The Bégueri Resolution

Peter J. Haine July 17, 2020

Let 푆 be a scheme and 퐺 a commutative, finite locally free 푆-group scheme. The pur- pose of this note is to explain how to resolve 퐺 by smooth affine 푆-groups. Specifically, we give an exposition of Bégueri’s result [1, Proposition 2.2.1] that there exists a (functorial) short exact sequence of fppf sheaves

0 퐺 R(퐺) Q(퐺) 0 , where R(퐺) and Q(퐺) are commutative, smooth affine 푆-groups. This resolution is a key technical ingredient used in Česnavičius and Scholze’s recent work on purity for flat cohomology [3, §§5.2–5.3]. The reason why the Bégueri resolution is so useful is that it allows one to transfer many questions about fppf cohomology with coefficients in a commutative, finite locally free 푆-group into questions about fppf cohomology with coefficients in a commutative, smooth affine 푆-group. Since fppf cohomology with smooth coefficients agrees with étale chomology, one can often deduce the desired results about fppf cohomology from existing results about étale cohomology. To explain Bégueri’s construction, it will be easier to work with the Cartier dual ∨ 퐺 = Hom푆(퐺, 퐆m,푆) of 퐺. The reason for this is that there is a natural choice of a ∨ larger group R(퐺 ) to embed Hom푆(퐺, 퐆m,푆) into: the sheaf Mor푆(퐺, 퐆m,푆) with the pointwise group structure. This is actually a reasonable choice of R(퐺∨) because the sheaf Mor푆(퐺, 퐆m,푆) is the Weil restriction of 퐆m,퐺 and there are standard existence and smoothness results for Weil restrictions. We fix some notation that we’ll use throughout this note and then recall the facts about Weil restrictions that we need. 1 Notation. Let 푿 be a topos, 푈 and 푉 objects of 푿, and 퐴 and 퐵 abelian group objects of 푿.

(1) We write 푉푈 = 푉 × 푈 for the pullback of 푉 to 푿/푈.

(2) We write Mor푿(푈, 푉) ∈ 푿 for the sheaf of morphisms 푈 → 푉 in 푿. If 푉 is an abelian group object of 푿, then we regard Mor푿(퐴, 퐵) as an abelian group object of 푿 with the pointwise group structure coming from the group structure on 퐵.

(3) We write Hom푿(퐴, 퐵) ⊂ Mor푿(퐴, 퐵) for the subsheaf of group homomorphisms 퐴 → 퐵.

푛 th (4) For each integer 푛, we write Ext푿(퐴, 퐵) ∈ Ab(푿) for the 푛 sheaf of extensions of 퐴 by 퐵.

1 푛 th (5) For each integer 푛, we write H퐗(푈; 퐴) ∈ Ab(푿) for the 푛 cohomology sheaf of 푈 with coefficients in 퐴.

Let 푆 be a scheme. When 푿 = 푆fppf is the topos of sheaves on 푆 with respect to the fppf topology, we simply use the subsectipt ‘S’ rather than a subscript ‘푆fppf’. 2 Recollection (Weil restriction). Let 푓∶ 푇 → 푆 be a morphism of schemes and 푌 a scheme over 푇. If the image of 푌 by the pushforward functor

푓⋆ ∶ PSh(Sch푇) → PSh(Sch푆)

퐹 ↦ [푈 ↦ 퐹(푈 ×푆 푇)] is representable by an 푆-scheme, we call the scheme 푓⋆(푌) the Weil restriction of 푌 along 푓, and write Res푇/푆(푌) ≔ 푓⋆(푌). We only recall two facts about Weil restrictions that we need here, and refer the reader to [2, §7.6; 4; 6, Chapter 4] for basic properties of Weil restriction as well as existence criteria. 3 Remark. Let 푓∶ 푇 → 푆 be a morphism of schemes and let 푋 be an 푆-scheme. If the Weil restriction of 푇 ×푆 푋 along 푓 exists, then by the universal property we have that

Res푇/푆(푇 ×푆 푋) ≅ Mor푆(푇, 푋) . The following is the standard existence result for Weil restrictions that we need: 4 Proposition ([2, §7.6, Theorem 4; 6, Proposition 4.4]). Let 푓∶ 푇 → 푆 be a finite locally free morphism of schemes. Then for every smooth affine 푇-scheme 푌, the Weil restriction Res푇/푆(푌) exists and is a smooth affine 푆-scheme.

5 Example. Since 퐆m,퐺 is a smooth affine 퐺-group scheme, Remark 3 and Proposition 4 show that Mor푆(퐺, 퐆m,푆) ≅ Res퐺/푆(퐆m,퐺) is a commutative, smooth affine 푆-group scheme. Thus in order to construct the desired resolution of 퐺∨, it suffices to show that the cokernel of the inclusion

Hom푆(퐺, 퐆m,푆) ↪ Mor푆(퐺, 퐆m,푆) is a smooth affine 푆-scheme. If we can show that this cokernel is representable by an affine 푆-group, the smoothness comes for free from the fact that if the quotient of a smooth group by a flat subgroup is representable, then it is also smooth:

6 Proposition ([SGA 3i, Exposé VIb, Proposition 9.2 (xii)]). Let 푆 be a scheme and let 푝 0 퐾 퐻 푄 0 be a short exact sequence of commutative 푆-group schemes. Then: (1) The 푆-group 퐾 is flat and locally of finite presentation over 푆 if and only if the morphism 푝∶ 퐻 → 푄 is faithfully flat and locally of finite presentation. (2) If 퐾 is flat and locally of finite presentation over 푆 and 퐻 is smooth over 푆, then the quotient 푄 is smooth over 푆.

2 Identification of the quotient

By Example 5 and Proposition 6, in order to construct the Bégueri resolution of 퐺∨, we’re reduced to showing that the cokernel of the inclusion Hom푆(퐺, 퐆m) ↪ Mor푆(퐺, 퐆m) is representable by an affine 푆-scheme. The first thing to notice is that we can identify the cokernel in the more general setup where we replace the fppf topos of 푆 with an arbitrary topos, and 퐺 and 퐆m by arbitrary abelian group objects. The correct term is a variant of an Ext group where the short exact sequences admit local sections on the level of sheaves of sets. Let 푿 be a topos. In the following we identify the category Ab(푿) of abelian group objects of 푿 with the category of limit-preserving functors 푿op → Ab. If one prefers, the following construction can be presented in terms of sheaves of abelian groups on a site; here we have chosen present the invariant formulation that does not make use of sites. 7 Construction. Let 푿 be a topos and 퐴 and 퐵 abelian group objects of 푿. Define 1 sec an abelian group object Ext퐗(퐴, 퐵) of 푿 as follows. For each object 푈 ∈ 푿, write E(퐴, 퐵)(푈) for the set of of isomorphism classes of short exact sequences

푝 (8) 0 퐵푈 퐶 퐴푈 0 in Ab(푿/푈) equipped with a morphism 푠∶ 퐴푈 → 퐶 in 푿/푈 that is a section of 푝. We simply write (퐶, 푠) ∈ E(퐴, 퐵)(푈) for the isomorphism class of the short exact sequence (8) equipped with the section 푠. We give the set E(퐴, 퐵)(푈) the structure of an abelian group as follows. Given ele- ments (퐶, 푠), (퐶′, 푠′) ∈ E(퐴, 퐵)(푈), the sum of (퐶, 푠) and (퐶′, 푠′) is given by forming the Baer sum of the short exact sequences with middle terms 퐶 and 퐶′, and noting that the map (푠, 푠′)∶ 퐴 → 퐶 ⊕ 퐶′ factors through the pullback 퐶 × 퐶′ ⊂ 퐶 ⊕ 퐶′, hence 푈 퐴푈 induces a section of the Baer sum of 퐶 and 퐶′. 1 sec Finally, we write Ext퐗(퐴, 퐵) for the sheafification of the presheaf of abelian groups given by the assignment 푈 ↦ E(퐴, 퐵)(푈) . Note that there is a natural morphism of abelian group objects

1 sec 1 Ext푿(퐴, 퐵) → Ext푿(퐴, 퐵) that simply forgets the additional data of the section. Define a group homomorphism d∶ Mor푿(퐴, 퐵) → E(퐴, 퐵) as follows: for each object 푈 ∈ 푿, the map d sends a map 푓∶ 퐴푈 → 퐵푈 to the split short exact sequence

pr1 0 퐵푈 퐴푈 ⊕ 퐵푈 퐴푈 0 equipped with the section (id, 푓)∶ 퐴푈 → 퐴푈 ⊕ 퐵푈. We also simply write d for the composite of the morphism d∶ Mor푿(퐴, 퐵) → E(퐴, 퐵) with the sheafification map 1 sec E(퐴, 퐵) → Ext푿(퐴, 퐵) .

3 1 sec With the somewhat involved definition of Ext푿(퐴, 퐵) in place, the fact that the following sequence is exact amounts to an exercise in the definitions. 9 Lemma ([1, Lemme 2.1.1]). Let 푿 be a topos and 퐴 and 퐵 abelian group objects of 푿. Then there is a functorial exact sequence

d 1 sec 1 1 0 Hom푿(퐴, 퐵) Mor푿(퐴, 퐵) Ext푿(퐴, 퐵) Ext푿(퐴, 퐵) H퐗(퐴; 퐵) .

To use Lemma 9 to construct the Bégueri resolution, we now make use of the following standard vanishing result of Ext groups. 10 Lemma ([5, Lemme 6.2.2]). Let 푆 be a scheme and 퐺 a commutative, finite locally free 1 푆-group. Then Ext푆(퐺, 퐆m) = 0. Thus we have a short exact sequence of fppf sheaves

∨ 1 sec 0 퐺 Res퐺/푆(퐆m) Ext푆(퐺, 퐆m) 0 .

1 sec Hence we will have succeeded if we can show that Ext푆(퐺, 퐆m) is representable by a 1 sec smooth affine 푆-group. To do this, we provide a cocycle description of Ext푆(퐺, 퐆m) . 11 Construction (symmetric 2-cocycles). Let 푆 be a scheme and 퐺 a commutative, finite 2 sym locally free 푆-group. We define the commutative, affine 푆-group scheme Z푆(퐺; 퐆m) of symmetric 2-cocycles on 퐺 with values in 퐆m as follows. First notice by Remark 3, the fppf sheaf Mor (퐺 × 퐺, 퐆 ) ≅ Res (퐆 ) 푆 푆 m (퐺×푆퐺)/푆 m 2 sym is a commutative, smooth affine 푆-group scheme. The group scheme Z푆(퐺; 퐆m) is the closed scheme of Mor푆(퐺 ×푆 퐺, 퐆m) of morphisms 푓 satisfying the symmetry condition 푓(푥, 푦) = 푓(푥, 푦) and the cocycle condition 푓(푥, 푦)푓(푥푦, 푧) = 푓(푦, 푧)푓(푥, 푦푧) .

2 sym Note that since closed immersions are affine, Z푆(퐺; 퐆m) is also affine over 푆. 12 Proposition ([1, Proposition 2.2.1]). Let 푆 be a scheme and 퐺 a commutative, finite 1 sec locally free 푆-group scheme. The fppf sheaf of abelian groups Ext푆(퐺, 퐆m) is representable 2 sym by the commutative, affine 푆-group Z푆(퐺; 퐆m) . Proof sketch. We define a morphism presheaves of abelian groups

2 sym 휃∶ E(퐺, 퐆m,푆) ⥲ Z푆(퐺; 퐆m) as follows. For each 푆-scheme 푈 and element (퐶, 푠) ∈ E(퐺, 퐆m,푆)(푈), write 휃(푈)(퐶, 푠) for the morphism (informally) defined by the assignment

퐺푈 ×푈 퐺푈 → 퐆m,푈 (푥, 푦) ↦ 푠(푥)푠(푦)(푠(푥푦))−1 .

4 It is not difficult to check that 휃 is an isomorphism with inverse given by sending a 2 sym symmetric 2-cocycle 푐 ∈ Z푆(퐺; 퐆m) (푈) to the class of the short exact sequence

pr1 0 퐆m,푈 퐺푈 ×푈 퐆m,푈 퐺푈 0 with multiplication on 퐺푈 ×푈 퐆m,푈 defined (informally) by (푥, 푎) ⋅ (푦, 푏) ≔ (푥푦, 푐(푥, 푦)푎푏) , and section 푠∶ 퐺푈 → 퐺푈 ×푈 퐆m,푈 defined (informally) by 푠(푥) ≔ (푥, 1). Combining Proposition 6, Lemma 9, Lemma 10, and Proposition 12 provides us with the Bégueri resolution: 13 Corollary (Bégueri resolution). Let 푆 be a scheme and 퐺 a commutative, finite locally free 푆-group scheme. Then there are functorial short exact sequences of commutative 푆- group schemes

∨ d 2 sym 0 퐺 Res퐺/푆(퐆m) Z푆(퐺; 퐆m) 0 and d 2 ∨ sym 0 퐺 Res퐺∨/푆(퐆m) Z푆(퐺 ; 퐆m) 0 . Moreover, the second and hence the third terms in each sequence are smooth and affine over 푆.

References

SGA 3i Schémas en groupes. I: Propriétés générales des schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962–64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 151. Berlin: Springer-Verlag, 1962–64, pp. xv+564. 1. L. Bégueri, Dualité sur un corps local à corps résiduel algébriquement clos, Mém. Soc. Math. France (N.S.), no. 4, p. 121, 1980/81. 2. S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Ergebnisse der Math- ematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1990, vol. 21, pp. x+325, isbn: 3-540-50587-3. doi: 10.1007/ 9783642514388. 3. K. Česnavičius and P. Scholze, Purity for flat cohomology, Preprint available at arXiv: 1912.10932, Dec. 2019. 4. L. Ji, S. Li, P.McFaddin, D. Moore, and M. Stevenson, Weil restriction for schemes and beyond, Preprint available at wwwpersonal.umich.edu/~stevmatt/weil_restriction.pdf, Sep. 1997. 5. M. Raynaud, Spécialisation du foncteur de Picard, Inst. Hautes Études Sci. Publ. Math., no. 38, pp. 27–76, 1970. 6. C. Scheiderer, Real and étale cohomology, Lecture Notes in Mathematics. Springer- Verlag, Berlin, 1994, vol. 1588, pp. xxiv+273, isbn: 3-540-58436-6. doi: 10 . 1007 / BFb0074269.