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The Lci Locus of the Hilbert Scheme of Points & The The lci locus of the Hilbert scheme of points & the cotangent complex Peter J. Haine April 2, 2020 Abstract Weintroduce the basic algebraic geometry background necessary to understand the main results of the work of Elmanto, Hoyois, Khan, Sosnilo, and Yakerson on motivic infinite loop spaces [5]. After recalling some facts about the Hilbert func- tor of points, we introduce local complete intersection (lci) and syntomic morphisms. We then give an overview of the cotangent complex and discuss the relationship be- tween lci morphisms and the cotangent complex (in particular, Avramov’s charac- terization of lci morphisms). We then introduce the lci locus of the Hilbert scheme of points and the Hilbert scheme of framed points, and prove that the lci locus of the Hilbert scheme of points is formally smooth. Contents 1 Hilbert schemes 2 2 Local complete intersections via equations 3 Relative global complete intersections ...................... 3 Local complete intersections ........................... 4 3 Background on the cotangent complex 6 Motivation .................................... 6 Properties of the cotangent complex ....................... 7 4 Local complete intersections and the cotangent complex 9 Avramov’s characterization of local complete intersections ........... 11 5 The lci locus of the Hilbert scheme of points 13 6 The Hilbert scheme of framed points 14 1 1 Hilbert schemes In this section we recall the Hilbert functor of points from last time as well as state the basic representability properties of the Hilbert functor of points. 1.1 Recollection ([STK, Tag 02K9]). A morphism of schemes 푓∶ 푌 → 푋 is finite lo- cally free if 푓 is affine and 푓⋆풪푌 is a finite locally free 풪푋-module. This is equivalent to saying that 푓 is finite, flat, and locally of finite presentation. If 푋 is locally noetherian, the condition that 푓 be locally of finite presentation is implied by the finiteness of 푓. 1.2 Definition. Let 푆 be a scheme and 푋 ∈ Sch푆. The Hilbert functor of points is the fin op functor Hilb (푋/푆)∶ Sch푆 → Set defined by sending 푌 ∈ Sch푆 to the set of closed subschemes 푍 ⊂ 푋 ×푆 푌 that are finite and locally free over 푌. 1.3. The degree of a finite locally free morphism induces a decomposition fin fin Hilb (푋/푆) ≃ ∐ Hilb푑 (푋/푆) 푑≥0 of product-preserving presheaves on Sch푆. We have the following representability properties of the Hilbert functor of points: 1.4 Theorem (see [5, Lemma 5.1.3]). Let 푆 be a scheme, 푋 ∈ Sch푆, and 푑 ≥ 0. (1.4.1) If 푋 → 푆 is separated, then Hilbfin(푋/푆) is representable by a separated algebraic space over 푆, which is: (1.4.1.1) Locally of finite presentation if 푋 → 푆 is. (1.4.1.2) A scheme if every finite set of points of every fiber of 푋 → 푆 is contained in an affine open in 푋 (e.g., 푋 → 푆 is locally quasi-projective). (1.4.2) If 푋 → 푆 is finite presented and locally (resp., strongly) quasi-projective, then fin Hilb푑 (푋/푆) is locally (resp., strongly) quasi-projective over 푆. Proof. For assertion (1.4.1), combine [13, Theorem 1.1; 16, Theorem 4.1; STK, Tag 0B9A]. Assertion (1.4.2) is [1, Corollaries 2.7 & 2.8]. 1.5 Remark. Nitsure gives a nice introduction to Hilbert and Quot functors [12]. 1.6 Theorem (Fogarty [6, Theorem 2.4 & Corollary 2.6]). Let 푘 be a field and 푋 a smooth fin surface over 푘. Then the Hilbert scheme Hilb푑 (푋/푘) is smooth of dimension 2푑 and bira- tional to Sym푑(푋). 1.7. Although the Hilbert scheme of points of a smooth surface is smooth, Hilbert schemes of points of smooth schemes are generally not smooth. In this talk we’ll de- scribe an open subscheme that is smooth, by considering the locus of points ‘cut out by the minimal number of equations’. 2 2 Local complete intersections via equations In this section we define and give examples of relative global complete intersections and local complete intersections, which make precise the notion of ‘being cut out by the min- imal number of equations’. Relative global complete intersections 2.1 Definition. A morphism of affine schemes 푓∶ Spec(푆) → Spec(푅) is a relative global complete intersection if there exists a presentation 푆 ≅ 푅[푥1, …, 푥푛]/(푓1, …, 푓푐) , such that all of the nonempty fibers of 푓 have dimension 푛 − 푐. 2.2 Example. Let 푘 by a field and 퐼 ⊂ 푘[푥] an ideal such that 푘[푥]/퐼 is a 0-dimensional scheme (equivalently, 푘[푥]/퐼 is finite-dimensional as a 푘-algebra). Since 푘[푥] is a princi- pal ideal domain, 퐼 = (푓) for some polynomial 푓 ∈ 푘[푥]. Thus 푘[푥]/퐼 is a relative global complete intersection over 푘. 2.3 Example. Let 푘 be a field, and 푒1, …, 푒푛 positive integers. Then the ring 푒1 푒푛 푘[푥1, …, 푥푛]/(푥1 , …, 푥푛 ) is 0-dimensional (since (0) is the only prime), hence a relative global complete intersec- tion over 푘. The following commuative algebra lemmas give some basic facts about relative global complete intersections that we use repeatedly. 2.4 Lemma ([STK, Tag 00SV]). Let 푅 be a ring and 푆 ≔ 푅[푥1, …, 푥푛]/(푓1, …, 푓푐) a relative global complete intersection. Let 픭 ∈ Spec(푆), and write 픭′ for the corresponding prime of 푅[푥1, …, 푥푛]. Then: (2.4.1) The sequence 푓1, …, 푓푐 is a regular sequence in the local ring 푅[푥1, …, 푥푛]픭′ . (2.4.2) For each 1 ≤ 푖 ≤ 푐, the ring 푅[푥1, …, 푥푛]픭′ /(푓1, …, 푓푖) is flat over 푅. (2.4.3) The ring 푆 is flat over 푅. 2 (2.4.4) The conormal module (푓1, …, 푓푐)/(푓1, …, 푓푐) is a free 푆-module with basis given by the classes of 푓1, …, 푓푐. 2.5 Lemma ([STK, Tag 07CF]). Let 푅 be a ring, and 퐼 ⊂ 푅[푥1, …, 푥푛] a finitely gen- 2 erated ideal. If the conormal module 퐼/퐼 is free over 푅[푥1, …, 푥푛]/퐼, then there exists a presentation 푅[푥1, …, 푥푛]/퐼 ≅ 푅[푦1, …, 푦푚]/(푓1, …, 푓푐) 2 such that (푓1, …, 푓푐)/(푓1, …, 푓푐) is free with basis given by the classes of 푓1, …, 푓푐. In this case, 푅[푥1, …, 푥푛]/퐼 is a relative global complete intersection over 푅. 3 The following lemma gives a method for producing relative global complete inter- sections. First we recall the conormal sequence, which we make repeated use of. 2.6 Recollection (conormal sequence [STK, Tag 01UZ]). Let 푖∶ 푍 ↪ 푋 be a closed immersion of 푆-schemes, with ideal sheaf 퐼. The differential d∶ 퐼 ⊂ 풪푋 → Ω푋/푆 maps 2 2 퐼 ⊂ 퐼 to 퐼Ω푋/푆, hence induces an 풪푋/퐼-linear map d∶ 퐼/퐼 → Ω푋/푆/퐼Ω푋/푆. That is, d ⋆ ⋆ pulls back to a map d∶ 풩푖 → 푖 Ω푋/푆 to a map from the conormal sheaf of 푖 to 푖 Ω푋/푆. Moreover, we have a conormal exact sequence d ⋆ 풩푖 푖 Ω푋/푆 Ω푍/푆 0 . 푛 2.7 Lemma (see [STK, Tag 00ST]). Let 푋 be an affine scheme, 푝∶ 푈 → 퐀푋 an affine étale morphism, and 푍 ⊂ 푈 a closed subscheme cut out by 푐 equations. If the nonempty 푛 fibers of 푍 → 퐀푋 → 푋 have dimension 푛 − 푐, then 푍 → 푋 is a relative global complete intersection. 푛 Proof. Factor 푝∶ 푈 → 퐀푋 as 푖 푚+푛 푈 퐀푋 푝 푛 퐀푋 , where 푖 is a closed immersion. Since 푝 is étale, Ω푝 ≅ 0, so the conormal sequence ⋆ 0 풩 푖 (Ω 푚+푛 푛 ) Ω 0 푖 퐀푋 /퐀푋 푝 provides an isomorphism ⋆ ⊕푚 풩 ≅ 푖 (Ω 푚+푛 푛 ) ≅ 풪 . 푖 퐀푋 /퐀푋 푈 푚+푛 ⊕푚 Choose functions 푓1, …, 푓푚 on 퐀푋 lifting to generators of 풩푖 ≅ 풪푈 . By Nakayama’s 푚+푛 lemma, there is a function ℎ on 퐀푋 such that 푈 is cut out by 푓1, …, 푓푚 in the local- 푚+푛 푚+푛 ization (퐀푋 )ℎ of 퐀푋 at ℎ. But then 푈 is cut out by the 푚 + 1 equations 푓1, …, 푓푚, 푚+푛+1 푚+푛+1 and ℎ푥푛+푚+1 − 1 in 퐀푋 . Hence 푍 is cut out by 푐 + 푚 + 1 equations in 퐀푋 , so (by definition) 푍 → 푋 is a relative global complete intersection. Local complete intersections In order to introduce local complete intersections, we recall some preliminaries on reg- ularity conditions for immersions of schemes. 2.8 Recollection (the Koszul complex). Let 푅 be a ring and 푟 ∈ 푅. The Koszul complex Koz(푟) of 푟 is the complex 0 푅 ⋅푟 푅 0 concentrated in degrees 0 and 1. Note that there is an augmentation Koz(푟) → 푅/(푟). Given a sequence of elements 푟1, …, 푟푛 ∈ 푅, the Koszul complex Koz(푟1, …, 푟푛) of the sequence 푟1, …, 푟푛 is the tensor product of complexes Koz(푟1, …, 푟푛) ≔ Koz(푟1) ⊗푅 ⋯ ⊗푅 Koz(푟푛) . 4 Hence there is an induced augmentation Koz(푟1, …, 푟푛) → 푅/(푟1, …, 푟푛) . 푝+1 푛 Explicitly, Koz푝(푟1, …, 푟푛) = Λ (푅 ) with differential given by 푝 d(푒 ∧ ⋯ ∧ 푒 ) = ∑(−1)푘푟 푒 ∧ ⋯ ∧ ̂푒 ∧ ⋯ ∧ 푒 . 푖0 푖푝 푖푘 푖0 푖푘 푖푝 푘=0 2.9 Recollection (regularity conditions for ideals [STK, Tag 07CU]). Let 푅 be a ring. A sequence of elements 푟1, …, 푟푛 ∈ 푅 is Koszul-regular if the Koszul complex Koz(푟1, …, 푟푛) is a resolution of 푅/(푟1, …, 푟푛), i.e., H푖(Koz(푟1, …, 푟푛)) = 0 for 푖 ≠ 0 . A regular sequence is necessarily Koszul-regular. If 푅 is a noetherian ring, then every Koszul-regular sequence is also regular; see [STK, Tag 063I].
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