Tannakian duality over Dedekind rings and applications

Abstract

We establish a duality between flat affine group schemes and rigid tensor categories equipped with a neutral fiber functor (called Tannakian lattice), both defined over a Dedekind ring. We use this duality and the known Tannakian duality due to Saavedra to study morphisms between flat affine group schemes. Next, we apply our new duality to the category of stratified sheaves on a smooth scheme over a Dedekind ring R to define the relative differential fundamental group scheme of the given scheme and compare the fibers of this group scheme with the fundamental group scheme of the fibers. When R is a complete DVR of equal characteristic we show that this category is Tannakian in the sense of Saavedra.

Appendix A. Tannakian duality for flat coalgebras over Dedekind rings

In this appendix we give a quick, complete and self-contained proof of Theorem 1.2.2. First we will recall the notion ind-category of an abelian category. The two equivalent descriptions of the ind-category will play a crucial role in Saavedra’s proof. A category ${\mathcal {I}}$ is called a filtered category if to every pair i, j of objects in ${\mathcal {I}}$ there exists an object k such that $\mathsf{Hom}(i, k)$ and $\mathsf{Hom}(j, k)$ are both not empty, and for every pair $u, v : i \longrightarrow j,$ there exists a morphism $w: j \longrightarrow k$ such that $wu = wv.$

Definition A.1.1

Ind-categories. Let ${\mathcal {C}}$ be an abelian category. The category $\mathsf{Ind}({\mathcal {C}})$ consists of functors $X: {\mathcal {I}} \longrightarrow {\mathcal {C}},$ where ${\mathcal {I}}$ is a filtering category. We usually denote $X_i$ for $X(i), i \in {\mathcal {I}},$ an write

$$\begin{aligned} X=\varinjlim _{i\in {\mathcal {I}}} X_i. \end{aligned}$$

For two objects $X= \varinjlim _{i\in {\mathcal {I}}}X_i$ and $Y = \varinjlim _{j\in {\mathcal {J}}}{Y_j }$ their hom-set is defined to be

$$\begin{aligned} \mathsf{Hom}(X,Y):= \displaystyle \varprojlim _{i\in {\mathcal {I}}} \displaystyle \varinjlim _{j\in {\mathcal {J}}} \mathsf{Hom}(X_i, Y_j). \end{aligned}$$

$\square$

Let $\omega : {\mathcal {C}} \longrightarrow {\mathcal {D}}$ be a functor. The extension of $\omega , \mathsf{Ind}(\omega ): \mathsf{Ind({\mathcal {C}})} \longrightarrow \mathsf {Ind}({\mathcal {D}})$ is defined by

$$\begin{aligned} \mathsf{Ind}(\omega )(\displaystyle \varinjlim _i {X_i}) := \displaystyle \varinjlim _i {\omega (X_i)}. \end{aligned}$$

There is an alternative description of $\mathsf{Ind}({\mathcal {C}})$. Denote $\mathsf{Lex}({\mathcal {C}}^{op}, {\mathcal Sets})$ the category of left exact functors from ${\mathcal {C}}^{op}$ to the category of sets. For $X= \varinjlim _i X_i$ we define functor

$$\begin{aligned} \displaystyle \varinjlim _i h_{X_i}(-):= \displaystyle \varinjlim _i \mathsf{Hom}(-, X_i) \in \mathsf{Lex}({\mathcal {C}}^{op}, {\mathcal Sets}). \end{aligned}$$

This yields a functor $\mathsf {Ind({\mathcal {C}})} \longrightarrow \mathsf{Lex}({\mathcal {C}}^{op}, {\mathcal Sets})$ which is an equivalence (cf.[2], I.8.3.3). Recall that the Hom-sets for objects of $\mathsf{Lex}({\mathcal {C}}^{op}, {\mathcal Sets})$ are by definition the sets of natural transformations. For simplicity, we shall use the notation $\mathsf{Hom}(F,G)$ instead of $\mathsf{Nat}(F,G)$ for objects of this category.

A.1.2. Suppose that ${\mathcal {C}}$ is an R-linear Noetherian abelian category. Let $\mathsf{Lex}_R({\mathcal {C}}^{op}, \mathsf{Mod}(R))$ be category of R-linear left exact functors from ${\mathcal {C}}^{op}$ to the category of modules $\mathsf{Mod}(R)$. Then the natural functor

$$\begin{aligned} \mathsf{Lex}_R({\mathcal {C}}^{op}, \mathsf{Mod}(R)) \xrightarrow {\ \simeq \ }\mathsf{Lex}({\mathcal {C}}^{op}, {\mathcal Sets}) \end{aligned}$$

is an equivalence (cf. Gabriel [15, II]). Thus, for an R-linear Noetherian abelian category we have an equivalence

$$\begin{aligned} \mathsf{Ind}({\mathcal {C}})\simeq \mathsf{Lex}_R({\mathcal {C}}^{op}, \mathsf{Mod}(R)),\quad X=\varinjlim _iX_i\longmapsto \displaystyle \varinjlim _i h_{X_i}(-). \end{aligned}$$

Further the category $\mathsf{Ind}({\mathcal {C}})$ is locally Noetherian and the inclusion ${\mathcal {C}}\longrightarrow \mathsf{Ind}({\mathcal {C}})$ identifies ${\mathcal {C}}$ with the full subcategory of Noetherian objects in $\mathsf{Ind}({\mathcal {C}})$, [15, II, 4, Thm. 1].

The following are our main examples.

Example A.1.3

The category $\mathsf{Mod}_{\mathrm{f}}(R)$ of finitely generated R-modules, where R is a Noetherian ring, is a Noetherian category. Its Ind category is precisely the category $\mathsf{Mod}(R)$ of all R-modules. This is obvious.

Example A.1.4

Let L be a coalgebra over a commutative ring R. Denote by $\mathsf{Comod}(L)$ the category of right L-comodules and by $\mathsf{Comod}_{\mathrm{f}}(L)$ the subcategory of comodules which are finitely generated as R-module. Then:

• (i) If L is flat over R then $\mathsf{Comod}(L)$ is an abelian category. In fact, the flatness of L implies that the kernel of a homomorphism of L-comodules is equipped with a natural coaction of L. In particular, the forgetful functor from $\mathsf{Comod}(L)$ to $\mathsf{Mod}(R)$ is exact. The converse is also true: if the forgetful functor preserves kernels then L is flat over R.

• (ii) Assume that L is flat over R and R is Noetherian. According to Serre [25, Cor. 2] each L-comodule is the union of its R-finite subcomodules. Consequently, $\mathsf{Comod}(L)$ is locally Noetherian and $\mathsf{Comod}_{\mathrm{f}}(L)$ is the full subcategory of Noetherian objects.

Let ${\mathcal {C}}$ be an R-linear abelian category, and $\omega : {\mathcal {C}} \longrightarrow \mathsf{Mod}_{\mathrm{f}}(R)$ be an R-linear exact faithful functor. Suppose that there exists a full subcategory of definition ${\mathcal {C}}^{\mathrm{o}}$ in ${\mathcal {C}}.$ Our aim is to show that there exists a flat R-coalgebra L such that $\omega$ induces an equivalence between $\mathsf{Comod}_{\mathrm{f}}(L)$ and ${\mathcal {C}}$, and between $\mathsf{Comod}(L)$ and $\mathsf{Ind}({\mathcal {C}})$.

The functor $\omega$ induces a functor $\mathsf{Ind}( {\mathcal {C}})\longrightarrow \mathsf{Mod}(R),$ which we, by abuse of language, will denote simply by $\omega$. Recall that we identify $\mathsf{Ind}({\mathcal {C}})$ with $\mathsf{Lex}({\mathcal {C}}^{op}, \mathsf{Mod}(R))$, the category of left exact functors on ${\mathcal {C}}^{op}$ with values in $\mathsf{Mod}(R)$. The key technique is to use alternatively these two equivalent descriptions of one category.

A.1.5. For any R-algebra A, we define functor

$$\begin{aligned} F^A: {\mathcal {C}}^{op} \longrightarrow \mathsf{Mod}(A),\quad X \longmapsto \mathsf{Hom}( \omega (X), A). \end{aligned}$$

Then $F^A$ is an object of $\mathsf{Lex}({\mathcal {C}}^{op}, \mathsf{Mod}(R))$. Set $F:=F^R$. There is a natural A-linear transformation $A\otimes F\longrightarrow F^A$:

$$\begin{aligned} \theta _X:A\otimes \mathsf{Hom}(\omega (X),R)\longrightarrow \mathsf{Hom}(\omega (X),A),\quad a\otimes f\longmapsto af. \end{aligned}$$

Lemma A.1.6

The A-linear transformation $\theta :A\otimes F\longrightarrow F^A$ given above is an isomorphism.

Proof

For any $K, G \in \mathsf{Lex}({\mathcal {C}}^{op}, \mathsf{Mod}(R))$ we denote $K^{\mathrm{o}}, G^{\mathrm{o}}$ their restrictions to $({\mathcal {C}}^{\mathrm{o}})^{op}$, respectively. We claim that

$$\begin{aligned} \mathsf{Hom}(K,G) \simeq \mathsf{Hom}(K^{\mathrm{o}}, G^{\mathrm{o}}). \end{aligned}$$

(14)

Indeed, let $\theta \in \mathsf{Hom}(K^{\mathrm{o}}, G^{\mathrm{o}})$, that is we have a family $\theta _X:K^{\mathrm{o}}(X)\longrightarrow G^{\mathrm{o}}$ for $X\in {\mathcal {C}}^{\mathrm{o}}$ commuting with morphism in ${\mathcal {C}}^{\mathrm{o}}$. Since each object of ${\mathcal {C}}$ can be represented as a cokernel of a morphism $X_1\longrightarrow X_2$ in ${\mathcal {C}}^{\mathrm{o}}$, we see that $\theta$ extends uniquely to a natural transformation $K\longrightarrow G$ (as these functors are left exact on ${\mathcal {C}}^{\mathrm{op}}$).

For $X\in {\mathcal {C}}^{\mathrm{o}}$, $\omega (X)$ is finite projective over R, hence

$$\begin{aligned} F^A(X) =\mathsf{Hom}(\omega (X), A) \simeq \mathsf{Hom}(\omega (X), R) \otimes A = A \otimes F(X). \end{aligned}$$

Therefore, for any $G\in \mathsf{Lex}({\mathcal {C}}^{op}, \mathsf{Mod}(R))$, we have

$$\begin{aligned} \mathsf{Hom}((F^A)^{\mathrm{o}}, G^{\mathrm{o}}) = \mathsf{Hom}((A\otimes F)^{\mathrm{o}}, G^{\mathrm{o}}) \end{aligned}$$

(15)

and (14) yields

$$\begin{aligned} \mathsf{Hom}(F^A, G) = \mathsf{Hom}(A\otimes F, G). \end{aligned}$$

(16)

So we have $F^A\simeq A \otimes F.$ $\square$

We will show that $L:=\omega (F)$ is the coalgebra to be found. To show this, first we will need

Lemma A.1.7

For any $X\in \mathsf{Lex}({\mathcal {C}}^{op},\mathsf{Mod}(R))$ and R-algebra A we have the following A-linear isomorphism:

$$\begin{aligned} \mathsf{Hom}(X, F^A) \simeq \mathsf{Hom}_A(A\otimes \omega (X), A)=\mathsf{Hom}_R( \omega (X), A). \end{aligned}$$

(17)

Proof

Every $X \in \mathsf{Lex}({\mathcal {C}}^{op},\mathsf{Mod}(R))$ can be represented as $X= \displaystyle \varinjlim _i h_{X_i} ( X_i \in {\mathcal {C}}),$ where $h_{X_i}$ is a functor over ${\mathcal {C}}$, defined by $h_{X_i}(-) := \mathsf{Hom}_{{\mathcal {C}}}( -, X_i).$ Hence we have

$$\begin{aligned} \mathsf{Hom}(X, F^A)= & {} \mathsf{Hom}( \varinjlim h_{X_i},F^A)\\= & {} \varprojlim \mathsf{Hom}(h_{X_i}, F^A)\\= & {} \varprojlim F^A (X_i) \\= & {} \mathsf{Hom}_R( \varinjlim \omega (X_i), A)\\= & {} \mathsf{Hom}_R(\omega (\varinjlim h_{X_i}), A)\\= & {} \mathsf{Hom}_R( \omega (X), A). \end{aligned}$$

It is easy to see that all isomorphisms are A-linear. $\square$

Isomorphism (17) for $A=R$ and $X=F$ reads $\mathsf{Hom}(F,F)\simeq \mathsf{Hom}_R(\omega (F),R)$. We denote $L:=\omega (F)$ and let $\varepsilon :L\longrightarrow R$ be the map on the right hand side that corresponds to the identity transformation on the left hand side of this isomorphism. The next lemma shows that one can replace the algebra A in (17) by any R-module M to get R-linear isomorphisms.

Lemma A.1.8

There exists a natural R-linear isomorphism extending (17)

$$\begin{aligned} \Phi _{X,M}: \mathsf{Hom}(X,M\otimes F) \simeq \mathsf{Hom}_R(\omega (X), M), \end{aligned}$$

(18)

which is given explicitly by

$$\begin{aligned} \Phi _{X,M}(f)= (\mathsf{id}_M\otimes \varepsilon )\circ \omega (f). \end{aligned}$$

Proof

For any R-module M, we can make $R\oplus M$ into an R-algebra by letting M be an ideal with square null. Hence the isomorphism (18) is a direct consequence of (17). By definition $\Phi _{F,R}$ is given by

$$\begin{aligned} \Phi _{F,R}(f)=\varepsilon \circ \omega (f). \end{aligned}$$

Each R-linear map $\iota :R\longrightarrow M$ induces by functoriality the commutative diagram

Now, the identity on F yields the equality:

$$\begin{aligned} \iota \circ \varepsilon =\Phi _{F,M}(\iota \otimes \mathsf{id}_{\omega (F)}):\omega (F)\longrightarrow M. \end{aligned}$$

Hence, for $m=\iota (1)$, we have $\Phi _{F,M}(l)=\varepsilon (l)m$, $l\in \omega (F)$. Thus the claim holds for $X=F$. Since the $\omega$ and Hom-functor in the first variant commute with direct limits we conclude that the claim hold of $X=N\otimes F$ for any R-module N. Now the general case follows from the following diagram

applied for the identity of $M\otimes F$:

$$\begin{aligned} \Phi _{X,M}(f)=\Phi _{F,M}(\mathsf{id})\circ \omega (f)=(\mathsf{id}_M\otimes \varepsilon )\circ \omega (f). \end{aligned}$$

$\square$

Proposition A.1.9

Let $L:=\omega (F)$. Then it is a coalgebra with $\varepsilon$ being the counit and $\omega$ factors though a functor

$$\begin{aligned} \mathsf{Ind}({\mathcal {C}})\longrightarrow \mathsf{Comod}(L). \end{aligned}$$

Proof

Choose $M= \omega (X)$ in (18) we have a morphism ${\sigma }_X :X \longrightarrow \omega (X) \otimes F$ which corresponds to the identity element $\mathsf{id}_{\omega (X)}$ under the isomorphism $\Phi _{X,\omega (X)}$ of Lemma A.1.8, thus we have

$$\begin{aligned} (\mathsf{id}_{\omega (X)} \otimes \varepsilon ) \circ \omega (\sigma _{X}) = \mathsf{id}_{\omega (X)}. \end{aligned}$$

(19)

For any morphism $\lambda : X \longrightarrow Y$ in $\mathsf{Ind}( {\mathcal {C}})$, according to A.1.8 we have the following equalities:

$$\begin{aligned}&\Phi _{X,\omega (Y)}\left( (\omega (\lambda ) \otimes \mathsf{id}_ {F}) \circ \sigma _X\right) =\omega (\lambda ),\\&\Phi _{X,\omega (Y)}\left( \sigma _Y \circ \lambda \right) = \omega (\lambda ). \end{aligned}$$

Thus $(\omega (\lambda ) \otimes \mathsf{id}_{F}) \circ \sigma _X= \sigma _Y \circ \lambda ,$ i.e, the following diagram commutes:

(20)

For $Y=\omega (X)\otimes F$ and $\lambda =\sigma _X$, we get

(21)

Applying $\omega$ on this diagram we obtain a commutative diagram in $\mathsf{Mod}(R)$:

(22)

where $\Delta :=\omega (\sigma _F)$. Together with (19), this diagram for $X=F$ gives a coalgebra structure on L with $\Delta$ being the coproduct and hence, for any X, it gives a comodule structure of L on $\omega (X)$. $\square$

Proof (of Theorem 1.2.2)

Let L be defined as in Proposition A.1.9. We consider $\omega$ as a functor ${\mathcal {C}}\longrightarrow \mathsf{Comod}_{\mathrm{f}}(L)$. It is to show that $\omega$ is an equivalence of category. By definition it is faithful. To see the fullness, suppose $X, Y \in {\mathcal {C}}$ and $\alpha : \omega (X) \longrightarrow \omega (Y)$ is a homomorphism of L-comodules, i.e., we have

$$\begin{aligned} (\alpha \otimes \mathsf{id})\circ \omega (\sigma _X)=\omega (\sigma _Y)\circ \alpha :\omega (X)\longrightarrow \omega (Y)\otimes L. \end{aligned}$$

Then is the image under $\omega$ of the morphism

Notice that (22) (for X replaced by Y) yields a split exact sequence

(23)

where the second homomorphism is $\delta =\mathsf{id}\otimes \Delta -\omega (\sigma _X)\otimes \mathsf{id},$ and the splitting is given by $\mathsf{id}\otimes \varepsilon :\omega (Y)\otimes L\longrightarrow \omega (Y).$ This sequence is the similar image under $\omega$ of the sequence coming from (21):

$$\begin{aligned} 0\longrightarrow Y\longrightarrow \omega (Y) \otimes F\longrightarrow \omega (Y) \otimes L \otimes F. \end{aligned}$$

Hence the latter sequence is also exact. On the other hand, it follows from the faithfulness of $\omega$ that the composed map

is the zero morphism (since its image under $\omega$ is zero by means of (22) and the fact that $\alpha$ is a homomorphism of L-comodules). Consequently, the morphism factor through a morphism $f:X\longrightarrow Y$ and the morphism $\sigma _Y$. Applying $\omega$ on the composition of these maps we conclude $\omega (f)=\alpha$, as $\omega (\sigma _Y)$ is injective. Thus $\omega$ is full.

It remains to show that $\varphi$ is essentially surjective. For any L-comodule $(E,\rho _E)$ let $E^{\mathrm{o}} \in C$ be such that the sequence

$$\begin{aligned} 0\longrightarrow E^{\mathrm{o}}\longrightarrow E \otimes F\xrightarrow {\quad \delta } E \otimes L \otimes F. \end{aligned}$$

is exact, where $\delta =\rho _E\otimes \mathsf{id}-\mathsf{id}\otimes \sigma _F$. Applying $\omega$ to this sequence and comparing with (23) we conclude that $\omega (E^{\mathrm{o}})=E$.

Thus $\omega :{\mathcal {C}}\longrightarrow \mathsf{Comod}_{\mathrm{f}}(L)$ is an equivalence of categories. Thus the forgetful functor $\mathsf{Comod}_{\mathrm{f}}(L)\longrightarrow \mathsf{Mod}(R)$ is exact, hence L is flat over R. $\square$

Remarks A.1.10

(i) Under the equivalence of Theorem 1.2.2, L, with the right coaction of itself given by the coproduct, corresponds to F. Indeed, this follows from the natural isomorphism

$$\begin{aligned} \mathsf{Hom}^L(E,L)\simeq \mathsf{Hom}_R(E,R),\quad f\mapsto \varepsilon \circ f. \end{aligned}$$

(ii) There is another way to determine L from the category of its comodules as follows. We claim that there is a natural isomorphism

$$\begin{aligned} \mathsf{Nat}(\omega ,\omega \otimes M)\simeq \mathsf{Hom}_R(L,M), \end{aligned}$$

(24)

for any R-module M. Indeed, we have

$$\begin{aligned} \mathsf{Hom}_R(L,M)\simeq \mathsf{Hom}(F,F\otimes M)\simeq \mathsf{Hom}(\mathsf{Hom}(\omega ,R),\mathsf{Hom}(\omega ,R)\otimes M). \end{aligned}$$

By means of (14), it suffices to show the isomorphism

$$\begin{aligned} \mathsf{Nat}(\omega (X),\omega (X)\otimes M)\simeq \mathsf{Hom}(\mathsf{Hom}(\omega (X),R),\mathsf{Hom}(\omega (X),R)\otimes M) \end{aligned}$$

for any $X\in {\mathcal {C}}^{\mathrm{o}}$. Since for such X, $\omega (X)$ is finitely generated projective over R, the last isomorphism is obvious. L is usually referred to as the Coend of $\omega$, denoted $\mathsf{Coend}(\omega )$.

(iii) If ${\mathcal {C}}=\mathsf{Comod}_{\mathrm{f}}(L)$ and $\omega$ is the forgetful functor from ${\mathcal {C}}$ to $\mathsf{Mod}(R)$, then the isomorphism (24) implies that $\mathsf{Coend}(\omega )\simeq L$. Thus a flat coalgebra over R can be reconstructed from the category of its comodules. $\square$

Remarks A.1.11

Let $({\mathcal {C}},\omega )$ and $({\mathcal {C}}',\omega ')$ be two categories satisfying the condition of Theorem 1.2.2 and let $\eta :{\mathcal {C}}\longrightarrow {\mathcal {C}}'$ be an R-linear functor such that $\omega '\eta =\omega$. Then $\eta$ induces a coalgebra homomorphism $f:L\longrightarrow L'$. This can be seen from (24) as follows. The coaction of $L'$ on $\omega '(X')$ defines a natural transformation $\delta ':\omega '\longrightarrow \omega '\otimes L'$. Combine this with $\eta$ we obtain a natural transformation $\delta :\omega \longrightarrow \omega \otimes L'$. Thus (24) yields a linear map $L\longrightarrow L'$, which satisfies the following commutative diagram:

$\square$