Étale Covers and Fundamental Groups of Schematic Finite Spaces

We introduce the category of finite étale covers of an arbitrary schematic space X and show that, equipped with an appropriate natural fiber functor, it is a Galois Category. This allows us to define the étale fundamental group of schematic spaces. If X is a finite model of a scheme S, we show that the resulting Galois theory on X coincides with the classical theory of finite étale covers on S, and therefore, we recover the classical étale fundamental group introduced by Grothendieck. To prove these results, it is crucial to find a suitable geometric notion of connectedness for schematic spaces and also to study their geometric points. We achieve these goals by means of the strong cohomological constraints enjoyed by schematic spaces.


Introduction
Schematic (finite) spaces are finite ringed spaces (not locally ringed in most cases of interest) admitting a «good »theory of quasi-coherent sheaves with minimal natural conditions. They were first constructed in [7]. These spaces can be used to study the category of quasi-compact and quasi-separated (qcqs) schemes via the construction of «finite models»: that is, a projection π : S → X from a scheme S to a schematic space X inducing an equivalence of quasi-coherent sheaves. The scheme S is reconstructed from X via a colimit that we denote Spec(X) and, actually, qcqs schemes are embedded into a suitable localization-by qc-isomorphisms (defined before Proposition 2.4)-of the category of schematic spaces SchFin. However, SchFin contains Proof. The «only if»part is proven in [7,Theorem 5.26]. For the converse: from the equivalence of categories, we have To prove that f −1 (U y ) is affine for every y ∈ Y , we have to check that taking global sections induces an equivalence of categories. This follows from the fact that U y is affine, that O Y,y O X (f −1 (U y )) by the previous argument, and that the equivalence of the hypothesis restricts to Qcoh(f −1 (U y )) Qcoh(U y ) by the extension theorem for quasi-coherent sheaves [7,Theorem 4.4].

Connectedness
Topological-combinatorial-connectedness of a schematic space X does not reflect the geometry of the locally ringed space Spec(X) that it represents. We enhance it in this section.
Definition 3.1. Let X be a schematic space. We say that • X is top-connected if its underlying poset is connected. • X is well-connected if it is connected and pw-connected.
Recall that the prime spectrum of a non-zero ring is connected if and only if the ring has no non-trivial idempotent elements. A schematic space X is connected if and only if Spec(X) is connected in the usual sense, since X and Spec(X) have the same global sections. The empty set ∅ is not considered connected. In particular, a pw-connected space has non-zero stalks.  Let π 0 : Top → Set denote the connected components functor. For any finite ringed space X, we construct a pw-connected space as follows: • As a set, pw(X) = x∈X π 0 (Spec(O X,x )). This set comes with a natural projection π : pw(X) → X. • pw(X) is endowed with the partial order defined as follows: for every x ≤ y in X, let ϕ xy : π 0 (Spec(O X,y )) → π 0 (Spec(O X,x )) denote the map induced by the restriction morphism. Now, for α, β ∈ pw(X) with π(α) = x and π(β) = y, we define α ≤ β ⇐⇒ x ≤ y and ϕ xy (β) = α.
For every x ∈ X, consider π −1 (x) = {α 1 , . . . , α n } = π 0 (Spec(O X,x )). Each connected component is affine (connected components of a locally Noetherian topological space are open), so α i corresponds to a non-zero ring with connected spectrum: Furthermore, for each x ≤ y, since the continuous image of a connected topological space is connected, the morphism ϕ xy induces a unique ring homomorphism A αi x → A βj y for every β j ∈ π −1 (y) and α i = ϕ xy (β j ). • We endow pw(X) with the sheaf of rings that for all α i with π(α i ) = x O pw(X),αi = A αi x , and, for every α i ≤ β j with π(α i ) = x and π(β j ) = y, we define the restriction morphism as the ring map A αi x → A βj y described above. • The projection π : pw(X) → X becomes a morphism of (finite) ringed spaces with π # αi : O X,x → A αi x being the natural projection for all α i with π(α i ) = x. If X is pw-connected, π is clearly the identity.

MJOM
This construction is functorial: given a morphism f : X → Y , the maps for every x ∈ X. Their disjoint union defines a continuous map pw(f ): pw(X) → pw(X ), such that the following diagram is commutative: We turn this into a commutative diagram of ringed spaces by endowing pw(f ) with the morphism of sheaves of rings, such that, with the previous notation, is the unique ring homomorphism defined by the same connectedness argument as the restriction morphism.
To prove that π is schematic it suffices to see that Rπ * M is quasicoherent for any quasi-coherent module M [7, Theorem 5.6], i.e., that for all is an isomorphism for all i. Since quasi-coherent modules on the affine space ,α and, as before, the only non-zero components of this tensor product are those with α ≥ α j .
Proof. The proof is routinary and similar to the last part of the previous one. If π X and π Y are the corresponding natural projections, we have to see that

Now, as in Proposition 3.4, one relates the sections of the standard resolution
Finally, the proof concludes using the hypothesis that f is schematic and the decomposition of the stalk rings of O X and O Y at each point. Explicit computations are left to the dedicated reader.
Let SchFin pw denote the subcategory of pw-connected schematic spaces. We have seen that there is a functor pw : SchFin → SchFin pw .
Theorem 3.6. The functor pw : SchFin → SchFin pw is right adjoint to i : SchFin pw → SchFin. The map pw(X) → X is a qc-isomorphism for all X and verifies pw • i = Id. In particular, SchFin pw qc SchFin qc .
Proof. It only remains to check the adjunction. If Y is pw-connected and X arbitrary, there is a bijection Hom SchFin (Y, X) Hom SchFin pw (Y, pw(X)).
Indeed, given f : Y → X, we apply pw and obtain Y = pw(Y ) → X. Conversely, any g : Y → pw(X) induces π X •g : Y → X, where π X : pw(X) → X is the natural projection. Proof. It follows from the definition of pw, Proposition 3.3 and the fact that qc-isomorphisms preserve global sections.
The topological connected components of pw(X) are called the well-connected components of X. We denote by π wc 0 (X):=π 0 (pw(X)) the set of well-connected components of X.
We can adapt this construction for sheaves of quasi-coherent algebras: let A be a quasi-coherent O X -algebra and (X, A) → X the corresponding schematic space, which is affine over X. The same argument gives us a decomposition A k A k . Definition 3.9. (Well-connected components of a sheaf of algebras) The algebras A k just introduced are called the well-connected components of A.
By the hypothesis, only one of them is nonzero.

Geometric Points
We introduce geometric points of schematic spaces. In this paper, we employ a straightforward analogy with Scheme Theory rather than the topos-theoretic approach. From now on, we consider spaces over ( , k) with k a field. The functor of points of a schematic space X is its image It is a simply consequence of the definition of schematic morphism for this particular case. The only non-trivial condition is that Ω ⊗ OX,x O X,x = 0 for every x < x . In other words, that the fiber of the morphism of schemes x ) is empty. In terms of rings, this is exactly the «lifting»condition of the proposition.  , p), where x ∈ X and p ⊆ O X,x is a prime ideal. For every such pair, we define its residue field κ(x, p) as (4.1) X is schematic and (x, p), (x , p ) are pairs, such that x ≤ x and p = r −1 xx (p ), then the natural map κ(x, p) → κ(x , p ) is an isomorphism.
We define the following binary relation in the set of these pairs: We prove that it is the equivalence relation realizing Spec(X) as a quotient set of x∈X Spec(O X,x ). This relies on X being schematic.
Lemma 4.4. For any (x, p) and (y, q), such that x, y ≥ s for some s ∈ X and by Proposition 4.3, the underlying set of this scheme-theoretic fiber product is the set-theoretic fiber product, which contradicts r −1 sx (p) = r −1 sx (q). Next, we see that U x ∩ U y ⊆ U s is affine: by [7,Corollary 4.11], it suffices to prove that it is acyclic ( is surjective, so there is a point z ≥ x, y and a prime r ∈ Spec(O X,z ), such that R (r) = (p, q). The pair (z, r) verifies (4.2).      κ(x, p). Conversely, for every schematic point (x, p), we define ( , κ((s, p))) → X as → x at the level of sets and as the natural map O X,x → κ(x, p) at the level of rings; thus, composing with ( , Ω) → ( , κ(x, p)), we obtain the desired morphism. The second part follows from the fact that every schematic point (x, p) has a unique maximal representative, so the morphism of ringed spaces constructed using this representative is a geometric point according to Proposition 4.2.
We see that schematic points are analogous to points in the sense of schemes. Let S be a qcqs scheme and π : S → X a finite model. Proof. Let s ∈ S be a point and p s ∈ Spec(O S (U s )) the ideal it defines as an element of the affine open U s . Then, (π(s), p s ) is a schematic point of X. Conversely, given a schematic point (x, p x ) of X, we have that p x ⊆ O X (U x ) = O S (U s ) for any s ∈ π −1 (x). This ideal determines a point of S independent of the chosen representative of (x, p x ). These correspondences are mutually inverse. From the definitions, it also follows that κ(s) = κ(π(s), p s ).
Since Spec(Ω) = ( , Ω) in LRS, the second bijection is explicitly given by composition with S → X, with inverse defined by the Spec functor.

The Category of Finite Étale Covers
Étale morphisms and ring maps are of finite presentation, and hence not wellsuited to work with general schematic spaces if one intends to obtain results that mimic scheme theory (this is because the notion is not stable under qcisomorphisms, so functors will not factor through SchFin qc ). There are two initial ways of avoiding this problem: working with weakly étale morphisms and defining the pro-étale topology (à la Scholze and Bhatt [1]), or exploiting the fact that finite étale covers are affine and thus can be interpreted as sheaves of algebras, which will behave well, because the restriction maps of our spaces are local isomorphisms. We proceed with the second approach. It is well known that the fibers of an étale morphism f are disjoint unions of spectra of finite separable field extensions of the residue field (which actually characterizes étale morphisms among all flat morphisms that are of finite presentation). Clearly, if f has compact fibers, these disjoint unions are finite. In particular, finite morphisms of schemes have compact fibers.
Let S be a scheme and denote by Fet S the category of finite and étale morphisms to S, also known as the category of finite étale covers of S, which is a full and faithful subcategory of Schemes /S ; and let Qcoh alg (S) denote the category of quasi-coherent O S -algebras (we shall use this notation for any ringed space S with its natural topology). The classical relative spectrum functor Spec S : Qcoh alg (S) → (Schemes Affine /S ) op induces an isomorphism of categories between Qcoh alg (S) and the opposite category of affine morphisms to S, whose inverse is given by

. Let S be a scheme. A quasi-coherent algebra A lies in Qcoh fet (S) iff it is finite, flat and A s ⊗ OS,s κ(s) is a finite étale κ(s)-algebra for every s ∈ S. If this holds, then for every affine U ⊆ S, O S (U ) → A(U ) is étale.
Remark 5.2. It is well known that Qcoh fet (S) Fet op S Loc(S et ), where the latter is the category of finite, locally constant sheaves of sets on the small étale site of S (which may also be considered as sheaves of Z-modules).
We will partially bring the last point of view back in Sect. 5.2.
Definition 5.5. Let X be a schematic space. A ∈ Qcoh alg (X) is an étale cover sheaf (or simply an étale cover ) if it is finite, flat, and for each schematic By Remark 2.2, this notion is well-defined. Since we are assuming quasicoherence, finiteness is a condition at stalks (c.f. [7, Theorem 2.8]).
Given a schematic space X, let Qcoh fet (X) denote the subcategory of Qcoh alg (X) whose objects are étale cover sheaves.
Proof. It follows from Proposition 4.8, Lemma 5.4, the definition of étale cover sheaf and a straightforward computation at stalks.

Finite Locally Free Sheaves on Schematic Finite Spaces
Let A → B be a morphism of rings, such that B is finitely generated and locally free over A. Once again, if A is Noetherian, this is equivalent to B being finite and flat. The degree (or rank) of A → B is classically defined to be the map deg(B) : Spec(A) → Z + with p → rank Ap B p . It is continuous and locally constant, and hence can be identified with a non-negative integer if Spec(A) is connected. We wish to extend this notion to schematic spaces. Now, let X be a schematic A ∈ Qcoh alg (X). We know that A x → A y (x ≤ y) are flat epimorphisms of rings, hence local isomorphisms. We leave Lemma 5.7 below as an algebra exercise, which states that they are also local isomorphisms with respect to localization at primes of O X,y .

Étale Cover Sheaves Are Locally Trivial
We use ideas of Lenstra [10,Proposition 5.2.9,p. 155] as a guide, proceeding fiberwise with an eye on connectedness and quasi-coherence. Definition 5.9. Let X be schematic. We say that B ∈ Qcoh alg (X) is a «covering»if it is finite and faithfully flat (O X,x → B x faithfully flat for all x ∈ X). Equipped with this family of coverings, Qcoh alg (X) becomes a cosite X fppf Qcoh .
Remark 5.3. Cosite means that the opposite category is a site and X fppf Qcoh is the analogue of the fppf site of (Noetherian) schemes.
Let X be schematic and O 1 , . . . , O k the well-connected components of O X (Definitions 3.8, 3.9). Given a finite set F , consider the constant functor: F : X fppf Qcoh → Set (with A → F ) and its (co)sheafification, denoted F # , which sends A to π wc 0 (X,A) F , where π wc 0 (X, A) is the set of well-connected components (Definition 3.8) of (X, A). Just as in the case of constant covers in ordinary theories, it follows that F # is representable (in the sense of the covariant functor of points) by 1≤j≤k O ×n X , which is called a constant object of cardinality F (or degree n = #F ).
· · · X → X. Proof. The «if»part is trivial. For the converse, recall that well-connectedness implies top-connectedness and pw-connectedness (Proposition 3.3). Using the notation of Definition 5.10, assume that n = 2, so pw(X, B) has two connected components (for n > 2, the idea is identical, but the argument is longer). We denote by B 1 and B 2 the corresponding connected components of B (Definition 3.9). We have to prove that n 1 = n 2 . Notice that, since B is faithfully flat and X is pw-connected, (X, B) has non-zero stalks.
Claim 1 There exists some x ∈ X, such that both B 1x and B 2x are non-zero. Proof of Claim 1. Consider π : B → X for i = 1, 2 and that π is surjective, because B has nonzero stalks. Let us prove that it is injective. Assume that either B 1x = 0 or B 2x = 0 for all x ∈ X. Note that both cannot be zero at the same time, because B x = 0. In this case π −1 (x) has exactly one element for all x ∈ X by construction, so π is injective and thus a homeomorphism. In particular, X would be homeomorphic to a disjoint union of non-empty topological spaces, contradicting the top-connectedness of X. We conclude that there is some x ∈ X, such that both B 1x , B 2x = 0.
Proof of Claim 2. Let f x : O X,x → B 1x × B 2x denote the natural faithfully flat structure morphism and let f ix : O X,x → B ix (for i = 1, 2) denote its composition with the natural projections. We have a surjective morphism with closed and open image for i = 1, 2. By Claim 1, there exists some contradicting the pw-connectedness of X.
from which it follows that n 1 = n 2 , which proves the lemma. Proposition 5.13. (Local triviality of étale cover sheaves) Let X be well-connected and schematic, and then, A ∈ Qcoh alg (X) is a degree n étale cover sheaf iff it is finite locally constant (of cardinality n) as an object of X fppf Qcoh . Proof. Assume A ⊗ OX B B ×n for some n > 0 (otherwise A = 0) and some faithfully flat algebra B ∈ Qcoh alg (X) (Lemma 5.11). Note that this also implies that A x = 0 for all x ∈ X.
Since for every x ∈ X, O X,x → B x is finite locally free and finiteness is a local condition, we can assume that it is finite and free, B x O ⊕m X,x . Since A x ⊗ OX,x B x is a finite O X,x module and it is also a finite A x -module, A x is a finite O X,x -module for all x ∈ X. Finally, for a schematic point (x, p), Conversely, if A = 0 is finite étale, it is locally trivial of constant positive degree n. By induction over n: if n = 1, A O X and there is nothing to say. In general, if I = ker(A ⊗ OX A → A), which is a quasi-coherent sheaf of ideals (Qcoh(X) is abelian) both as a sheaf of O X -modules and of Amodules; I/I 2 is the sheaf of relative differentials of O X → A, which is trivial (at stalks) by étaleness. Since A x is finite and finitely presented, I x is finitely generated, and hence, Lemma 5.12 applies and I x = (e x ) for some non-trivial idempotent e x ∈ O X,x and A x ⊗ OX,x A x A x × C x for some ring C x . Since both the natural morphism A x → A x ⊗ OX,x A x and A x × C x → C x are finite étale, C x is a finite étale A x -algebra for all x ∈ X.
We sheafify the situation: define a sheaf of ideals J by J x :=(1 − e x ) and setting, for every x ≤ y, J x → J y to be the obvious map induced by I x → I y (note that A = I ⊕J as O X -modules). Since I is quasi-coherent, J is readily seen to be quasi-coherent. Now, we consider the quotient C:=(A ⊗ OX A)/J , which is quasi-coherent and verifies C x = C x , so it is an étale cover sheaf.
By construction, C (since X is well-connected) has constant degree n−1 as a finite locally free A-algebra (and n 2 − n as an O X -algebra). Consider the qc-isomorphism pw(X, A) → (X, A) and denote by C the pullback of C, which also has constant degree n − 1. By induction, there is a faithfully flat finite algebra B in Qcoh(pw(X, A)) with C ⊗ O pw(X,A) B B ×n−1 . Let B be its push-forward to (X, A), which verifies C ⊗ A B B ×n−1 . Finally, Proof. Pick x ∈ X and consider f x : A x → B x . Since O X,x is connected (it has no non-trivial idempotents), it is an exercise of basic algebra to check that f x is indeed induced by a morphism φ : F → E [5, ex. 5.11(d)]. This pointwise argument is compatible with restrictions maps.

Stability by qc-Isomorphisms
Let X be schematic. This section will reduce our study to SchFin pw .