An a priori proof that projective objects are torsion-free in Serre's category of proalgebraic groups

In 'Groupes proalgébriques' (1960), Serre introduced the abelian category $\mathscr{P}$ of commutative proalgebraic groups over an algebraically closed field $k$ — the pro-category of the quasi-algebraic groups, i.e. commutative algebraic groups with radicial isogenies inverted — and developed its homological algebra: enough projectives, an exact universal covering functor, vanishing of all homotopy objects $\pi_i$ for $i \geq 2$, and projective dimension at most 2 for every object (at most 1 in characteristic 0). The projective objects are classified: every projective is a product of indecomposable projectives, and the indecomposable ones are the projective envelopes of the elementary groups $Z/lZ$, $G_m$, $G_a$, and the simple abelian varieties. Serre determined these envelopes explicitly — the $l$-adic groups $Z_l$; the universal covering of $G_m$, a pro-torus with character group $\mathbb{Q}$; $G_a$ itself in characteristic 0 and the universal Witt covering $W$ in characteristic $p$; and a pro-linear extension $X(A)$ of the abelian variety — and observed that in each case the envelope is torsion-free, concluding that every projective object of $\mathscr{P}$ is torsion-free. He then asked (§3, p. 26): 'y a-t-il une démonstration a priori de ce résultat ?', i.e. whether there is a proof of this torsion-freeness that does not go through the classification of projective objects. No subsequent work (Demazure–Gabriel 1970; Oort, Milne, and Oort–Oda on homological dimension; Brion's 2016–2020 program on isogeny categories and fundamental groups of commutative algebraic groups) is known to state or address this question, although related soft results exist in adjacent categories — notably Brion's unique divisibility of the projective cover $P(A)$ of an abelian variety in the pro-category of commutative algebraic groups. The question remains open: produce a classification-independent proof that all projective objects of $\mathscr{P}$ are torsion-free, valid over any algebraically closed field in both characteristics.

Background

This problem comes from the homological algebra of algebraic groups. Throughout, $k$ denotes an algebraically closed field and 'algebraic group' means a commutative group scheme of finite type over $k$. By the classical structure theory, every connected algebraic group is an extension of an abelian variety (a projective connected algebraic group, of which elliptic curves are the one-dimensional examples) by a linear (affine) algebraic group, and a connected linear one decomposes into a part of multiplicative type (a torus) and a unipotent part (built from copies of the additive group $G_a$ and, in positive characteristic, the finite constant groups $Z/pZ$).

The category of commutative algebraic groups over $k$ is abelian in characteristic 0 but not in characteristic $p$: purely inseparable ('radicial') isogenies, such as the relative Frobenius, have trivial kernel and cokernel without being isomorphisms. Serre's mémoire repairs this by inverting exactly these morphisms. The resulting isogeny category of 'quasi-algebraic groups' is abelian, and its pro-category — formal filtered inverse limits — is the category $\mathscr{P}$ of (commutative) proalgebraic groups. The category $\mathscr{P}$ is abelian, admits arbitrary products, filtered inverse limits in it are exact, and it has enough projectives.

A projective object $P$ of $\mathscr{P}$ is one for which the functor $\operatorname{Hom}(P, -)$ is exact, equivalently $\operatorname{Ext}^1(P, G) = 0$ for all $G$. Every object $G$ admits a projective envelope: an essential surjection $X \to G$ with $X$ projective, unique up to isomorphism (the dual of Gabriel's theory of injective envelopes). Serre proves (Theorem 1 of §3) that every projective object of $\mathscr{P}$ is a product of indecomposable projectives, and (Theorem 2 of §3) that every indecomposable projective is the projective envelope of an 'elementary group', where the elementary groups are, up to isomorphism, the finite constant groups $Z/lZ$ ($l$ prime), the multiplicative group $G_m$, the additive group $G_a$, and the simple abelian varieties.

The mémoire then determines the envelopes explicitly, one family at a time: the envelope of $Z/lZ$ is the group $Z_l$ of $l$-adic integers (§4); the envelope of $G_m$ is its universal covering, the pro-torus with character group $\mathbb{Q}$ (§7); the envelope of $G_a$ is $G_a$ itself in characteristic 0 — in fact every unipotent quasi-algebraic group is then projective — and in characteristic $p$ the universal covering $W$ of the Witt group, a pro-$p$ group built from the Witt vectors, the coordinate system on which the additive group of a characteristic-$p$ field is best expressed (§8); the envelope of an abelian variety $A$ is a pro-linear extension $X(A) \to A$ (a universal affine extension) whose kernel is of the form $T \times U \times \pi_1(A)$, with $T$ of multiplicative type, $U$ a $k$-vector group of dimension $\dim A$ in characteristic 0 and $0$ in characteristic $p$, and $\pi_1(A)$ a profinite group whose $l$-primary part is the $l$-adic Tate module of $A$, i.e. $(Z_l)^{2\dim A}$ for $l$ different from the characteristic and $(Z_p)^s$ with $0 \leq s \leq \dim A$ for $l$ equal to it (§6.5 and §9).

A proalgebraic group is torsion-free ('sans torsion') if multiplication by $n$ is injective for every positive integer $n$, equivalently if it contains no copy of $Z/lZ$. For special subcategories the paper gives structure-theoretic characterizations: a zero-dimensional proalgebraic group is projective if and only if it is a product of $Z_l$'s, if and only if it is torsion-free, if and only if its Cartier dual is divisible (§4.4, prop. 5) — here the Cartier dual is the duality between a profinite commutative group and its discrete group of characters; a group of multiplicative type is connected if and only if its character group is torsion-free (§7.3, prop. 3). But for the whole category the only available statement is the one obtained from the classification: each of the four explicit envelopes is inspected directly and seen to be torsion-free, and Theorem 2 transfers this to all projective objects.

This matters because the main results of the mémoire flow through the projectives: the universal covering functor is exact; the homotopy objects $\pi_i$ — the left derived functors of the component functor $\pi_0$, in analogy with the homotopy groups of a topological space — vanish for all $i \geq 2$ and all objects; every object of $\mathscr{P}$ has projective dimension at most 2, and at most 1 in characteristic 0 (the additive group has projective dimension exactly 2 in characteristic $p$). Whether these consequences can be reached without the case-by-case classification is a natural structural question about the framework itself.

Later literature. Chapter V of Demazure–Gabriel (1970) develops the general theory of pro-artinian categories (pro-categories built from artinian categories), of which $\mathscr{P}$ is a prototype, and determines the projective objects of the pro-category of linear algebraic groups over a perfect field [DG70, V.3.7], by the same structure-theorem techniques; the book also contains an appendix by Hazewinkel realizing the local class field theory applications announced in the introduction of the 1960 paper (realized first by Serre's own 1961 paper). The homological dimension of categories of commutative algebraic groups was studied by Oort (1966) and determined over perfect fields by Milne (1970), and the higher extension groups of abelian varieties vanish (Oort–Oda 1968). Brion's program (2016–2020) reworks this circle over arbitrary fields with representation-theoretic and soft homological methods: isogeny categories presented as module categories; homological dimensions of these categories computed without the case-by-case extension analysis of the earlier literature; and a description of the indecomposable projectives of the pro-category Pro(C) of commutative algebraic groups (radicial isogenies not inverted), including the soft result that the projective cover $P(A)$ of an abelian variety is uniquely divisible. None of this later literature states or addresses the torsion-freeness of all projectives of $\mathscr{P}$ without the classification.

Problem Statement

Serre asks, at §3, p. 26 of 'Groupes proalgébriques' (1960), after noting that each of the four explicitly determined projective envelopes is torsion-free and that therefore (by the classification of projectives) every projective object is torsion-free: 'y a-t-il une démonstration a priori de ce résultat ?' — is there an a priori proof of this result?

Concretely: let $k$ be an algebraically closed field (of either characteristic), and let $\mathscr{P}$ denote the abelian category of (commutative) proalgebraic groups over $k$ — the pro-category (category of filtered inverse limits) of the isogeny category of commutative algebraic groups over $k$, i.e. of commutative algebraic groups with radicial isogenies inverted. Does there exist a proof that every projective object $P$ of $\mathscr{P}$ is torsion-free — i.e. that multiplication by $n$ is injective on $P$ for every positive integer $n$ — which does not pass through the classification of the projective objects? The known proof uses Theorem 1 of §3 (every projective is a product of indecomposable projectives, via Gabriel's structure theorem), Theorem 2 of §3 (the indecomposable projectives are the projective envelopes of the elementary groups $Z/lZ$, $G_m$, $G_a$, and the simple abelian varieties), and the explicit determination of those envelopes ($Z_l$; the universal covering of $G_m$; $G_a$ in characteristic 0 and the universal Witt covering $W$ in characteristic $p$; the pro-linear extensions $X(A)$ of abelian varieties), in each case observing torsion-freeness by inspection. An a priori proof is one that does not invoke these classification results or any equivalent determination of which objects are projective, and it is valid in both characteristic 0 and characteristic $p$.

The question concerns the full statement (all projective objects, both characteristics, arbitrary algebraically closed base field), and the proof sought is one independent of the classification. It concerns all projective objects as a whole, not the zero-dimensional or the multiplicative-type cases alone.

The term 'a priori' is the author's and is not formally defined in the source; here it is fixed by the two closed lists below.

Prohibited inputs. An a priori proof is one that does not invoke: (i) Theorem 1 of §3 (every projective object of $\mathscr{P}$ is a product of indecomposable projectives, via Gabriel's structure theorem); (ii) Theorem 2 of §3 (the indecomposable projectives are the projective envelopes of the elementary groups $Z/lZ$, $G_m$, $G_a$, and the simple abelian varieties); (iii) the explicit determinations of those envelopes ($Z_l$; the universal covering of $G_m$; $G_a$ in characteristic 0 and the universal Witt covering $W$ in characteristic $p$; the pro-linear extensions $X(A)$ of abelian varieties), including the description of the abelian-variety envelope $X(A) \to A$ and its kernel $T \times U \times \pi_1(A)$ recalled in the Background above; or (iv) the equivalent determination of the projective objects as 'a product of projective envelopes of elementary groups' (equivalently, a retract of such a product).

Permitted inputs. The following are not excluded and may be used freely: the definitions of projective object and of torsion-freeness, and the lifting criterion for projectivity (equivalently $\operatorname{Ext}^1(P, -) = 0$); elementary homological algebra of the abelian category $\mathscr{P}$ (the long exact $\operatorname{Ext}$ sequence and the exactness of filtered inverse limits); the fact that $\mathscr{P}$ has enough projectives and that every object admits a projective envelope, unique up to isomorphism — existence and uniqueness only, not the identification of the envelope of an elementary group; and the general structure theory of commutative algebraic groups (the Chevalley decomposition, the arithmetic of Witt vectors, the structure of divisible abelian groups).

References

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