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-Set; 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 monoid (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 forgetful functor 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 group(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 -module or an abelian group 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 sequence 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 homomorphism 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 ring 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