A topos-theoretic view of difference algebra

Ivan Tomašic´

Queen Mary University of London

Toposes Online 28/06/2021

1 / 29 Ritt-style difference algebra

Ritt 1930s Difference algebra is the study of rings and modules with added endomorphisms.

2 / 29 Difference categories à la Ritt

Let C be a category. Define its difference category

σ-C

I objects are pairs (X, σX ),

where X ∈ C , σX ∈ C (X,X); I a morphism f :(X, σX ) → (Y, σY ) is a commutative diagram in C f X Y

σX σY f X Y

3 / 29 Examples

We will consider: I σ-; I σ-Gr; I σ-Ab; I σ-Rng. Given R ∈ σ-Rng, consider I R-Mod, the category of difference R-modules.

4 / 29 The topos of difference sets

Let N be the category associated with the (N, +). Note σ-Set ' [N, Set] ' BN op is a Grothendieck topos (as the presheaf category on N ' N), the classifying topos of N.

5 / 29 The topos of difference sets Canonical geometric morphism γ : σ-Set → Set σ-Set

γ∗ γ! = Quo a a γ∗ = Fix

Set

The b c : σ-Set → Set. is the inverse image of an essential point Set → σ-Set Set

bc d| a a |e

σ-Set

6 / 29 Topos theory view of difference algebra

Old adage of topos theory: A topos can serve as a universe for developing mathe- matics.

Motto Difference algebra is the study of algebraic objects internal in the topos σ-Set.

We study ordinary algebra in the internal logic of σ-Set, and then externalise to observe what actually happened.

7 / 29 Topos theory view of difference algebra

Indeed,

σ-Gr ' Gr(σ-Set) σ-Ab ' Ab(σ-Set) σ-Rng ' Rng(σ-Set).

For R ∈ σ-Rng,

R-Mod ' Mod(σ-Set,R)

is the category of modules in a ringed topos.

8 / 29 Difference homological algebra

Let R ∈ σ-Rng. Then R-Mod is monoidal closed

HomR(A ⊗ B,C) ' HomR(A, [B,C]R).

Fact (in any Grothendieck topos) R-Mod = Mod(σ-Set,R) is abelian with enough injectives and enough internal injectives.

Cohomology of difference modules is an instance of topos cohomology; we have i I ExtR(M,N); i i i I H (σ-Set,M) = R γ∗(M) = ExtR(R,N).

9 / 29 Ext of difference modules

Let R ∈ σ-Rng, and M,N ∈ R-Mod, with M étale. Then  R-Mod(M,N), i = 0, i  ExtR(M,N) = [M,N]σ, i = 1, 0, i > 1,

In particular,

 σ Fix(N) = N i = 0, i  R Fix(N) = N/Im(σ − id) = Nσ, i = 1, 0, i > 1.

10 / 29 Difference algebraic geometry

Motto Difference algebraic geometry is algebraic geometry over the base topos σ-Set.

Plan

1. Pursue the development of algebraic geometry over a base topos S : I start with Hakim’s monograph; I extend to relative étale cohomology and fundamental (oid). 2. Specialise to S = σ-Set.

11 / 29 Hakim spectra Hakim-Cole Zariski spectrum There is a 2-adjunction

Top.LocRng

i a Spec.Zar

Top.Rng

For a ringed topos (S ,A),

Spec.Zar(S ,A)

is a locally ringed topos equipped with a morphism of ringed topoi π Spec.Zar(S ,A) −−→Zar (S ,A).

12 / 29 Hakim spectra

For any ‘scheme topology’ τ (Zariski, étale, fppf, fpqc etc), using Hakim’s techniques, we can define the τ-spectrum

π Spec .τ(S ,A) −→τ (S ,A).

Construction sketch, in terms of Caramello-Zanfa If S = Sh(C ,J),

Spec .τ(S ,A) = Sh((C ,J) o ([SchA, τ])),

where the relative site is indexed over (C ,J) by

U [SchA, τ] = (SchA(U), τ).

13 / 29 Tierney’s internal construction

Zariski Spectrum via internal sites Joyal/Tierney’s internal frame of radical ideals gives an internal site in S = Sh(C ,J) (A, J) so that

Spec.Zar(S ,A) = ShS (A, J) ' Sh((C ,J) o (A, J)).

14 / 29 Relative schemes

Let (S , k) be a ringed topos. We define Sch(S ,k) as the stack completion of

U 7→ SchΓ(U,k).

Hakim considers its fibre over 1 as the category of S -schemes over k.

15 / 29 Cohomology of S -schemes

Given an S -scheme X and a scheme topology τ, we have a τ-realisation of X,

πτ Xτ S γ γτ Set

If M is an Oτ - or an in Xτ , we define the τ-cohomology and the relative τ-cohomology groups as the topos cohomology groups

i i i i H (Xτ ,M) = R γτ,∗(M), and H (Xτ /S ,M) = R πτ,∗(M),

the latter being abelian groups in S .

16 / 29 Cohomology of quasi-coherent sheaves

Let A ∈ Rng(S ), M ∈ A-Mod, let (X, OX ) = Spec.Zar(S ,A) and M˜ = M ⊗A OX . Hakim: for i > 0, Hi(X/S , M˜ ) = 0.

17 / 29 Étale cohomology of relative schemes Kummer theory If n is invertible in an S -scheme X, we have a short exact in Xét

()n 1 → µn → Gm −−→ Gm → 1,

and we can study the resulting long exact cohomology sequence.

Artin-Schreier theory If X is an S -scheme of characteristic p > 0, we have an exact sequence in Xét

F −id 0 → Z/pZ → Oét −−−→ Oét → 0,

where F − id is associated to a of additive groups f 7→ f p − f. 18 / 29 Magid-Janelidze Galois theory in S

Let S be a topos. Consider

A

S a C

P

where I A = Rng(S )op; I P = StoneLoc(S ) (compact zero-dimensional locales). I S is the Pierce spectrum functor

S(A) = the locale with frame Idl(Idemp(A)).

19 / 29 Magid-Janelidze Galois theory in S

Janelidze’s Galois theory f For X −→ Y of relative Galois descent, we obtain a P-internal groupoid Gal[f] = (S(X ×Y X) S(X)) , and an equivalence of categories

SplitY (f) ' [Gal[f], P],

an S -version of Magid’s separable Galois theory for commutative rings.

20 / 29 Étale fundamental groupoid of a relative scheme

Let X be an S -scheme. Then Xproét is a P-determined topos over S , so [Bunge 08]

given a cover U in Xproét, get a locally discrete S -localic groupoid GU so that the associated fundamental P-pushout is equivalent to its classifying topos,

GU (Xproét) ' B(GU ).

21 / 29 Spectra of a difference Let A ∈ σ-Rng and τ a scheme topology. Write

Sτ ,→ Sch/Spec(bAc) for the classical τ-site of Spec(bAc), so that

bXcτ = Sh(Sτ ) is the classical τ-topos of Spec(bAc). Let  : Sτ → Sτ

be the base change functor along Spec(σA). The data Sτ = (Sτ , ) defines an internal site in σ-Set, and the τ-spectrum is the topos of internal sheaves

Xτ = Shσ-Set(Sτ ).

22 / 29 Difference spectra vs classical spectra

Explicitly,  Xτ ' bXcτ ,

the category of -equivariant sheaves in bXcτ , i.e., those F ∈ bXcτ equipped with a morphism

F → F ◦ .

There is a natural geometric morphism

πτ Xτ −→ σ-Set.

We can recover the difference ring as

πZar∗OX ' A.

23 / 29 Étale cohomology of a difference field

We obtain (Hilbert 90)

1 × H (két, Gm) ' Pic(k) ' (k )σ,

and 2 σ H (két, Gm) ' Br(k) ' Br(bkc) . Kummer theory gives

× σ 1 × 0 → ((k ) )n → H (k, µn) → n((k )σ) → 0,

and × 1 × σ 0 → (n(k ))σ → H (k, µn) → ((k )n) → 0.

24 / 29 Étale fundamental groupoid of a difference scheme

Difficulties: 1. a difference scheme X can be topologically totally disconnected, yet indecomposable; 2. X may not have geometric points; 3. the base topos σ-Set is not Boolean.

25 / 29 Étale fundamental groupoid of a difference scheme

Let k¯ be a separable closure of k ∈ σ-Rng, i.e., bk¯c is the separable closure of bkc in the sense of Magid. Fundamental difference groupoid [T-Wibmer 21] Janelidze’s Galois theory applied to f : k¯ → k in σ-Rngop gives a groupoid in σ-Prof

π1(k, k¯) = Gal[k¯ → k]

which classifies the category of difference locally étale k-algebras.

Forthcoming work A localic version in the StoneLoc(σ-Set)-determined context.

26 / 29 Difference Galois theory and dynamics

Connection to symbolic dynamics [T-Wibmer] If k is a difference field,

G = π1(k, k¯)

is a difference profinite group, and the Galois correspondence restricts to

{finitely σ-presented ind-étale k-algebras} '{subshifts of finite type with G-action}

limit degree ! entropy.

27 / 29 Difference-differential algebra

Iannazzo-T: spectrum of a differential ring

I Explicit site for Set[TLocalDifferentialRing]. I Construction of Spec(A, δ) using the locale of radical differential ideals. I Relation to [Carrà Ferro 90].

Keigher: differential rings in a topos E DRng(E ).

Difference-differential rings δ-σ-Rng ' DRng(σ-Set).

28 / 29 Why pursue this programme? 1. Ariadne’s thread.

2. Generalisations: we work over the base topos BN, but one can replace N by an arbitrary monoid, group, category etc to obtain the corresponding ‘equivariant’ geometry.

29 / 29