Automorphisms, forms and parabolic subgroups

Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Original content dedicated to the public domain under CC0.

A root datum determines a pinned split reductive group, but it does not remove the geometry of its subgroups or the possibility of descent. This lesson treats both. Parabolic subgroups provide the geometry needed to move Borel pairs; the pinned isomorphism theorem then determines all automorphisms; torsors record how these local descriptions descend.

Throughout, a reductive group over a scheme is smooth and affine, with connected reductive geometric fibres. A split group has a constant root datum and trivial root lines. Statements with one fixed datum are made on the corresponding open and closed part of the base. We use the preceding lessons on maximal tori, torus centralizers, root groups and the pinned isomorphism theorem. The supporting descent results are the supporting lessons Faithfully flat descent and Quotients and torsors: effective descent of affine schemes and group laws, and the classification of torsors by nonabelian first cohomology. We give the group-specific arguments here.

1. Parabolic subgroups over a field

Let first $k$ be algebraically closed and $G/k$ reductive. A smooth closed subgroup $P\subset G$ is parabolic if $G/P$ is proper. Equivalently, it contains a Borel subgroup. Indeed, the Borel fixed-point theorem applied to the action of a Borel $B$ on the proper quotient gives a fixed point $gP$, and hence $g^{-1}Bg\subset P$. Conversely, the classification below identifies every subgroup containing $B$ with a cocharacter parabolic, whose quotient is projective.

Fix $T\subset B$, positive roots $\Phi^+$, and simple roots $\Delta$. For $J\subset\Delta$, choose a cocharacter $\lambda$ such that

Such a cocharacter exists by taking a rational point of the indicated face and clearing denominators. Put $P_J=P_G(\lambda)$, the subgroup on which conjugation by $\lambda(a)$ has a limit at $a=0$. The root-coordinate construction in Root data, Weyl chambers and the Bruhat decomposition, Section 8, gives

where $L_J$ has roots $\Phi\cap\mathbf ZJ$, and $U_J$ has the positive roots outside that subsystem. In particular $P_J$ is smooth and connected and contains $B$.

Theorem 1.1. The parabolic subgroups containing $B$ are exactly the $P_J$, for the subsets $J\subset\Delta$. Each is its own scheme-theoretic normalizer. Its quotient in $G$ is smooth and projective.

Proof of the classification. Suppose $P\supset B$. By Bruhat decomposition, if a point of the cell $Bn_wB$ belongs to $P$, then $n_w\in P$ and the entire cell belongs to $P$. Thus

is a subgroup of $W$, and $P$ is the union of its Bruhat cells. If $w=s_{i_1}\cdots s_{i_m}$ is reduced, the closure of its cell contains the cells of each $s_{i_j}$. Here is the geometric reason, including the fact needed about the closure. Form the successive contracted products of the rank-one minimal parabolics $P_{i_j}$ over $B$. Each rank-one quotient $P_i/B$ is $\mathbf P^1$: the positive roots outside the rank-one Levi are in the shared unipotent radical, so the quotient is the rank-one Borel quotient proved in the third lesson. Their quotient by the last $B$ is consequently an iterated $\mathbf P^1$-bundle. Multiplication gives a proper map to $G/B$. The open rank-one cells in the factors multiply isomorphically to $Bn_wB/B$: their conjugated roots are exactly the distinct inversion roots of the reduced word, and the ordered root-product theorem applies. The image of this proper map is therefore the cell closure. Setting all factors but the $j$-th to the identity includes $n_{s_{i_j}}B$ in the image. Since $P/B$ is closed, every such simple reflection belongs to $W_P$.

Let $J=\{\alpha\in\Delta:s_\alpha\in W_P\}$. The preceding argument shows $W_P=W_J$, the subgroup generated by these reflections. On the other hand, $n_w\in P_G(\lambda)$ precisely when $w\lambda=\lambda$. To check this, the torus part of $\lambda(a)n_w\lambda(a)^{-1}$ is the cocharacter $w^{-1}\lambda-\lambda$; it extends to a point of the closed torus at zero only when it is zero. The stabilizer of $\lambda$ in $W$ is $W_J$. One can see this without a separate stabilizer theorem: if $w\lambda=\lambda$, choose a simple inversion $\delta$ of a nonidentity $w$. Dominance of $\lambda$ and negativity of $w\delta$ force $\langle\delta,\lambda\rangle=0$. Hence $\delta\in J$, and multiplying by $s_\delta$ lowers the length while preserving $\lambda$. Induction proves the assertion. Consequently $P$ and $P_J$ have the same geometric points. They are reduced closed subgroups, so they are equal.

For the normalizer, let $g\in N_G(P)(k)$. The Borels $B$ and $gBg^{-1}$ of the connected smooth affine group $P$ are conjugate in $P$, by the field conjugacy theorem. After multiplication by an element of $P$, $g$ normalizes $B$, so belongs to $B\subset P$. This proves equality on geometric points. It also proves equality as group schemes: the infinitesimal quotient $\operatorname{Lie}N_G(P)/\operatorname{Lie}P$ consists of $P$-fixed vectors in $\mathfrak g/\mathfrak p$: infinitesimal normalization gives $\operatorname{Ad}(p)X-X\in\mathfrak p$ for every test-valued $p\in P$. Its invariants under $\lambda$ are zero, since all the weights in that quotient are strictly negative. Thus the tangent spaces of the normalizer and $P$ agree. Its reduced group is $P$, and its local dimension and tangent dimension at the identity agree; it is smooth there and, by translation, everywhere. Hence $N_G(P)=P$.

Finally $G/P_J$ is proper: the map $G/B\to G/P_J$ is surjective, and $G/B$ is proper by the field argument in the first lesson. Apply the quotient and projectivity lemma in Roots and reductive groups of rank one, Section 5: a smooth connected self-normalizing subgroup with complete homogeneous fibres has a scheme quotient, and the character $\det(\operatorname{Lie}P_J)^{-1}$ defines an ample line bundle. This proves projectivity. Smoothness follows from the smooth $P_J$-torsor $G\to G/P_J$. $\square$

The ordering is $P_J\subset P_K$ when $J\subset K$. The empty subset gives $B$; the full subset gives $G$. Thus allowing $G$ itself as a parabolic adds one point to the parameter scheme.

2. Parabolics and their parameter scheme over a base

Over $S$, it is useful initially to allow a smooth affine monomorphism $P\to G$ whose geometric fibres are parabolic subgroups. The theorem below shows that this is automatically a closed immersion. This definition therefore does not hide a closedness hypothesis in the conclusion.

Theorem 2.1. A parabolic subgroup of a reductive $S$-group is, étale locally on $S$, a standard cocharacter parabolic. It is closed, has connected fibres, and satisfies $N_G(P)=P$. The functor of parabolic subgroups is represented by a smooth projective $S$-scheme $\operatorname{Par}(G)$, and formation of this scheme commutes with every base change.

Proof of the local description. Apply the torus-lifting and conjugacy argument of the first lesson to the smooth affine group $P$. Étale locally, a maximal torus of a chosen geometric fibre lifts to a torus $T\subset P$. It is maximal also in $G$, by the field classification. Shrink and split $T$, and make the root datum and root lines constant as in the third and fourth lessons.

The differential of the monomorphism $P\to G$ is a fibrewise injection of vector bundles; hence its image is a locally direct summand of $\operatorname{Lie}G$. As a $T$-module it contains $\operatorname{Lie}T$ and either the whole of each root line or none of it. The selection is locally constant. Choose $\lambda$ from the parabolic pattern in the chosen fibre. All weights of $\operatorname{Lie}P$ have nonnegative pairing with $\lambda$. The dynamic subgroup $P_P(\lambda)\subset P$ is smooth and has exactly those nonnegative tangent weights, by the graded-coordinate proof in the third lesson. Its inclusion is therefore étale; on every geometric fibre it is an isomorphism, since that fibre is the connected parabolic with this pattern. It follows that $P_P(\lambda)=P$. The limit functor now factors $P\to G$ through $P_G(\lambda)$. This is a monomorphism between smooth schemes, with an isomorphism on every geometric fibre; its differential is an isomorphism, so it is an étale, surjective monomorphism, and hence an isomorphism. In particular the original $P\to G$ is closed.

The formal separation and normalizer lemmas in the rank-one lesson show that $N_G(P)$ is closed of finite presentation and commutes with every base change. Their proof uses a finite prime filtration, Artin–Rees and the determinant trick to detect subgroup equations over a universal Noetherian coefficient ring; it therefore also applies on test schemes with nilpotents. Its geometric fibres are the normalizers calculated in Theorem 1.1.

We use the flat-source criterion proved in the rank-one lesson directly over the original base. Locally a finitely presented closed immersion \(X\hookrightarrow Y\) over \(\operatorname{Spec}R\) has a finitely generated ideal \(I\subset C=\mathcal O(Y)\). If \(C/I\) is \(R\)-flat, tensoring \(0\to I\to C\to C/I\to0\) with \(\kappa(s)\) makes \(I\otimes_R\kappa(s)\to C\otimes_R\kappa(s)\) injective. Equality of every geometric fibre makes this ideal zero, first after a faithful field extension and then over \(\kappa(s)\). For a prime \(x\subset C\) over \(s\), this says \(I_x=\mathfrak p_sI_x\); the finitely generated \(C_x\)-module \(I_x\) vanishes by Nakayama. Hence \(I=0\). Apply this to \(P\hookrightarrow N_G(P)\), using the fibre equality in Theorem 1.1 and flatness of \(P/S\). Neither Noetherianity of the base nor flatness of the normalizer is required.

The quotient $G/P$ is therefore smooth and projective by the quotient lemma just used, now over $S$. Its ample line bundle comes canonically from $\det(\operatorname{Lie}P)^{-1}$.

For a pinned split $G$ of constant datum, we obtain

Here a morphism to the disjoint union may choose different $J$'s on open and closed parts. To see that (2.1) represents the subgroup functor on every test scheme, a section of $G/P_J$ gives a conjugate of $P_J$ after a local lift to $G$. Conversely the preceding local classification supplies such lifts for any parabolic. Two lifts give the same subgroup exactly when their quotient lies in $N_G(P_J)=P_J$, so the sections glue uniquely. This proves the functorial assertion, including uniqueness over bases with nilpotents.

For an arbitrary reductive \(G\), the local schemes just constructed have unique identifications on overlaps, because they represent the same subgroup functor. They satisfy the cocycle condition. The universal subgroup \(\mathcal P\) gives the canonical line bundle \(L=\det(\operatorname{Lie}\mathcal P)^*\); its identifications also satisfy the cocycle. We show directly that these projective schemes with their line bundle descend to a projective scheme.

Work on a quasi-compact affine part of \(S\) and take finitely many affine étale charts \(S_i\to S\) carrying the split descriptions. Their disjoint union is a faithfully flat quasi-compact cover. On each chart the Grassmannian construction of AG-RG-03 Lemma 5.A, applied in its Lemma 5.1, gives a positive power of \(L_i\) and a specific closed Plücker embedding on every parabolic component. Choose a common exponent and place these finitely many components in disjoint coordinate blocks of one projective bundle, using the direct sum of their embedding modules. This gives a closed projective embedding of their disjoint union. Descend the charts, projective schemes, these specified embeddings, line bundles and their identifications to a common Noetherian coefficient stage. S02 Proposition 3.2 retains smoothness and the line bundles, and finite-presentation descent retains the closed embeddings and their finitely many identities. Thus ampleness on the model is witnessed by retained embeddings, rather than inferred from fibres of the original base.

Choose a common multiple \(b>0\) of the finitely many embedding exponents. Its sufficiently large multiples \(m\) are still very ample, by the polynomial Veronese maps. S01 Theorem 8.5 makes the positive cohomology of \(L_i^m\) vanish over each affine Noetherian model chart for sufficiently large \(m\). To pass to all geometric fibres, use the universal finite projective complex of S02 Lemma 5.1, in degrees \([0,N]\), for this smooth projective family. Its positive cohomology is zero over that affine chart. Starting at the last term, its exact positive tail splits because the last nonzero term is projective; repeat to remove the whole positive tail. The remaining degree-zero module is finite projective, and the splitting survives every tensor. Thus \(H^q(X_{i,\bar s},L_i^m)=0\) for every \(q>0\) on every geometric model fibre. A single sufficiently large common multiple works on the finitely many charts. Pull this choice back to the original cover. S02 Proposition 5.2 gives \(E_i=f_{i*}L_i^m\) finite locally free, with every base change. The overlap identifications of schemes and \(L_i\) therefore identify the \(E_i\) compatibly. Faithfully flat module descent produces a finite locally free \(E/S\).

Evaluation defines compatible closed immersions \(X_i\hookrightarrow\mathbf P(E)|_{S_i}\). Their complete section systems contain the sections of the chosen very ample systems, so on each generating target chart the coordinate-ring map is still surjective. This first gives an immersion; the map is also proper by the graph argument of S01 Lemma 8.3, since \(X_i/S_i\) is projective. Its image is closed, and the chartwise closed immersions together with the empty inverse image off that closed image make the immersion closed globally. On affine charts of \(\mathbf P(E)\), the ideal sheaves of these immersions have compatible faithfully flat descent data. Module descent gives their ideal submodules, and multiplication preserves them because that fact is checked on the faithfully flat cover. Their quotient algebras define a closed \(S\)-subscheme \(X\subset\mathbf P(E)\), whose pullback is each \(X_i\). The represented subgroup functor and smoothness descend through the same cover, so \(X=\operatorname{Par}(G)\) is smooth and projective. The construction is independent of charts by its representing identity and hence glues over \(S\). Finally the subgroup functor itself commutes with every base change; Yoneda identifies the pullback of its representing scheme with \(\operatorname{Par}(G_{S'})\). \(\square\)

For a nonsplit group, subsets of a chosen simple-root set need not be globally labelled. The finite étale type scheme is the descent of the finite set of subsets of $\Delta$. The map from $\operatorname{Par}(G)$ to this scheme remembers the type. Borel subgroups are its component of minimal parabolics, corresponding locally to $J=\varnothing$.

3. Automorphisms of a pinned split group

Fix a pinned split group

of based root datum $\mathcal R_b$. Let $\Gamma=\operatorname{Aut}(\mathcal R_b)$, including its action on the full character and cocharacter lattices. Regard it as a constant group scheme. The pinned isomorphism theorem gives a unique pinned automorphism $s_\gamma$ inducing each $\gamma\in\Gamma$; uniqueness gives $s_\gamma s_\delta=s_{\gamma\delta}$.

Write $G^{\mathrm{ad}}=G/Z(G)$. This is the adjoint semisimple quotient: central torus directions disappear, but they remain present in $\Gamma$.

Theorem 3.1. On all $S$-schemes, including nonreduced ones, the automorphism functor is represented by

This group scheme is smooth, separated and locally of finite presentation. It need not be of finite type.

Proof. Put $q:G\to G^{\mathrm{ad}}$ and $T^{\mathrm{ad}}=q(T)$. The scheme-centre calculation in RG03 Section 1 gives $Z(G)=D_S(X/\mathbf Z\Phi)$. The central quotient theorem of RG03 Theorem 1.1 and RG04 Theorem 7.1 therefore gives the fppf quotient $q$, the torus $T^{\mathrm{ad}}=D_S(\mathbf Z\Phi)$ and the images of all root parameters with their unchanged frames. In particular $\mathbf Z\Phi=\bigoplus_{\alpha\in\Delta}\mathbf Z\alpha$ is its character lattice. Conjugation by a section of $G^{\mathrm{ad}}$ is well defined on $G$: lift the section through $q$ fppf locally, conjugate there, and descend the group morphisms by S04 Lemma A1.5. Two lifts differ by a section of the scheme centre and give the same morphism on every test scheme. Each $s_\gamma$ preserves this centre and induces $s_\gamma^{\mathrm{ad}}$ on the quotient. The morphism in (3.1) respects multiplication, since

Let $a$ be an automorphism after any base change $S'\to S$. Theorem 2.1 identifies the Borel $aB$ with a section of $G/B$. Pulling back the fppf $B$-torsor $G\to G/B$ permits us locally to conjugate $aB$ to $B$. Its transported $aT$ is a maximal torus of $B$. The torus transporter in this smooth affine $B$ is smooth by RG01 Lemma 3.1; its geometric fibres are nonempty by the solvable-group torus-conjugacy proof in RG01 Section 2. It is consequently a smooth surjection, and its own tautological section supplies an fppf cover on which a further conjugation in $B$ takes this torus to $T$. We have reduced to an automorphism $b$ preserving both $T$ and $B$.

RG01 Lemma D.1 identifies $b|_T$ with a locally constant automorphism of the full integral character lattice, including its central directions. Preservation of $B$ and the canonical root and coroot maps identifies its value with a section $\gamma$ of $\Gamma_{S'}$. A locally constant value may be treated on its open-and-closed fibres, without assuming that there are finitely many such values. Set $\beta=s_\gamma^{-1}b$. This automorphism induces the identity on the entire based datum. On each framed simple root line it has the form $d\beta(E_\alpha)=u_\alpha E_\alpha$, with $u_\alpha\in\mathcal O_{S'}^\times$: an automorphism of a free rank-one module is multiplication by a unit. Because the simple roots are a basis of $X^*(T^{\mathrm{ad}})$, the rule $\alpha(t)=u_\alpha^{-1}$ defines one and only one section $t$ of $T^{\mathrm{ad}}$. Conjugation by it has precisely these multipliers on the root frames, by the quotient root-equivariance formula of RG04 Theorem 7.1. Thus $\operatorname{Int}(t)\beta$ fixes the pinning and induces the identity on its full based datum. RG05 Theorem 5.1 makes it the identity automorphism. Undoing the conjugations and using the displayed multiplication identity writes $a=\operatorname{Int}(g)s_\gamma$ fppf locally. This proves local surjectivity.

We prove injectivity on the same arbitrary test scheme. Suppose $\operatorname{Int}(h)s_\gamma=1$. Then $\operatorname{Int}(h)=s_\gamma^{-1}$ preserves $T$, $B$ and therefore their quotient torus and Borel in $G^{\mathrm{ad}}$. RG04 Theorem 4.2, with the pinned representatives, identifies the entire normalizer of $T^{\mathrm{ad}}$ with the copies $n_wT^{\mathrm{ad}}$ labelled by the constant Weyl sheaf. Hence $h$ has a locally constant Weyl label $w$. Preservation of the Borel preserves its positive root lines, so this label preserves the positive chamber. RG04 Lemma 3.A gives $w=1$. The identity-labelled copy is the torus itself as a scheme; thus $h\in T^{\mathrm{ad}}(S')$, including when $S'$ has nilpotents. Its inner automorphism is the identity on $T$, so $\gamma=1$ on the entire datum. The equation now says that every simple frame is fixed, and therefore $\alpha(h)=1$ for every $\alpha\in\Delta$. Their character-basis property gives $h=1$. The kernel is trivial on every test scheme, which proves injectivity of the group-functor map.

The locally constructed factors of $a$ consequently agree on double overlaps. S04 Lemma A1.5 glues them uniquely as sections of $G^{\mathrm{ad}}$ and $\Gamma_S$, and the same lemma descends their equality with $a$. Every construction commutes with base change, so these sections give the inverse natural transformation to (3.1). The representing scheme is the disjoint union, indexed by $\Gamma$, of copies of the smooth affine finitely presented $G^{\mathrm{ad}}$. It is smooth and locally of finite presentation. Its relative diagonal is closed: on each pair of indexed components it is either the closed diagonal of $G^{\mathrm{ad}}$ or empty. Thus it is separated as well. This proves all the assertions. $\square$

For a semisimple datum, $\Gamma$ is finite: its roots span the rational character space, so their images determine a rational linear map and its restriction to the free character lattice. An automorphism is therefore determined by its permutation of the finite root set, which injects $\Gamma$ into that finite permutation group. For a torus of rank $r$, $G^{\mathrm{ad}}=1$ and $\Gamma=\operatorname{GL}_r(\mathbf Z)$, which is infinite for $r\geq2$. Thus the finite-type assertion cannot be added in general.

Even a small central direction matters. On $\operatorname{GL}_2$,

is an involutive automorphism. On the diagonal torus it sends $(t_1,t_2)$ to $(t_2^{-1},t_1^{-1})$. It preserves the positive root and reverses the scalar direction, so is visible in the full based datum. It is not inner, since it acts nontrivially on the centre.

4. Forms and torsors

A form of $G_0/S$ is an $S$-group which becomes isomorphic to $G_0$ on an étale cover. Let $A=\operatorname{Aut}(G_0)$.

Proposition 4.1. Isomorphism classes of forms of $G_0$ are in natural bijection with $H^1_{\mathrm{et}}(S,A)$.

Proof. Choose local isomorphisms $\phi_i:G_0|_{S_i}\to G|_{S_i}$. On overlaps,

Changing the $\phi_i$'s changes this cocycle by a coboundary. Equivalently, $\operatorname{Isom}(G_0,G)$ is a right $A$-torsor: after choosing a local isomorphism, every other one is its composite with a unique automorphism of $G_0$. Conversely an $A$-torsor has local sections on an étale cover; their transition automorphisms give a descent datum on the affine scheme $G_0$. S04, Lemmas A1.2 and A1.5, give effective descent of its coordinate algebras and compatible morphisms. The group operations descend with these morphisms, and their identities follow by faithful descent. Thus one obtains an affine $S$-group with the prescribed local isomorphisms. Its finite presentation and smoothness descend by S04, Lemma A1.3a.

For the remaining fibre assertion, take a geometric point $\operatorname{Spec}\Omega\to S$, with $\Omega$ algebraically closed. The pulled-back étale cover is nonempty. A nonempty affine chart of it is of finite type over $\Omega$, so a maximal ideal gives an $\Omega$-point by AG-CA-06, Theorem 2.1, whose field-algebra proof is Theorem 1.3 there. Pulling a covering isomorphism back along that point identifies the descended geometric fibre with $(G_0)_\Omega$. Hence that fibre is connected and reductive. This proves reductivity over every geometric point of the arbitrary base.

The two constructions recover each other's cocycles and local isomorphisms. Compatibility and uniqueness in S04, Lemma A1.5, identify their descended objects and morphisms, so they are mutually inverse on isomorphism classes. This is the concrete torsor argument from Quotients and torsors, applied to the automorphism sheaf $A$. $\square$

For split pinned $G_0$, an inner form with inner structure is obtained from a $G_0^{\mathrm{ad}}$-torsor by conjugation. These structures are classified by $H^1(S,G_0^{\mathrm{ad}})$. Forgetting the inner structure need not be injective on isomorphism classes. The exact sequence split by the pinning is

The fibre over the neutral element of $H^1(S,\Gamma_S)$ is the set of $\Gamma(S)$-orbits on $H^1(S,G_0^{\mathrm{ad}})$. Indeed a trivialization of the induced $\Gamma$-torsor reduces the original cocycle to $G_0^{\mathrm{ad}}$; changing that trivialization by $\gamma$ changes the inner cocycle by its $\gamma$-action. This also explains why a marked inner structure and the underlying group are different classification problems.

5. Quasi-split forms

A reductive group is quasi-split if it admits a Borel subgroup over the base. Twisting the pinned $G_0$ by a $\Gamma$-torsor gives a quasi-split form: the pinned automorphisms preserve $T$, $B$ and the simple-root frames as an indexed collection, so these data descend.

Over a field, this accounts for every quasi-split form of $G_0$, up to the corresponding based-datum form. We supply the cohomological step because the torus which occurs in it is usually twisted.

Proposition 5.1. For a field $k$, the forms of $G_0$ that admit a $k$-Borel are exactly the forms obtained by twisting its pinning by $\Gamma$-torsors. Their based-datum classes lie in $H^1(k,\Gamma)$.

Proof. Start with its local group isomorphisms. By Theorem 2.1 the transported Borel is locally a conjugate of $B$; the smooth Borel torsor has étale local sections by S07 Lemma 2.1. Refining the affine cover and making these conjugations gives isomorphisms of the pairs. Their overlap automorphisms preserve $B$. In Theorem 3.1 write such an automorphism as $\operatorname{Int}(g)s_\gamma$. The pinned factor preserves $B$, so the inner factor must too. The inverse image of $B^{\mathrm{ad}}$ in $G_0$ is exactly $B$, since the centre is in its torus and the quotient root maps are unchanged. The all-test normalizer equality of Theorem 2.1 applied to the adjoint Borel therefore forces $g\in B_0^{\mathrm{ad}}$. Conversely its sections locally lift to $B_0$ and preserve $B$. This proves the entire stabilizer identity, including nonreduced overlap algebras, namely

Fix its $\Gamma$-torsor. Relative to the canonical pinned twist, the possible reductions are torsors under the corresponding twisted $B_0^{\mathrm{ad}}$. Its torus has character lattice freely based by $\Delta$, with the descent action permuting that basis. Descend the finite set $\Delta$, or equivalently the algebra of functions on it: on a splitting affine cover this algebra is the finite product of copies of the covering ring, with overlap permutations of the factors. S04 Lemmas A1.2–A1.3 descend it to a finite étale $k$-algebra $E$. The complete finite-étale field calculation of S07 Lemma 5.3A writes $E=\prod_i k_i$ for finite separable fields $k_i/k$. The group of units of $E\otimes_k R$ is represented by the determinant-invertibility open in the affine space of its underlying finite-dimensional vector space: multiplication by an element is invertible exactly when that determinant is a unit. On the splitting cover it is the product of multiplicative groups, with exactly the same permutation descent datum as the torus. S04 Lemma A1.5 therefore identifies the whole torus functor with this unit group. In particular the torus is

We prove the vanishing needed for this torus. A torsor under $\operatorname{Res}_{k_i/k}\mathbf G_m$ has transition functions which are units in $k_i\otimes_k R$ on a splitting cover $\operatorname{Spec}R\to\operatorname{Spec}k$. The same functions glue a rank-one module over $k_i$, by faithfully flat module descent. Conversely a rank-one $k_i$-module has precisely these local frames and transition functions, so these two constructions are inverse. Every rank-one module over the field $k_i$ has a basis. Its frame torsor has a section, hence is trivial. Taking the product gives $H^1(k,T)=1$. This is the multiplicative descent statement often called Hilbert 90, with its proof and its restriction-of-scalars application included here.

The root-height filtration from RG04 Section 5 descends to the twisted unipotent radical $U$, since a based-datum automorphism preserves positive-root height. The root-coordinate commutator proof makes each quotient $U_h/U_{h+1}$ the additive group of the direct sum of its height-$h$ root lines, with linear permutation and frame transition maps. S04 Lemma A1.2 descends this module to a finite-dimensional $k$-vector space $V_h$. Thus the quotient really is a vector group, rather than an assumed split form of a unipotent group.

We prove additive torsor vanishing directly on a faithfully flat affine cover $k\to R$. For a vector group $V$ its transition cocycle is $c\in V\otimes_k R\otimes_k R$, satisfying the alternating unit-insertion equation $d c=0$ in $V\otimes_k R^{\otimes3}$. The entire augmented complex in S04 Lemma A1.1 is exact in every degree: after tensoring with $R$, multiplication of the extra factor with the first covering factor is its displayed contracting homotopy, and faithful flatness detects exactness downstairs. Consequently $c=d b$ for $b\in V\otimes_k R$. Translation of the local torsor section by $-b$ makes its two restrictions equal; they descend to a global section. Hence every vector-group torsor over $k$ is trivial in every characteristic. This uses no division by a Galois-group order and no unproved Galois-cohomology comparison.

Apply this successively to a $U$-torsor. On an affine faithfully flat cover splitting the root data and trivializing the torsor, its cocycle lies in $U(R\otimes_k R)$. Its image in $V_1$ is $d b_1$ by the preceding calculation. The ordered root coordinates give a lift of $b_1$ to $U(R)$: put its height-one entries in their root factors and every higher entry zero. Changing the local section by this lift puts its cocycle in $U_2$. At height $h$ the same split coordinate product lifts every $b_h\in V_h(R)$ to $U_h(R)$, and removes that quotient cocycle. All filtration subgroups and maps have already descended, so each changed cocycle lies in the stated subgroup on the entire overlap algebra. The finite filtration ends at one. The final equal local sections descend, proving $H^1(k,U)=1$. The same affine cover can be used throughout; if initially several étale affine charts are needed, their finite disjoint union supplies one faithfully flat algebra. Such a finite choice exists because their nonempty images cover the one-point scheme $\operatorname{Spec}k$.

First project a twisted Borel cocycle to $T$, whose torsor has already been proved trivial. On a splitting affine cover the chosen torus coboundary lifts through the split semidirect product $B^{\mathrm{ad}}=T\ltimes U$; changing the section by this lift leaves a $U$-cocycle. Its vanishing just proved trivializes the whole Borel torsor. Thus $H^1(k,B_0^{\mathrm{ad}})=1$ for the Borel twisted by the fixed $\Gamma$-torsor.

There is therefore no further Borel-preserving inner twist. The $\Gamma$-class is recovered from the canonical quotient of the full automorphism group, so it is independent of the chosen Borel. The pinning provides a section on cohomology, and the vanishing just proved leaves exactly that section as the quasi-split class. Thus the based-datum class determines the quasi-split group. $\square$

Over a general base, the analogous vanishing can fail. Here is the whole construction of the example. Choose local frames of a line bundle $L$, with unit transitions $\ell_{ij}$. Glue the all-characteristic $\mathrm{PGL}_2$ of Lemma 6.0 by conjugation with $\operatorname{diag}(1,\ell_{ij})$. These conjugations satisfy the triple cocycle, so affine algebra/group-law descent in S04 makes a smooth affine reductive group denoted $\mathrm{PGL}(\mathcal O_S\oplus L)$. The upper triangular subgroups and diagonal tori are preserved and descend too. The former stabilize $\mathcal O_S$ and are Borels by their geometric fibres. The two off-diagonal tangent entries are the homomorphism bundles $\operatorname{Hom}(L,\mathcal O_S)=L^{-1}$ and $\operatorname{Hom}(\mathcal O_S,L)=L$: in local frames conjugation multiplies $E_{12}$ by $\ell_{ij}^{-1}$ and $E_{21}$ by $\ell_{ij}$. These are exactly the two canonical root lines, by Lemma 6.0. A pinning for this particular Borel pair requires a global generator of its positive root line. Such a generator would identify that line with $\mathcal O_S$, by the local unit criterion; if $L$ is nontrivial neither $L$ nor its inverse admits it. This example concerns the displayed Borel pair and does not assert that every other pair lacks a pinning. The field proof used triviality of rank-one modules over a field, which fails for general line bundles.

6. Inner forms of the special linear group

A central simple algebra over a field $k$ is a finite-dimensional simple algebra with centre $k$. Its matrix form and the separability of its splitting field will be proved below. We first identify the automorphisms that carry its descent data.

Lemma 6.0 (the matrix groups in every characteristic). Over any base, the matrix groups $\mathrm{GL}_n$, $\mathrm{SL}_n$ and the scalar quotient $\mathrm{PGL}_n$ are split reductive with their diagonal tori and their usual $A_{n-1}$ roots and coroots. The determinant-one group has character lattice $\mathbf Z^n/\mathbf Z(1,\ldots,1)$ and cocharacter lattice $\{(a_i):\sum a_i=0\}$; the projective group interchanges them.

Proof. The determinant open is a smooth affine group over $\mathbf Z$. On the determinant-one equation the cofactors form the entries of the inverse transpose, so some cofactor is a unit locally; solve for its matrix entry on that open. This is the full one-equation smooth Jacobian chart over every base, as in AG-CA-17 Theorem 5.1. All matrix operations and determinant identities are the finite polynomial calculations of S06 Section 1. On every algebraically closed geometric field, determinant-one row and column elimination uses a nonzero pivot, the two-coordinate swap from RG05 (1.1), and elementary additions to clear its row and column. Induction leaves a determinant-one diagonal. Each diagonal pair $\operatorname{diag}(d,d^{-1})$ is the four-root product in RG05 Section 10. Thus $\mathrm{SL}_n$ is generated on these field points by $I+uE_{ij}$. Every such one-parameter image is connected and contains one. The full component and Nullstellensatz argument in RG05 Lemma 9.A therefore proves geometric connectedness; the determinant open for $\mathrm{GL}_n$ is irreducible because its polynomial coordinate ring embeds in a field, with intersecting nonempty principal opens, by the prime/irreducibility proof of AG-CA-08.

The diagonal Laurent algebras and GS05 Theorem 2.2 give the stated character and cocharacter lattices. Their full torus weights on the tangent matrices are $e_i-e_j$ on $E_{ij}$, with diagonal zero-weight part, trace zero for $\mathrm{SL}_n$. A smooth connected normal unipotent subgroup $N$ has trivial scheme intersection with the torus by RG01 Lemma 2.A and GS05 Theorem 4.2. On dual numbers the tangent intersection is the tangent space of this scheme intersection, since its equations are the sum of the two ideals. Thus $\operatorname{Lie}N$ has zero diagonal intersection. Its torus character projections, given by the complete exponent-basis grading of GS05 Lemma 2.1, either give zero or include some $E_{ij}$. Normality makes it a Lie ideal: conjugate $1+\epsilon X$ by $1+\eta Y$, with $\epsilon^2=\eta^2=0$, and compare the $\epsilon\eta$ coefficient, exactly as in RG05 Section 6. Bracketing $E_{ij}$ with $E_{ji}$ gives $E_{ii}-E_{jj}$, nonzero even in characteristic two, contradicting the zero diagonal intersection. Hence $\operatorname{Lie}N=0$. S07 Lemma 5.3A makes the smooth connected zero-dimensional group trivial. This proves reductivity in every characteristic.

A torus containing the diagonal one has its Lie algebra in the displayed zero-weight space, so the ranks are equal. The closed inclusion of these tori corresponds by GS05 Theorem 4.2 to a surjection of free character groups of equal rank. Its kernel is a free subgroup of rank zero, by the integer subgroup proof in RG05 Theorem 10.1, and is zero. Thus the tori coincide. The two-coordinate $\mathrm{SL}_2$ maps, polynomial over $\mathbf Z$, give the root parameters $I+uE_{ij}$ and coroot $e_i-e_j$; RG03 Theorems 4.1 and 7.1 identify them with the canonical maps, including on nonreduced tests. The upper triangular subgroup is the Borel for the displayed positive roots by RG04 Section 5, and the $E_{i,i+1}$ give its global simple-root frames.

The central scalar $\mathbf G_m$ is closed on coordinate equations. RG03 Theorem 1.1 constructs its smooth affine reductive quotient $\mathrm{PGL}_n$. RG04 Theorem 7.1 gives its diagonal quotient torus, with character lattice $\{(a_i):\sum a_i=0\}$ and dual lattice $\mathbf Z^n/\mathbf Z(1,\ldots,1)$, and the same root maps with the coroot images. These operations commute with arbitrary base change. For $n=1$, $\mathrm{GL}_1=\mathbf G_m$, while the determinant-one and scalar quotient groups are trivial, so the same statement has empty roots. $\square$

Lemma 6.1. The automorphism functor of the matrix algebra is $\operatorname{PGL}_n$, over any base ring. Thus $\operatorname{PGL}_n$-torsors correspond to degree-$n$ Azumaya algebras.

Proof. Let $\phi$ be an algebra automorphism of $\operatorname{Mat}_n(R)$. Its images $p_i=\phi(E_{ii})$ of the diagonal matrix units are orthogonal idempotents, and $R^n=\bigoplus_i p_iR^n$. Each summand is finitely generated projective, since it is the image of an idempotent. It has rank one on every fibre: after a residue-field extension, the corner $p_i\operatorname{Mat}_n p_i$ is the endomorphism algebra of the corresponding vector-space image, whereas $\phi$ identifies this corner with $E_{ii}\operatorname{Mat}_n E_{ii}$, of dimension one. If the image has dimension $r_i$, its corner has dimension $r_i^2$, so $r_i=1$. Local freeness of finite projective modules, proved in the preceding algebra lesson, now makes these rank-one locally direct summands over $R$ itself. Locally choose a generator $v_1$ of the first summand, and put $v_i=\phi(E_{i1})v_1$. The matrix-unit identities show that the $v_i$'s are a basis and that $\phi(E_{ij})$ sends $v_j$ to $v_i$ and kills the other $v_h$'s. Hence $\phi$ is conjugation by the basis matrix. Two such matrices differ by a scalar, because a matrix commuting with every matrix unit is scalar. This proves the functor identity, with local lifts and descent through $\operatorname{GL}_n\to\operatorname{PGL}_n$. Twisting the matrix algebra by a torsor gives its Azumaya algebra; conversely the sheaf of local algebra isomorphisms is that torsor. Faithfully flat descent of the algebra and its multiplication proves effectivity. $\square$

Theorem 6.2 (separable matrix splitting). Let $k$ be any field and let $A$ be a finite-dimensional simple $k$-algebra with centre $k$. Then $\dim_k A=n^2$ for some $n$, and there is a finite separable extension $K/k$ with $A\otimes_k K\simeq\operatorname{Mat}_n(K)$. Conversely, an algebra obtained by faithfully flat descent of $\operatorname{Mat}_n$ over a field is central simple. Its algebra-isomorphism torsor has group $\operatorname{PGL}_n$.

Proof. We first prove the finite-dimensional density calculation needed here. Let $R$ act on a finite-dimensional simple $k$-module $V$, and suppose $\operatorname{End}_R(V)=k$. For linearly independent $v_1,\ldots,v_m$, the map

is surjective. For $m=1$ this is simplicity. Induct on $m$. The image $L\subset V^m$ projects onto $V^{m-1}$. Its kernel in the last summand is an $R$-submodule of $V$, hence is either $V$ or zero. The first case gives surjectivity. In the second case $L$ is the graph of an $R$-linear map $V^{m-1}\to V$. Each component of this map is an $R$-endomorphism of $V$, hence multiplication by a scalar $c_i$. Applying the graph equation to $r=1$ gives $v_m=\sum_{i<m}c_iv_i$, a contradiction. Taking a basis of $V$ proves that the action map $R\to\operatorname{End}_k(V)$ is surjective.

Apply this calculation to $R=A\otimes_k A^{\mathrm{op}}$ acting on $V=A$ by $(a\otimes b)x=axb$. Its submodules are the two-sided ideals of $A$, so $V$ is simple. An endomorphism commuting with left multiplication is $x\mapsto xc$, where $c$ is its value at $1$; commuting also with right multiplication forces $c\in Z(A)=k$. Density and equality of dimensions therefore give an isomorphism

Tensoring this linear isomorphism with any field extension $E/k$ gives the same isomorphism for $A_E$. A two-sided ideal of $A_E$ is stable under every endomorphism of its underlying $E$-vector space, so is zero or all of $A_E$. Also $Z(A_E)=E$: centrality is the kernel of the finite family of linear commutator maps with a $k$-basis of $A$, and kernels commute with field extension. Thus central simplicity survives every field extension.

Now take $E=\overline{k}$. Choose a proper left ideal of $A_E$ of largest vector-space dimension; such a choice exists among the integers from zero to $\dim_E A_E-1$, and zero is proper. Its quotient $V$ is nonzero and simple, since a larger proper submodule would give a larger proper ideal. An endomorphism of $V$ has submodules as its kernel and image; if nonzero, simplicity makes it bijective, and its linear inverse remains $A_E$-linear by applying its defining intertwining identity. Thus its endomorphism algebra is a finite-dimensional division algebra over the algebraically closed field $E$. Every one of its elements is scalar: its characteristic polynomial as an $E$-linear operator has a root $c$, so the operator minus $c$ is singular; in a division algebra this makes it zero. Density gives $A_E\to\operatorname{End}_E(V)$ surjective, and simplicity makes its kernel zero. Consequently $A_E\simeq\operatorname{Mat}_n(E)$, where $n=\dim_E V$, and $\dim_k A=n^2$.

To obtain a separable splitting, consider the scheme $I=\operatorname{Isom}_{k\text{-alg}}(A,\operatorname{Mat}_n)$. Choose bases of the two algebras. A linear isomorphism is a matrix with invertible determinant; preservation of $1$ and of the products of basis vectors is a finite list of polynomial equations. Thus $I$ is an affine scheme of finite presentation. Its base change to \(\overline k\) has a point by the preceding paragraph. Composition with that point identifies the entire scheme \(I_{\overline k}\) with the automorphism scheme of the matrix algebra, hence with \(\operatorname{PGL}_{n,\overline k}\) by Lemma 6.1. With the quotient convention for projective bundles, \(\operatorname{PGL}_n\) is the determinant-nonvanishing open in \(\mathbf P(\operatorname{Mat}_n^\vee)\), which parameterizes lines in the matrix module: the determinant is a homogeneous section of degree \(n\), and invertible matrices modulo a unit scalar have exactly these projective matrix classes. On a chart where a matrix entry is a unit, normalize that entry to one; this gives a local section of \(\operatorname{GL}_n\to\operatorname{PGL}_n\), and its inverse image is the product of that chart with \(\mathbf G_m\). Thus the open projective scheme represents the quotient on all test rings. It is smooth over every base because projective-space charts are affine spaces and smoothness is preserved under open restriction. In particular \(I_{\overline k}\) is smooth. The faithful field-extension descent proof in Supporting group-scheme proofs, Lemma G.2.1, makes \(I/k\) smooth, without a perfectness or characteristic hypothesis.

A nonempty smooth finite-type scheme over $k$ has a point over a finite separable extension. Here is the precise reason, including for imperfect $k$. Over a separable closure $k^s$, choose a nonempty open with an étale map to affine space, as supplied by smooth coordinates. Its image is open. The separably closed field $k^s$ is infinite: if it had $q$ elements, multiplication by a nonzero $a$ permutes all nonzero elements and cancellation of their product gives $a^{q-1}=1$. Thus every element satisfies $a^q=a$. The polynomial $X^q-X-1$ would have no root, although its derivative is $-1$ and any irreducible factor defines a finite separable extension, a contradiction. This infinite field has a rational point in every nonempty open of affine space: a nonzero polynomial cannot vanish on all its tuples, by induction on the number of variables. The nonempty étale fibre over this point has a closed point finite separable over $k^s$, hence rational because $k^s$ is separably closed. All coordinates of this point and of the chosen charts lie in a finite separable subextension of $k^s/k$. Applying this to $I$ supplies the asserted $K$ and splitting.

Finally, for an algebra $B$ that becomes a matrix algebra after a faithfully flat field extension, a nonzero proper two-sided ideal would remain a nonzero proper ideal after extension. Matrices have no such ideal: multiplying a nonzero entry on its two sides by matrix units produces all matrix units. Centrality again consists of linear equations, so $Z(B)\otimes E=Z(B_E)=E$ and $Z(B)=k$. The isomorphism sheaf is locally the automorphism group of matrices, and is therefore the $\operatorname{PGL}_n$-torsor of Lemma 6.1. Effective descent of its algebra operations is precisely the affine descent already proved in Quotients and torsors. This proves both directions, without a perfectness or characteristic hypothesis. $\square$

This is the finite-dimensional density method for central simple algebras, followed by smooth descent of their matrix-isomorphism scheme. Compare the Stacks authors' splitting-field theorem, which also proves existence of a separable splitting field. The argument above proves the precise input needed for inner forms; Tsen's theorem and a computation of a field's entire Brauer group are not needed here.

Over the finite separable splitting extension in Theorem 6.2, $A$ becomes $\operatorname{Mat}_n$. The determinant descends to its reduced norm $\operatorname{Nrd}:A^\times\to\mathbf G_m$. Put

Conjugation of matrices acts simultaneously on the algebra and on $\operatorname{SL}_n$. Twisting both by the same torsor shows that every marked inner form is (6.1), and every such algebra gives one. The reduced norm is well defined because the determinant is invariant under conjugation, so its local definitions agree on overlaps. This proves the assertion in arbitrary characteristic.

For $n\geq3$, transpose-inverse represents the nontrivial diagram class of $\operatorname{SL}_n$. Indeed it negates the diagonal character lattice and interchanges the upper and lower Borels; conjugating by a longest-element representative returns the upper Borel and reverses the $A_{n-1}$ simple-root chain. Any automorphism of this chain is identity or reversal: the image of one end fixes each successive neighbour, so determines the entire chain. Its simple roots span the rational character space; hence their images determine the entire lattice automorphism. The full outer group is therefore this order-two group. The all-ring rank-two and noninner calculations are written in Exercise 1. Its action on the algebra torsor replaces $A$ by $A^{\mathrm{op}}$. Indeed, transpose identifies matrix multiplication with the opposite multiplication, and conjugation by $g$ becomes conjugation by $g^{-T}$. Thus, if inner markings are forgotten, the resulting algebras are identified by $A\leftrightarrow A^{\mathrm{op}}$. The underlying groups are isomorphic by $a\mapsto a^{-1}$, viewed as a homomorphism from $A^\times$ to $(A^{\mathrm{op}})^\times$. For $n=2$ transpose-inverse is inner; the marked classification is still by the degree-two algebras.

7. Flags and the general linear group

Let $n=n_1+\cdots+n_r$, with all $n_i>0$, be an ordered composition. The corresponding standard parabolic consists of invertible block upper triangular matrices with block sizes $n_1,\ldots,n_r$. It stabilizes the partial flag with dimensions

The order matters: the subset of cuts in $\{1,\ldots,n-1\}$, not an unordered partition, is the parabolic type containing the fixed upper triangular Borel.

For an $S$-scheme $S'$, let $\operatorname{Flag}_{d_1,\ldots,d_{r-1}}(S')$ consist of chains

of locally direct summands of the indicated ranks, with locally free successive quotients. Use the subbundle Grassmannians, obtained by dualizing the quotient Grassmannians of AG-RG-03 Lemma 5.A. In their product impose the vanishing of each bundle map \(F_{d_j}\to\mathcal O^n/F_{d_{j+1}}\). In local frames its finitely many entries generate the ideal of a closed incidence subscheme. Its test points have exactly the desired inclusions. The quotient \(F_{d_{j+1}}/F_{d_j}\) is automatically locally free: the exact sequence \[ 0\to F_{d_{j+1}}/F_{d_j}\to \mathcal O^n/F_{d_j}\to\mathcal O^n/F_{d_{j+1}}\to0 \] splits locally because its last term is locally free. This verifies the incidence functor over arbitrary rings. The product of the closed Plücker embeddings is projective, so its closed incidence subscheme is projective too.

The open chart where \(F_{d_j}\to\mathcal O^{d_j}\) is an isomorphism for every \(j\), using projection to the first \(d_j\) coordinates, is represented by block lower unitriangular matrices with diagonal blocks the identity. Unit-pivot elimination gives the matrix uniquely from a flag and conversely sends its first blocks to that flag; it only inverts the indicated invertible projection determinants. Its free blocks give affine space of dimension \[ \sum_{i<j}n_in_j.\tag{7.1} \] The corresponding charts for coordinate permutations cover the scheme. On a residue-field flag, row-pivot elimination chooses nested coordinate sets \(I_j\), of sizes \(d_j\), for which all those projections are isomorphisms; one permutation sends them to the initial sets. Their determinant conditions define open charts on the original scheme, and coverage of all residue-field points is coverage of its underlying space. Thus these actual affine charts prove smoothness over \(S\), including when \(S\) is nonreduced. Finally every flag admits an adapted basis Zariski locally: successively split its locally free quotients and choose bases of the summands.

Hence $\operatorname{GL}_n$ acts transitively as a sheaf, with stabilizer the block upper triangular $P$. It follows directly on the functor of points that

Taking every cut gives the full flag scheme, representing Borel subgroups; its relative dimension is $n(n-1)/2$. Omitting every cut gives the one-point scheme representing the whole group as a parabolic. This is the linear-algebra model of Theorem 2.1.

Schubert cells for split flag schemes

Theorem 7.1 (relative Schubert cells). Let $G/S$ be pinned split of constant datum, let $J\subset\Delta$, and let $P_J$ be its standard parabolic. Let $W_J$ be the Weyl subgroup generated by $J$, and $W^J$ the minimal-length representatives for the right cosets $W/W_J$. Then $G/P_J$ has a finite disjoint stratification by locally closed subschemes

each isomorphic to $\mathbf A_S^{\ell(w)}$. These cells commute with arbitrary base change. In particular the theorem applies over $\mathbf Z$ and supplies both a torification by coordinate tori and, over a finite field, the count $|(G/P_J)(\mathbf F_q)|=\sum_{w\in W^J}q^{\ell(w)}$.

Proof. Let $U_J^-$ have the negative roots outside the root subsystem generated by $J$. The relative cocharacter open-cell theorem identifies $U_J^-\times P_J$ with an open subscheme of $G$. Quotienting this open set by its right $P_J$-action gives an open immersion $U_J^-\to G/P_J$. To justify the quotient assertion, its inverse image in the $P_J$-torsor $G\to G/P_J$ is the indicated right-stable open product, and the quotient of that product is $U_J^-$. Openness and the isomorphism descend through this faithfully flat torsor.

We first prove all minimal-coset facts used here. Reflections in $J$ change only the simple-root coefficients indexed by $J$. A positive root outside $\Phi_J$ retains a positive coefficient outside $J$ under every such reflection, so remains positive; a negative such root remains negative. Thus an element of $W_J$ has inversions only in the Levi root system. That system has simple roots $J$ by the positive coefficient and cocharacter-face calculation of Theorem 1.1. The entire chamber/length proof of RG04 Section 3 applied to this subsystem gives a reduced word using $J$ and its length equals the full length, since the inversion sets agree. Starting with any $v\in W$, repeatedly remove a right descent from $J$. Each step lowers full length by one, so this ends at $w$ with $w\alpha$ positive for every $\alpha\in J$. A positive Levi root is a nonnegative sum of these simple roots, and its image is consequently positive. The removed word gives $v=wz$, $z\in W_J$. If its number of letters is $m$, then $\ell(v)=\ell(w)+m$ and $\ell(z)\le m$. The length triangle inequality, proved by concatenating reduced words, forces $m\le\ell(z)$, so equality holds. Conversely, for any such $w$ and a reduced $J$-word for $z$, its successive positive inversion roots are Levi roots and their images by $w$ are positive. The length-one inversion rule therefore gives $\ell(wz)=\ell(w)+\ell(z)$, making $w$ minimal in its coset. It is unique: if $w'=wz$ also sends every positive Levi root to a positive root and $z\ne1$, a positive Levi inversion $\beta$ of $z$ would make $w'\beta=w(z\beta)$ negative, a contradiction. This proves the indexed set $W^J$, the unique length-additive factorization and its inversion-root union used below.

Minimality of $w$ therefore means $w(\Phi_J^+)\subset\Phi^+$. Thus $w^{-1}\Psi_w$ is negative and contains no Levi root: a negative Levi root $-\beta$ would give the positive root $w(-\beta)$, contradicting that condition. The ordered negative-root product makes $n_w^{-1}U_{\Psi_w}n_w$ a closed subscheme of $U_J^-$, by setting the other coordinates to zero. Translating the previous open chart by $n_w$ proves that the displayed map is a locally closed immersion. Its source is the product of $\ell(w)$ framed root lines, hence affine space of that dimension over $S$.

For coverage on a geometric fibre, write \(v=wz\), with \(w\in W^J\), \(z\in W_J\) and \(\ell(v)=\ell(w)+\ell(z)\). Then \(\Psi_v=\Psi_w\sqcup w\Psi_z\). The arbitrary-order root-product theorem of AG-RG-04 Section 5, with the first set first, gives \[ U_{\Psi_v}=U_{\Psi_w}\,n_wU_{\Psi_z}n_w^{-1} \] as a product of schemes. The second factor and \(n_z\) are absorbed on the right in \(P_J\), so the image of the Bruhat cell for \(v\) is the displayed cell for \(w\). Field Bruhat coverage therefore covers every geometric point of the quotient.

Each displayed cell is the entire \(B\)-orbit of \(n_wP_J\). Indeed the decomposition \(U^+=U_{\Psi_w}U_{\Theta_w}\) of AG-RG-04 Section 5 has \(w^{-1}\Theta_w\subset\Phi^+\), so \(U_{\Theta_w}\) fixes this coset, as does \(T\). We also verify the needed indexing of \(T\)-fixed geometric cosets. If \(gP_J\) is fixed, \(g^{-1}Tg\subset P_J\). The field conjugacy theorem for maximal tori of the connected smooth affine \(P_J\) supplies \(p\in P_J\) with \(gp\in N_G(T)\). Hence every fixed coset is \(n_vP_J\). Two representatives give the same coset exactly when their Weyl elements differ on the right by the image of \(N_G(T)\cap P_J\), which is \(W_J\) by the Bruhat description in Theorem 1.1. Thus the fixed cosets are precisely \(W/W_J\).

Choose an integral cocharacter positive on every positive root. On \(U_{\Psi_w}\simeq\mathbf A^{\ell(w)}\) its root coordinates have strictly positive weights, so its fixed point, and consequently its only \(T\)-fixed point, is the origin \(n_wP_J\). If two displayed cells met on a geometric fibre, their entire \(B\)-orbits would coincide; their unique \(T\)-fixed points would then coincide, contradicting the distinct cosets in \(W/W_J\). The images are therefore pairwise disjoint. Every point of the relative quotient admits a geometric point above it, so this gives disjoint coverage of the underlying space over \(S\). The locally closed subschemes themselves were constructed over \(S\) before this fibre argument.

Every map, root-coordinate closed immersion and open chart used here commutes with base change, as does \(G/P_J\) by Theorem 2.1. Thus the locally closed stratification commutes with every base change, including nonflat changes to nonreduced schemes. Over \(\mathbf F_q\), each rational point belongs to exactly one cell, and its affine coordinates give \(q^{\ell(w)}\) points; summing gives the stated count. Over \(\mathbf Z\), partition each affine cell coordinate by zero versus invertible. The resulting locally closed pieces are split tori and partition geometric points and points over every field, giving the claimed torification. A stratification does not assert that the coproduct of its strata equals the original fppf sheaf: a point over a ring with nilpotents can cross a stratum boundary and need not factor through one stratum. All constructions here still represent their stated subfunctors on every scheme test. \(\square\)

For $G=\operatorname{GL}_n$ and block type $(n_1,\ldots,n_r)$ this applies to the flag scheme just constructed, with $W=S_n$ and $W_J=S_{n_1}\times\cdots\times S_{n_r}$. Indeed each simple-root reflection swaps adjacent coordinates. These swaps generate every permutation: move its last coordinate into place by adjacent swaps and induct on the remaining coordinates. Restricting to the simple roots within each block gives exactly the displayed product of block permutation groups. It retains the ordered block type and the locally direct-summand condition on flags. For a nonsplit form, these globally indexed affine cells are not asserted: étale-local splitting of its parameter scheme alone does not give such a global stratification.

8. Exercises

Exercise 1 $easy$. Show that $\tau(g)=(g^T)^{-1}$ is outer on $\operatorname{SL}_n$ for $n\geq3$, and is conjugation by $\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)$ on $\operatorname{SL}_2$.

Exercise 2 $medium$. Classify the inner forms of $\operatorname{SL}_n$ over a field, specifying what changes when the inner structure is forgotten.

Exercise 3 $medium$. Classify the parabolics of $\operatorname{GL}_n$ containing the upper triangular Borel.

Exercise 4 $hard$. Prove that the Borel subgroup functor of $\operatorname{GL}_n$ is represented by a smooth projective flag scheme over an arbitrary base.

Exercise 5 $medium$. Explain why the automorphism group of a split torus need not be of finite type, and verify the central automorphism (3.2).

Solutions

Solution 1

Reversing the order twice makes transpose-inverse a homomorphism; it is an involution. If it were inner by $h$, its preservation of the diagonal torus would put $h$ in that torus's normalizer. Its action on $X^*(T)=\mathbf Z^n/\mathbf Z(1,\ldots,1)$ is $-1$, whereas the Weyl action is a coordinate permutation. For three distinct indices, a permutation carrying $e_i-e_j$ to $e_j-e_i$ for every pair would have to exchange both $(i,j)$ and $(i,k)$, which is impossible. Thus $-1$ is not a Weyl action for $n\geq3$. For $n=2$, direct multiplication, using $ad-bc=1$, gives

The identities hold over every ring, including characteristic two.

Solution 2

Its adjoint group is $\operatorname{PGL}_n$. Lemma 6.1 sends a torsor to a degree-$n$ central simple algebra $A$; applying that torsor to the determinant-one matrix group gives $\operatorname{SL}_1(A)$. Conversely the local splitting isomorphisms of $A$ give the same torsor. This proves the classification with inner structure. For $n\geq3$, forgetting it divides by the diagram involution, which sends $A$ to $A^{\mathrm{op}}$, as the transpose calculation in Section 6 shows. The inversion map supplies the corresponding group isomorphism. The distinction concerns the marking; it does not create an additional family of inner groups.

Solution 3

The simple roots are $e_i-e_{i+1}$, for $1\leq i<n$. A subset $J$ joins consecutive indices into blocks: put a cut at $i$ precisely when $e_i-e_{i+1}\notin J$. The resulting ordered block sizes are positive and sum to $n$. The roots in the Levi are exactly differences within blocks; the other permitted root entries are above the blocks. Thus Theorem 1.1 gives exactly the block upper triangular groups. Recovering their cuts recovers $J$, so there is neither repetition nor omission. For $n=3$, the four types are $(1,1,1),(1,2),(2,1),(3)$.

Solution 4

Form the closed incidence subscheme in $\prod_{d=1}^{n-1}\operatorname{Gr}(d,n)$ defined by $F_d\subset F_{d+1}$. Its test points are exactly full locally split flags. The Plücker embeddings and the incidence equations prove projectivity. Adapted bases give local lifts to $\operatorname{GL}_n$; their transition matrices preserve the flag exactly when upper triangular, so this scheme represents $\operatorname{GL}_n/B$. The lower unitriangular charts have $n(n-1)/2$ free coordinates and prove smoothness. By Theorem 2.1 a Borel is locally a conjugate of $B$, and its normalizer is $B$. Hence it determines exactly one quotient section, with descent on overlaps. This identifies the flag and Borel functors on all schemes, rather than merely on field-valued points.

Solution 5

A torus automorphism is an invertible integral change of its character basis, so the representing scheme is the disjoint union of copies of the base indexed by $\operatorname{GL}_r(\mathbf Z)$. For $r\geq2$ it has infinitely many components, since the matrices $\left(\begin{smallmatrix}1&m\\0&1\end{smallmatrix}\right)$ are distinct. On $\operatorname{GL}_2$, centrality of the determinant factor proves that (3.2) is a homomorphism. Its determinant is $\det(g)^{-1}$, so applying it twice returns $g$. On scalar matrices it sends $zI$ to $z^{-1}I$, whereas every inner automorphism fixes them. Thus it is a nontrivial outer automorphism arising from the central lattice direction.

The course prerequisite guide records the exact supporting statements, their full proof routes, and the hypotheses needed in their applications.

References and exact prerequisite proofs

• Michel Demazure and Alexander Grothendieck, Schémas en groupes, freely accessible Gille–Polo re-edition of 13 October 2024: Exposé XXIV and Exposé XXVI, for comparison with the automorphism, form and parabolic arguments proved here.
• Brian Conrad, Reductive group schemes, §§5.2 and 7.1–7.2. The proofs here use the preceding lessons' root-coordinate and pinned classification results.
• Milne’s freely accessible Algebraic Groups, version 2.00 (2015). This exact free author edition provides comparison material; its citations do not replace the programme proofs.

The preceding written supporting proofs are Algebra and sheaf cohomology, §§3–8, for the finite-cover, affine and projective calculations; Projective cohomology and smooth affine models, Proposition 3.2 and Proposition 5.2, for retaining smooth projective models and line bundles and for direct images commuting with every base change; and Supporting group-scheme proofs, Lemma G.2.1, for smoothness descent over any field. The projective quotient and scheme-theoretic normalizer arguments are AG-RG-03 §5, using the earlier written descent, Zariski Main and flat-quotient proofs. AG-RG-03 Lemma 5.A supplies the quotient Grassmannian charts and closed Plücker immersion, with subbundle charts obtained by dualizing. S04 Lemmas A1.1–A1.3a and A1.5 supply the entire augmented faithfully flat complex, effective module/algebra descent, finite presentation and smoothness descent, and morphism descent. Sections 4–5 here prove the actual torsor/cocycle classification and both multiplicative and additive vanishing used for quasi-split forms. The normalizer Lie calculation is explicitly given in Theorem 1.1 and uses the already proved torus weight projections. The separable matrix-splitting argument needed for inner forms is Theorem 6.2 of this lesson itself; no Tsen theorem or computation of an entire Brauer group is consumed. The proofs above establish the remaining parabolic, automorphism, form, flag and Schubert assertions. This identifies the consumed proof chain without claiming that unrelated parts of the programme are proof-complete.