# The algebraic fundamental group of a reductive algebraic group

For a connected reductive algebraic group $G$ over a field $k$, other than the \'etale fundamental group of $G$ (regarded just as a scheme), there seems to be another notion, usually called the algebraic fundamental group of $G.$ I am not sure of its definition, but I guess (at least when $G$ is split) it might be something like $I/\Gamma,$ where $I$ is the "topological fundamental group" of a maximal torus $T$ (if it makes sense), and $\Gamma$ is the subgroup generated by the inverse roots. In particular, for $GL_n$ this should produce $\mathbb Z,$ which agrees with the topological fundamental group when $k=\mathbb C.$

Could anyone give some references on this?

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Dear shenghao: In pos. char. the concepts are very different (formation of etale $\pi_1$ doesn't commute with products, "universal cover" as limit scheme has no $k$-gp structure). The alg. fund. gp "classifies" central isogenies, which may not be etale! Central tori complicate things, so focus on ss case. Composition of central isogenies is central, and there's a unique central $k$-isogeny $f:G' \rightarrow G$ where $G'$ has no non-trivial central isogenous cover. That is the "simply conn'd (central) cover", with ker($f$) of mult. type. Its etale Cartier dual is (by defn) the alg. fund. gp. – BCnrd Dec 13 '10 at 17:42
Dear shenghao: In general a conn'd ss group that is simply connected in the sense of alg. groups (having no nontrivial central isogenous cover over the ground field) retains this property after a ground field extension. Over alg. closed fields of char. 0 the alg. fundamental gp and etale fundamental gp are duals. Indeed, the formation of etale fund. gp commutes with direct products in char. 0, and more specifically the "universal cover" as an affine limit scheme has a group scheme structure. Exhausting by Hopf subalgebras of finite type, it's exhausted by etale isogenous covers, so voila. – BCnrd Dec 13 '10 at 17:55
@BCnrd: Je suis aussi exhausted. Maybe your comments could be expanded a little into an answer? The comparison of algebra and topology, mixed in with some algebraic geometry, gets complicated to explain but is important for achieving some unity of language when treating semisimple algebraic groups over fields of many kinds. – Jim Humphreys Dec 13 '10 at 19:14

At Jim's request, here's an expanded version of my comments above. I will have to use some facts from the topological theory of complex algebraic varieties, but out of stubbornness I will not use any such facts which are part of the theory of Lie groups (the maximal compact subgroup, facts specific to complex semisimple Lie algebras, etc.)

Let $X$ be a smooth connected affine scheme over a field $k$ (the case of interest being a connected semisimple $k$-group). Consider the collection of connected finite \'etale covers of $X$. This is an inverse system of affine schemes (with coordinate rings that are domains). Consider the inverse limit (i.e., Spec of direct limit of coordinate rings), call it $\widetilde{X}$. This is an algebraist's analogue of a universal cover: it is Spec of a (typically huge, not finite type over $k$, nor noetherian) domain, so it is connected, and one can show by "standard" direct limit arguments that it has no nontrivial connected finite etale cover. So every finite etale cover of $X$ is totally split by pullback over $\widetilde{X}$, "as if" it were a universal cover in topology.

The automorphism group of $\widetilde{X}$ over $X$ is (by one definition) the opposite group of the etale fundamental group of $X$ (upon fixing some geometric point of $X$ as the base point, and lifting it to a geometric point of $\widetilde{X}$; if $k$ were separably closed then we could take a point in $X(k)$ as the base point and lift it to a $k$-point of $\widetilde{X}$). Is $\widetilde{X} \rightarrow X$ a finite-degree covering, say when $k$ is separably closed? This is the same to ask that $\widetilde{X}$ is finite type over $k$. In characteristic $p > 0$ one can use the Artin-Schreier method to make infinitely many pairwise non-isomorphic degree-$p$ connected finite etale Galois covers of $X$ if $\dim X > 0$, so in positive characteristic $\widetilde{X}$ is never of finite type over $k$ when $\dim X > 0$. (To verify infinitude, one approach involves working with finite generically-etale maps to an affine space to essentially reduce the problem to the more familiar case of affine spaces of positive dimension.) But sometimes in char. 0 it is of finite type (such as algebraic varieties over $\mathbf{C}$ whose complex points are simply connected in the topological case; we'll come to some examples below).

Let's call a connected scheme $S$ simply connected if it has no nontrivial connected finite etale covers; e.g, $\widetilde{X}$ above (see, no noetherian hypotheses). Now there arises a question (inspired by the topological case): is $\widetilde{X} \times_{{\rm{Spec}}(k)} \widetilde{X}$ simply connected (assuming it is at least connected, which is automatic when $k$ is separably closed)? This amounts to asking if the natural map $\pi_1(X \times X) \rightarrow \pi_1(X) \times \pi_1(X)$ is an isomorphism (with $X \times X$ connected). The Artin-Schreier method shows that this fails in char. $> 0$ when $\dim X > 0$, even if $k = k_s$. But if $k$ is alg. closed of char. 0 then it is true. (Here is a sketch of a proof. The content is to show that a cofinal system of connected finite etale covers of $X \times X$ is given by products of such covers of the factors, and then group theory handles the rest. By specialization arguments typically called "Lefschetz principle", we can assume $k = \mathbf{C}$. Then the known result on the topological side reduces the task to checking that $E \rightsquigarrow E(\mathbf{C})$ sets up an equivalence from the category of finite \'etale covers of $X$ to the category of finite-degree covering spaces over $X(\mathbf{C})$. This is the so-called Riemann Existence Theorem, and is proved in section 5 of Exp. XII of SGA1 via resolution of singularities; can also be proved via alterations. Maybe there is an more elementary algebraic proof of the product compatibility by exploiting tame ramification in char. 0, but if so then it is escaping my memory at the moment.)

So when $k$ is alg. closed of char. 0, if $G$ is a smooth connected affine $k$-group then by simple connectedness (and connectedness) of $\widetilde{G} \times \widetilde{G}$ we can copy the same argument with Lie groups to uniquely equip $\widetilde{G}$ with a $k$-group scheme structure over that of $G$ making a chosen $k$-rational base point on $\widetilde{G}$ over the identity of $G$ as the identity point and the covering map a $k$-homomorphism. Continuing with such $k$, the coordinate ring $\widetilde{A}$ of $\widetilde{G}$ is a Hopf algebra over $k$, yet it is constructed as a directed union of $k$-subalgebras that are finite etale over the coordinate ring $A$ of $G$. By a general fact from the land of Hopf algebras (proved in Waterhouse's book on affine group schemes, for example), $\widetilde{A}$ is a directed union of finite type $k$-subalgebras $A_i$ that are also Hopf subalgebras. Since $A$ is finite type over $k$, by considering only "big enough" $i$, we may assume that every $A_i$ contains $A$. But $A_i$ is finitely generated over $k$, hence over $A$, yet $\widetilde{A}$ is a directed union of finite \'etale $A$-algebras. Thus, each $A_i$ is contained in a finite $A$-algebra and hence is itself module-finite over $A$. In other words, $G_i := {\rm{Spec}}(A_i) \rightarrow G$ is an isogeny between smooth connected affine $k$-groups. This isogeny is necessarily etale (as we're in char. 0), and hence the kernel is central by connectedness of the $k$-groups, and each $G_i$ is necessary semisimple when $G$ is.

To summarize, if $k$ is alg. closed of char. 0 and $G$ is connected semisimple, then $\widetilde{G}$ is an inverse limit of connected semisimple $k$-groups equipped with an etale isogeny over $G$. But by the theory of connected semisimple groups over general fields, the collection of central isogenous covers of $G$ has a single maximal element that dominates all others (called the simply connected member of the central isogeny class, and characterized by the property that it admits no non-trivial central isogenous covers by another smooth connected semisimple $k$-group; more on this dude below). Voila, so for $k$ alg. closed of char. 0 the collection of $G_i$'s is actually finite and terminates at $\widetilde{G}$. That is, for such $k$ the "abstract" $\widetilde{G}$ coincides with the "simply connected" central cover of $G$ in the sense of algebraic groups, so we conclude that the etale fundamental group is actually finite and coincides with the Cartier dual of the algebraic fundamental group (as the latter is by definition of the Cartier dual of the kernel of the central isogeny from the simply connected central cover; more on this over general fields below). In particular, $G$ is simply connected in the sense of algebraic groups if and only if it is simply connected as a scheme. In the special case $k = \mathbf{C}$, we recover the fact that a connected semisimple $\mathbf{C}$-group $G$ is simply connected in the sense of algebraic groups if and only if $G$ is simply connected as a scheme, and (by the Riemann Existence Theorem) also if and only if $G(\mathbf{C})$ is simply connected in the sense of topology. This latter "if and only if" rests on the fact that when $G(\mathbf{C})$ is not simply connected then it has a nontrivial connected cover of finite degree, which is a consequence of the topological fundamental group being commutative and (as for any algebraic variety over $\mathbf{C}$) finitely generated.

Meanwhile, as indicated above with Artin-Schreier coverings (with details left to the interested reader), in characteristic $p > 0$ (say assuming $k = k_s$) the etale fundamental group of a positive-dimensional smooth affine $k$-scheme is always infinite. But the etale fundamental group is an entirely different creature from the algebraic fundamental group over such $k$, as is most easily seen by noting that ${\rm{PGL}}_p$ is not simply connected in the sense of algebraic groups due to the non-etale central isogeny ${\rm{SL}}_p \rightarrow {\rm{PGL}}_p$ of degree $p$.

Finally, let's address the characteristic-free theory of the "simply connected central cover" for connected semisimple groups over any field, and the related notion of "algebraic fundamental group". A connected semisimple group $G$ over a field $k$ is simply connected if any central $k$-isogeny $f:G' \rightarrow G$ from a connected semisimple $k$-group is necessarily an isomorphism. (By "central isogeny" I mean that the scheme-theoretic kernel of $f$ is contained in the scheme-theoretic center of $G'$; see Definition A.1.10 and preceding discussion in "Pseudo-reductive groups".) Since every maximal $k$-torus in a connected semisimple $k$-group is its own scheme-theoretic centralizer, the finite scheme-theoretic center of such $k$-groups is contained in a $k$-torus and hence is of multiplicative type. Together with properties of "multiplicative type" groups under central extensions, this underlies the fact that a composition of central isogenies between connected semisimple groups is again central: the crux is that even when the kernels are not etale, their automorphism schemes are always etale. (Beyond the connected reductive setting, over any field $k$ of char. $p > 0$ there exists a pair of central $k$-isogenies $G \rightarrow G'$ and $G' \rightarrow G''$ whose composition has kernel that is not central in $G$. For example, over $\mathbf{F}_p$ let $G$ and $G''$ be the standard upper-triangular unipotent subgroup of ${\rm{SL}}_3$, whose scheme-theoretic center is the upper-right $\mathbf{G}_a$. Take $G \rightarrow G''$ to be the Frobenius homomorphism, and take $G'$ to be the intermediate quotient of $G$ by the unique central $\alpha_p$ from the upper-right entry.)

Then the real theorem is the existence and uniqueness (up to unique isomorphism) of a simply connected central cover of any connected semisimple $k$-group, and the compatibility of its formation with respect to any extension of the base field. By Galois descent, the strong uniqueness requirements reduce the proof of this assertion to the case $k = k_s$, so all connected semisimple $k$-groups are split. Hence, we can appeal to the Existence and Isomorphism/Isogeny Theorems with root data to conclude. (For further discussion, see Corollary A.4.11 in "Pseudo-reductive groups" and back-references in its proof.)

If $G$ is a connected semisimple group over a field $k$ and $\pi:\widetilde{G} \rightarrow G$ is its simply connected central cover in the sense of algebraic groups, then ${\rm{ker}}(\pi)$ is a finite $k$-group scheme of multiplicative type (since it is central in the connected semisimple $\widetilde{G}$) and hence its Cartier dual is a commutative finite \'etale $k$-group. That is called the algebraic fundamental group $\pi_1(G)$ in the sense of algebraic groups. (So by definition, the algebraic fundamental group is trivial if and only if $G$ is simply connected in the sense of algebraic groups.) As we saw above, if $k$ is alg. closed of char. 0 then this is "dual" to the etale fundamental group of the variety $G$, and in characteristic $p > 0$ the example $G = {\rm{PGL}}_p$ shows that it is really quite unrelated to the usual etale fundamental group in the sense of schemes (even when $k = k_s$).

Jim, what were you saying about being exhausted? :)

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Bravo ! Infatigable ! – Chandan Singh Dalawat Dec 14 '10 at 4:30
Merci beaucoup, Brian! – shenghao Dec 16 '10 at 18:23
Je t'en prie, shenghao. – BCnrd Dec 17 '10 at 19:55

One reference: The section "The algebraic fundamental group of a reductive group", in "Abelian Galois Cohomology of Reductive Groups", by Mikhail Borovoi. This has exactly what you're looking for, I think.

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This reference is written in characteristic 0 (even though some of the constructions work more generally, with suitable formulation). It is unclear if shenghao has a problem with that (though knowing him, I guess he prefers something applicable in all characteristics). – BCnrd Dec 13 '10 at 23:29
@Marty: Besides seconding Brian's comment, I'd urge you (and others) to include more explicit references or links to papers. In this case I seem to have reviewed the work in question: MR1401491 (98j:20061) 20G10 (11E72 14E20 18G50), Borovoi, Mikhail (IL-TLAV), Abelian Galois cohomology of reductive groups. Mem. Amer. Math. Soc. 132 (1998), no. 626, viii+50 pp. – Jim Humphreys Dec 15 '10 at 14:41
It's even online: math.tau.ac.il/~borovoi/papers/galofile.pdf – Dylan Moreland Dec 15 '10 at 19:52
@Jim: I'll try to be more explicit in my references. I was just sending out a very quick reference, in response to the OP's "I am not sure of its definition" comment, and due to the fact that Borovoi's paper is so easy to find with Google. But I suppose a proper reference is most professional. – Marty Dec 15 '10 at 20:16
Thanks to you all for the references. In fact, I was looking for the definition of the "algebraic fundamental group" because it is involved in Kisin's result on integral models of Shimura varieties, where the group is defined over a number field. So I guess char. 0 is fine. But Brian's discussion on char. p is quite helpful, and I am grateful to him for giving such a complete answer. – shenghao Dec 16 '10 at 18:16

In addition to Marty's reference, I would recommend to look into §6 "Le groupe fondamental algébrique des groupes algébriques linéaires connexes via les resolutions flasques" of Colliot-Thélène's paper MR2404747, Colliot-Thélène, Jean-Louis, Résolutions flasques des groupes linéaires connexes, J. Reine Angew. Math. 618 (2008), 77–133. This reference is in any characteristic.

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