Notes from Brian Conrad's course on Linear Algebraic Groups at Stanford, Winter 2010 typed by Sam Lichtenstein July 19, 2017 Contents 1 January 4 3 1.1 Some Definitions . 3 1.2 Smoothness . 4 1.3 Connectedness . 5 1.4 Examples . 5 2 January 6 6 2.1 Translations . 6 2.2 Homomorphisms . 7 2.3 Normal subgroups . 7 2.4 Further discussion of examples . 7 2.5 How far is a general smooth connected algebraic group from being either affine (i.e. linear algebraic) or projective (i.e. an abelian variety)? . 8 3 January 8 9 3.1 How do linear algebraic groups arise in nature? . 9 3.2 Zariski closures of subgroups . 9 4 January 11 (Substitute lecture by A. Venkatesh) 11 4.1 Tori . 11 4.2 Maximal split tori . 12 4.3 Building up groups from tori . 13 4.3.1 Construction of Φ ...................................... 13 5 January 13 14 5.1 Mapping Ga and Gm into a reductive group . 14 5.2 Classification of 1-dimensional connected linear algebraic groups . 14 5.3 Examples of non-split 1-dimensional linear algebraic k-groups . 15 5.4 Start of proof of the classification theorem . 15 6 January 20 (Substitute lecture by A. Venkatesh) 16 6.1 End of proof of classification of 1-dimensional connected linear algebraic groups . 16 6.2 Smoothness criteria . 16 7 January 22 18 7.1 More on forms . 18 7.2 A loose end concerning smoothness . 18 7.3 How to apply Grothendieck's smoothness criterion . 19 7.4 How to show algebraic k-groups are smooth . 19 7.4.1 Smoothness of GLn; SLn in coordinates, via matrices . 20 1 7.4.2 Smoothness of GL(V); SL(V) without using coordinates (much) . 20 8 January 25 20 8.1 How to Show Connectedness . 20 8.2 Connectedness of SLn ........................................ 21 8.2.1 Method 1 - Flag varieties . 21 8.2.2 Method 2 - Projective Space . 22 8.3 Smoothness of Sp2n ......................................... 22 9 January 29 24 9.1 A little more about Sp2n ...................................... 24 9.2 Actions, Centralizers and Normalizers . 25 9.3 Closed orbit lemma . 26 9.4 Start of proof of closed orbit lemma (Theorem 9.3.5) . 27 10 February 1 27 10.1 Conclusion of proof of closed orbit lemma (Theorem 9.3.5) . 27 10.2 Criterion for a k-homomorphism of k-groups to be a closed immersion . 28 11 February 3 28 11.1 Proof of Proposition 10.2.1 . 28 11.2 Embedding linear algebraic groups in GLn ............................. 30 12 February 5 30 12.1 Actions of affine finite type k-groups on coordinate rings of affine k-schemes . 30 12.1.1 Application to embedding a smooth linear algebraic group into GLn . 31 12.2 Proof of Theorem 12.1.1 . 31 13 February 8 32 13.1 Jordan decomposition . 32 14 February 10 34 14.1 All subgroups of a smooth finite type k-group are stabilizers of a line . 34 15 February 12 35 15.1 Unipotent groups . 35 15.2 Proof of Theorem 15.1.8 on representations of unipotent groups . 36 15.3 Remaining ingredients necessary for structure theory . 37 16 February 17 37 16.1 Some motivation: a key calculation on SL2(k) ........................... 37 16.2 Subgroups generated by connected varieties . 38 16.3 Proof of Proposition 16.2.1 . 39 16.4 Improvements on Proposition 16.2.1 . 40 17 February 19 41 17.1 Conclusion of proof of Proposition 16.4.2 . 41 17.2 Solvable groups . 41 17.3 Structure of smooth connected commutative affine k-groups . 42 17.4 Coset spaces for closed subgroups (that is, quotients) . 43 18 February 22 45 18.1 Existence of quotients (of smooth affine groups) . 45 18.2 Lie algebras . 46 18.2.1 Functoriality of Lie(·) .................................... 46 2 19 February 24 47 19.1 Linear representations of split tori, over general rings . 47 19.2 Proof of Theorem 19.1.2 . 48 20 February 26 49 20.1 k-split solvable groups . 49 20.2 Borel Fixed Point Theorem . 51 21 March 1 52 21.1 Remark on quotients . 52 21.2 Lie-Kolchin Corollary 20.2.4 . 52 21.3 Structure theory of solvable groups . 52 21.4 Unipotent radical . 53 21.5 Reductive groups . 54 22 March 3 55 22.1 Borel subgroups . 55 23 March 5 57 23.1 Properties of parabolics . 57 23.2 Conjugacy of maximal tori . 58 24 March 8 59 24.1 Proof of Proposition 23.2.1 . 59 24.2 Cartan subgroups . 60 24.3 PG(λ);UG(λ);ZG(λ) for a 1-parameter subgroup λ ........................ 60 25 March 10 61 25.1 Proof of Lemma 24.3.3 on PG;UG;ZG ............................... 61 25.1.1 Proof that (i)+(ii) (iii) [except unipotence of UG(λ)].................. 61 25.1.2 Proof of (i) . 62 25.1.3 Proof of (ii) and unipotence) of UG(λ) ........................... 62 26 March 12 63 26.1 Classification of split reductive groups of rank 1 . 63 1 January 4 1.1 Some Definitions Let k be a field. Definition 1.1.1. A group variety over k is a smooth k-scheme G (possibly disconnected!) of finite type, equipped with k-morphisms m : G × G G; i : G G, and a rational point e 2 G(k) satisfying the group axioms, in the sense that the following diagrams commute: ! ! 1 ×m (1G;i) (1G;e) G × G × G G / G × G G / G × G G / G × G m×1G m (e;1G) m $ (i;1 ) m G × G m / G G Spec k G × G m / G e $ G × G m / G 3 Remark 1.1.2. The rational point e can be regarded as a section of the structure map G Spec k. In the third diagram, the morphisms e : G G are the \constant maps to the identity", i.e. the compositions e G Spec k G. ! Remark 1.1.3. By HW1 Exercise 4(i),! a connected group variety G over k is [geometrically connected and] geometrically! ! irreducible. By some people's usage, this justifies the term \variety" in the name. Definition 1.1.4. If we relaxe smoothness in Definition 1.1.1 to “finite type" but keep everything else, then G is called an algebraic k-group scheme. By HW1 Exercise 4(iii), over imperfect k reducedness does not imply geometric reducedness even for connected algebraic k-group schemes. Definition 1.1.5. A group variety G over k is called linear algebraic if it is affine. Remark 1.1.6. If G is an algebraic k-group scheme, then one can show that G is affine if and only if it is a k-subgroup scheme (cf. Definition 1.1.7) of GLn for some n. (See Example 1.4.1 below for the definition of GLn.) This is special to the case of fields in the sense that it is not known over more general rings (e.g., not even over the dual numbers over a field), though it is also true (and useful) over Dedekind domains by a variation on the argument used for fields. Definition 1.1.7. Let G be a group variety over k, with corresponding multiplication map mG, inversion map iG, and identity section eG.A k-subgroup H ⊂ G is a closed subscheme such that there exist factorizations mH iH Spec k H × H / H H / H _ _ _ _ eG eH " H / G G × G / G G / G mG iG By Yoneda's lemma, a k-group scheme is the same as a k-scheme G such that the Yoneda functor G h = Homk-Sch(-;G): Sch=k Sets is equipped with a factorization through the forgetful functor Groups Sets. This is useful! Note that this is the same as the requirement that G(R) is a group, functorially in R, for all k-algebras!R, not just fields K=k. By the same reasoning, a closed subscheme H ⊂ G is a k-subgroup! if and only if H(R) ⊂ G(R) is a[n abstract] subgroup for all k-algebras R. Remark 1.1.8. Beware that over every imperfect k there exist connected non-reduced G with Gred not k-subgroup (see [SGA3, VIA, 1.3.2(2)]). 1.2 Smoothness Let X be a k-scheme which is [locally] of finite type. Here is one definition of many for what it means for X to be smooth. 1 Definition 1.2.1. We say that X is smooth if and only if Xk is regular, meaning that all of the local rings are regular local rings, which can be checked via the Jacobian criterion. Remark 1.2.2. For the purpose of checking the Jacobian criterion, one may freely go up and down between algebraically closed fields containing k. For example, one might care about both XQ and XC in the case k = Q. Remark 1.2.3. If k happens to be perfect, then smoothness is the same as regularity. Let G be a k-group scheme [of finite type]. The group G(k) acts on Gk by.