Frobenioids 1. Frobenioids 1. Weronika Czerniawska. The Univeristy of Nottingham

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1 Weronika Czerniawska The Univeristy of Nottingham

2 A Frobenioid is a category that is meant to encode the theory of divisors and line bundles on coverings i.e. normalizations in various finite separable extensions of the function field of a given normal integral scheme. Having a sketchy idea of how to formulate IUT, Mochizuki developed the theory of Frobenioids which provided a unified, intrinsic, category theoretic language to encode the theory of divisors and line bundles in appropriate categories, general enough to fit in whatever would be developed.

3 Plan Plan 1. Motivating examples 2. Basic definitions 3. Model Frobenioids

4 Examples Example (Frobenioid of geometric origin) V proper normal variety over k K the function field Div K the set of Q-Cartier divisors on V For a finite extension L of K put Div L prime divisors of the normalization V [L] that map into Div K

5 Examples effective Cartier divisors of V [L] with support in Div L Φ(L) Mon (a subgroup of) Cartier divisors Φ gp (L) Grp Mon the group of rational functions on V [L] with zeroes and poles belonging to Div L B(L) principal divisors homomorphism B(L) Φ gp (L)

6 Examples Let K be a Galois extension of K ( can be infinite ) with the Galois group which has a natural profinite topology. G def = Gal( K /K ) The connected objects of the category of finite sets with continuous G-action (those which don t split into a disjoint union of non-empty G-sets) D def = B(G) 0 can be identified with finite extensions K L K.

7 Examples We can consider a category of pairs (L, L) where K L K is finite and L is a line bundle on V [L] with morphisms consisting of φ : (L, L) (M, M) Spec(L) Spec(M) morphism over Spec(K ) d N 1 L d M V [L] morphism of line bundles whose zero locus is a Cartier divisor supported in Div L

8 Examples Example (Frobenioid of arithmetic origin) L: a number field V(L): the set of valuations of L, L v : the completion of L at v V(L), O v def = { z = 1}, O v def = {0 < z 1} { ord(l v ) def = L v /O v Z, if v nonarchimedean = R, if v archimedean { ord(ov ) def = Ov /O v Z 0, if v non-archimedean = R 0, if v archimedean ord(l v ) = ord(o v ) gp.

9 Examples effective arithmetic divisors on L Φ(L) def = vv(l) ord(o v ) arithmetic divisors on L multiplicative group of L Φ(L) gp = vv(l) B(L) def = L ord(l v ) principal divisor homomorphism B(L) Φ(L) gp

10 Examples Let F be a number field and let F /F be a Galois extension with Galois group G has a natural profinite topology. G def = Gal( F /F ) The connected objects of the category of finite sets with continuous G-action D def = B(G) 0 can be again identified with finite extensions F L F.

11 Examples We can consider a category of pairs (L, L) where F L F is finite and L is an arithmetic line bundle on Spec(O L ) with morphisms φ : (L, L) (M, M) consisting of Spec(L) Spec(M) morphism over Spec(F ) d N 1 L d M L morphism of arithmetic line bundles on L.

12 Examples A Frobenioid is a category C which consists of the following data category imposing linebundle like structure category of coverings, (base category) C F Φ projection D Φ Mon covariant functor contravariant functor

13 Definitions For a commutative monoid M Mon M ± submonoid of invertible elements of M M char = M/M ± M gp groupification of M Definition A monoid M Mon is called 1. sharp if M ± = 0 2. integral if ι : M M gp is injective 3. saturated if for a M gp if na ι(m) for n N 1 then a ι(m) 4. of characteristic type if fibres of M M char are torsors over M ± 5. group-like if M char is trivial

14 Definitions Definition A monoid is called pre-divisorial if it is integral, saturated and of characteristic type divisorial if it is pre-divisorial and sharp Definition A morphism M N in Mon is called characteristically injective if it is injective and the induced morphism is also injective. M char N char

15 Definitions Definition A category is called connected if its associated graph vertices edges objects morphisms is connected. A category is called totally epimorphic if every morphism in this category is an epimorphism.

16 Definitions Definition Let C be a category. An arrow β : B A is called fiberwise-surjective if for every arrow γ : C A there exist arrows δ B : D B and δ A : D A such that the following diagram β γ B A C δ B D δ A commutes. FSM-morphism if it is a fiberwise-surjective monomorphism.

17 Definitions Definition Let D be a category. A monoid on D is a contravariant functor Φ : D Mon such that for every morphism α : B A in D where α : Φ(A) Φ(B) is characteristically injective if α is FSM-morphism then α is an isomorphism of monoids, α Φ(A) Φ(B) := Φ(α : B A).

18 Definitions Elementary Frobenioids Definition (Elementary Frobenioid) Let Φ be a monoid on a category D. Elementary Frobenioid associated to Φ is a category F Φ which objects are just objects of the category D and morphisms φ : A B are triples where φ D : A B is a morphism of D, Div(φ) Φ(A) is the zero-divisor of φ, φ = (φ D, Div(φ), deg Fr (φ)) deg Fr (φ) N 1 is the Frobenius degree of φ. The composite of two morphisms φ = (φ D, Z φ, n φ ) : A B, ψ = (φ D, Z ψ, n ψ ) : B C is given as ψ φ = ( ψ D φ D, ψ D (Z ψ) + n ψ Z φ, n ψ n φ ) : A C.

19 Definitions Elementary Frobenioids Example Let s consider the elementary Frobenioid F ΦM associated to the functor Φ M :{ } Mon M on the one-morphism category { }. We have F M := F ΦM = M N 1.

20 Definitions Elementary Frobenioids Indeed, the monoid of morphisms consists of triples where a M and n N 1. (id { }, a, n) The composition of (id { }, a 1, n 1 ) and (id { }, a 2, n 2 ) can be seen as a multiplication (a 1, n 1 ) (a 2, n 2 ) = (a 1 + n 1 a 2, n 1 n 2 ) in the semi-direct product M N 1.

21 Definitions Pre-Frobenioids Definition (Pre-Frobenioid) Let Φ : D Mon be a monoid on a connected, totally epimorphic category D. Let be a connected, totally-epimorphic category. C We say that C is a pre-frobenioid if we have a covariant functor C F Φ.

22 Model Frobenioids Model Frobenioids Let s consider the following data D a connected a totally epimorphic category Φ : D Mon a divisorial monoid B : D Mon a group-like monoid Div B : B Φ gp a homomorphism of monoids

23 Model Frobenioids Proposition We have a well defined category C constructed in the following way the objects of C are pairs of the form (A D, α) where A D Ob(D) and α Φ(A D ) gp a morphism φ : (A D, α) (B D, β) is a collection of data deg Fr (φ) N 1 Base(φ) : A D B D Div(φ) Φ(A) u φ B(A) such that deg Fr α + Div(φ) = (Φ gp (Base(φ)))(β) + Div B (u φ )

24 Model Frobenioids For given two morphisms φ(a D, α) (B D, β), ψ : (B D, β) (C D, γ) Mor(C) the composition data ψ φ = ( deg Fr (ψ φ), Base(ψ φ), Div(ψ φ), u ψ φ ) is defined as follows deg Fr (ψ φ) = deg Fr (ψ) deg Fr (φ) Base(ψ φ) = Base(ψ) Base(φ) Div(ψ φ) = ( Φ(Base(φ)) ) (Div(ψ)) + deg Fr (ψ) Div(φ) u ψ φ = B ( Base(ψ) ) (u φ ) + deg Fr (ψ) u φ

25 Model Frobenioids There is a natural functor given by C F Φ (A D, α) A D φ = ( deg Fr (φ), Base(φ), Div(φ), u φ ) ( Base(φ), Div(φ), degfr (φ) ) so model Frobenioids are in particular pre-frobenioids.

26 Model Frobenioids Example ( Frobenioid of geometric origin) V nice variety, K the function field and K its Galois extension with G := Gal( K /K ). D := B(G) 0 divisorial monoid Φ : D L Mon Div L group-like monoid B : D Mon L L homomorphism of monoids Div B : B L Φ gp PDiv L

27 Model Frobenioids We get a model Frobenioid C K /K Φ C K /K F Φ D Mon (L, L) L Div L which is exactly the Frobenioid of geometric origin described earlier.

28 Model Frobenioids Example ( Frobenioid of arithmetic origin) F a number field and F its Galois extension with G := Gal( F /F ). D := B(G) 0 divisorial monoid Φ : D L Mon vv(l) ord(o v ) group-like monoid B : D Mon L L homomorphism of monoids Div B : B L Φ gp PDiv L

29 Model Frobenioids We get a model Frobenioid C F/F Φ C F/F F Φ D Mon (L, L) L Div L which is the Frobenioid of arithmetic origin described earlier.

30 Model Frobenioids Plan for tomorrow 1. Torsor-theoretic approach to model Frobenioids. 2. Frobenioids in IUT. 3. The Main Theorem about reconstruction of the functor C F Φ. that gives C structure of a Frobenioid can be reconstructed from C as a category.