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 2 C , σX 2 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 2 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 dj a a je 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 2 σ-Rng, R-Mod ' Mod(σ-Set;R) is the category of modules in a ringed topos. 8 / 29 Difference homological algebra Let R 2 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 2 σ-Rng, and M; N 2 R-Mod, with M étale. Then 8 >R-Mod(M; N); i = 0; i < ExtR(M; N) = [M; N]σ; i = 1; :>0; i > 1; In particular, 8 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 2 Rng(S ), M 2 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 2 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 2 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 2 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 ffinitely σ-presented ind-étale k-algebrasg ' fsubshifts of finite type with G-actiong 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.