Mackey-functor structure on the Brauer groups of a finite Galois covering of schemes
Past studies of the Brauer group of a scheme tells us the importance of the interrelationship among Brauer groups of its finite \'etale coverings. In this paper, we consider these groups simultaneously, and construct an integrated object "Brauer-Mack…
Authors: ** - **히로유키 나카오카 (Hiroyuki NAKAOKA)** – 도쿄대학 수학과 대학원, 일본 *(논문에 기재된 유일한 저자이며, 연구비 지원은 JSPS에서 제공받음.)* --- **
Mac k ey-functor structu r e on the Brauer groups of a finite Galois co v ering of sc hemes Hiroyuki NAKA OKA 1 , 2 Gr ad u ate Sch o ol of Mathema tic al Scienc es, The University of T okyo 3-8-1 Komab a, Me gur o, T okyo, 153-8914 Jap an Abstract P ast studies of the Brauer group of a sc heme tells u s the imp ortance of the in terrelationship among Brauer groups of its finite ´ etale co verings. In this p ap er, we consider these groups simultaneously , and construct an in tegrated ob ject “Brauer- Mac k ey functor”. W e r ealize this as a c ohomo lo gic al Mackey functor on the Galois category of finite ´ etale co verings. F or any finite ´ etale cov erin g of schemes, w e can associate tw o homomorphisms for Brauer groups , namely the pu ll-bac k and the norm map. These homomorphisms m ake Brauer group s into a biv arian t fu nctor (= Mac k ey fun ctor) on the Galois categ ory . As a corolla r y , Restricting to a finite Ga lois cov ering of schemes, w e obtain a cohomologi cal M ack ey functor on its Galois group. This is a generalization of the result for rings by F ord [5]. Moreo ver, applying Bley and Boltje’s theorem [1], we can deriv e certain isomorph ism s for the Brauer groups of intermediate co verings. Key wor ds: Mac k ey fun ctor, Brauer group, Galo is category 1 In tro duction In this pap er, any sc heme X is assumed to b e No etherian. π ( X ) denotes its ´ etale fundamental group. Any morphism is locally of finite t yp e, unless otherwise sp ecified. As in [10], X et denotes t he small ´ etale site, consisting of Email addr ess: deutsche @ms.u-t okyo.ac. jp (Hiro yuki NAKAOKA). 1 The au th or is supp orted by JSPS. 2 The au th or w ish es to thank Professor T oshiyuki K atsura for his encouragemen t. Preprint submitted to Elsevier 4 Nov em b er 2018 ´ etale mor phisms of finite t yp e ov er X . If U = ( U i f i − → X ) i ∈ I is a cov ering in this site, w e write as U ∈ Co v et ( X ). U ≺ V means U is a refinemen t of V . As for the finite ´ etale cov ering, the ´ etale fundamen tal g roup and the Galois category , we fo llow the terminolog y in [11]. F or example a finite ´ etale co ve ring is just a finite ´ etale mo r phism of sche mes. Our aim is to make the follo wing generalization o f the result b y F ord [5], whic h w as sho wn f o r rings. Corollary (Corollary 7.2) L et π : Y → X b e a finite Galois c ove ri n g of schemes with Galois gr oup G . Assume X satisfies Assumption 5. 1 . Then the c o rr esp ondenc e H ≤ G 7→ Br( Y /H ) forms a c ohomolo gic al Mackey functor on G . Her e, H ≤ G me ans H is a sub gr oup o f G . This follows from our main theorem: Theorem (Theorem 6.6) L et S b e a c on n e cte d sc heme satisfying Assump- tion 5.1. L et (F Et /S ) deno te the c ate gory of finite ´ etale c overings over S . Then, the Br auer gr o up functor Br forms a c ohomolo gic al Mackey functor on (FEt /S ) . As in Definition 6.1 , a Mack ey functor is a biv arian t pa ir of functors Br = (Br ∗ , Br ∗ ). F or any morphism π : Y → X , the con trav ariant part Br ∗ ( π ) : Br( X ) → Br( Y ) is the pull-back, and the cov ariant part Br ∗ ( π ) : Br( Y ) → Br( X ) is the norm map defined later. By applying Bley and Boltje’s theorem (F act 8.2) to Corollary 7.2, w e can obtain certain relations b et w een Bra uer groups of intermediate cov erings: Corollary (Corollary 8.3) L et X b e a c onne cte d scheme satisfying As- sumption 5 . 1 , an d π : Y → X b e a finite Galois c o vering with Gal( Y /X ) = G . (i) L et ℓ b e a p rime numb er. If H ≤ G is not ℓ -hyp o elementary, then ther e is a natur al isomorp h ism of Z ℓ -mo dules M U = H 0 < ···
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