24 VICTORIA HOSKINS 3.6. Moduli spaces and quotients. Let us give one result about the construction of moduli spaces using group quotients. For a moduli problem , a family over a schemeS has the local universal property if for any other family overM a schemeT Fand for anyk-pointt T, there exists a neighbourhoodU oft inT and a morphismG f:U S such that ∈ f . → G| U ∼U ∗F Proposition 3.35. For a moduli problem , let be a family with the local universal property over a schemeS. Furthermore, suppose thatM thereF is an algebraic groupG acting onS such that twok-points s, t lie in the sameG-orbit if and only if . Then F t ∼F s a) any coarse moduli space is a categorical quotient of theG-action onS; b) a categorical quotient of theG-action onS is a coarse moduli space if and only if it is an orbit space. Proof. For any schemeM, we claim that there is a bijective correspondence natural transformationsη: h G-invariant morphismsf:S M { M→ M } ←→{ → } given byη η S( ), which isG-invariant by our assumptions about theG-action onS. The inverse of this�→ correspondenceF associates to aG-invariant morphismf:S M and a family overT a morphismη ( ):T M by using the local universal property→ of overS. More G T G → F precisely, we can coverT by open subsetsU i such that there is a morphismh i :U i S and h . Foru U U , we have → i∗F∼ Ui G|Ui ∈ i ∩ j (h ∗ ) (h ∗ ) Fhi(u) ∼ i F u ∼G u ∼ j F u ∼F hj (u) and so by assumptionh i(u) andh j(u) lie in the sameG-orbit. Sincef isG-invariant, we can glue the compositionsf h i :U i M glue to a morphismη T ( ):T M. We leave it to the reader to verify that this◦ determines→ a natural transformationη G(that→ is, this is functorial with respect to morphisms) and that these correspondences are inverse to each other. Hence, if (M,η: h M ) is a coarse moduli space, thenη S( ):S M isG-invariant and universal amongstM→ allG-invariant morphisms fromS, by the universalityF → ofη. This proves statement a). Furthermore, theG-invariant morphismη ( ):S M is an orbit space if and S F → only ifη Speck is bijective. This proves statement b). � 4. Affine Geometric Invariant Theory In this section we consider an action of an affine algebraic groupG on an affine schemeX of finite type overk and show that this action has a good quotient whenG is linearly reductive. The main references for this section are [25] and [31] (for further reading, see also [2], [4] and [32]). LetX be an affine scheme offinite type overk; then the ring of regular functions (X) is afinitely generatedk-algebra. Conversely, for anyfinitely generatedk-algebraA, the spectrumO of prime ideals SpecA is an affine scheme offinite type overk. The action of an affine algebraic groupG on an affine schemeX given by a morphism σ:G X X × → corresponds to a homomorphism ofk-algebrasσ ∗ : (X) (G X) ∼= (G) k (X), which gives aG-co-module structure on the (typically infiniteO dimensional)→O × k-vectorO space⊗ O(X). This co-module structure in turn determines a linear representationG GL( (X)). Concretely,O on the level ofk-points, the action ofg G(k) onf (X) is given by → O ∈ ∈O 1 (g f)(x)=f(g − x). · · The ring ofG-invariant regular functions onX is G (X) := f (X):σ ∗(f)=1 f . O { ∈O ⊗ } AnyG-invariant morphismϕ:X Z of schemes induces a homomorphismϕ ∗ : (Z) (X) whose image is contained in the→ subalgebra ofG-invariant regular functions (X)O G.→ ThisO leads us to an interesting problem in invariant theory which wasfirst consideredO by Hilbert. MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 25 4.1. Hilbert’s 14th problem. For a rational action of an affine algebraic groupG on afinitely generatedk-algebraA, Hilbert asked whether the algebra ofG-invariantsA G isfinitely gener- ated. The answer to Hilbert’s 14th problem is negative in this level of generality: Nagata gave an example of an action of an affine algebraic group (constructed using copies of the additive groups) for which the ring of invariants is notfinitely generated (see [27] and [29]). However, for reductive groups (which we introduce below), the answer is positive due to a Theorem of Nagata. The proof of this result is beyond the scope of this course. However, we will prove that for a rational action of a ‘linearly reductive’ group on an algebra, the subalgebra of invariants is finitely generated, using a Reynolds operator, which essentially mimics Hilbert’s 19th century proof that, over the complex numbers, a rational action of the general linear group GLn on an algebra has afinitely generated invariant subalgebra. 4.2. Reductive groups. In this section, we will give the definition of a reductive group, a lin- early reductive group and a geometrically reductive group, and explain the relationship between these different notions of reductivity. Our starting point is the Jordan decomposition for affine algebraic groups overk. Wefirst recall the Jordan decomposition for GL : an elementg GL (k) has a decomposition n ∈ n g=g ssgu =g ugss whereg ss is semisimple (or, equivalently, diagonalisable, ask is algebraically closed) andg u is unipotent (that is,g I n is nilpotent). For any affine algebraic− groupG, we would like to have an analogous decomposition, and we can hope to make use of the fact thatG admits a faithful linear representationG� GL n. However, this would require the decomposition to be functorial with respect to closed immersions→ of groups. Definition 4.1. LetG be an affine algebraic group overk. An elementg is semisimple (resp. unipotent) if there is a faithful linear representationρ:G� GL n such thatρ(g) is diagonalisable (resp. unipotent). → Theorem 4.2 (Jordan decomposition, see [23] X Theorem 2.8 and 2.10). LetG be an affine algebraic group overk. For everyg G(k), there exists a unique semisimple elementg and a ∈ ss unique unipotent elementg u such that g=g ssgu =g ugss. Furthermore, this decomposition is functorial with respect to group homomorphisms. In particu- lar, ifg G(k) is semisimple (resp. unipotent), then for all linear representationsρ:G GL n, the element∈ ρ(g) is semisimple (resp. unipotent). → Letρ:G GL(V ) be a linear representation of an affine algebraic groupG on a vector spaceV and→ letρ :V (G) V denote the associated co-module. Then a vector subspace ∗ →O ⊗ k V � V isG-invariant ifρ ∗(V �) (G) k V � and a vectorv V isG-invariant ifρ ∗(v)=1 v. We⊂ letV G denote the subspace⊂ ofOG-invariant⊗ vectors. ∈ ⊗ Definition 4.3. An affine algebraic groupG is unipotent if every non-trivial linear representa- tionρ:G GL(V ) has a non-zeroG-invariant vector. → Proposition 4.4. For an affine algebraic groupG, the following statements are equivalent. i)G is unipotent. ii) For every representationρ:G GL(V) there is a basis ofV such thatρ(G) is contained → in the subgroupU GL(V) consisting of upper triangular matrices with diagonal entries equal to 1. ⊂ iii)G is isomorphic to a subgroup of a standard unipotent groupU n GL n consisting of upper triangular matrices with diagonal entries equal to 1. ⊂ 26 VICTORIA HOSKINS Proof. i) ii): Ife 1, . , en is a basis ofV such thatρ(G) U, thene 1 isfixed byρ. Conversely ⇐⇒ ifρ:G GL(V ) is a representation of a unipotent group⊂ G, then we can proceed by induction on→ the dimension ofV . AsU is unipotent, the linear subspace ofG-fixed points G G G V is non-zero; lete 1, e m be a basis ofV . Then there is a basis em+1,..., en ofV/V such that the induced representation··· has image in the upper triangular matrices with diagonal entries equal to 1. By choosing liftse m+i V of em+i, we get the desired basis ofV. ii) iii): As every affine algebraic∈ groupG has a faithful representationρ:G GL , ⇐⇒ → n we see that ii) implies iii). Conversely, any subgroup ofU n is unipotent (see [23] XV Theorem 2.4). � Remark 4.5. IfG is a unipotent affine algebraic group, then everyg G(k) is unipotent. The converse is true if in additionG is smooth (for example, see [23] XV∈ Corollary 2.6 or SGA3 XVII Corollary 3.8). Example 4.6. (1) The additive groupG a is unipotent, as we have an embeddingG a � U 2 given by → 1c c . �→ 0 1 � � (2) In characteristicp, there is afinite subgroupα p G a where we define the functor of ⊂ points ofα p by associating to ak-algebraR, p αp(R) := c G a(R):c = 0 . { ∈ } p This is represented by the scheme Speck[t]/(t ) and soα p is a unipotent group which is not smooth. Definition 4.7. An algebraic subgroupH of an affine algebraic groupG is normal if the conjugation actionH G G given by (h, g) ghg 1 factors throughH� G. × → �→ − → Definition 4.8. An affine algebraic groupG overk is reductive if it is smooth and every smooth unipotent normal algebraic subgroup ofG is trivial. Remark 4.9. In fact, one can define reductivity by saying that the unipotent radical ofG (which is the maximal connected unipotent normal algebraic subgroup ofG) is trivial; however, to define the unipotent radical carefully, we would need to prove that, for a groupG, the subgroup generated by two smooth algebraic subgroups ofG is also algebraic (see [22] Proposition 2.24). Exercise 4.10. Show that the general linear group GLn and the special linear group SLn are reductive.