h-TOPOLOGIES OLIVER E. ANDERSON

Abstract. The aim of these notes is to give precise proofs of refinement results for the h, qf h and cdh-topologies. We also take a brief look at representable sheaves in the h-topologies.

Contents Acknowledgements 1. h-topology 1.1. (Universal) Topological epimorphisms 1.2. The h-topology 1.3. Refinements of qf h-coverings 1.4. Refinements of h-coverings 1.5. The cdh-topology 1.6. Refinements of cdh-coverings 2. Representable sheaves 2.1. Morphisms inducing isomorphisms of representable sheaves 2.2. Sections of representable sheaves at fields 2.3. Cofiltered limits and representable sheaves A. Grothendieck pretopologies and sheaves A.1. Grothendieck pretopologies A.2. Sheaves A.3. Sheafification References Index

1 1 1 2 4 5 6 7 9 10 12 14 16 16 17 17 18 19

Acknowledgements The author would like to thank his advisor Vladimir Guletski˘ı for introducing him to the h-topologies and for pointing out good litterature on this topic. He would also like to thank Paul Arne Østvær for directing him to the paper [Ryd07] which has been useful to the author in many ways. There are also many more people who the the author would like to mention that have been kind enough to discuss and answer questions of the author related to this topic. From the top of my head they are: Anwar Alameddin, Benjamin Antieau, Stefano Nicotra, Adeel A. Khan, Jason Van Zelm, Martin Gonzalez, Remy van Dobben de Bruyn, Jonas Irgens Kylling and Jarle Stavnes. 1. h-topology 1.1. (Universal) Topological epimorphisms. Definition 1.1. A morphism of schemes p : X → Y is called a topological epimorphism if the underlying topological space of Y is a quotient space of the underlying topological space of X. That is , p is surjective and a subset U of Y is open if and only if p−1 (U ) is open in X. Date: December 9, 2017. 1

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A topological epimorphism p : X → Y is called a universal topological epimorphism if for any morphism f : Z → Y the projection Z ×Y X → Z is a topological epimorphism. Example 1.2. (1) Using only point-set topology, we see that any open or closed surjective morphism is a topological epimorphism. (2) Since any flat morphism locally of finite presentation is universally open (See [Stacks, Tag 01UA]) and since surjectivity is stable under base change, we see that surjective flat morphisms locally of finite type are universal topological epimorphisms (This is also the case for surjective quasi-compact flat morphisms, see [Stacks, Tag 02JY]). (3) Proper morphisms are by definition universally closed, hence surjective proper morphisms are universal topological epimorphisms. Lemma 1.3. Any composition of (universal) topological epimorphisms is a (universal ) topological epimorphism. Proof. The composition of quotient maps (in topology) is a quotient map, hence a composition of topological epimorphisms is a topological epimorphism. Now use this fact together with the fact that towers of fiber diagrams are fiber diagrams to see that a composition of universal topological epimorphisms is a universal topological epimorphism.  1.2. The h-topology. Definition 1.4. The h-pre-topology is the Grothendieck pre-topology on the category of schemes with coverings of the form {pi : ` Ui → X} where {pi } is a finite family of morphisms of finite type such that the induced morphism Ui → X is a universal topological epimorphism. If we in addition require the pi to be quasi-finite then we get the coverings of the qf h-pretopology. The h-topology (resp. qf h-topology) is the Grothendieck topology associated to the h-pre-topology (respectively the qf h-pre-topology). Example 1.5. (1) It follows immediately from the definitions that a coproduct of flat morphisms is flat, hence any flat covering (of finite type) is an h-covering. Moreover if a scheme S is quasicompact and quasi-separated then it follows easily from Theorem 1.16 that a flat covering of S is a covering in the saturation of the qf h-pretopology (See Definition A.4). (2) Since open embeddings are flat, it follows from (1) that open finite coverings are h-coverings, hence on the category of Noetherian schemes the Zariski-topology is coarser than the htopology. (3) Any surjective proper morphism is an h-covering. (4) Since the fibered product of a coproduct with a scheme over some other scheme, is the coproduct of the fibered products, we see that if {fi : Ui → X}i∈I is a jointly surjective family of proper morphisms, then this is an h-covering. In particular joinly surjective families of closed embeddings amd finite morphisms are h-coverings. s (5) Consider the affine plane A2k . Let C denote the x−axis C = V (y) and let U be the complement A2k \ C = D(y) the canonical morphism a π:U C → A2k is clearly surjective, however π −1 (C) is open while C is not open, hence π is not an h-covering. (6) Voevodsky gives an example in [Voe96] of a surjective morphism which is not an h-covering by blowing-up a surface at a closed point x and removing a closed point from the preimage of this point, and so we get a morphism p : U → X which can be shown to be surjective but not a topological epimorphism. Example 1.6. Consider the two k-algebra maps φ1 , φ2 : k[t] → k[t]/(t2 ) where the first is the canonical quotient and φ2 is given by t 7→ 0.When we compose either of these two maps with the canonical map q : k[t]/(t2 ) → k we get the map k[t] → k given by t 7→ 0. This shows that HomSch /k (Spec(k[t]/(t2 )), A1k ) → HomSch /k (Spec(k), A1k )

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is not injective as we have Spec(φ1 ) ◦ Spec(q) = Spec(φ2 ) ◦ Spec(q). Note however that the closed embedding Spec(q) : Spec(k) → Spec(k[t]/(t2 )) is a qf h-covering, thus the presheaf hA1 is not a sheaf k in the qf h-topology. Which implies that both the qf h and the h topologies are not subcanonical. Part (5) of 1.5 is a consequence of the following more general statement: pi

Proposition 1.7. of a Noetherian scheme ` [Voe96, Proposition 3.1.3] Let {Ui → X} be an h-covering ` X. Denote by j Vj the disjoint union of irreducible components of Ui such that for any j there ` exists an irreducible component Xi of X which is dominated by Vj . Then the morphism q : Vj → X is surjective. Proof. Suppose first that X is irreducible. Let x ∈ X be a point of X. We want to prove that x lies in the image of q. By considering the base change along the natural morphism Spec(OX,x ) → X, we may suppose that X is the Spectrum of a local ring (note that since there is a bijection between irreducible components between Spec(OX,x ) and irreducible components of X, we get that the spectrum of our local ring is irreducible by our assumption that X is irreducible) and x is the closed ` point of X. Denote by Z the closure of the image Z 0 of those irreducible components of Ui which are not 0 is a constructible subset of X by Chevalley’s theorem, we have that dominant over X. Since Z ` Z 0 = k U 0 k ∩ Ck where U 0 k are open and Ck are closed. Hence we have that [ U 0 k ∩ Ck ⊂ ∪Ck Z = Z0 = k

and since

Z0

does not contain the generic point, none of the Ck contain the generic point, thus Z ( X.

From [GD64, (10.5.5) (i)] we have that the set of points x in X such that {x} is finite, is a dense subset X0 of X. Note that since we are working in the spectrum of a local ring it follows that the closure of one-dimensional points consists of two points, hence the set of one dimensional points is contained in X0 . Now if q is in X0 , then we have that V (q) is finite and by [GD64, (10.5.3)] this implies that V (q) the dimension of V (q) is less than or equal to one, hence the set X0 is exactly the set of one dimensional points and the closed point, and so we have that the set of one-dimensional points of X is dense in X. Therefore there exists a one-dimensional point y which does not belong to Z. If x does not lie in the image of q then we have q −1 (V (y)) = q −1 ({y} ∪ {x}) = q −1 ({y}) and so q −1 ({y}) is closed which implies that p−1 i ({y}) is closed as well but {y} is not closed in X, giving us a contradiction that {pi } is an h-covering. Suppose now that X is an arbitrary scheme and let Xred = ∪Xk be the decomposition of the maximal reduced subscheme of X into the union of its irreducible components. Consider the natural morphism Xk → X and let {Ui ×X Xk → Xk } be the preimages ` ` of our h-covering. Then the morphisms Vj,k → Xk , where Vj,k are the X X ` irreducible components of` Ui ×` `k which are dominant over Xk are surjective, implying that Vj → X is surjective since Vj = Vj,k .  Corollary 1.8. Let Z be a closed subscheme of an integral scheme X and b : XZ → X the blow up with center Z. Suppose that for any open subscheme U ,→ XZ , the composition U → XZ → X is an h-covering. Then U = XZ . Proof. Let us consider the base change along the projection XZ → X, XZ ×X XZ . Now since b is a birational morphism we have some open subset V of X such that b−1 (V ) → V is an isomorphism and b−1 (V ) ∼ = b−1 (V ) ×X b−1 (V ) ,→ XZ ×X XZ . Thus the closure of b−1 (V ) ×X b−1 (V ) in XZ ×X XZ denoted by T , is the unique irreducible component of XZ ×X XZ dominating XZ . Now we claim that T is contained in the diagonal ∆. To see this, just note that the diagonal morphism ∆XZ /X restricted to b−1 (V ) yields the isomorphism b−1 (V ) ∼ = b−1 (V ) ×X b−1 (V ), hence the diagonal is a closed subscheme containing T . Now from this it follows from Proposition 1.7 that (U ×X XZ )∩∆ → XZ is a surjection, and so we must have U = XZ .  In the case where X is a normal connected Noetherian scheme Proposition 1.7 has a converse.

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Proposition 1.9. [Voe96, Prop. 3.1.4] Let {pi : Ui → X} be a finite family of quasi-finite morphisms over a normal connected Noetherian scheme X. Then {pi } is a qfh-covering ` if and only if the subfamily {qj } consisting of those pi which are dominant over X is such that qj is surjective. In that case {qj } is also a qfh-covering of X. Proof. The ”only if” part follows immediately from Proposition 1.7 . The ”if” part follows because in the case of a normal Notherian connected (hence irreducible ) scheme X, it is explained in [Gro71, p.24] that a dominant quasi-finite morphism is universally open.  1.3. Refinements of qf h-coverings. As one might expect, Zariski’s main theorem can be used to give neat refinements of qf h-coverings. Lemma 1.10. [SV96, Lemma 10.3] Let X be a normal connected quasi-compact and quasi-separated scheme such that the normalization of X in a finite field extension L of k(X) is a finite morphism1 Then any qf h-covering of X admits a refinement of the form {Vi → V → X}i∈I where V is the normalization of X in a finite normal extension of its field of functions and {Vi → V } is a Zariski open covering of V . Proof. Let {pj : Uj → X}j∈J be a qfh-covering. Replacing each Uj by its irreducible components, we may assume that all Uj are integral and by replacing each Uj by an appropriate open covering we may also assume that the morphism pj are separated. Using Proposition 1.9 we may assume that Uj dominates X for every j. According to Zariski’s main theorem ([Stacks, Tag 05K0], [GD64, Thm 8.12.6.I]) we have a factorization of pj of the form Uj

aj ◦

Uj

pj

X,

Where aj is open and pj finite, and by replacing Uj by the irreducible component containing the image of Uj we may assume that Uj is integral and pj is finite and surjective. Let E be the composite of the normal closures of the fields k(Uj ) over k(X). Let q : V → X be the normalization of X in E. For each j the morphism q factors through Uj ([Stacks, Tag 035I]) and we will denote by Vj the inverse image of Uj in V . Note that since we don’t know if the Vj form an open cover of V we cannot conclude the proof at this point. Let G = Gal(E/k(X)) and note that it acts on X-automorphisms of V in a canonical way (use for instance [Stacks, Tag 035I]). Consider the set I = J × G and for i = (j, σ) ∈ I set Vi = σ(Vj ). It follows from [Bou64, Ch.5, Sec. 2, n.3, Prop. 6] that G-acts transitively on the fibers of q, and since the pi are jointly surjective we then have that ∪i∈I Vi = V . Finally since the group G acts on X-automorphisms of V we have that the composition q

Vi = σ(Vj ) → V → X coincides with σ −1 |V

q

Vi → i Vj → V → X Thus {Vi → V → X} is a refinement of the original covering {Uj → X}j∈J .



Corollary 1.11. Let X be a reduced Noetherian Nagata scheme. Then any qf h-covering of X admits a refinement {Vk → V → X} where {Vk → V }k is an open covering of V and V → X is finite and surjective. Proof. Suppose {Uj → X}j∈J is a covering of X. The normalization of X which we denote Xnorm is a finite disjoint union of integral normal schemes (see [Stacks, Tag 035Q]) which we denote by Xi . The covering {Xi ×X Uj → Xi }j∈J is a qfh-covering of Xi and by Lemma 1.10 this covering admits a refinement {Vα(i) → Vi → Xi }α(i)∈A(i) 1this is in particular the case if X is of finite type over a field, or of finite type over the ring of integers in a number

field, or more generally if X is excellent as this is stated by Suslin and Voevodsky or even more generally if X is Nagata.

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where Vα(i) → Vi is an open embedding and Vi is the normalization of Xi in a finite normal extension of k(Xi ). Now let A = ∪i A(i), then the covering a a {Va → Vi → Xi = Xnorm → X}a∈A i

i

is a refinement of the original covering {Uj → X} of the desired form.



1.4. Refinements of h-coverings. For reasonably nice schemes h-covers also have neat refinements. Before we formulate and prove this we need to introduce basic notation for blow-ups and breifly recall the strict transform. Definition 1.12. Let Z be a closed subscheme of a scheme X. We denote by pZ : XZ → X the blowup of X with center in Z . For a scheme Y → X over X, denote by p˜Z (Y ) (or just Y˜ when p and Z are clear from the context) the scheme theoretic closure in Y ×X XZ of the open subscheme Y ×X XZ \ pr2−1 (p−1 ˜Z (Y ) is called the strict transform or proper transform of Y Z (Z)). The scheme p with respect to pZ . Lemma 1.13. Suppose that X is a scheme and f : Y → X be any scheme over X. For any closed subscheme Z of X let pZ : XZ → X be the blow-up of X with center Z. Then the proper transform p˜Z (Y )

/

Y ×X XZ

is an isomorphism over the open subscheme XZ \ p−1 Z (Z). Hence an isomorphism over all the generic points of XZ . Proof. Let i : Y ×X XZ \ pr2−1 (p−1 Z (Z)) → Y ×X XZ denote the open embedding. We have that the proper transform p˜Z (Y ) is cut out by a subsheaf of   Ker i# : OY ×X XZ → i∗ OY ×X XZ \pr−1 (p−1 (Z)) 2

Z

XZ \ p−1 Z (Z).

which obviously vanishes over The last statement because p−1 Z (Z) is an effective cartier divisor and thus contains no generic points of XZ .  pi

Definition 1.14. A finite family of morphisms {Ui → X} is called an h- covering of normal form if the morphisms pi admit a factorization of the form pi = s ◦ f ◦ ini , where {ini : Ui → U } is an open covering, f : U → XZ is a finite surjective morphism and s : XZ → X is the blowup of a closed subscheme in X. Theorem 1.15. [Voe96, Theorem 3.1.8] Platification by blowup. Let f : Y → X be a morphism of finite type, which is flat over an open subset U ⊂ X. Then there exists a closed subscheme Z ⊂ X disjoint with U such that the strict transform p˜Z (Y ) is flat over XZ . Proof. See [RG71, Sec. 5.2].



Theorem 1.16. Suppose that f : X → S is a faithfully flat morphism locally of finite presentation. Then there exists a morphism g : S 0 → S which is faithfully flat, locally of finite presentation and locally quasi-finite and an S-morphism S 0 → X. If S is quasi-compact (resp. quasi-compact and quasi-separated which in particular is the case if S is Noetherian), then S 0 can be taken to be an affine scheme (resp. S 0 can be taken to be affine and g can be taken to be quasi-finite). Proof. See [GD64, Corollaire 17.16.2].



Remark 1.17. (Fun implication) Hilbert’s Nullstellensatz may be stated as follows: For any finite type morphism X → Spec k where k is a field, there exists a finite field extension k 0 /k and a morphism of k-schemes Spec k 0 → X. As quasi-finite morphisms over a field are finite (by for example Chevalley’s theorem), we get the Nullstellensatz as a special case of Theorem 1.16. pi

Theorem 1.18. [Voe96, Theorem 3.1.9] Let {Ui → X} be an h-covering of a reduced Noetherian Nagata scheme X. Then there exists an h-covering of normal form, which is a refinement of {pi }.

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Proof. Suppose that {pj : Uj → X} is an h-covering ` of X. ` By [Stacks, Tag 052B] there is a dense open subscheme X0 of X such that the morphism j pj : Uj → X is flat over X0 (hence pj is also flat over X0 ). Now from Theorem 1.15 we can find a closed subscheme Z disjoint with X0 such that the strict transform ` ` f : p˜Z ( Uj ) XZ ×X ( Uj ) XZ /

is flat. Now consider the following qf h-covering of XZ : {f, C → XZ } ` ` where C denotes the closure of the complement of the strict transform p˜Z ( Uj ) in XZ ×X Uj . By Lemma 1.13 it follows that C lies over p−1 Z (Z) thus C does not dominate XZ , hence by Proposition 1.7 it follows that f is faithfully flat. We can now apply Theorem 1.16 to obtain a faithfully-flat quasi-finite morphism U 0 → XZ which factors through f . From Corollary 1.11 we get a refinement of U 0 → XZ of the form {Vk → V → XZ }k∈K . By letting Wj,k denote the preimage of Uj in Vk , we obtain an h-covering {Wj,k → V → XZ → X}(j,k)∈J×K of X of normal form which is a refinement of the original covering {Uj → X}.  1.5. The cdh-topology. Another non-subcanonical topology used in the work of Voevodsky is the cdh-topology. It is coarser than the h-topology and incomparable with the qf h-topology. Definition 1.19. A morphism of schemes f : Z → Y satisfies the Nisnevich condition if for every point y ∈ Y there is some point z ∈ Z lying over y such that the induced morphism of residue fields k(y) → k(z) is an isomorphism. Definition 1.20. Recall that the Nisnevich pre-topology on Sch /S is the Grothendieck-pretopology on Sch /S with coverings ´etale coverings {pi : Ui → X}i≤n such that the induced morphism a (pi ) Ui → X satisfies the Nisnevich condition. The cdh-topology on Sch /S is defined as the saturation (Definition A.4) of the minimal pretopology on Sch /S for which the following two families of morphisms are coverings: (1) Nisnevich coverings ` (p 0 ,pZ ) (2) Coverings of the form X 0 Z X→ X such that pX 0 is a proper morphism, pZ is a closed embedding and the induced morphism p−1 X 0 (X \ pZ (Z)) → X \ pZ (Z) is an isomorphism. Remark 1.21. Since coverings/maps of the form ((1)) and ((2)) of Definition 1.20 are preserved under pullback, it follows that the minimal Grothendieck pretopology in which (1) and (2) are coverings is the pretopology where coverings are finitely many composites (See [Vis04, Def. 2.24 (iii)]) of coverings of the form` (1) and (2). From this it is clear that if {Ui → X} is a cdh-covering of X, then the induced morphism Ui → X satisfies the Nisnevich condition. Lemma 1.22. For a scheme X the closed embedding Xred → X is a cdh-covering and if X is Noetherian then the covering of X by its irreducible components (considered as closed integral subschemes) is also a cdh-covering. Proof. For the first statement simply note that a {Xred Xred → Xred → X} ` is a refinement of {Xred → X} and Xred Xred → X is a covering of the form ((2)) in Definition 1.20. For the second statement we induct on the number of irreducible components of X and we may assume that X is reduced from what we have already proved. Thus if X is irreducible there is nothing to prove, hence suppose we have n-irreducible components X1 , . . . , Xn . The proper morphism ∪i6=1 Xi → X is an isomorphism away from X1 thus from ((2)) and ((1)) it follows that {X1 → X, ∪i6=1 Xi → X} is a cdh-covering and the result follows immediately now from the inductive hypothesis.



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1.6. Refinements of cdh-coverings. Before understanding refinements of cdh-coverings we need to introduce cdp-covers: Definition 1.23. (cdp-covering) A morphism f : Y → X is called a cdp-covering (or morphism) if it is proper and satisfies the Nisnevich condition. Remark 1.24. Such morphisms are also referred to as proper cdh-covers such as in [SV00a]. Lemma 1.25. Suppose that p : Y → X is a cdp morphism. If X is a reduced scheme then p is an epimorphism in the category of schemes. Proof. It is enough to check that p is surjective and p# : OX → p∗ OY is injective. The first is satisfied by definition, for injectivity of p# suppose that U is any affine open subset of X and a ∈ Γ(U, OX ) is such that p# (U )(a) = 0. For any x ∈ U there is some y ∈ Y lying over x with the induced map of residue fields px : k(x) → k(y) being an isomorphism. From this it follows that the image of a in any of the residue fields k(x), with x ∈ U , is zero. Since Γ(U, OX ) is a reduced ring it then follows that a = 0.  Lemma 1.26. [SV00a, Lemma 5.8] Suppose that X is a Noetherian scheme and p : X 0 → X is a cdp morphism then p is a cdh-covering. Proof. By Noetherian induction we may assume that the induced morphism X 0 ×X T → T is a cdhcovering for every proper closed subscheme T of X. If X is not an integral scheme then by Lemma 1.22 it follows easily that X 0 → X can be refined by {Xi × X 0 → Xi → X}i X

which is a cdh-cover of X. Hence we may assume that X is an integral scheme. Since X 0 → X is a cdp-morphism there exists a closed integral subscheme X 00 ⊂ X 0 such that the composition p00 = (X 00 → X 0 → X) is a birational morphism. Let T ⊂ X be a proper closed subscheme for which p00 is an isomorphism over X \ T . It follows immediately from the definitions that {p00 : X 00 → X, T 



/

/X }

is a cdh-covering. From the universal property of fibre products it follows immediately that the map prX 00 : X 0 ×X X 00 → X 00 admits a section from which we immediately conclude that prX 00 is a cdhcovering. Furthermore by our induction hypothesis the morphism X 0 ×X T → T is a cdh-covering. Hence the following    / / 0 00 00 0 0 X , X ×X → X → X → X T ×X → T X

X

is a cdh-covering and it is clear that this refines p : X 0 → X thus p is indeed a cdh-covering.



We will also need some auxiliary results concerning flat morphisms. Proposition 1.27. ([SV00b, Prop.2.1.8]) Let f : X → S be a flat morphism of finite type between Noetherian schemes. Then f is flat of relative dimension r if and only if Xf (g) is equidimensional of pure dimension r for every generic point g ∈ X. Proof. The proof that we give here was pointed out to the author by Remy van Dobben de Bruyn. One way is obvious. For the other suppose that we have some x ∈ X with dim Xf (x) > r. Then f (x) is in the set U = {s ∈ S : dim Xs > r} which by [Stacks, Tag 0D4H] is open, but then the set f −1 (U ) is open and non-empty hence it contains a generic point of X which contradicts our assumption. Thus for any x ∈ X we have that dim Xf (x) ≤ r. On the other hand the set {x ∈ X : dimx (Xf (x) ) ≥ r}, where dimx (Xf (x) ) denotes the dimension of the scheme Xf (x) at the point x defined as the number inf dim(U ) where the infimum runs over all open subsets of Xf (x) containing the point x, contains all

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the generic points of X by assumption and is closed by Chevalley’s theorem (See [GD64, Thm. 13.1.3]) hence equal to the underlying set of X. Thus any irreducible component of Xf (x) has dimension at least r for any point x of X which now completes the proof.  Lemma 1.28. Let f : X → S be a flat birational morphism of finite type between reduced Noetherian schemes. Then the morphism f is quasi-finite and if f is also proper then f is an isomorphism. Proof. From [Stacks, Tag 0BAB] and Proposition 1.27 we conclude that f is of relative dimension zero, hence the first claim follows. Suppose now that f is also proper. From what we have already shown and [Stacks, Tag 02OG] it follows that the morphism f is finite. From [Vak13, Ex. 24.4.H] we have that f∗ OX is a locally free sheaf on S and the function (s ∈ S) 7→ dimk(s) (f∗ OX )s ⊗OS,s k(s) = dimk(s) Γ(Xs , OXs ) is locally constant. Since the generic fibers of f are isomorphisms we conclude that f∗ OX is locally free of rank 1 hence the map f # : OS → (f∗ OX ) is an isomorphism which completes the proof.  From this point on we will assume that all schemes are separated and change all the relevant definitions (such as Definition 1.20) to only concern separated schemes. Proposition 1.29. [SV00a, Prop.5.9] Every cdh-covering of a Noetherian scheme X has a refinement of the form p pi {Ui → X 0 → X}ni=1 where {pi : Ui → X 0 }ni=1 is a Nisnevich cover and p : X 0 → X is a cdp morphism. Proof. Thorought this proof we shall call a covering of the type given in the statement of this proposition a special cdh-covering. Recall from Remark 1.21 that any cdh-covering can be refined to a finite numbers of composites of coverings where each covering appearing is either of the form ((1)) or ((2)) of Definition 1.20. By inducting ` on the number of composites it is clear that it is enough to prove it for` a covering of the form {(Ui Zi ) → Ui → X} where {Ui → X}i is a Nisnevich cover and the maps Ui Z → X are cdp coverings of the form ((2)). Note that this induces maps a a a T = (Ui Zi ) → U = Ui → X i

where the first morphism is a cdp-morphism and the second is Nisnevich hence gives a cdh-covering of X. If this aforementioned covering has a refinement of special form {Vj → X 0 → X} with maps Vj → T for every j then it is clear that a {Vj ×(Ui Zi ) → Vj → X 0 → X}i,j T

gives the desired refinement of special form. Hence we have reduced the proof to proving that any covering of the from p q {T → U → X} where p is a cdp-covering and q is a Nisnevich covering admits a refinement of special form. By Noetherian induction it is enough to prove that if we have that for any proper closed subscheme Z ⊂ X the induced covering of Z admits a special refinement then so does our given covering of X. From Lemma 1.22 we see that we may assume that X is an integral scheme. By [GD64, Prop. ` (17.4.9)] we have that every connected component of U is open and thus we have U = Ui where the Ui are open (and closed) subschemes of U and note that there are finitely many of them since U is quasi-compact. Let Uij denote the irreducible components of each Ui . For the generic point of Ui,j we know by the cdp-property of p that we can find some point ti,j ∈ T such that its scheme theoretic image in T denoted Ti,j is birational to Ui,j .

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By generic flatness we can find an open dense subset U of X such that all the Tij are flat over U . Furthermore by Theorem 1.15 we can find a closed subscheme Z of X disjoint with X such that the proper transform T˜ij is flat over the blow-up XZ with center Z which we now illustrate in a diagram: T˜ij /

XZ ×X Tij

Tij

XZ ×X Uij

Uij

/

/

XZ ×X Ui

Ui

◦

◦

XZ ×X U

U q

XZ

X

Note that since T˜ij → XZ is flat and XZ ×X Ui → XZ is unramified thus has open diagonal ([Stacks, Tag 02GE]) it follows (see [Vak13, Thm. 10.1.19]) that the composition  T˜ij → XZ ×X Uij 

/

/ XZ ×X Ui

is flat. We can now conclude that XZ ×X Uij is an open and closed subscheme of XZ ×X Ui hence by  [Stacks, Tag 04PW] and [Stacks, Tag 0819] the closed embedding XZ ×X Uij  / / XZ ×X Ui is also an open embedding yielding that the morphism T˜ij → XZ ×X Uij is flat. Note also that since XZ ×X Ui → XZ is flat and since XZ → X is an isomorphism over a dense open set it follows easily now that the scheme X 0 ×X Uij is integral and that the morphism T˜ij → XZ ×X Uij is birational. Since the aforementioned morphism is also clearly proper and flat from what we have proved above we conclude from Lemma 1.28 that it is in fact an isomorphism. Hence the covering {XZ × Uij → XZ → X} X

is a refinement of our original covering q ◦ p and {XZ ×X Uij → XZ } is a Nisnevich covering. Furthermore according to our inductive hypthesis there exists a special refinement {Vk → Z 0 → Z} of the induced covering T ×X Z → U ×X Z → Z and hence a a {Vk → Z 0 XZ → X, XZ × Uij → Z 0 XZ → X}k,ij X

is a special refinement of the covering q ◦ p which completes the proof.  2. Representable sheaves In this last section of these notes we shall study representable sheaves in the h-topology and give a detailed proof of ([Voe96, Lemma 3.2.3]) where an explicit description is given of the sections of an hrepresentable functor over the spectrum of a field. For more on representable sheaves in the h-topology the reader is adviced to take a look at Section 3.2 of [Voe96] which is our main reference for this topic. Another nice source is [Ryd07] where sections of presheaves h-represented by an algebraic space are studied. Furthermore for descriptions of the sections of representable sheaves in the cdh-topology see [HK17], [And17].

10

OLIVER E. ANDERSON

The schemes considered in Section 3.2 of [Voe96] are of finite type over a base scheme S. We shall in general not make this assumption and let Sch /S denote the category of separated schemes over a Noetherian separated scheme S and we will explicitly state when we need finite type assumptions. Notation 2.1. When we are working with the sheafification of presheaves we shall use the notation and construction given in the proof of Theorem A.6 with the exception that for a Grothendieck topology t on Sch /S we let Lt (X) denote the t-sheafification of the representable functor hX and(Lt (X))0 denote the separation of the presheaf hX (See Definition A.10). For a morphism f : X → Y ∈ Sch /S we let Lt (f ) : Lt (X) → Lt (Y ) be the morphism induced by the morphism h(f ) : hX → hY . Finally for a scheme X and a set of morphisms S = {Ui → X} we shall write S ∈ Covt (X) if S is a t-covering. 2.1. Morphisms inducing isomorphisms of representable sheaves. We start by proving Lemmas [Voe96, Lemma 3.2.1] and [Voe96, Lemma 3.2.2] in several small steps. Unless otherwise stated t shall always denote an arbitrary Grothendieck topology on the category of Noetherian separated schemes over a base scheme S. Lemma 2.2. Suppose that the morphism f : X → Y is a t-covering of Y . Then the induced morphism Lt (f ) : Lt (X) → Lt (Y ) is an epimorphism. Proof. It is enough to show that the morphism h(f ) : hX → hY is locally an epimorphism. Suppose that g ∈ hY (U ) for some U ∈ Sch /S. Then the projection prU : (f ∗ (g) = X ×Y U ) → U is a t-covering of U and prU∗ (g) = g ◦ prU = f ◦ prX which is the image of prX under h(f ).  Corollary 2.3. Suppose that f : X → Y is a monomorphism in Sch /S and also a t-covering of Y . Then the induced morphism Lt (f ) : Lt (X) → Lt (Y ) is an isomorphism. In particular for any topology t where the closed embedding is i : Xred → X is a covering (i.e. the qf h, cdp, h topologies) we have that Lt (i) : Lt (Xred ) → Lt (X) is an isomorphism. Proof. Follows from the fact that L is a left exact functor together with Lemma 2.2.



Lemma 2.4. Suppose that f : X → Y is a morphism such that the diagonal morphism ∆X/Y : X → X × X Y

induces an isomorphism

∼ =

Lt (∆) : Lt (X) → Lt (X × X) Y

then Lt (f ) : Lt (X) → Lt (Y ) is a monomorphism. Proof. Since Lt commutes with finite limits this follows immediately.



Lemma 2.5. Let t be any Grothendieck pretopology where for any X ∈ Sch /S where the closed embedding Xred → X is a covering . Then for any radicial morphism f : X 0 → X we have that L(f ) : L(X 0 ) → L(X) is a monomorphism. Proof. Since X 0 → X is radicial it follows that the diagonal ∆X 0 /X : X 0 → X 0 ×X X 0 is surjective. hence (X 0 )red → (X 0 ×X X 0 )red is an isomorphism. Thus we have have a commutative diagram 0 ) Lt (Xred

∼ =

∼ =

Lt ((X 0 ×X X 0 )red )

Lt (X 0 ) Lt (∆X 0 /X )

∼ =

Lt (X 0 ×X X 0 )

h-TOPOLOGIES

11

thus Lt (∆X 0 /X ) is an isomorphism hence by Lemma 2.4 Lt (f ) is a monomorphism.



Lemma 2.6. Let t be any Grothendieck pretopology where universal homeomorphisms of finite type are coverings. Then if f : X → Y is a universal homeomorphism of finite type we have that Lt (f ) : Lt (X) → Lt (Y ) is an isomorphism. Proof. By Lemma 2.2 it follows that Lt (f ) is an epimorphism and by Lemma 2.5 it is also a monomorphism.  Lemma 2.7. Let t be any Grothendieck pretopology where coverings are assumed to be jointly surjective. Then for any reduced scheme X ∈ Sch /S and any scheme Y ∈ Sch /S the canonical map HomS (X, Y ) → Hom(Lt (X), Lt (Y )) is injective. Proof. Let f1 , f2 ∈ HomS (X, Y ) with f1 6= f2 . Since X is reduced there is some x ∈ X such that the composition f1 ◦ qx 6= f2 ◦ qx where qx : Spec(kx ) → X is the canonical morphism. If Lt (f1 ) = Lt (f2 ) then this implies in particular that the images of f1 ◦ qx and f2 ◦ qx in Lt (Y )(Spec(kx )) coincide. In other words there exists some t-covering {pi : Ui → Spec(k)} such that f1 ◦ qx ◦ pi = f2 ◦ qx ◦ pi for all i, but since Spec(kx ) is the spectrum of a field and Ui is not the empty scheme we easily see that for any i the morphism pi is an epimorphism hence we get the equality f1 ◦ qx = f2 ◦ qx which is a contradiction.



Corollary 2.8. Suppose that t is a Grothendieck pretopology on Sch /S satisfying the assumption of Lemma 2.7 then the canonical map hX (Yred ) → (Lt )0 (X)(Yred ) is a bijection. Proof. By appyling Yoneda Lemma we get a commutative diagram: hX (Yred ) = HomS (Yred , X)

Hom((Lt )0 (Y ), (Lt )0 (X)) ∼ =

Hom(hYred , (Lt )0 (X)) ∼ =

(Lt )0 (X)(Yred ) the desired result now follows from Lemma 2.7.



Remark 2.9. [Ryd07, Remark 8.10] In the text following Lemma [Voe96, Lemma. 3.2.2] it is claimed that for the h-topology one can infact use [Voe96, Lemma 3.2.1] (Corollary 2.3) to deduce that hX (Yred ) = (Lh )0 (X)(Y ). This is incorrect because if hX (Y ) → hX (Yred ) is not surjective then we can not have that (Lh )0 (X)(Y ) → (Lh )0 (X)(Yred ) is surjective. Taking X = Yred for any scheme Y such that the canonical map Yred → Y does not have a retraction gives a counterexample. For such an example consider Y = Spec(Z/(36)), then Yred = Spec(Z/(6)) and there are no morphisms of rings from Z/6 to Z/36.

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OLIVER E. ANDERSON

2.2. Sections of representable sheaves at fields. Before moving on, let us briefly recall some field theory. Lemma 2.10. Let K be a perfect field and let L be any finite extension of K. If an element x ∈ L satisfies x ⊗ 1 = 1 ⊗ x ∈ L ⊗K L then x ∈ K. Proof. Since K is perfect the extension L/K must necessarily be separable. By the theorem of primitive element we have some polynomial P (T ) ∈ K[T ] such that L ∼ = K[T ]/(P (T )). Hence ∼ L ⊗K L = L[T ]/P (T )L[T ] and under this isomorphism the element x 7→ x ⊗ 1 mapsto to a constant polynomial with coefficients in L while 1 ⊗ x maps to a polynomial in T with coefficients in K of degree strictly less that P (T ) thus it is clear that the two images coincide if and only if x ∈ K.  Lemma 2.11. (Notation as in the construction of the sheafifcation given in A.6) Let t be any Grothedieck pretopology where for any scheme X and {fi : Ui → X}i∈I ∈ Covt (X) then all fi are morphisms of finite type. Suppose further for any field E the collection Covt (Spec(E)) contains all singleton of the form {Spec(L) → Spec(E)} where L/E is any finite field extension. Then for any morphism Spec(K) → S with K a field we have that any element in ∈ L(Y )(Spec(K)) has a representative of the form ({Spec(L) → Spec(K)}, [f ] ∈ (Lt )0 (Y )(Spec(L))) where L/K is a finite field extension of K. Proof. Let f ∈ Lt (Y )(Spec(K)) and pick a representative f = (S = {pi : Ui → Spec(K)}i∈I , {[fi ]}) ∈ Lt (Y )(Spec(K)). By Nullstellensatz it is clear that the covering S admits a refinement of the form Spec(L) → Spec(K) where L/K is a finite field extension. Then there exists an i ∈ I such that Spec(L) → Spec(K) factors through Ui as γi pi Spec(L) → Ui → Spec(K) and one checks easily that the pair ({pi ◦ γi : Spec(L) → Spec(K)}, [fi ◦ γi ]) is a representative of f .



Lemma 2.12. Let t be a Grothendieck pretopology satisfying the conditions of Lemma 2.11. Then for any morpism Spec(K) → S where K is a perfect field one has Lt (Y )(Spec(K)) = HomS (Spec(K), Y ). Proof. Let g be any element of Lt (Y )(Spec(K)). By Lemma 2.11 g has a representative of the form g = ({Spec(L) → Spec(K)}, [f ] ∈ (Lt )0 (Y )(Spec(L))) ∈ Lt (Y )(Spec(K)) where L/K is a finite field extension and by definition we have the equality [f ◦ pr1 ] = [f ◦ pr2 ] ∈ (Lt )0 (Y )(Spec(L)

×

Spec(L)).

Spec(K)

Since K is perfect the field extension L/K is separable thus by applying the theorem of primitive element we easily see that the scheme Spec(L) ×Spec(K) Spec(L) is reduced hence it follows from Corollary 2.8 that f ◦ pr1 = f ◦ pr2 . We now claim that the morphism f factors through Spec(K). To this extent we may assume that Y is an affine scheme Spec(A) and that f is induced by a morphism of rings ϕ : A → L. Furthermore by assumption we have that for any a ∈ A we have the equality ϕ(a) ⊗ 1 = 1 ⊗ ϕ(a) ∈ L ⊗K L.

h-TOPOLOGIES

13

Lemma 2.10 now yields that ϕ(a) ∈ K for every a ∈ A thus f : Spec(L) → Y does indeed factor through Spec(K). Thus we have proved that Y (K) → Lt (Y )(Spec(K)) is surjective and by Lemma 2.7 it is also injective.  Definition 2.13. Let X, Y ∈ Sch /S and f ∈ Hom(L(X), L(Y )). We say that a set of morphisms ` (pi ) {pi : Ui → X} satisfying that Ui → X weakly realizes f if there exists morphisms fi : Ui → Y such that L(fi ) = f ◦ L(pi ) for all i, or equivalently by the sheafification adjunction and Yoneda Lemma we say that {pi } weakly realizes f ∈ L(Y )(X) if p∗i (f ) = L(Y )(Ui ) is the image of some fi ∈ hY (Ui ). A set of morphisms {pi : Ui → X} is said to t-realize f ∈ Hom(L(X), L(Y )) (or equivalently in L(Y )(X)) if {pi } is a t-covering and {pi } weakly realizes f . If the Grothendieck pretopology t is clear from context we shall simply just say that {pi } realizes f . Lemma 2.14. Let A ⊂ B be an integral extension of rings and let b ∈ B. Then the extension A ⊂ A[b] = A0 induces a universal homeomorphism Spec(A0 ) → Spec(A) if and only if b ⊗ 1 = 1 ⊗ b ∈ (B ⊗A B)red . Proof. A clear proof can be found in [Ryd07, Lemma B.5].



Lemma 2.15. Let t be a Grothendieck pretopology satisfying the conditions of Lemma 2.11. Then for any S-scheme of the form Spec(K)/S with K a field and any element a ∈ Lt (Y )(Spec(K)) there is a representative of a of the form ({γ : Spec(E) → Spec(L)}, [f ] ∈ (Lt )0 (Y )(Spec(E))) where E/K is a purely inseparable extension of K. Furthermore the cover {γ} realizes f . Proof. By Lemma 2.11 we can represent a as
{Spec(L) → Spec(K)}, [f 0 ] ∈ (Lt )0 (Y )(Spec(L)) where L/K is a finite field extension. Let i be the morphism i : (Spec(L)

× Spec(K)

Spec(L))red → Spec(L)

×

Spec(L)

Spec(K)

then from the equality [f 0 ◦ pr1 ] = [f 0 ◦ pr2 ] ∈ (Lt )0 (Y )(Spec(L)

×

Spec(L))

Spec(K)

we must also have [f 0 ◦ pr1 ◦ i] = [f 0 ◦ pr2 ◦ i] hence by Corollary 2.8 we have f 0 ◦ pr1 ◦ i = f 0 ◦ pr2 ◦ i. Note also that we can assume that Y is an affine scheme Spec(A) and so f 0 is induced by a morphism of rings ϕ : A → L satisfying ϕ(a) ⊗ 1 = 1 ⊗ ϕ(a) ∈ (L ⊗K L)red for all a ∈ A. By Lemma 2.14 and [Stacks, Tag 01S4] it follows that ϕ(a) is purely inseparable over K for all a ∈ A, thus the all elements of the subring ϕ(A) ⊂ L are purely inseparable over K and one easily checks that the field of fractions E := (ϕ(A))(0) ⊂ L is purely inseparable over K. This gives us a morphism f : Spec(E) → Y which f 0 factors through. One easily checks that the pair ({γ : Spec(E) → Spec(K)}, [f ]) represents a and that the image of f under the map hY (Spec(E)) → L(Y )(Spec(E)) coincides with γ ∗ (a). 

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OLIVER E. ANDERSON

2.3. Cofiltered limits and representable sheaves. We shall describe sections of representable sheaves in the h-topology in terms of filtered colimits of sets of morphisms. Before doing so we should at least recall the relevant notions: Definition 2.16. We say that a category I is filtered if it has the following properties (1) the category I has at least one object, (2) for every pair of objects x, y of I there exists an object z and morphisms x → z, y → z, and (3) for every pair of objects x, y of I and every pair of morphisms a, b : x → y of I there exists a morphism c : y → z of I such that (c ◦ a) = (c ◦ b) . For a filtered diagram F : I → C is a functor from a filtered category and the colimit (if it exists) of a filtered diagram is called a filtered colimit. Dually we say that an index category I is cofiltered if I op is filtered. Then a cofiltered limit is the limit of a diagram F : I → C where I is cofiltered. The perfect closure of a field K is an example of a filtered colimit: Lemma 2.17. [Stacks, Tag 046W] For every field K there exists a purely inseparable extension K ⊂ K 0 such that K 0 is perfect. The field extension K ⊂ K 0 is unique up to unique isomorphism. Proof. Consider the filtered diagram F : (N, ≤) → Rings given by F (n) = K for all n and for a morphism f : n → m in N we take F (f ) = (FrK )m−n where FrK denotes the Frobenius endomorphism. Let K 0 be the filtered colimit of F . The ring K 0 can be described as classes of the form [x, m] with n−m x ∈ K and m ∈ N where [x, m] = [y, n] iff xp = y. It is clear that K 0 is a field and for [x, m] we m p 0 have that [x, 0] = [x, m] thus K ⊂ K is a purely inseparable extension. Furthermore for [x, m] ∈ K 0 r we have that [x, m + r]p is a pr ’th root of [x, m] hence K 0 is also perfect by [Stacks, Tag 030Z].  Definition 2.18. Let K be a field. The field extension K ⊂ K 0 of Lemma 2.17 is called the perfect ∞ closure of K. We shall denote the perfect closure of K by K −p . Before moving on we will also need the following auxiliary result: Lemma 2.19. Let p : X 0 → X be a morphism in Sch /S which is radicial. For any pretopology t where surjective closed embeddings are coverings and {p} ∈ Covt (X) the map HomS (X 0 , Y ) → Lt (Y )(X 0 ) Factors through p∗ : Lt (Y )(X) → Lt (Y )(X 0 ) that is we have a commutative diagram HomS (X 0 , Y )

γX 0 /X

Lt (Y )(X) p∗

Lt (Y )(X 0 ) q0

p

Furthermore if {q : X 00 → X} ∈ Covt (X) with q radicial and the morphism q factors as X 00 → X 0 → X then the following diagram commutes HomS (X 0 , Y ) (q 0 )∗

HomS (X 00 , Y )

γX 0 /X γX 00 /X

Lt (Y )(X)

Proof. Consider an element f ∈ HomS (X 0 , Y ). Since X 0 → X is radicial it follows that the diagonal ∆X 0 /X : X 0 → X 0 ×X X 0 is surjective hence the following pair
{X 0 → X}, [f ] ∈ (Lt )0 (Y )(X) is an element of Lt (Y )(X). Thus we get a morphism γX 0 /X : HomS (X 0 , Y ) → Lt (Y )(X) and it is easily checked that it satisfies the desired properties.



h-TOPOLOGIES

15

Let now t be a Grothendieck pretopology in which universal homeomorphisms of finite type are coverings. For a fixed quasi-compact scheme X in Sch /S suppose that I is a cofiltered category with a terminal object 0 and F : I → Sch /S is a cofiltered diagram with F (0) = X and the morphisms to the terminal object in I are mapped to universal homeomorphisms of finite type. We use the notation Xλ to denote the image F (λ). Suppose further that for any non-reduced Xλ there is a morphism β → λ in I such that Xβ = Xred and F (β → λ) is the closed embedding Xred → X. By Lemma 2.19 we have morphisms HomS (Xλ , Y ) → Lt (Y )(X) for each λ and by the same Lemma it follows that we get induced a morphism from the filtered colimit colimλ HomS (Xλ , Y ) := colim(hY ◦ F op ) → Lt (Y )(X) Lemma 2.20. (Notation and assumptions as above) Suppose that any section f ∈ Lt (Y )(X) there exists some λ ∈ I such that F (λ → 0) = (Xλ → X) realizes f . Then the map colimλ HomS (Xλ , Y ) := colim(hY ◦ F op ) → Lt (Y )(X) is a bijection. Proof. For surjectivity: Let f ∈ Lt (Y )(X) be a section. By assumption we have a universal homeomorphism of finite type Xλ → X that realizes f thus the sheaf property together with Lemma 2.19 gives us surjectivity. For injectivity: Suppose [f ], [g] ∈ colim HomS (Xλ , Y ) have the same image in Lt (Y )(X). Since the colimit is filtered we can find some λ ∈ I such that the classes [f ] and [g] are images of morphisms f, g ∈ HomS (Xλ , Y ) and by our assumption on the diagram we can also assume that Xλ is a reduced scheme. We have that ({Xλ → X}, [f ]) = ({Xλ → X}, [g]) ∈ Lt (Y )(X) and thus [f ◦ pr1 ] = [g ◦ pr2 ] ∈ (Lt )0 (Xλ × Xλ ) X

thus by applying (Lt )0 to the diagonal we get [f ] = [g] ∈ (Lt )0 (Xλ ) and by Corollary 2.8 we must have f = g.



Corollary 2.21. ([Voe96, Lemma 3.2.3]) Let Spec(K) → S be a morphism where K is a field. Suppose that Y is of finite type over S. Then for any Grothendieck pretopology t satisfying the conditions of Lemma 2.11 we have ∞ Lt (Y )(Spec(K)) = HomS (Spec(K −p ), Y ), ∞ where K −p is the perfect closure of K (Definition 2.18) ∞

Proof. Recall that K −p is a filtered colimit of the diagram where the objects are the field K and the transition maps are given by Frobenius (See 2.17). By applying the Spec functor to the aforementioned diagram in rings we get a filtered diagram in schemes where we denote the objects by Spec(Kλ ). By Lemma 2.15 and Lemma 2.20 we have that colimλ HomS (Spec(Kλ , Y )) → Lt (Y )(Spec(K)) is a bijection. To finish the proof we need to show that the aforementioned colimit is in bijection with ∞ HomS (Spec(K −p , Y )). To this extent note that we clearly have a map ∞

colimλ HomS (Spec(Kλ , Y )) → HomS (Spec(K −p , Y )) For injectivity: suppose that two classes [f ], [g] ∈ colimλ HomS (Spec(Kλ ), Y )) have the same image ∞ in HomS (Spec(K −p ), Y ). Then since the colimit is filtered we may assume that the classes [f ], [g] are ∞ represented by morphisms f, g : Spec(Kλ ) → Y and letting γλ : Spec(K −p ) → Spec(Kλ ) being the projection we have that f ◦ γλ = g ◦ γλ by assumption. We can assume that Y and S are affine schemes

16

OLIVER E. ANDERSON

Spec(B) and Spec A respectively with A → B a morphism of finite type and since A is Noetherian by assumption the morphism A → B is in fact of finite presentation so we may suppose that B = A[x1 , . . . , xr ]/(f1 , . . . , fn ). Furthermore we can assume that f, g are induced by ring morphisms ϕ, ψ : B → Kλ . Consider now the morphisms ϕ

B

Kλ

ψ

colim Kλ = K −p

∞

are determined by where the generators xi map to it is clear that we must have [f ] = [g] ∈ colimλ HomS (Spec(Kλ ), Y ). ∞

For surjectivity: Consider an element f ∈ HomS (Spec(K −p ), Y ). We may again assume that Y, S are affine Spec B and Spec A respectively with A → B of finite presentatition, say B = ∞ A[x1 , . . . , xr ]/(f1 , . . . , fn ). Then the map B → K −p is determined by where the generators xi are ∞ mapped and as the polynomials fi map to 0 in K −p = colimλ Kλ it is clear that there is some Kλ in ∞ which the morphism B → K −p factors through.  Actually the following propositions allows us to vastly generalize Corollary 2.21. Proposition 2.22. [Stacks, Tag 01ZC] Let f : X → S be a morphism of schemes. The following are equivalent: (1) The morphism f is locally of finite presentation. (2) For any directed set I, and any inverse system (Ti , fii0 ) of S-schemes over I with each Ti affine, we have MorS (lim Ti , X) = colimi MorS (Ti , X) i

(3) For any directed set I, and any inverse system (Ti , fii0 ) of S-schemes over I with each fii0 affine and every Ti quasi-compact and quasi-separated as a scheme, we have MorS (lim Ti , X) = colimi MorS (Ti , X) i

Corollary 2.23. Under the assumptions of Lemma 2.20 we have that Lt (Y )(X) = colimλ HomS (Xλ , Y ) = HomS (lim Xλ , Y ). λ

A. Grothendieck pretopologies and sheaves We assume the reader to already be familiar with this topic and we only recall here the definitions and the construction of sheafification as given in [Fan+05],[Vis04]. A.1. Grothendieck pretopologies. Definition A.1. Let C be a category. A Grothendieck pretopology τ on C is the assignment to each object U of C a collection of sets of arrows {Ui → U }, called coverings of U , so that the following conditions are satisfied. (i) If V → U is an isomorphism, then the set {V → U } is a covering. (ii) If {Ui → U } is a covering and V → U is any arrow, then the fibered products {Ui ×U V } exist, and the collection of projections {Ui ×U V → V } is a covering. (iii) If {Ui → U } is a covering, and for each index i we have a covering {Vi,j → Ui } (here j varies on a set depending on i), the collection of composites {Vi,j → Ui → U } is a covering of U . We denote the collection of coverings of an object U ∈ C by Cov(U ). A category with a Grothendieck pretopology is called a site and we denote it by the pair (C, τ ). We denote the category of presheaves on C by Psh(C). Definition A.2. Let (C, τ ) be a site and suppose that {Ui → U }i∈I ∈ Cov(U ). A refinement of {Ui → U }i∈I is a covering {Va → U }a∈A such that for each index a ∈ A there is some index i ∈ I such that Va → U factors through Ui → U .

h-TOPOLOGIES

17

A.2. Sheaves. Let (C, τ ) be a site, F ∈ Psh(C). Given a covering {pi : Ui → U }i∈I . Let a : F(U ) → Q i∈I F(Ui ) be the unique map such that F(pi ) = pri ◦ a Q

where prj : i∈I F(Ui ) → F(Uj ) is projection onto the j’th factor. Further for each ordered pair of indices (i, j) let pr1,i,j : Ui ×U Uj → Ui be the projection Q to Ui andQpr2,i,j : Ui ×U Uj → Uj be the projection to Uj . For k = 1, 2 let bk be the unique map i F(Ui ) → i,j F(Ui ×U Uj ) such that pr(i,j) ◦ bk = F(prk,i,j ) ◦ pri for each i, j where pr(i,j) is the projection onto the (i, j)’th factor Y pr(i,j) : F(Ul × Um ) → F(Ui × Uj ). l,m

U

U

Definition A.3. A presheaf F : C op → Sets is a separated presheaf if for every covering {Ui → U }i∈I the map Y a F(U ) → F(Ui ) i∈I

is injective. It is a sheaf if the following diagram F(U )

a

Q

i∈I

F(Ui )

b1 b2

Q

i,j

F(Ui ×U Uj )

is an equalizer. We denote the category of sheaves on C by Sh(C). Definition A.4. ([Vis04, Def. 2.52]) A pretopology T on a category C is called saturated if a set of arrows {Ui → U } which has a refinement that is in T is also in T . If T is a pretopology on C, the saturation T of T is the set of sets of arrows which have a refinement in T . Proposition 2.53 in [Vis04] tells us that the saturation of a pretopology is saturated and that a presheaf is a sheaf with respect to T if and only if it is a sheaf with respect to the saturation of T . A.3. Sheafification. Definition A.5. Let (C, τ ) be a site, and F ∈ Psh(C). A sheafification of F is a sheaf Fτ ∈ Sh(C), together with a natural transformation F → Fτ , such that (i) given an object U of C and two elements ξ and η of F(U ) whose images ξ a and η a in Fτ (U ) are the same, there exists a covering {σi : Ui → U } such that σi∗ (ξ) := F(σi )(ξ) = σi∗ (η), and (ii) for each object U of C and each ξ ∈ Fτ (U ), there exists a covering {σi : Ui → U } and elements ξi ∈ F(Ui ) such that ξia = σi∗ ξ. Theorem A.6. Let (C, τ ) be a site, F ∈ Psh(C). (i) If Fτ ∈ Sh(C) is a sheafification of F, any morphism from F to a sheaf factors uniquely through Fτ . (ii) There exists a morphism F → Fτs where Fτs is a separated presheaf, such that any morphism from F to a separated presheaf factors uniquely through Fτs . (iii) There exists a sheafification F → Fτ , which is unique up to a canonical isomorphism. (iv) The natural transformation F → Fτ is injective if and only if F is separated. Sketch of proof. Part (iv) follows easily from the definition. For (i) and (ii) and (iii) we only provide the constructions. For (i): Let φ : F ¸ → G be a natural transformation from F to a sheaf G ∈ Sh(C). Given an element ξ ∈ Fτ (U ) we want to define the image of ξ in G(U ). There exists a covering {σi : Ui → U } and elements ξi ∈ F(Ui ), such that the image of ξi in Fτ (Ui ) is σi∗ (ξ). Set ηi = φ(Ui )(ξi ) ∈ G(Ui ). The pullbacks pr1∗ ξi and pr2∗ ξj in F(Ui ×U Uj ) both have as their image in Fτ (Ui ×U Uj ) the pullback of ξ; hence there is a covering {Ui,j,α → Ui ×U Uj }α such that ξi and ξj both pullback to the same element in F(Ui,j,α ) for every α. Using this together with the fact that G is separated we get that ηi and ηj

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both pullback to the same element in G(Ui ×U Uj ) and since G is a sheaf we get that the ηi glue to give an element η ∈ G(U ). We now let φτ : Fτ → G be given by φτ (U )(ξ) = η. For (ii): For each object U of C, we define an equivalence relation ∼ on F(U ) as follows: Given two elements a and b of F(U ), we write a ∼ b if there is a covering {Ui → U }i∈I such that the pullbacks of a and b to each Ui coincide. We define Fτs (U ) := F(U )/ ∼. If V → U is a morphism in C, then the pullback F(U ) → F(V ) is compatible with the equivalence relations, yielding a pullback Fτs (U ) → Fτs (V ). This defines the functor Fτs with the surjective morphism F → Fτs . For an element a ∈ F(U ) we denote its image in Fτs (U ) by [a] and since F → Fτs is surjective we denote any element in Fτs (U ) in this way. The presheaf Fτs is separated, and every natural transformation from F to a separated presheaf factors uniquely through Fτs . For (iii): To construct Fτ , we take for each object U of C the set of pairs ({Ui → U }, {[ai ]}), where {Ui → U } is a covering, and {[ai ]} is a set of elements with [ai ] ∈ Fτs (Ui ), such that the pullback of [ai ] and [aj ] to Fτs (Ui ×U Uj ), along the first and second projection coincide. On this set we impose an equivalence relation, by declaring ({Ui → U }, {[ai ]}) to be equivalent to ({Vj → U }, {[bj ]}) when the restrictions of [ai ] and [bj ] to F s (Ui ×U Vj ), along the first and second projection respectively, coincide. For each U , we denote by Fτ (U ) the set of equivalence classes. If V → U is a morphism, we define a function Fτ (U ) → Fτ (V ) by associating with the class of a pair ({Ui → U }, {[ai ]}) in Fτ (U ) the class of the pair ({Ui ×U V }, p∗i ([ai ]), where pi : Ui ×U V → Ui is the projection. This gives a sheaf Fτ . There is also a natural transformation Fτs → Fτ , obtained by sending an element [a] ∈ Fτs (U ) into ({U = U }, [a]) and the composition F → Fτs → Fτ is the sheafification.



ˇ 0 (V, F s ) where V is a covering, H ˇ 0 (V, F s ) denotes the Remark A.7. Note that Fτ (U ) = colimV H τ τ equalizer of the obvious diagram and the colimit is a filtered colimit ordered by the relation ”refinement”. Remark A.8. (Explicit computation of the induced morphism from the sheafification). Suppose η : F → G is a morphism from the presheaf F to the sheaf G. Let Fτ be the sheafification as constructed in the proof of Theorem A.6 and let Sh(η) denote the induced map Sh(η) : Fτ → G. Then for any element f = ({Ui → X}i∈I , fi ∈ Fτs (Ui )) ∈ Fτ (X), where fi is the image of fi ∈ F(Ui ) in Fτs (Ui ) we have that Sh(η)(X)(f ) is the ”gluing of the elements” η(Ui )(fi ). Remark A.9. Suppose that φ : F → G is a morphism of presheaves. Then the induced morphism φτ : Fτ → Gτ is given by φτ (U )({pi : Ui → U }, {[ai ]}) = ({pi : Ui → U }, {[φ(Ui )(ai )]}) Definition A.10. (Notation and assumptions as in Theorem A.6) we will call the separated presheaf Fτs constructed in the proof of Theorem A.6 the separation of the presheaf F. References [And17]

Oliver Eivind Anderson. Cdh-sheaves via semi-normalization. 2017. url: https://sites. google.com/site/olivereivindanderson/cdhsn.pdf. ´ ements de math´ematique. Fasc. XXX. Alg`ebre commutative. Chapitre 5: [Bou64] N. Bourbaki. El´ Entiers. Chapitre 6: Valuations. Actualit´es Scientifiques et Industrielles, No. 1308. Hermann, Paris, 1964, p. 207. [Fan+05] Barbara Fantechi et al. Fundamental algebraic geometry. Vol. 123. Mathematical Surveys and Monographs. Grothendieck’s FGA explained. American Mathematical Society, Providence, RI, 2005, pp. x+339. isbn: 0-8218-3541-6. ´ ements de g´eom´etrie alg´ebrique IV. Vol. 20, [GD64] Alexander Grothendieck and Jean Dieudonn´e. El´ ´ 24, 28, 32. Publications Math´ematiques. Institute des Hautes Etudes Scientifiques., 19641967.

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