Tannakian categories.

*(English)*Zbl 0810.18008
Jannsen, Uwe (ed.) et al., Motives. Proceedings of the summer research conference on motives, held at the University of Washington, Seattle, WA, USA, July 20-August 2, 1991. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 55, Pt. 1, 337-376 (1994).

The notion of Tannakian category grew out of Grothendieck’s theory of motives in his search for a universal cohomology theory for algebraic varieties. A definitely satisfactory theory of motives does not yet exist, the obstruction being Grothendieck’s standard conjectures on algebraic cycles. However, for varieties over finite fields one knows, by work of Deligne, Jannsen et. al., that the corresponding category of motives is in fact a (semi-simple, \(\mathbb{Q}\)-linear) Tannakian category. Roughly speaking this means that these motives are the objects of a \(k\)- linear category \({\mathcal C}\) (here e.g. \(k = \mathbb{Q}\), in general \(k\) may be any field) having tensor products, a unit and inverses and satisfying suitable axioms to make it a so-called \(k\)-lineown such that \({\mathcal C}\) is equivalent to the category of finite-dimensional representations in \(k\)-vector spaces of the group \(G\) of tensor automorphisms of the fibre functor \(\omega\).

The general case is much more difficult and it was settled only in the late 1980’s by P. Deligne in his contribution to the Grothendieck Festschrift [Vol. II, Prog. Math. 87, 111-195 (1990; Zbl 0727.14010)]. In a surprisingly ingenious proof of about fourty pages Deligne showed that for a general Tannakian category \({\mathcal C}\) over the field \(k\) the fibre functors form an affine gerbe over \(\text{Spec} (k)\) in the fpqc- topology and \({\mathcal C}\) becomes equivalent to the category of representations of this gerbe.

In the first three sections of the underlying paper these results and related notions are discussed. First, a proof of the neutral case along the lines set out in Deligne’s general proof is sketched. The importance of the Barr-Beck theorem giving the equivalence between the categories of certain naturally defined modules and of specific finite type comodules under a \(k\)-coalgebra determined by the fibre functor, thus making possible a co-end construction, is stressed. This coalgebra \(B\) is shown to be a commutative Hopf algebra representing the group functor \(G = \operatorname{Aut} (\omega)\) and \({\mathcal C}\) becomes equivalent to \(\text{Rep}(G)\). Next, the formalism of gerbes and nonabelian second cohomology is extensively discussed. It is shown that a gerbe \({\mathcal G}\) over a site \({\mathcal S}\) with final object \(e\), locally neutralised by an object \(x\) in \({\mathcal G}_ S\) for some covering \(S \to e\), leads to a \((p_ 1^* G, p_ 2^* G)\)-bitorsor \(E = \text{Isom} (p_ 2^*x, p_ 1^*x)\), where \(G = \operatorname{Aut} (x)\) and \(p_ 1, p_ 2 : S \times S \to S\) are the projections, satisfying nice ‘cocycle’ conditions \(\psi\). The pair \((E, \psi)\) is called the bitorsor cocycle. This description determines a transitive groupoid \(\Gamma : (E \twoheadrightarrow S)\). Conversely, one may introduce the notion of torsor under a groupoid \(\Gamma\). These form a stack Tors\((\Gamma)\), and Tors\((\Gamma)\) is a gerbe for transitive \(\Gamma\). In fact, the construction \(\Gamma\to\text{Tors}(\Gamma)\) is quasi-inverse to that which associates to a locally neutralised gerbe \({\mathcal G}\) its bitorsor cocycle (viewed as transitive groupoid \(\Gamma:(E\twoheadrightarrow S))\).

Now the stage is set for the proof of the fact that for a Tannakian category \({\mathcal C}\) the stack \(\text{FIB} ({\mathcal C})\) of fibre functors is indeed a gerbe, i.e. any two fibre functors are locally isomorphic in the fpqc-topology (the missing point in Saavedra’s proof of the main theorem), or in other words, the associated groupoid \(\Gamma : (\text{Isom} (p_ 2^* \omega, p_ 1^* \omega) \twoheadrightarrow S)\) is transitive in the fpqc-topology. The proof of this transitivity is based on Deligne’s construction of a tensor product of tensor categories and the fact that this tensor product of two Tannakian categories is itself a tensor category. The proof of the main theorem now follows the lines of the one for neutral Tannakian categories with the coalgebra \(B\) replaced by a coalgebroid \(L\) and the group \(G = \text{Spec} (B) \to \text{Spec} (k)\) replaced by the groupoid \(\Gamma : (E = \text{Spec} (L) \twoheadrightarrow S)\).

The fourth section deals with gerbes in the étale topology. One considers a Tannakian category \({\mathcal C}\) over \(k\) admitting a fibre functor with values in \(K\)-vector spaces, where \(K\) is a finite separable field extension of \(k\). The gerbe \({\mathcal G} = \text{FIB} ({\mathcal C})\) is then a gerbe on \(\text{Spebes} {\mathcal E}\) as group extensions of the form \[ 1 \to G (k^ s) \to {\mathcal E} @> \phi>> \text{Gal} (k^ s/k) \to 1, \] split by a local section \(j : \text{Gal} (k^ s/K) \to {\mathcal E}\), where \(k^ s\) is a separable closure of \(k\) and where \(G\) is some algebraic group scheme. Conversely, one recovers the entire bitorsor cocycle from the associated Galois gerbe. As a matter of fact one obtains a bijection between the set of smooth affine gerbes over \(\text{Spec} (k)_{ \text{é}t}\) neutralized over \(\text{Spec} (K)\) and the set of \(K/k\)-Galois gerbes.

The final section gives a very explicit description of a gerbe \({\mathcal G}\), locally neutralized by an object \(x\) in \({\mathcal G}_ S\) and giving a section \(u\) in the induced bitorsor cocycle \(E\), in terms of nonabelian 2-cocycles with values in the \(S\)-group \(G = \operatorname{Aut} (x)\). This leads to a nonabelian \(H^ 2\), not completely identical to Giraud’s definition in terms of the band associated to a gerbe.

For the entire collection see [Zbl 0788.00053].

The general case is much more difficult and it was settled only in the late 1980’s by P. Deligne in his contribution to the Grothendieck Festschrift [Vol. II, Prog. Math. 87, 111-195 (1990; Zbl 0727.14010)]. In a surprisingly ingenious proof of about fourty pages Deligne showed that for a general Tannakian category \({\mathcal C}\) over the field \(k\) the fibre functors form an affine gerbe over \(\text{Spec} (k)\) in the fpqc- topology and \({\mathcal C}\) becomes equivalent to the category of representations of this gerbe.

In the first three sections of the underlying paper these results and related notions are discussed. First, a proof of the neutral case along the lines set out in Deligne’s general proof is sketched. The importance of the Barr-Beck theorem giving the equivalence between the categories of certain naturally defined modules and of specific finite type comodules under a \(k\)-coalgebra determined by the fibre functor, thus making possible a co-end construction, is stressed. This coalgebra \(B\) is shown to be a commutative Hopf algebra representing the group functor \(G = \operatorname{Aut} (\omega)\) and \({\mathcal C}\) becomes equivalent to \(\text{Rep}(G)\). Next, the formalism of gerbes and nonabelian second cohomology is extensively discussed. It is shown that a gerbe \({\mathcal G}\) over a site \({\mathcal S}\) with final object \(e\), locally neutralised by an object \(x\) in \({\mathcal G}_ S\) for some covering \(S \to e\), leads to a \((p_ 1^* G, p_ 2^* G)\)-bitorsor \(E = \text{Isom} (p_ 2^*x, p_ 1^*x)\), where \(G = \operatorname{Aut} (x)\) and \(p_ 1, p_ 2 : S \times S \to S\) are the projections, satisfying nice ‘cocycle’ conditions \(\psi\). The pair \((E, \psi)\) is called the bitorsor cocycle. This description determines a transitive groupoid \(\Gamma : (E \twoheadrightarrow S)\). Conversely, one may introduce the notion of torsor under a groupoid \(\Gamma\). These form a stack Tors\((\Gamma)\), and Tors\((\Gamma)\) is a gerbe for transitive \(\Gamma\). In fact, the construction \(\Gamma\to\text{Tors}(\Gamma)\) is quasi-inverse to that which associates to a locally neutralised gerbe \({\mathcal G}\) its bitorsor cocycle (viewed as transitive groupoid \(\Gamma:(E\twoheadrightarrow S))\).

Now the stage is set for the proof of the fact that for a Tannakian category \({\mathcal C}\) the stack \(\text{FIB} ({\mathcal C})\) of fibre functors is indeed a gerbe, i.e. any two fibre functors are locally isomorphic in the fpqc-topology (the missing point in Saavedra’s proof of the main theorem), or in other words, the associated groupoid \(\Gamma : (\text{Isom} (p_ 2^* \omega, p_ 1^* \omega) \twoheadrightarrow S)\) is transitive in the fpqc-topology. The proof of this transitivity is based on Deligne’s construction of a tensor product of tensor categories and the fact that this tensor product of two Tannakian categories is itself a tensor category. The proof of the main theorem now follows the lines of the one for neutral Tannakian categories with the coalgebra \(B\) replaced by a coalgebroid \(L\) and the group \(G = \text{Spec} (B) \to \text{Spec} (k)\) replaced by the groupoid \(\Gamma : (E = \text{Spec} (L) \twoheadrightarrow S)\).

The fourth section deals with gerbes in the étale topology. One considers a Tannakian category \({\mathcal C}\) over \(k\) admitting a fibre functor with values in \(K\)-vector spaces, where \(K\) is a finite separable field extension of \(k\). The gerbe \({\mathcal G} = \text{FIB} ({\mathcal C})\) is then a gerbe on \(\text{Spebes} {\mathcal E}\) as group extensions of the form \[ 1 \to G (k^ s) \to {\mathcal E} @> \phi>> \text{Gal} (k^ s/k) \to 1, \] split by a local section \(j : \text{Gal} (k^ s/K) \to {\mathcal E}\), where \(k^ s\) is a separable closure of \(k\) and where \(G\) is some algebraic group scheme. Conversely, one recovers the entire bitorsor cocycle from the associated Galois gerbe. As a matter of fact one obtains a bijection between the set of smooth affine gerbes over \(\text{Spec} (k)_{ \text{é}t}\) neutralized over \(\text{Spec} (K)\) and the set of \(K/k\)-Galois gerbes.

The final section gives a very explicit description of a gerbe \({\mathcal G}\), locally neutralized by an object \(x\) in \({\mathcal G}_ S\) and giving a section \(u\) in the induced bitorsor cocycle \(E\), in terms of nonabelian 2-cocycles with values in the \(S\)-group \(G = \operatorname{Aut} (x)\). This leads to a nonabelian \(H^ 2\), not completely identical to Giraud’s definition in terms of the band associated to a gerbe.

For the entire collection see [Zbl 0788.00053].

Reviewer: W.W.J.Hulsbergen (Haarlem)

##### MSC:

18G50 | Nonabelian homological algebra (category-theoretic aspects) |

14L15 | Group schemes |

20G05 | Representation theory for linear algebraic groups |

11E72 | Galois cohomology of linear algebraic groups |

14A20 | Generalizations (algebraic spaces, stacks) |

14F99 | (Co)homology theory in algebraic geometry |

18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |