwedge ^{r(n-r)}\Omega _{{\rm Gr}/\mathbb {Z}}\) be the canonical bundle of the Grassmannian. Then
\[ \omega _{\rm Gr} \cong \left(\bigwedge \nolimits ^r{\mathscr U}\right)^{\otimes n} \]

Proof

The computation of \(\Omega _{{\rm Gr}/\mathbb {Z}}\) give us, denoting the highest exterior power of a locally free sheaf \({\mathscr E}\) by \(\det ({\mathscr E})\),

\begin{align*} \omega _{\rm Gr} & \cong \det \left({\mathscr Q}^\vee \otimes {\mathscr U}\right)\\ & \cong \det ({\mathscr Q})^{\vee , \otimes r}\otimes \det ({\mathscr U})^{\otimes (n-r)}\\ & \cong \det \left({\mathscr U}\right)^{\otimes n}, \end{align*}

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 \({\mathscr Q}\)) \(\det \left({\mathscr Q}\right)^{\vee }\cong \det \left({\mathscr U}\right)\).

Lemma 5.14
Le \(X\) be a ringed space and let \({\mathscr E}\), \({\mathscr F}\) be locally free sheaves on \(X\) of finite ranks \(e\) and \(f\), respectively. Write \(\det (-)\) for the highest exterior power of a locally free sheaf of finite rank. Then
\[ \det ({\mathscr E}\otimes {\mathscr F})\cong \det ({\mathscr E})^{\otimes f}\otimes _{{\mathscr O}_X}\det ({\mathscr F})^{\otimes e}. \]

Sketch of proof

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

\[ (b_1\otimes c_1)\wedge (b_1\otimes c_2)\wedge \cdots (b_e\otimes c_f)\mapsto (b_1\wedge \cdots \wedge b_e)^{\otimes f}\otimes (c_1\wedge \cdots c_f)^{\otimes e}. \]

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.

Proposition 5.15
Let \(k\) be a field, and let \(0 {\lt} r {\lt} n\). Let \(\operatorname{Grass}_{r, n}\) be the Grassmannian of \(r\)-dimensional subspaces in \(n\)-dimensional space over \(k\), and let \({\mathscr U}\) denote the universal object as above. Then pullback under the Plücker embedding maps \({\mathscr O}(-1)\) to \(\bigwedge ^r {\mathscr U}\) and induces an isomorphism \(\operatorname{Pic}(\operatorname{Grass}_{r, n})\cong \operatorname{Pic}(\mathbb {P}^N_k) = \mathbb {Z}\).

Remarks on the proof

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.

(5.4) The flag variety

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

Proposition/Definition 5.16
Let \(n\ge 1\).
  1. 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 \(\prod _{i=1}^{n-1} \operatorname{Grass}_{i,n}\) which we also denote by \({\rm Flag}\). The scheme \({\rm Flag}\) is projective over \(\mathbb {Z}\).

  2. 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 \(F_\nu \in \operatorname{Grass}_{i_\nu , n}\).

Sketch of proof

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

\[ S\mapsto \{ (F, F')\in \operatorname{Grass}_{r, n}(S)\times \operatorname{Grass}_{r', n}(S);\ F\subseteq F' \} \]

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 \(J':=\{ 1,\dots , n\} \setminus I'\), 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.

Proposition 5.17
Fix a non-empty set \(I = \{ i_1 {\lt} \cdots {\lt} i_s\} \subseteq \{ 1,\dots , n\} \). The (partial) flag variety \({\rm Flag}_I\) admits an open cover by affine spaces. It is a smooth projective \(\mathbb {Z}\)-scheme of relative dimension \(i_1(n-i_1) + (i_2-i_1)(n-i_2) + \cdots + (i_s-i_{s-1})(n-i_s)\).

Proof

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

\[ G^w_i = \langle e_{w(n)}, \dots , e_{w(n-i+1)}\rangle . \]

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

\[ {\rm Flag}^w(S) := \{ (F_i)_i\in {\rm Flag}(S);\ \forall i\colon G^w_{n-i} \to {\mathscr O}_S^n/F_i\ \text{is an isomorphism} \} . \]

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.)

Remark 5.18

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.

(5.5) The Gauß-Bruhat algorithm

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 \(W\) 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.

Proposition 5.19
Let \(w\in S_n\) be a permutation. Then the number
\[ \ell (w) := \# \{ (i,j);\ i{\lt}j,\ w(i) {\gt} w(j)\} \]
of inversions of \(w\) is equal to the minimal number of factors in a product of simple transpositions that equals \(w\). We call this number the length of \(w\).

Proof

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.

Lemma 5.20

(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.)

  1. We have \(G = \bigcup _{w\in W} BwB\) (disjoint union of \(B\)-double cosets), where \(W\subset GL_n(k)\) denotes the subgroup of permutation matrices. This is called the Bruhat decomposition of \(GL_n(k)\).

  2. 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.

  3. 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 \({\rm Flag}(k)\).

Proof

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:

  1. Add a multiple of a row of the matrix \(g\) to another row lying above the given one.

  2. Add a multiple of a column of the matrix \(g\) to another column lying to the right of the given one.

  3. 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. As before, \(W\subset GL_n(k)\) denotes the subgroup of permutation matrices.

Jan. 31,
2024

Proposition 5.21

(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.

  1. 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.

  2. 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 \(GL_n\).

  3. 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.

  4. 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.

Definition 5.22
Let \(k\) be a field. For flags \(F_\bullet \), \(F'_\bullet \) in \(k^n\) we write
\[ \operatorname{inv}(F_\bullet , F'_\bullet ) = w\in S_n, \]
if for all \(i,j\) we have
\[ \dim (F_i\cap F'_j) = \# \{ \nu \in \{ 1,\dots , j\} ;\ w(\nu )\le i \} \]
and call \(w\) the relative position of the flags \(F_\bullet \) and \(F'_\bullet \).

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

\[ \operatorname{inv}\colon {\rm Flag}(k)\times {\rm Flag}(k)\to S_n. \]

Example 5.23

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 \(\operatorname{inv}(F_\bullet , F'_\bullet ) = s\) 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,

\[ C(w)(k) = \{ F\in {\rm Flag}(k);\ \operatorname{inv}(E_\bullet , F) = w\} . \]

Remark 5.24
  1. 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 \(k^n\).

  2. 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 ] .

(5.6) Schubert varieties

Let \(k\) be a field. We denote by \({\rm Flag}\), \(C(w)\) the flag variety and Schubert cells over \(k\).

Definition 5.25
Let \(w\in S_n\). The Schubert variety \(X(w)\) attached to \(w\) is the reduced closed subscheme of \({\rm Flag}\) whose underlying space is the closure of \(C(w)\).

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.

Example 5.26
We have \(X(\operatorname{id})=C(\operatorname{id})\cong \operatorname{Spec}(k)\). For \(s\in \mathbb S\), \(X(s)\cong \mathbb {P}^1_k\). For the longest element \(w_0\in W\), \(C(w_0)\) is (open and) dense in \({\rm Flag}\), whence \(X(w_0)={\rm Flag}\).

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

\[ X(w) = \bigcup _{v\le w} C(v), \]

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.

(5.7) The Demazure resolution

We work over a fixed field \(k\) (i.e., \({\rm Flag}\) denotes the flag variety over \(k\), …).

Lemma 5.27
Let \(w\in S_n\) and \(s\in \mathbb S\) such that \(\ell (ws) {\gt} \ell (w)\). Let \(F_i\in {\rm Flag}(k)\).
  1. If \(\operatorname{inv}(F_1, F_2) = w\), \(\operatorname{inv}(F_2, F_3) = s\), then \(\operatorname{inv}(F_1, F_3) = ws\).

  2. Given \(F_1\), \(F_3\) with \(\operatorname{inv}(F_1, F_3) = ws\), there exists a unique \(F_2\) with \(\operatorname{inv}(F_1, F_2) = w\), \(\operatorname{inv}(F_2, F_3) = s\).

Proof

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.

Proposition 5.28

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 \(\prod _{\nu =1}^\ell {\rm Flag}\) such that for every \(k'/k\) we have

\[ D(i_1,\dots , i_\ell )(k') = \left\{ (F_\nu )_{\nu =1,\dots , \ell } \in \prod _{\nu =1}^\ell {\rm Flag}(k');\ \forall \nu \colon \operatorname{inv}(F_{\nu -1}, F_\nu ) \in \{ \operatorname{id}, s_{i_\nu }\} \right\} . \]

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

\[ D(i_1,\dots , i_\ell )^\circ = \left\{ (F_\nu )_{\nu =1,\dots , \ell } \in \prod _{\nu =1}^\ell {\rm Flag};\ \forall \nu \colon \operatorname{inv}(F_{\nu -1}, F_\nu ) = s_{i_\nu } \right\} . \]

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 \((F_\nu )_\nu \mapsto F_\ell \).

Proposition 5.29
With notation as above
  1. 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 \(\mathbb {P}^1_k\)).

  2. The scheme \(D(i_1,\dots , i_\ell )\) is a smooth projective \(k\)-scheme

  3. The restriction to \(\pi ^{-1}(C(w))\) induces an isomorphism \(\mathbb {A}^{\ell (w)}\cong D(i_\bullet )^\circ \overset {\sim }{\to }C(w)\).

  4. The map \(\pi _{i_\bullet }\) has image \(X(w)\).

Proof

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).

(5.8) Demazure resolutions are rational resolutions

Using the theory of “Frobenius splittings” (see  [ Ra ] , [ BK ] ) one can show the following result.

Proposition 5.30
The morphism \(\pi :=\pi _{i_\bullet }\colon D(i_\ell )\to X(w)\) is a rational resolution of \(X(w)\), i.e.,
  1. \(\pi _*{\mathscr O}_{D(i_\bullet )} = {\mathscr O}_{X(w)}\),

  2. \(R^q\pi _*{\mathscr O}_{D(i_\bullet )} = 0\) for all \(q {\gt} 0\),

  3. \(R^q\pi _*\omega _{D(i_\bullet )} = 0\) for all \(q {\gt} 0\).

Corollary 5.31
The Schubert variety \(X(w)\) is Cohen-Macaulay.

Proof

Problem 43. (See  [ Ra ] .)