\documentclass[11pt,a4paper,openany]{report} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=23mm]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{longtable,booktabs,array,calc} \newcounter{none} \usepackage{graphicx,xcolor,tikz} \usepackage{microtype} \usepackage[unicode,breaklinks,colorlinks,linkcolor=blue!45!black,urlcolor=blue!45!black]{hyperref} \urlstyle{same} \providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} \providecommand{\passthrough}[1]{#1} \providecommand{\NL}{\operatorname{NL}} \providecommand{\colim}{\operatorname*{colim}} \setlength{\parindent}{0pt} \setlength{\parskip}{.4em} \setlength{\emergencystretch}{3em} \setcounter{tocdepth}{1} \setcounter{secnumdepth}{0} \renewcommand{\chaptername}{Lesson} \DeclareUnicodeCharacter{03BB}{\ensuremath{\lambda}} \DeclareUnicodeCharacter{0393}{\ensuremath{\Gamma}} \DeclareUnicodeCharacter{2192}{\ensuremath{\rightarrow}} \DeclareUnicodeCharacter{2011}{-} \DeclareUnicodeCharacter{202F}{\,} \DeclareUnicodeCharacter{2212}{\ensuremath{-}} \DeclareUnicodeCharacter{03B1}{\ensuremath{\alpha}} \DeclareUnicodeCharacter{03B2}{\ensuremath{\beta}} \DeclareUnicodeCharacter{03BC}{\ensuremath{\mu}} \DeclareUnicodeCharacter{03A6}{\ensuremath{\Phi}} \DeclareUnicodeCharacter{2260}{\ensuremath{\ne}} \DeclareUnicodeCharacter{00D7}{\ensuremath{\times}} \DeclareUnicodeCharacter{2082}{\ensuremath{_2}} \DeclareUnicodeCharacter{2228}{\ensuremath{\vee}} \DeclareUnicodeCharacter{2032}{\ensuremath{^{\prime}}} \DeclareUnicodeCharacter{220E}{\ensuremath{\square}} \title{Regular elements and centralizers} \author{GPT-6.1 Sol (OpenAI)\\Codex, Ultra setting} \date{October 2026} \begin{document} \hypersetup{pageanchor=false} \maketitle \pagenumbering{roman} \hypersetup{pageanchor=true} \noindent Written and self-checked by the writing AI. No human or independent review is claimed. Text is dedicated under CC0 1.0. Human mathematical sources are cited in the lessons. \par\medskip \noindent Course sources: \url{https://github.com/KokunoYumeto/open-math-courses/tree/main/docs/courses/AG-RG}. \begingroup\small\tableofcontents\endgroup \clearpage \pagenumbering{arabic} \setcounter{chapter}{1} \chapter{Regular elements and centralizers}\label{AG-RG-02-regular-elements-and-centralizers} \emph{Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Public domain (CC0).} A maximal torus describes a reductive group by diagonal symmetries. A regular semisimple element describes the same torus by one element: the torus is the identity component of its centralizer. This observation is useful in families, where the centralizer of a section is easier to specify than a choice of eigenvectors. We use the torus deformation and conjugacy arguments of \href{https://github.com/KokunoYumeto/open-math-courses/blob/main/docs/courses/AG-RG/AG-RG-01.md}{Tori, maximal tori and their conjugacy}. The classical geometric input concerning Borel subgroups is used below to prove the centralizer theorem, rather than importing that theorem from a reference. All centralizers are scheme-theoretic. Over a field, a geometric assertion means an assertion after algebraic closure. \section{1. The geometry behind the centralizer theorem}\label{AG-RG-02-the-geometry-behind-the-centralizer-theorem} We first work over an algebraically closed field \(k\). A Borel subgroup of a smooth connected affine group \(H\) is a maximal smooth connected solvable subgroup. Write \(B=U\rtimes T\) for a Borel subgroup and one of its maximal tori. The solvable-group construction in the preceding lesson identifies \(U\) with the unipotent radical of \(B\). We shall need several consequences of that construction. Every smooth connected solvable subgroup lies in a Borel: among those containing it choose one of largest dimension, bounded by \(\dim H\). A proper inclusion would strictly increase dimension, since both groups are reduced connected and hence integral by RG01 Lemma 2.L, and the earlier domain dimension theorem excludes a proper closed subset of the same dimension. Thus the chosen subgroup is maximal. RG01 Lemma 2.1 proves that \(H/B\) is projective and that all Borel subgroups are conjugate. The scheme-theoretic normalizer theorem is proved below, after the centralizer and Borel-intersection arguments needed for its induction. \textbf{Lemma 1.A (unipotent radicals).} A smooth connected affine \(H/k\) has a unique largest smooth connected normal unipotent subgroup \(R_u(H)\). It is preserved by every automorphism. Every smooth connected normal unipotent subgroup lies in every Borel. If \(K\) is a smooth connected normal solvable subgroup of \(H\), then \(R_u(K)\) is normal in \(H\). \textbf{Proof.} We first prove the precise affine image assertion used here. If \(f:E\to H\) is a homomorphism with \(E\) smooth connected affine of finite type, then \(k[E]\) is a domain: Lemma 1.0 below gives irreducibility, and smooth local rings are regular and therefore domains by AG-CA, \emph{Smooth algebras over a field and the Jacobian criterion}, Theorem 2.1, and the regular-parameter theorem. Let \(C\) be the image of \(k[H]\to k[E]\). This is a finitely generated domain. Comultiplication, inversion and counit restrict to it, because they commute with \(f\) and tensoring vector-space inclusions over \(k\) is injective. Their identities therefore restrict as well. Thus \(J=\operatorname{Spec}C\) is a closed subgroup of \(H\), and \(E\to J\) is schematically dominant. The complete monic-division and translation argument of AG-GS, \emph{Quotients and torsors}, Lemma 11.1c makes it faithfully flat and of finite presentation, in precisely its integral-target case. The group \(J\) is smooth and connected. Put \(r=\operatorname{trdeg}_k\operatorname{Frac}C\). Noether normalization, proved in AG-CA, \emph{Krull dimension and Noether normalization}, Corollary 3.2, makes this field finite over a rational function field with \(r\) parameters. S02 Lemma 3.F gives its differential dimension \(r\), since \(k\) is perfect. At the generic point the local dimension is zero and the residue transcendence degree is \(r\), so the earlier Jacobian Theorem 2.1 makes that point smooth. Its Corollary 2.2 gives a nonempty smooth open. This open contains a \(k\)-point by the weak Nullstellensatz. Translation by the rational points of \(J\) makes every rational point smooth; a nonempty closed complement of the smooth locus would again have a rational point. Hence \(J\) is smooth everywhere. Its coordinate domain makes it irreducible and connected. This proves the entire finite-type affine image assertion needed here, without consuming an image theorem for arbitrary group schemes. For smooth connected normal unipotent \(N_1,N_2\), the semidirect product \(N_1\rtimes N_2\) has underlying scheme \(N_1\times N_2\). It is smooth and integral by Lemma 1.0, the regular-local-domain calculation above and S08 Lemma G.4.1, so is connected. Apply the preceding image proof to multiplication into \(H\); denote its closed image by \(N_1N_2\). It contains both factors. Normality holds schematically: conjugation by any \(H\)-point preserves both factors, hence their multiplication image; pulling back equations along the faithfully flat cover \(H\times(N_1\rtimes N_2)\to H\times(N_1N_2)\) detects their vanishing. Thus their product is a smooth connected normal closed subgroup. This product is unipotent. In any nonzero finite-dimensional representation \(W\), AG-RG-01, Lemma 2.A makes \(W^{N_1}\ne0\). Normality makes this subspace \(N_2\)-stable, so AG-RG-01, Lemma 2.A supplies a nonzero vector fixed by both groups, and hence by their product: the latter equality follows faithfully flatly from the surjective multiplication map. Applying the same argument to successive quotients gives a trivial-quotient full flag. A faithful representation of the affine image is therefore unitriangular. Choose a smooth connected normal unipotent subgroup of maximal dimension, possible since the dimensions are bounded by \(\dim H\). Its product with any other such subgroup is again of this type. Maximal dimension forces that product to equal it: a proper integral closed subgroup of an integral finite-type group has strictly smaller dimension, by the strict-specialization and dimension proofs in AG-CA, \emph{Krull dimension and Noether normalization}, Lemma 4.1 and Theorem 4.2. This proves greatestness and uniqueness. Automorphisms preserve its defining property, so preserve the subgroup. For a Borel \(B\) and such a normal \(N\), the product \(NB\) is similarly the smooth connected closed image of \(N\rtimes B\). It is solvable directly on all test rings. Projection to the solvable \(B\) makes every sufficiently iterated commutator word in \(N\rtimes B\) take values in \(N\). The unitriangular entry-distance filtration, with its polynomial commutator identities proved in RG01 Lemma 2.J, makes finitely many further derived commutators equal to the identity. Every tuple of points in the image lifts fppf locally through the multiplication map; hence the same commutator identities hold for \(NB\). Thus maximality of \(B\) gives \(NB=B\), and \(N\) lies in \(B\). If \(K\) is normal in \(H\), every conjugation by \(H(k)\) gives an automorphism of \(K\) and preserves its unique \(R_u(K)\). The whole conjugation morphism \(H\times R_u(K)\to H\) factors through that closed subgroup: the source is reduced by AG-RG-S08, Lemma G.4.1, and its defining equations vanish at all closed points. To justify the required detection, a nonzero element \(a\) of a reduced finite-type \(k\)-algebra has a nonzero localization at \(a\), since it is not nilpotent. That localization is still finite type and has a maximal ideal, whose residue field is \(k\) by the weak Nullstellensatz. It gives a rational point where \(a\) is invertible. Therefore an element vanishing at every rational point is zero. Apply this calculation to the affine source of conjugation. This proves scheme normality in \(H\). In the solvable \(K\), the decomposition constructed in AG-RG-01, Theorem 2.K, \(K=U\rtimes T\), identifies its radical with \(U\): every smooth connected unipotent subgroup, including one not assumed normal in \(K\), projects trivially to \(T\) by AG-RG-01, Lemma 2.A and is contained in \(U\). \(\square\) \textbf{Lemma 1.0 (components of an affine group over an algebraically closed field).} Let \(J\) be an affine group scheme of finite type over an algebraically closed field \(k\). Its irreducible components are pairwise disjoint and open and closed. In particular a connected such group is irreducible. No smoothness or reducedness hypothesis is needed for this topological assertion. \textbf{Proof.} The Hilbert basis theorem, proved in AG-CA, \emph{Noetherian and Artinian rings}, Theorem 2.1, makes \(k[J]\) Noetherian. Its spectrum has finitely many irreducible components: if a closed subset failed to be a finite union of irreducible closed subsets, choose a minimal such subset using the descending chain condition on closed sets. It is reducible, hence the union of two proper closed subsets, both finite unions by minimality, a contradiction. If \(C_1,C_2\) are components containing the identity, give them their reduced closed structures. They are integral finite-type schemes, so their product is integral by the complete coefficient-and-Nullstellensatz proof in the earlier S08 Lemma G.4.1. The closure of its multiplication image in \(J\) is irreducible and contains both \(C_1\) and \(C_2\), by multiplication with the identity in either factor. It is contained in a component, and maximality of \(C_1\) and \(C_2\) forces them to be that same component. Thus exactly one component contains the identity. If two different components met, their closed intersection would contain a \(k\)-point by the earlier Nullstellensatz, AG-CA Theorem 2.1 of \emph{The Nullstellensatz and Jacobson rings}. Translation by the inverse of that point is a scheme automorphism of \(J\). It would carry the two distinct components to two distinct components containing the identity, contradicting the preceding paragraph. Hence all components are disjoint. There are only finitely many, so each has closed complement and is open as well as closed. Connectedness leaves exactly one component, proving irreducibility. This is the finite-type algebraically closed case of the earlier AG-GS, \emph{Group schemes over a field}, Section 1 and Theorem 2.2; the proof just given supplies precisely the case consumed here. \(\square\) \textbf{Lemma 1.1.} Let \(H\) be a smooth connected affine group of finite type over an algebraically closed field \(k\), and let \(A\subset H\) be a torus. Its schematic centralizer \(C_H(A)\) is smooth and connected. \textbf{Proof.} The complete fixed-subgroup proof in AG-GS, \emph{Diagonalizable groups and groups of multiplicative type}, Theorem 3.4 supplies smoothness. We give explicitly the cocharacter reduction needed for connectedness. Over the algebraically closed \(k\), the torus \(A\) is split. Indeed its étale splitting cover, given by that lesson's Theorem 7.7, has a nonempty finite-type affine chart; the earlier Nullstellensatz supplies a \(k\)-point, and pulling the splitting back along that point splits \(A\). Write \(A=\mathbf G_m^r\) and its character lattice as \(\mathbf Z^r\). Choose finitely many algebra generators of \(k[H]\) and replace them by all their finitely many homogeneous components for conjugation by \(A\). The complete weight decomposition of AG-GS Theorem 3.1 shows that the resulting homogeneous list still generates. Let \(\Xi\subset\mathbf Z^r\) be its finite set of nonzero weights. If \(r>0\), associate to \(\chi=(a_1,\ldots,a_r)\in\Xi\) the nonzero polynomial \[ p_\chi(Z)=a_1+a_2Z+\cdots+a_rZ^{r-1}\in\mathbf Q[Z]. \] Each has only finitely many roots: division by \(Z-c\) removes a root, and induction on the degree bounds the number of distinct roots by that degree. This is also the earlier explicit polynomial calculation in S02 Lemma 4.A. Choose a positive integer \(M\) outside the finite union of their root sets, and put \(\lambda(t)=(t,t^M,\ldots,t^{M^{r-1}})\). Its pairing with every member of \(\Xi\) is nonzero. For an empty weight list choose \(M=1\), and for \(r=0\) use the unique cocharacter of the trivial torus. The centralizers \(C_H(A)\) and \(C_H(\lambda)\) are equal as schemes, including on nonreduced test algebras. To check this, use the universal conjugation action on each homogeneous algebra generator \(f\) of weight \(\chi\). A test point fixed by \(A\) must satisfy \(f(h)(e^\chi-1)=0\) in its group algebra. The distinct basis monomials force \(f(h)=0\) when \(\chi\ne0\), and impose no condition when \(\chi=0\). The corresponding condition for \(\lambda\) is \(f(h)(t^{\langle\chi,\lambda\rangle}-1)=0\) in the Laurent polynomial algebra. Our choice makes its exponent nonzero precisely for the nonzero weights on this generating list, so coefficient comparison imposes exactly the same equations. Checking the algebra generators checks the full action on every function. Thus the two represented fixed subgroups have the same defining ideal over \(k\) and the same points on every test algebra. Use the general cocharacter construction proved in the preceding lesson, Lemma 3.3. Its proof is a grading and smoothness argument for arbitrary smooth affine groups and uses no reductive centralizer theorem. Multiplication gives an open immersion \[ U_H(-\lambda)\times C_H(\lambda)\times U_H(\lambda)\longrightarrow H. \] The domain contains the triple of identity elements. Lemma 1.0 makes the connected finite-type \(H\) irreducible, so the nonempty open image, and hence the domain, is irreducible. Its projection onto \(C_H(\lambda)\) is surjective: insertion of the identity in the two unipotent factors gives a section. The centralizer is therefore irreducible and connected. Together with the fixed-subgroup smoothness proof and the schematic equality already established, this proves the lemma. \(\square\) The cocharacter lemma and the classical fixed-point arguments are independent of the root classification and of the reductive centralizer theorem proved below. \textbf{Lemma 1.B (fixed unipotent groups are smooth connected).} Let a diagonalizable \(D\) act on \(U\) as in AG-RG-01, Lemma 2.C. Then \(U^D\) is smooth connected. Moreover the sequence \[ 1\longrightarrow U_1^D\longrightarrow U^D \longrightarrow (U/U_1)^D\longrightarrow1 \] is exact as fppf sheaves. \textbf{Proof.} Fixed subschemes are affine finitely presented and smooth by the coordinate and infinitesimal proofs in AG-GS-05, Proposition 3.2 and Theorem 3.4, including a nonreduced \(D\). Let x be a \(D\)-fixed quotient section on any test algebra. Choose a lift u fppf locally. The defect \[ c(d)=u^{-1}d(u) \] lies in \(U_1\) and is a regular crossed homomorphism. Write it fppf locally as \(c(d)=v d(v)^{-1}\) by AG-RG-01, Lemma 2.C. Then u v is fixed, since \(d(u v)=u c(d)d(v)=u v\). This proves surjectivity of the last map of the displayed sequence, with the kernel equality evident on every test scheme. Differences of two fixed lifts belong uniquely to \(U_1^D\); consequently its action identifies \(U_1^D\times U^D\) with \(U^D\times_{(U/U_1)^D}U^D\) on every test scheme. A local fixed lift translates this torsor into the product with \(U_1^D\). For a vector line of character \(\chi\), its \(D\)-fixed subgroup is the whole line if \(\chi=0\), and is zero otherwise: comparison of the two distinct basis monomials \(e^\chi\) and 1 forces the vector coordinate to vanish, including over nonreduced tests. We induct on the given filtration length, simultaneously after every algebraically closed extension of \(k\); the length-zero group is the identity. The fixed kernel in the displayed sequence is smooth and geometrically connected by induction, and the quotient \(Q\) is either \(\mathbf G_a\) or 1. On a geometric fibre the torsor has a point by the earlier Nullstellensatz, and translation identifies it with that connected kernel. Hence all geometric fibres of \(q:U^D\to Q\) are connected. Here is an explicit descent proof that the source is connected. Both source and target are affine. Write \(A=k[Q]\), \(B=k[U^D]\). On each affine fppf trivializing chart \(\operatorname{Spec}A'\to\operatorname{Spec}A\), the algebra \(B\otimes_A A'\) is flat over \(A'\): it is the base change of the coordinate algebra of the smooth \(k\)-group \(U_1^D\), and AG-CA, \emph{Smooth algebras over a field and the Jacobian criterion}, Theorem 6.1 proves smooth flatness, with the flat base-change proof in the flat-module lesson, Proposition 3.2. For any prime \(\mathfrak p\) of \(A\), choose such a chart and a prime \(\mathfrak p'\) above it. The flat local map \(A_{\mathfrak p}\to A'_{\mathfrak p'}\) is faithfully flat by AG-CA, \emph{Faithful flatness and the local criterion for flatness}, Theorem 2.1. For an injection of \(A_{\mathfrak p}\)-modules, its tensor map with \(B_{\mathfrak p}\) becomes injective after this faithfully flat extension, because the extended algebra is flat. Flatness of the extension and faithful detection make the original tensor map injective. Thus \(B_{\mathfrak p}\) is flat, and local flatness detection in the flat-module lesson, Theorem 3.3, proves \(B\) flat over \(A\). The local torsor sections give a point above every prime; hence \(A\to B\) is faithfully flat by the same faithful-flatness Theorem 2.1. A partition of the source into open and closed pieces is an idempotent \(e\in B\): glue the function equal to one on one piece and zero on the other. Conversely an idempotent gives these pieces, since in every local ring one of \(e\) and \(1-e\) is a unit and their product is zero. Let \(e_1,e_2\in B\otimes_A B\) be the pullbacks of \(e\) from the two factors. On each geometric fibre of \(q\), connectedness makes that idempotent constant, so \(e_1\) and \(e_2\) have equal values at every geometric point of the fibre product. Their difference therefore belongs to every prime ideal and is nilpotent. Here is the required nilradical argument for any ring \(C\). If \(z\in C\) is not nilpotent, the powers \(1,z,z^2,\ldots\) avoid zero. An ideal maximal among those avoiding these powers exists by the union-of-chains argument. It is prime: if \(ab\) lies in it but neither \(a\) nor \(b\) does, both larger ideals obtained by adjoining \(a\) and \(b\) meet the powers; multiplying the two resulting equations puts a power of \(z\) in the original ideal, a contradiction. This prime avoids \(z\). Conversely a nilpotent belongs to every prime. Apply this to \(C=B\otimes_A B\). Two idempotents with nilpotent difference are equal: \(e_1(1-e_2)\) and \(e_2(1-e_1)\) are both idempotent and nilpotent, hence zero; thus \(e_1=e_1e_2=e_2\). The faithfully flat equalizer proved in the earlier S04 Lemma A1.1 now descends \(e\) to \(A\). The map \(A\to B\) is injective, so the descended element is also idempotent. But \(A\) is \(k[x]\) or \(k\), both domains, and has only the idempotents zero and one. Hence \(e\) was trivial. This proves connectedness, and the same argument after every algebraically closed extension proves geometric connectedness. The torsor and kernel equalities were identities on all test schemes, so the entire lemma retains the stated schematic exactness. \(\square\) For \(A\subset T\subset B=U\rtimes T\), these proofs give \[ C_B(A)=U^A\rtimes T, \] so this centralizer is smooth connected and solvable. \textbf{Lemma 1.2.} A unipotent group acting on an affine variety has closed orbits, with their full scheme structures. No smoothness or connectedness assumption on the group or the schematic stabilizer is needed. \textbf{Proof.} We first prove closedness for a smooth connected unitriangular group \(L\). Give the closure \(Y\) of one point orbit its reduced structure. It is integral. S08 Theorem G.4.2 makes the orbit open in \(Y\), and makes both \(Y\) and its reduced boundary \(D\) invariant under \(L\), using reducedness of the products and the strong Nullstellensatz. If \(D\ne\varnothing\), choose a nonzero function in its ideal. AG-GS-01 Lemma 5.1 puts that function in a finite-dimensional \(L\)-stable subspace of the ideal: in the coefficient construction all coefficient functions stay in the ideal, since its coaction maps into \(k[L]\otimes I_D\). RG01 Lemma 2.A gives a nonzero invariant function in this subspace. Its values are constant on the dense point orbit and hence on \(Y\). It vanishes on the nonempty \(D\), so the constant is zero, a contradiction. Thus the orbit is closed. S08 Theorem G.4.C represents the full quotient by its possibly nonreduced stabilizer as this reduced locally closed orbit. Since its support is closed, it is a closed immersion in the affine target and is affine. Now embed the given unipotent group \(N\) as a closed subgroup of the smooth connected group \(L=U_m\) of upper unitriangular matrices. Its coordinate ring is the polynomial ring in the entries above the diagonal, and the matrix formulas give its group operations. Fix \(x\) in the affine variety and retain the entire schematic stabilizer \(K=N_x\subset N\subset L\). The RG01 detecting-line construction supplies an \(L\)-representation and a line \(kv\) with full stabilizer \(K\). RG01 Lemma 2.A makes the character of \(K\) on this line trivial, including its infinitesimal points. The vector stabilizer is consequently \(K\) as well. The first paragraph and S08 Theorem G.4.C now construct an affine closed vector orbit \(P\), with an fppf \(K\)-torsor \[ q:L\longrightarrow P,\qquad L\times K\simeq L\times_P L. \] All these identities hold on arbitrary test algebras. The closed subscheme \(N\subset L\) descends along this torsor. Indeed, in the displayed relation the two projections are \((l,k)\mapsto l\) and \((l,k)\mapsto lk\). The inverse images of \(N\) coincide as schemes: right multiplication by any section of \(K\subset N\) preserves \(N\), and its inverse does too. This is the descent datum for its defining ideal. The module, ideal and algebra effectivity proofs of S04 Lemmas A1.1, A1.2 and A1.5 give a closed subscheme \(Z\subset P\) with \[ q^{-1}(Z)=N,\qquad N\longrightarrow Z \text{ an fppf }K\text{-torsor}. \] For clarity, ideal descent applies on the affine cover \(L\to P\): the compatible ideal descends as a submodule of \(\mathcal O_P\), multiplication preserves it after faithful pullback and hence before pullback, and its quotient algebra pulls back to \(\mathcal O_N\). Thus \(Z\) retains any nilpotents of \(N/K\); no reduction has been taken here. The torsor identity is the restriction of that for \(L\). The action map \(N\to X\), \(n\mapsto nx\), descends to \(i:Z\to X\), since its two pullbacks on \(N\times K\) agree. This is a finite-type morphism between affine schemes and is a monomorphism. To check the latter on every test scheme, lift two sections of \(Z\) fppf locally to \(n,n'\in N\). Equality of their images says exactly \(n^{-1}n'\in K\), so the two quotient sections agree. Morphism descent then gives their equality before the cover. Left translation by \(N\) likewise descends to an action on \(Z\), and \(i\) is equivariant. Let \(W\subset X\) be the schematic image of \(i\). It is invariant under the entire \(N\)-action. In affine coordinates the ideal of the schematic image is a kernel, and tensoring with the \(k\)-vector space \(k[N]\) preserves that kernel. Thus the schematic image of \(1_N\times i\) is \(N\times W\). The shear automorphisms \((n,z)\mapsto(n,nz)\) on \(N\times Z\) and \(N\times X\) commute with this map; the latter automorphism and its inverse consequently preserve \(N\times W\). This proves invariance with its original scheme structure. S08 Lemma G.4.E gives an open \(V\subset W\) over which \(i\) is an isomorphism. Choose \(z_0\in i^{-1}(V)(k)\). Every point of \(Z(k)\) lifts to \(N(k)\): a fibre of its fppf torsor is nonempty and of finite type over the algebraically closed field, so has a rational point. Hence \(N(k)\) acts transitively on \(Z(k)\). Translating \(V\) by these elements gives opens \(nV\subset W\) whose inverse images are isomorphic to them and cover all closed points of \(Z\). They cover \(Z\), since a nonempty closed complement in a finite-type scheme has a closed rational point. The inverses agree on overlaps. Therefore \(Z\) is isomorphic to the open union \(O=\bigcup_{n\in N(k)}nV\subset W\). Finally this open has closed support. RG01 Lemma 2.L gives the smooth connected group \(J=(N_{\mathrm{red}})^0\), finitely many component translates, and \(N(k)=N_{\mathrm{red}}(k)\). The first paragraph makes every \(J\)-point-orbit closed. The \(N(k)\)-orbit of \(x\) is the finite union of translates of such an orbit, so it is the set of rational points of a closed reduced subscheme \(D\subset X\). The locally closed image \(O\) has precisely these rational points. Its support equals \(|D|\): the difference in either direction, if nonempty, is constructible and has a rational point by S08 Lemma G.4.A. The schematic image \(W\) has this same support, because the image of \(i\) is dense in it. Thus the open \(O\subset W\) contains every point of \(W\), so \(O=W\) as an open subscheme. It follows that \(i:Z\to X\) is a closed immersion. This constructs the full possibly nonreduced orbit and proves the assertion. \(\square\) Put \(X=H/B\), initially as a homogeneous space with stabilizer \(B\). Choose a maximal torus \(T\) of \(H\) contained in \(B\), and put \(C=C_H(T)\). Such a torus exists: a maximal torus is contained in a Borel by the dimension argument of RG01 Theorem 2.2, and Borel conjugacy in RG01 Lemma 2.1 moves that Borel to \(B\). We justify both finiteness of \(X^T\) and the assertion that \(C\) fixes it, instead of assuming either one. \textbf{Lemma 1.C (a central maximal torus).} If a maximal torus \(T\) of a smooth connected affine group \(C\) is central, then \[ C=T\times U_C \] with \(U_C\) smooth connected unipotent. \textbf{Proof.} We construct the affine quotient by \(T\), without presupposing solvability of \(C\). Write \(T=D_k(M)\), with \(M\) a finite free character group. Right translation grades \(A=k[C]\) by \(M\); put \(B=A_0\). Choose finitely many homogeneous generators \(a_i\) of \(A\), of weights \(m_i\). The weight-zero monomials in these generators generate \(B\). They have a finite generating list. Here is the needed integer argument: in \(\mathbf N^r\) every infinite sequence has an infinite subsequence nondecreasing in every coordinate, by successive extraction in the finitely many coordinates. Thus an antichain is finite. The nonzero vectors \(v\) with \(\sum v_i m_i=0\) have finitely many minimal elements; subtracting a minimal element from a comparable vector of the same weight leaves weight zero and decreases the coordinate sum. Induction expresses every such vector as a sum of minimal ones. The corresponding finitely many monomials generate \(B\). Hilbert's basis theorem makes \(B\) finitely presented over \(k\). Centrality makes \(B\) a Hopf subalgebra: for \(b\in B\), the function \(b(c_1c_2)\) is invariant under right translation by \(T\) in either variable. The two exact weight projectors put its comultiplication in \(B\otimes B\). Inversion and the counit restrict for the same reason. Their identities restrict along the injective degree-summand inclusions. We obtain an affine group \(Q=\operatorname{Spec}B\) and a homomorphism \(q:C\to Q\). The projector \(A\to B\) is \(B\)-linear and retracts its inclusion. Consequently \(A\otimes_B\kappa(y)\ne0\) for every \(y\in Q\), so \(q\) is surjective. The closed embedding \(T\subset C\) gives a graded surjection \(A\to k[M]\). For a basis \(m_j\) of \(M\), lift \(e^{m_j},e^{-m_j}\) to \(a_j,b_j\) of those weights and put \(f=\prod_j a_jb_j\in B\). Its value at the identity is one. All these elements are units in \(A_f\). Their products supply a unit \(u_m\) of each weight \(m\), and \[ A_f=\bigoplus_{m\in M}B_f u_m. \] This is a faithfully flat \(B_f\)-module. The torsor algebra map sends \(a\otimes c_m\) to \(ac_m\otimes e^m\), and its inverse sends \(a\otimes e^m\) to \(a u_m^{-1}\otimes u_m\). Thus \(q\) is a \(T\)-torsor on \(D(f)\). Every \(Q(k)\)-point lifts to \(C(k)\), by the nonzero finite-type fibre and the weak Nullstellensatz. Translates of \(D(f)\) by these image points cover all closed points of \(Q\), and hence \(Q\). Translation in \(C\) gives the torsor calculation on each such open. Finite presentation follows from the finite-type Noetherian coordinate rings. It follows that \(q\) is an fppf \(T\)-torsor globally, representing \(C/T\) on all test schemes. The smooth connected \(C\) is integral, and its subring \(B\) is an integral finite-type \(k\)-algebra. The perfect-field differential-rank calculation of S02 Lemma 3.F and the Jacobian criterion give a nonempty smooth open of \(Q\); translating it covers all closed points, so \(Q\) is smooth. Surjectivity makes \(Q\) connected. Also \(q\) is smooth: its geometric fibres are translates of the smooth \(T\), and it is flat of finite presentation, so the proved geometric-fibre criterion applies. Suppose \(S\subset Q\) is a nontrivial torus and put \(E=q^{-1}(S)\). Then \(E\to S\) is a smooth surjective \(T\)-torsor. The group \(E\) is smooth and connected. Indeed its smooth components are disjoint, open and closed; its geometric fibres are translates of the connected \(T\). Its ordinary fibres are connected as well, because field-extension projections are surjective. By S08 Lemma G.4.F the images of these open components are open; connectedness of the fibres makes the images of distinct components disjoint. They cover the connected \(S\), so there is just one component. Its commutator maps land in the central \(T\), hence its next derived commutator is trivial. Thus \(E\) is solvable, and RG01 Theorem 2.K writes it as \(V\rtimes M_E\). The homomorphism \(V\to S\) is trivial because a unitriangular group has no character, by RG01 Lemma 2.A. Hence \(V\subset T\); a diagonalizable subgroup of a unitriangular group is trivial by that same lemma. Thus \(V=1\) and \(E\) is a torus. It properly contains \(T\), since its quotient \(S\) is nontrivial, contradicting maximality. We conclude that \(Q\) has no nontrivial torus. RG01 Lemma 2.D now makes \(Q\) unipotent. A sufficiently long derived commutator in \(C\) lands in \(T\), since its image in the unitriangular \(Q\) is trivial; the next derived commutator is one. Thus \(C\) is solvable. RG01 Theorem 2.K gives \(C=U_C\rtimes M_C\), with \(U_C\) smooth connected unipotent. By RG01 Lemma 2.N the maximal torus \(T\) is conjugate to \(M_C\). Centrality makes every conjugate of \(T\) equal to \(T\), so \(M_C=T\). The action of this central torus on \(U_C\) is trivial, giving the asserted direct product. \(\square\) Apply Lemma 1.C to \(C=C_H(T)\), which is smooth and connected by Lemma 1.1. The torus \(T\) is central and remains maximal in this subgroup. \textbf{Torus-fixed cosets.} The scheme \(X^T\) is a finite disjoint union of \(k\)-points, and \(C\) fixes it pointwise. \textbf{Proof.} RG01 Lemma 2.1 constructs the projective homogeneous orbit \(X=H/B\), and S08 Theorem G.4.C gives its full fppf \(B\)-torsor and its smooth scheme structure. The complete equivariant embedding of RG01 Lemma 2.1 realizes \(X\) as a closed subvariety of a projective space with a linear \(H\)-action. Choose a weight basis for the restricted \(T\)-representation, using the weight decomposition proved in RG01. Every standard projective chart given by a nonzero weight coordinate is \(T\)-stable. Its intersection with \(X\) is affine and smooth; these intersections cover \(X\). On each of these affine charts, AG-GS, \emph{Diagonalizable groups}, Proposition 3.2 constructs the fixed scheme by the ideal of nonzero-weight elements and proves that it is closed, of finite presentation and compatible with every base change. Its Theorem 3.4, with the universal coefficient calculation of Lemma 3.3, proves smoothness and identifies its tangent functor with the invariant tangent functor of the chart. The fixed schemes agree on chart overlaps because they represent the same universal fixed-point condition. They glue to the closed smooth scheme \(X^T\subset X\). Thus \(X^T\) is projective and complete. This uses the earlier full infinitesimal proof directly and includes nonreduced test schemes. At a fixed point \(x=gB\), put \(B'=gBg^{-1}\). The equality of its schematic stabilizer with \(B'\) shows that \(T\subset B'\). The quotient map \(H\to X\) is smooth: on the fppf torsor trivializations it is projection from the smooth group \(B\), and smoothness descends. Its differential is therefore surjective, with kernel the tangent space of the fibre \(B'\). It gives the exact \(T\)-representation sequence \[ 0\longrightarrow\mathfrak b'\longrightarrow\mathfrak h \longrightarrow T_xX\longrightarrow0. \] The exact weight projections of RG01 give \[ T_x(X^T)=(T_xX)^T =\mathfrak h^T/(\mathfrak b')^T =\operatorname{Lie}C/\operatorname{Lie}C_{B'}(T). \] The last identities are the fixed-subgroup tangent calculation of that same AG-GS Theorem 3.4, applied to conjugation on \(H\) and \(B'\). The differential of the orbit map \(C\to X^T\) is consequently surjective at the identity. Lemma 1.1 already proves that \(C\) is smooth and connected. S02 Lemma 3.A now makes this orbit map smooth near the identity, and translation makes it smooth everywhere. Its image is open by S08 Lemma G.4.F. S08 Theorem G.4.C supplies the full schematic orbit, its fppf stabilizer torsor and its immersion. Flatness of the smooth orbit map descends through the fppf map from \(C\) to its homogeneous orbit, so that immersion is flat. Locally it is a quotient \(A\to A/I\). Tensoring its ideal sequence with the flat \(A/I\) gives \(I/I^2=0\); at a point of the orbit the ideal is finite and lies in the maximal ideal, so the determinant Nakayama proof of S01 Lemma 1.2 gives \(I=0\) locally. Thus the orbit immersion is an open immersion, and the full scheme orbit is precisely this open subscheme. These disjoint open orbits cover \(X^T\). Indeed they contain every \(k\)-point, and a nonempty closed complement would have a \(k\)-point by the Nullstellensatz. Quasi-compactness gives a finite subcover, hence finitely many orbits. Each is also closed, since its complement is the union of the other open orbits. Thus every orbit is smooth and complete. Since \(C=T\times U_C\) and \(T\) fixes the entire fixed scheme, each is also the full schematic \(U_C\)-orbit of its chosen point. Such a \(U_C\)-orbit is affine, with the following explicit earlier proof route. Let \(K\subset U_C\) be its schematic stabilizer. The RG01 detecting-line construction gives a representation and a line \(kv\) whose schematic stabilizer in \(U_C\) is \(K\). The subgroup \(K\) is unitriangular, so RG01 Lemma 2.A makes its line character trivial, including for nonreduced \(K\). Its vector stabilizer is therefore exactly \(K\). S08 Theorem G.4.C identifies \(U_C/K\), retaining the entire \(K\), with the vector orbit in affine space. Lemma 1.2 makes that orbit closed there. The original orbit and this vector orbit represent the same fppf quotient by \(K\), so they are isomorphic. This proves affineness without replacing the stabilizer by its rational points. An orbit just obtained is integral: it is smooth, and its underlying space is the surjective image of the irreducible smooth connected \(U_C\). As a closed projective subscheme, its algebra of global functions is finite-dimensional over \(k\), by S01 Theorem 8.4 and its closed-immersion cohomology calculation. This algebra is a domain. Multiplication by any nonzero element is injective and hence surjective on that finite-dimensional vector space; thus it is a field, and algebraic closedness makes it \(k\). The orbit is affine, so it is \(\operatorname{Spec}k\). We conclude that \(X^T\) is a finite disjoint union of such reduced points. The action of \(C\) on each of these point orbits is trivial as a scheme morphism. In particular, at every \(gB\) fixed by \(T\), its full schematic stabilizer contains \(C\); equivalently \(C\subset B'\) for every Borel \(B'\) containing \(T\). \(\square\) The normalizer acts transitively on \(X^T\). If \(gB\) is fixed, conjugate \(g^{-1}Tg\) to \(T\) within \(B\) to obtain a representative in \(N_H(T)\). Its stabilizer is \(N_B(T)=C_B(T)=C\), since conjugation in the solvable decomposition induces the identity on \(B/R_u(B)\). Thus \(X^T\) is the set \(N_H(T)/C_H(T)\), with a simply transitive action of that quotient. A cocharacter separating the finitely many weights in an equivariant projective embedding has exactly the same fixed points as \(T\). \textbf{Lemma 1.D (Borel intersections).} For a torus \(A\subset B\), the full subgroup \(K=C_H(A)\cap B=C_B(A)\) is a Borel subgroup of \(C=C_H(A)\). Its full quotient \(C/K\) is a closed projective smooth orbit in \((H/B)^A\). \textbf{Proof.} Lemma 1.1 makes \(C\) smooth and connected, and Lemma 1.B applied to the solvable \(B\) makes \(K\) smooth and connected. The equivariant projective embedding and smooth quotient \(X=H/B\) are those of RG01 Lemma 2.1 and S08 Theorem G.4.C. Choose a weight basis for \(A\) in the embedding representation. The projective weight-coordinate charts are \(A\)-stable affine opens. AG-GS-05 Proposition 3.2 and Theorem 3.4 construct their closed smooth fixed schemes with invariant tangent spaces; they agree on overlaps by their universal fixed-point condition. They therefore give a closed smooth projective scheme \(X^A\subset X\), with its full fixed-point condition on arbitrary test schemes. For any \(x=gB\in X^A(k)\), put \(B'=gBg^{-1}\). Its full stabilizer contains \(A\). The map \(H\to X\) is smooth: it is a flat finitely presented \(B\)-torsor whose geometric fibres are translates of the smooth \(B\), so the geometric-fibre criterion applies. Its tangent sequence at \(x\) is exact. Exact weight projections therefore give \[ T_x(X^A)=(\mathfrak h/\mathfrak b')^A =\operatorname{Lie}C/\operatorname{Lie}C_{B'}(A). \] These centralizer tangent identities are the fixed-subgroup calculation of AG-GS-05 Theorem 3.4. The differential of \(C\to X^A\) is surjective at the identity. S02 Lemma 3.A makes the orbit map smooth near the identity, and translation makes it smooth throughout \(C\). Its image is open by S08 Lemma G.4.F. S08 Theorem G.4.C gives its full orbit immersion and its fppf stabilizer torsor. That immersion is flat, since its composite with the faithfully flat torsor is smooth and flatness descends. In a local closed-immersion presentation \(R\to R/I\), flatness gives \(I/I^2=0\). At a point of the orbit the finite ideal \(I\) lies in the maximal ideal, so Nakayama gives \(I=0\) locally. Hence the full orbit is an open subscheme of \(X^A\). The disjoint open \(C\)-orbits cover all rational points of \(X^A\). Their complement, if nonempty, would be a closed finite-type subscheme with a rational point, so they cover \(X^A\). Quasi-compactness gives finitely many such orbits; each is also closed, because its complement is the union of the others. In particular the orbit \(O\) of \(eB\) is closed and projective, smooth and integral, and its full stabilizer is exactly \(K\). S08 Theorem G.4.C identifies \(O\) with the full quotient \(C/K\), without passing to pointwise stabilizers. The smooth connected solvable \(K\) is contained in a Borel \(D\) of \(C\): choose a containing smooth connected solvable subgroup of largest dimension, and use the strict dimension drop for proper closed integral subgroups, as in RG01 Lemma 2.1. Its action on the complete nonempty \(O\) has a fixed point by RG01 Lemma 2.M. At \(cK\) this says \(D\subset cKc^{-1}\) schematically. Thus \(\dim D\le\dim K\). Since \(K\subset D\) and both are smooth connected integral groups, a proper inclusion would give a strict dimension increase. They are equal. Consequently \(K\) is a Borel of \(C\). Finally RG01 Lemma 2.1 gives conjugacy of Borels in \(C\), so all its Borels are conjugates of this intersection. \(\square\) \textbf{Theorem 1.E (Borel normalizers).} Every Borel subgroup of a smooth connected affine group \(H\) over an algebraically closed field satisfies \[ N_H(B)=B \] as group schemes. \textbf{Proof.} We first construct the full normalizer. In the conjugation representation on \(k[H]\), AG-GS-01 Lemma 5.1 gives a finite-dimensional stable space \(W\) containing a finite generating list for the ideal \(I_B\). Put \(J=I_B\cap W\). Its stabilizer in \(H\) is a closed finite-type subgroup: in a basis of \(W\) extending one of \(J\), preservation of \(J\) and of \(J\) by the inverse matrix is a finite list of equations in the representation matrix and its inverse. This stabilizer represents \(N_H(B)\). Indeed, \(J\) generates \(I_B\), so stabilizing \(J\) preserves that ideal in both directions. Conversely a normalizing section preserves \(I_B\) and \(W\), hence their intersection. These statements remain true over every \(k\)-algebra \(R\): exact scalar extension identifies \(J_R=I_{B,R}\cap W_R\), and the chosen ideal generators still generate. Thus this construction includes arbitrary nonreduced test schemes. Write \(N\) for the resulting subgroup. Choose \(T\subset B\) maximal in \(H\), as in the preceding fixed-coset argument. The torus-fixed-coset proof gives \(C_H(T)\subset B\), and hence \(\mathfrak h^T\subset\mathfrak b\). We compute the tangent space of \(N\) directly. An infinitesimal element with tangent vector \(X\in\mathfrak h\) normalizes \(B\) precisely when, for its universal section \(b\), infinitesimal conjugation stays in \(B\). Translating the resulting tangent at \(b\) to the identity gives \[ X-\operatorname{Ad}(b)X\in\mathfrak b\otimes_k k[B]. \] To check this criterion schematically, substitute the infinitesimal conjugation into every generator of \(I_B\): its constant coefficient is zero on \(B\), and its first-order coefficient is the pairing with this translated tangent vector. Vanishing of all such coefficients is exactly membership in the tangent space of \(B\); translation identifies that space with \(\mathfrak b\) on all test algebras. The inverse infinitesimal conjugation gives the negative coefficients, so the criterion gives preservation in both directions. Consequently \[ \operatorname{Lie}N/\mathfrak b=(\mathfrak h/\mathfrak b)^B \subset(\mathfrak h/\mathfrak b)^T=0, \] where the last equality uses exact weight projections and \(\mathfrak h^T\subset\mathfrak b\). Thus the tangent dimension of \(N\) at the identity is \(\dim B\). Its local dimension is at least \(\dim B\), since it contains the smooth \(B\), and at most its tangent dimension, by the local embedding-dimension bound. Equality makes this local ring regular; the perfect-field Jacobian criterion makes \(N\) smooth at the identity. Translation proves smoothness at every rational point, and hence everywhere. It will therefore suffice to show \(N(k)=B(k)\), since equations vanishing at all points of the reduced \(N\) vanish by the strong Nullstellensatz. We also need the following consequence of projectivity: every rational element of \(H\) lies in a Borel. Choose \(t\in T(k)\) avoiding the equations \(\chi(t)=1\) for every nonzero adjoint weight \(\chi\). Such a point exists, because these are finitely many proper closed subsets of the integral torus; the weak Nullstellensatz supplies a rational point of their open complement. For \(T=1\) the same calculation has no nonzero weights. The differential of \[ H\times C_H(T)\longrightarrow H,\qquad(g,c)\longmapsto gcg^{-1}, \] at \((1,t)\), with tangent spaces translated to the identity, is \((1-\operatorname{Ad}t)X+Y\). It is surjective: the first summand covers every nonzero weight and \(\operatorname{Lie}C_H(T)=\mathfrak h^T\) covers weight zero. S02 Lemma 3.A and S08 Lemma G.4.F give a nonempty open image, dense in the integral \(H\). The image lies among conjugates of \(B\), since \(C_H(T)\subset B\). The incidence of pairs \((h,x)\) with \(hx=x\) is closed in \(H\times H/B\), by the separated diagonal. Its projection to \(H\) has closed image by the projectivity of RG01 Lemma 2.1 and S01 Lemma 7.A and Corollary 7.B. This image contains the preceding dense open, so is all of \(H\). At any rational \(h\), its nonempty finite-type fibre has a rational fixed coset; lifting that coset through the full \(B\)-torsor gives \(h\in gBg^{-1}\) for some \(g\in H(k)\). In particular, if \(B\) is normal in \(H\), then \(B(k)=H(k)\) and reducedness gives \(B=H\) schematically. We now induct on \(\dim H\); the zero-dimensional smooth connected group is trivial. Let \(n\in N(k)\). RG01 Lemma 2.N conjugates \(nTn^{-1}\) back to \(T\) within \(B\). Multiplying \(n\) by that element of \(B\) preserves both its membership in \(N(k)\) and the question whether it belongs to \(B(k)\). We may assume that \(n\) normalizes \(T\). If \(T=1\), RG01 Lemma 2.D makes \(H\) unipotent, hence solvable; its Borel is then \(H\) itself. Otherwise consider the torus homomorphism \[ f:T\longrightarrow T,\qquad t\longmapsto ntn^{-1}t^{-1}. \] If \(f\) is not surjective, its character matrix has rank smaller than \(\dim T\). The integer basis calculation of S02 Lemma 3.M expresses its kernel as a diagonalizable group with a free character summand of positive rank. The corresponding split factor supplies a nontrivial torus \(A\subset\ker f\). Thus \(n\in C_H(A)\), and it normalizes \(K=C_H(A)\cap B\), a Borel of this smooth connected centralizer by Lemma 1.D. If the centralizer is proper in \(H\), its dimension is strictly smaller; induction gives \(n\in K\subset B\). If instead \(C_H(A)=H\), the torus \(A\) is central. The quotient construction in Lemma 1.C uses only centrality through its construction of the smooth connected affine \(Q=H/A\) and its full smooth \(A\)-torsor \(q\). S08 Lemma G.4.D gives \(\dim Q=\dim H-\dim A<\dim H\). The closed subgroup \(B\subset H\) descends along \(q\) to a closed subgroup \(\overline B\subset Q\): right multiplication by \(A\subset B\) preserves its defining ideal, so the ideal effectivity argument of S04 Lemmas A1.1 and A1.2 gives \[ q^{-1}(\overline B)=B, \qquad B\longrightarrow\overline B \text{ an fppf }A\text{-torsor}. \] Its coordinate ring injects into the reduced \(k[B]\), so \(\overline B\) is reduced and connected. RG01 Lemma 2.L makes it smooth; derived commutator identities descend through the torsor, so it is solvable. The central \(A\) acts trivially on \(H/B\), since \(a hB=h aB=hB\). Morphism descent gives an action of \(Q\) there with full stabilizer \(\overline B\), and the same local representatives identify its full quotient with the projective \(H/B\). Any containing Borel of \(Q\) has a fixed point on this quotient by RG01 Lemma 2.M, and the stabilizer inclusion and strict dimension argument of Lemma 1.D force it to equal \(\overline B\). Thus \(\overline B\) is a Borel. Induction in \(Q\) gives \(q(n)\in\overline B\), so the exact inverse-image identity gives \(n\in B\). It remains to handle surjective \(f\). The RG01 detecting-line construction gives an \(H\)-representation and a line \(kv\) with full stabilizer \(N\); write \(\chi:N\to\mathbf G_m\) for its character. This character kills commutators, so \(\chi(f(t))=1\) on every test algebra. Surjectivity of the torus homomorphism makes its character pullback injective, hence \(\chi|_T=1\). RG01 Lemma 2.A also makes \(\chi\) trivial on the unipotent radical of \(B\). The split-solvable decomposition therefore makes \(B\) fix \(v\) as a vector. The map \(H\to V\), \(h\mapsto hv\), descends along its full \(B\)-torsor to \(H/B\to V\). The global-function algebra of this integral projective quotient is finite-dimensional by S01 Theorem 8.4; being a domain over algebraically closed \(k\), it is \(k\). All coordinate functions of the descended map are constant, with value \(v\) at \(eB\). Thus \(H\) fixes \(v\) as a scheme morphism, and the detecting-line stabilizer gives \(H=N\). Now \(B\) is normal in \(H\), so the incidence argument above gives \(H=B\). This proves \(n\in B\) in the final case and completes the induction. The smooth normalizer and the strong Nullstellensatz turn equality of rational points into the required full scheme equality. \(\square\) We may consequently identify \(H/B\) with the variety of Borel subgroups. The preceding simply transitive action on its torus-fixed points is also the action on the Borels containing \(T\). \textbf{Lemma 1.3.} Let \(H\) be smooth connected affine over the algebraically closed field \(k\), and let \(T\subset H\) be a maximal torus. If \(\mathcal B^T\) is the collection of Borels containing \(T\), put \[ J=\bigcap_{B'\in\mathcal B^T}R_u(B'),\qquad I=(J_{\mathrm{red}})^0. \] Then \(I\subset R_u(H)\), as a schematic inclusion. \textbf{Proof.} Write \(X=H/B\), where \(B\) contains \(T\). The full normalizer theorem identifies its rational points with the Borel subgroups, and the torus-fixed-coset proof makes \(X^T\) a finite disjoint union of reduced points. Thus \(\mathcal B^T\) is finite and nonempty, and \(J\) is the full finite intersection of closed subgroup schemes. RG01 Lemma 2.L makes \(I\) smooth connected. It is unipotent, since it embeds in the unitriangular \(R_u(B)\). For each \(n\in N_H(T)(k)\), conjugation permutes the Borels in \(\mathcal B^T\) and their unipotent radicals, which are characteristic by Lemma 1.A. It preserves the full \(J\) and hence \(I\). This invariance under actual rational conjugating elements is sufficient for the chart transport below. RG01 Lemma 2.1 embeds \(X\) equivariantly as a closed subscheme of \(\mathbf P_{\mathrm{lines}}(V)=\mathbf P(V^\vee)\). Replace \(V\) by the span of its point lines. This span is \(H\)-stable on all test algebras: the matrix entries into its quotient vanish on \(H(k)\), and hence on the reduced \(H\) by the strong Nullstellensatz. The embedding factors through this span by the same argument on reduced \(X\). Thus \(X\) spans \(V\). Choose a cocharacter \(\lambda\) separating the distinct \(T\)-characters in \(V\). Here is an explicit choice when \(T=\mathbf G_m^r\), \(r>0\). Write each difference of two characters as \((a_1,\ldots,a_r)\). The nonzero integer polynomial \(\sum_i a_i M^{i-1}\) has finitely many integer roots by S02 Lemma 4.A. Choose one positive integer \(M\) outside these finitely many root sets and put \(\lambda(u)=(u,u^M,\ldots,u^{M^{r-1}})\). For \(r=0\) use the unique cocharacter. On weight-coordinate projective charts, the ideals of the \(T\)-fixed and \(\lambda\)-fixed schemes are generated by the same coordinates: a difference character is nonzero precisely when its pairing with \(\lambda\) is nonzero. The universal weight-ideal calculation of AG-GS-05 Proposition 3.2 therefore gives \(X^\lambda=X^T\), including arbitrary test algebras. Let \(V_0\) be the weight space of lowest \(\lambda\)-weight among those in \(V\). The open \(U\subset X\) where projection to \(V_0\) is nonzero is nonempty, since \(X\) spans \(V\). For a point in this open, multiply its weight coordinates by \(u^{w-w_0}\). All exponents are nonnegative, so this defines the limit at \(u=0\), in \(\mathbf P_{\mathrm{lines}}(V_0)\). The limit is in \(X\): on each projective chart its equations vanish in the Laurent polynomial ring and hence in the polynomial ring. Thus projection defines a morphism \[ U\longrightarrow X\cap\mathbf P_{\mathrm{lines}}(V_0)\subset X^T. \] The integral \(U\) has irreducible image in this finite reduced scheme, so that image is one point \(x=[v]\). Every nonzero projected point line is consequently \(kv\); projection of the point lines spanning \(V\) spans \(V_0\), so \(V_0=kv\). Choose the coordinate covector \(\ell\) with \(\ell(v)=1\) and zero on the other weight spaces. The chart \[ X_x=X\cap D(\ell) \] is affine, contains \(x\), is \(T\)-stable, and \(\lambda\) contracts every one of its rational points to \(x\). The group \(N_H(T)(k)\) is transitive on \(X^T(k)\). Indeed a fixed coset \(gB\) has \(g^{-1}Tg\subset B\); this is a maximal torus of \(B\). RG01 Lemma 2.N supplies \(b\in B(k)\) with \(b^{-1}g^{-1}Tgb=T\), so \(gb\in N_H(T)(k)\) represents that same coset. Apply this to two fixed cosets to obtain an element carrying one to the other. Transport \(X_x\) to an affine \(T\)-stable chart \(X_y\) containing each \(y\in X^T(k)\). These charts cover \(X\). For a rational point \(z\), the reduced closure of its \(T\)-orbit is projective and \(T\)-stable by S08 Theorem G.4.2. RG01 Lemma 2.M gives a fixed point \(y\) in this closure. If \(z\notin X_y\), its entire orbit and closure would be contained in the closed \(T\)-stable complement, contradicting \(y\in X_y\). Thus all rational points lie in these finitely many charts; the Nullstellensatz rules out a nonempty closed complement. We prove their invariance under \(I\). Every \(H(k)\)-orbit in \(\mathbf P_{\mathrm{lines}}(V^\vee)\) meets the affine chart of covectors nonzero on \(v\). Otherwise a nonzero covector would vanish on every translate of \(v\); these translates span \(V\), since \(H\) is transitive on \(X\). On this dual chart \(\lambda^{-1}\) contracts to \([\ell]\), by the same coordinate calculation. S08 Theorem G.4.2 gives a closed \(H\)-orbit in this projective space. It meets the dual chart, so its closedness includes \([\ell]\). It is therefore exactly the orbit \(H[\ell]\). S08 Theorem G.4.C gives this closed projective orbit with its full stabilizer \(P\) and its fppf torsor \(H\to H[\ell]\). The action of \(B\) on this nonempty projective orbit has a fixed point by RG01 Lemma 2.M. Its full stabilizer supplies an \(H\)-Borel \(B_1\subset P\). Both \(B_1\) and the torus \(T\), which fixes \([\ell]\), lie in the smooth connected \((P_{\mathrm{red}})^0\), by RG01 Lemma 2.L. A maximal torus \(T_1\) of \(H\) in \(B_1\), and \(T\), are maximal there too. RG01 Theorem 2.2 conjugates \(T_1\) to \(T\) by an element of this reduced identity component. It carries \(B_1\) to a Borel \(B_2\subset P\) containing \(T\). By its definition \(I\subset R_u(B_2)\subset P\) schematically. It therefore preserves the covector line \(k\ell\), and hence the chart \(D(\ell)\), on every test algebra. This proves invariance of \(X_x\). The previously proved invariance of \(I\) under each transporting normalizer element gives invariance of every \(X_y\). Neither smoothness nor connectedness of \(P\) has been assumed. For \(z\in X(k)\), the reduced projective closure of \(Iz\) is \(I\)-stable, by S08 Theorem G.4.2. The smooth connected solvable \(I\) has a fixed point \(z_0\) there, by RG01 Lemma 2.M. Choose an invariant affine chart \(X_y\) containing \(z_0\). If the orbit met its invariant closed complement, that complement would contain the whole point orbit and its closure, contradicting \(z_0\in X_y\). Hence the orbit lies in \(X_y\). Lemma 1.2 makes its full orbit closed in this chart, so it contains \(z_0\); transitivity makes it a point. S08 Theorem G.4.C gives its reduced scheme structure, so the full \(I\)-action fixes \(z\). This holds for every rational \(z\); the reducedness of \(I\times X\) and the strong Nullstellensatz give equality of the action with the projection on the whole scheme. For the final normality argument, form the schematic intersection \[ K=\bigcap_{g\in H(k)}gBg^{-1}. \] It is a closed finite-type subgroup: the sum of these conjugate ideals in the Noetherian \(k[H]\) is generated by finitely many elements, each from a finite subsum. The resulting finite intersection represents the same condition on every test algebra. The stabilizer of \(gB\) is the entire \(gBg^{-1}\), so the preceding scheme action gives \(I\subset K\). Put \(K_0=(K_{\mathrm{red}})^0\). It is smooth connected, contains \(I\), and is solvable because it is contained in \(B\). Every rational conjugation by \(H\) preserves \(K_0\); the reducedness of \(H\times K_0\) and the strong Nullstellensatz make \(K_0\) normal in \(H\) schematically. RG01 Theorem 2.K gives \(K_0=U\rtimes D\), with \(U=R_u(K_0)\) smooth connected unipotent and \(D\) a torus. Every character of \(I\) is trivial by RG01 Lemma 2.A, so its map to the torus quotient \(D\) is trivial and \(I\subset U\). The uniqueness of the unipotent radical makes \(U\) invariant under every rational conjugation by \(H\); reducedness of \(H\times U\) and the same equation argument make it normal schematically. Lemma 1.A then gives \(U\subset R_u(H)\), proving \(I\subset R_u(H)\). \(\square\) This is the geometric mechanism in Chevalley's description of the unipotent radical. It gives exactly the part needed for centralizers without presupposing roots. \section{2. Centralizers and weight zero}\label{AG-RG-02-centralizers-and-weight-zero} \textbf{Theorem 2.1.} If \(G\) is connected reductive over an algebraically closed field and \(T\) is maximal, then \(C_G(T)=T\). More generally, the centralizer of any torus in \(G\) is connected reductive. \textbf{Proof.} For a maximal \(T\), Lemmas 1.1 and 1.C give the full decomposition \[ C=C_G(T)=T\times U_C, \] where \(U_C\) is smooth connected unipotent. The torus-fixed-coset proof gives \(C\subset B'\) for every Borel \(B'\) containing \(T\). Its split-solvable quotient \(B'/R_u(B')\) is a torus by RG01 Theorem 2.K. The homomorphism from \(U_C\) to this quotient is trivial: every character pulled back to a unitriangular group is trivial by RG01 Lemma 2.A, and the torus characters generate its coordinate algebra. Thus \(U_C\subset R_u(B')\) on all test algebras. It lies in their full intersection \(J\), and its reduced connected identity-containing source factors through \(I=(J_{\mathrm{red}})^0\), by RG01 Lemma 2.L. Lemma 1.3 gives \(U_C\subset R_u(G)\). Reductivity makes this radical trivial, so \(C_G(T)=T\) as group schemes. Now let \(A\subset G\) be any torus. Choose a maximal torus \(T\supset A\): among containing tori choose one of largest dimension, bounded by \(\dim G\); every proper closed inclusion of integral tori increases dimension, so it is maximal. Put \(C=C_G(A)\), smooth connected by Lemma 1.1, and put \(R=R_u(C)\), whose existence and normality are Lemma 1.A. For every Borel \(B'\) of \(G\) containing \(T\), Lemma 1.D makes \(K'=C\cap B'\) a Borel of \(C\), with projective full quotient \(C/K'\). The action of the smooth connected solvable \(R\) on this quotient has a fixed point by RG01 Lemma 2.M. Lift that point through its fppf stabilizer torsor to \(c\in C(k)\); its full stabilizer gives \(R\subset cK'c^{-1}\). Since \(R\) is normal in \(C\), this is equivalent to \(R\subset K'\subset B'\). Its map to \(B'/R_u(B')\) is again trivial by the character argument, so \(R\subset R_u(B')\). Thus \(R\) factors through the same \(I\) attached to \(G,T\), and Lemma 1.3 gives \[ R\subset R_u(G)=1. \] The smooth connected affine \(C\) therefore has trivial unipotent radical, which is precisely reductivity over this field. This proves both assertions with their full scheme structures. \(\square\) The Borel-intersection step here is the complete projective fixed-orbit argument of Lemma 1.D; no additional closure or rigidity argument is needed. \textbf{Theorem 2.2.} If \(G\to S\) is reductive and \(T\) is a maximal \(S\)-torus, then \(C_G(T)=T\). For any subtorus \(A\), the group \(C_G(A)\) is reductive over \(S\). These centralizers commute with arbitrary base change. \textbf{Proof.} A torus splits étale locally by its definition. On a split affine chart, conjugation by \(A\) grades the affine coordinate algebra of \(G\). AG-GS-05 Proposition 3.2 represents the fixed group by the ideal generated by all nonzero-weight elements, proves finite presentation, and proves compatibility with every base change, including nonreduced bases. Its Theorem 3.4 proves smoothness of this fixed group when \(G\) is smooth. The fixed-point functor is the centralizer functor on every test scheme. Thus the constructions agree canonically on overlaps; the ideal and algebra effectivity in S04 Lemmas A1.1 and A1.2 gives the global affine closed subgroup \(C_G(A)\). Finite presentation and flatness descend by S04 Lemmas A1.3 and A1.3a; its geometric fibres are smooth by the split calculation, so the geometric-fibre criterion of AG-FSE-05 Theorem 3.1 proves global smoothness. The same canonical fixed condition and the local base-change calculation prove arbitrary base-change compatibility globally. Each geometric fibre is the connected reductive centralizer in Theorem 2.1. Consequently this smooth affine group of finite presentation is reductive by the definition of a reductive group scheme. In particular this proves the assertion for arbitrary \(A\) before treating maximal \(T\). For maximal \(T\), its closed inclusion \(i:T\to C_G(T)\) is an isomorphism on every geometric fibre by Theorem 2.1 and the just-proved base-change compatibility. Both groups are smooth of finite presentation over \(S\). Their relative tangent map is an isomorphism at every point of \(T\), since it becomes one over a geometric residue field; the exact tangent and differential scalar-extension calculation identifies that map with the one on the fibre. S02 Lemma 3.A therefore makes \(i\) étale everywhere. Its finite presentation follows also from the affine-pair presentation calculation of AG-MO-02 Theorem 3.1. Its étale smoothness makes it flat by AG-CA-18 Theorem 6.1. Here is the scheme equality over the original base, without a fibrewise-isomorphism shortcut. On an affine chart of \(C_G(T)\), the closed inclusion has algebra \(D\to D/I\), with \(I\) finitely generated by finite presentation. Flatness of \(D/I\) makes the tensor of \(I\hookrightarrow D\) injective; that tensor map is zero, so \[ I\otimes_D D/I=I/I^2=0. \] Every target point is in the image of \(i\): extend its residue field to an algebraic closure and use the fibre isomorphism. Thus, at every prime \(\mathfrak p\) of \(D\), the finite ideal \(I_{\mathfrak p}\) is contained in the maximal ideal and satisfies \(I_{\mathfrak p}=I_{\mathfrak p}^2\subset\mathfrak pD_{\mathfrak p}I_{\mathfrak p}\). The determinant Nakayama proof of S01 Lemma 1.2 gives \(I_{\mathfrak p}=0\). Localization detects the zero module, so \(I=0\). All these affine charts therefore give \(T=C_G(T)\), with the original base and every nilpotent retained. \(\square\) \textbf{Theorem 2.3.} A reductive group scheme admits maximal tori étale locally. Any two maximal tori are conjugate étale locally. \textbf{Proof.} Fix \(s\in S\). AG-RG-01 Theorem 4.2 gives a geometrically maximal torus \(M\subset G_s\) over \(\kappa(s)\) itself. AG-RG-01 Lemma 3.1 extends that specified closed embedding to a torus \(T\subset G_U\) on a pointed étale neighbourhood \((U,u)\to(S,s)\), with \(\kappa(u)=\kappa(s)\) and fibre embedding equal to \(M\subset G_s\). No separate residue extension or assertion about an unrepresented embedding functor is needed. Theorem 2.2, applied to this subtorus before it is known to be maximal on \(U\), makes \(C=C_{G_U}(T)\) a smooth affine reductive group of finite presentation and identifies every geometric fibre with the centralizer of the fibre torus. Write \(r=\dim M\). The torus rank is locally constant, so restrict to an open neighbourhood of \(u\) on which it is \(r\). The relative Lie module \[ \mathfrak c=e_C^*\mathcal T_{C/U} \] is finite locally free: the standard smooth-coordinate calculation in AG-RG-S02 Lemma 3.1 makes the relative differential bundle locally free, and dualizing and pulling back by the identity gives this module. Its scalar extension is the Lie algebra of the fibre centralizer, by the differential base-change calculation of AG-CA-15 Theorem 2.1. At a geometric point above \(u\), Theorem 2.1 gives \(C_{G_{\bar u}}(T_{\bar u})=T_{\bar u}\), so \(\mathfrak c\) has rank \(r\) there. A local free basis of \(\mathfrak c\) consequently gives an open neighbourhood of \(u\) on which its rank is \(r\). Restrict to that neighbourhood. For every geometric point \(v\) of this smaller \(U\), the smooth connected group \(C_v\) has dimension \(r\), since its dimension equals its tangent dimension at the identity. The closed subgroup \(T_v\) is a torus of the same dimension. A smooth connected finite-type group over an algebraically closed field is integral by the component calculation in AG-RG-01 Lemma 2.L. A proper closed subset of an integral finite-type scheme has smaller dimension, by AG-CA-09 Theorem 4.2. Thus \(T_v\) has the whole underlying space of \(C_v\). Its defining ideal is radical because \(T_v\) is reduced, and its radical is zero because \(C_v\) is reduced and the underlying space is the whole space. Therefore \(T_v=C_v\) as schemes. Any torus containing \(T_v\) centralizes it and hence lies in \(C_v=T_v\); it equals \(T_v\). Thus every fibre torus is maximal, which makes \(T\) a maximal \(U\)-torus. These open restrictions and étale neighbourhoods exist around every \(s\), proving local existence over the original, possibly nonreduced base. Now let \(T_1,T_2\subset G\) be maximal \(S\)-tori. On each geometric fibre they are conjugate by AG-RG-01 Theorem 2.2. The full equality transporter \[ P=\operatorname{Transp}_G(T_1,T_2) \] is represented by a scheme of finite presentation and is smooth by AG-RG-01 Lemma 3.1: its Laurent-coefficient equations test equality of the subgroup schemes, including on algebras with nilpotents. Each geometric fibre has a conjugating point, so \(P\to S\) is surjective. Apply AG-FSE, \emph{Smooth morphisms}, Theorem 5.2 to this smooth surjection. Its proof first chooses a point of the smooth fibre with finite separable residue extension, realizes that extension by a pointed étale neighbourhood, and then pulls an étale coordinate chart back along a section of affine space. The resulting étale neighbourhood \(V\to S\) has an \(S\)-map \(V\to P\). The represented transporter condition says exactly that its image \(g\in G(V)\) satisfies \(gT_{1,V}g^{-1}=T_{2,V}\) as subgroup schemes. This proves étale-local conjugacy. \(\square\) The statement is local around every point, and the conjugating element belongs to the group over the chosen neighbourhood. It does not assert a global section of the transporter. If \(T\) is split, write \[ \mathfrak g=\mathfrak t\oplus\bigoplus_{\alpha\in\Phi}\mathfrak g_\alpha. \] Here \(\Phi\) is the finite collection of nonzero weights on a neighbourhood where their ranks are constant. Theorem 2.2 says that the zero-weight summand is exactly \(\mathfrak t\). In the next lesson we prove that the nonzero summands have rank one and construct their root groups. The arguments about regularity below need the weight decomposition and weight zero; they do not need that rank-one conclusion. \section{3. Three uses of the word regular}\label{AG-RG-02-three-uses-of-the-word-regular} Let \(G\) be connected reductive over an algebraically closed field, with rank \(r=\dim T\). A group element is \textbf{dimension-regular} if its centralizer has the minimum dimension \(r\). A \textbf{regular semisimple} element is a semisimple element \(g\) such that \(C_G(g)^0\) is a maximal torus. A \textbf{strongly regular semisimple} element additionally has \(C_G(g)\) itself equal to that torus. This last condition is stronger. There is also the convention used in SGA 3, Exposé XIII: a group element is regular if the generalized eigenspace for eigenvalue \(1\) in its adjoint action has the minimum dimension, the nilpotent rank. For reductive groups this minimum is \(r\), and this condition selects exactly the regular semisimple elements. We verify this identification below. Thus dimension-regular unipotent elements are not regular in that convention. For \(t\in T\), define \[ T_{\mathrm{rs}}=\{t:\alpha(t)\ne1\text{ for every }\alpha\in\Phi\}. \] The inequalities mean invertibility of the indicated functions over a general base. Since every \(\alpha\) is a nontrivial character, its equation \(\alpha=1\) is a proper closed subset of a torus over an algebraically closed field. A finite union of such subsets cannot cover the irreducible torus. Hence \(T_{\mathrm{rs}}\) is open and dense, and is nonempty in every geometric fibre in the relative case. For a semisimple \(t\), its centralizer is smooth: the closure of the subgroup generated by \(t\) is of multiplicative type, and invariants for such a group are exact. Its Lie algebra is the fixed space of \(\operatorname{Ad}(t)\). Consequently \[ \operatorname{Lie}(C_G(t))=\mathfrak t\oplus \bigoplus_{\alpha(t)=1}\mathfrak g_\alpha. \] The identity centralizer equals \(T\) precisely on \(T_{\mathrm{rs}}\). Indeed when no moving weight is fixed, the smooth identity centralizer has the same tangent dimension as \(T\). Its closed torus inclusion has an isomorphic differential, so S02 Lemma 3.A and the étale-monomorphism proof of S04 make it open. It is also closed; connectedness makes it the whole identity component. If a moving weight is fixed, its nonzero summand gives strictly larger tangent dimension, and smoothness makes the identity centralizer strictly larger than \(T\). \textbf{Theorem 3.1.} The regular semisimple locus \(G_{\mathrm{rs}}\) is a dense open subset. Its elements lie in a unique maximal torus. \textbf{Proof.} First construct an open set of conjugates of torus elements. Consider the conjugation map \[ q:G\times T_{\mathrm{rs}}\longrightarrow G,\qquad(g,t)\longmapsto gtg^{-1}. \] After translations identify tangent spaces, its differential at \((1,t)\) is \[ (X,Y)\longmapsto (1-\operatorname{Ad}(t))X+Y. \] It is surjective: the second term covers \(\mathfrak t\), while the first is invertible on every nonzero weight summand. \href{AG-RG-S02.html\#smooth-map-differential}{AG-RG-S02, Lemma 3.A}, therefore makes \(q\) smooth, so its image is nonempty and open. We justify density precisely. The affine finite-type \(G\) is Noetherian by the Hilbert basis proof and has finitely many irreducible components. If two met, their nonempty closed intersection would have a rational closed point by the Nullstellensatz. Smoothness makes its local ring regular by the field Jacobian proof; AG-CA-14 Theorem 1.1 makes that local ring a domain, contradicting two component primes there. Thus the components are disjoint and each is open and closed. Connectedness leaves one component. Hence \(G\) is irreducible and this nonempty open image is dense. Here is a useful consequence of that density. Choose a Borel \(B\) containing \(T\). In \(G\times G/B\) the incidence scheme \[ I=\{(g,z):gz=z\} \] is closed, since the flag variety is separated. Its projection to \(G\) is proper: AG-RG-01, Lemma 2.1 and \href{AG-RG-S01.html\#projective-properness}{AG-RG-S01, Lemma 7.A and Corollary 7.B} make \(G/B\) proper, and Corollary 7.B preserves properness under this base change and restriction to the closed incidence. Its image contains the dense open just constructed, and is consequently all of \(G\). Over our algebraically closed field, a nonempty fibre has a rational point. Thus \textbf{every element of \(G\) belongs to a Borel subgroup}. Let b be a semisimple element of \(B=U\rtimes T\). In a faithful representation diagonalize b and take the reduced closure \(D\) of its integral powers. It is a closed subgroup of a diagonal torus, hence diagonalizable by the explicit Hopf-quotient proof in AG-GS-05, Theorem 4.2. Since it is reduced, its finite character torsion has order invertible in \(k\), by the equation calculation of AG-GS-05, Theorem 4.4. The intersection \(D\cap U\) is trivial by AG-RG-01, Lemma 2.A. Thus \(D\to T\) is a monomorphism and a closed immersion by a direct character calculation. Write \(D=D(M)\) and \(T=D(\mathbf Z^r)\). The kernel has character group \(M/\operatorname{im}(\mathbf Z^r\to M)\), by AG-GS-05 Theorem 4.1. A diagonalizable group is trivial only when its character group is zero, by its monomial basis. Thus the character map is surjective, and so is the group-algebra map defining \(D\to T\). Let \(A\) denote that image. In \(U\rtimes A\), the subgroup \(D\) is the graph of a regular crossed homomorphism \(A\to U\). AG-RG-01, Lemma 2.B supplies a \(T\)-stable filtration of \(U\), hence an \(A\)-stable one, and AG-RG-01, Lemma 2.C makes this cocycle principal. Conjugation by its correcting \(u\in U(k)\) places \(D\) and b in \(T\). The incidence proof just given puts every group element in a Borel. Its multiplicative Jordan factors belong to that same closed group, by the Jordan proof in AG-RG-01, Section 4. Applying the preceding paragraph to the semisimple factor proves that every semisimple group element belongs to a maximal torus. Hence the conjugation map \(G\times T_{\mathrm{rs}}\to G\), whose differential was computed above, has image exactly the regular semisimple locus. Any maximal torus containing \(g\) belongs to \(C_G(g)^0\), because a torus is connected. That identity centralizer is already a maximal torus. This proves uniqueness. \(\square\) Minimum centralizer dimension is indeed \(r\). On the dense open just constructed it is \(r\). Here is the complete local dimension bound needed for the converse. The universal centralizer \(H\to G\) is an affine scheme of finite presentation: in the finite coordinate generators of \(G\), impose equality of the two products with the universal element. Its identity is a section. Suppose at a rational point \(g_0\) its fibre has dimension \(d