Devissage and Localization for the Grothendieck Spectrum of Varieties
aa r X i v : . [ m a t h . K T ] A p r DEVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OFVARIETIES
JONATHAN A. CAMPBELL AND INNA ZAKHAREVICH
Abstract.
We introduce a new perspective on the K -theory of exact categories via the notion of a CGW-category . CGW-categories are a generalization of exact categories that admit a Qullen Q -construction, butwhich also include examples such as finite sets and varieties. By analyzing Quillen’s proofs of d´evissageand localization we define ACGW-categories , an analogous generalization of abelian categories for which weprove theorems analogous to d´evissage and localization. In particular, although the category of varietiesis not quite ACGW, the category of reduced schemes of finite type is; applying d´evissage and localizationallows us to calculate a filtration on the K -theory of schemes of finite type. As an application of this theorywe construct a comparison map showing that the two authors’ definitions of the Grothendieck spectrum ofvarieties are equivalent. Contents
1. Introduction 12. CGW-Categories 33. The K -theory of a CGW-category 74. Examples 115. ACGW-Categories 136. D´evissage 167. Relationship with the S • -construction 188. Localization of ACGW-categories 219. A comparison of models 2410. Proof of Theorem 8.5 28Appendix A. Checking that C\A is a CGW-category 39References 461.
Introduction
On August 16, 1964, Grothendieck wrote to Serre of a conjectured category of motives. Such a category(called M ( k )) would encode schemes up to decomposition (by cutting out subvarieties), but would itself bean abelian category capturing the cohomological structures involved.The sad truth is that for the moment I do not know how to define the abelian category ofmotives, even though I am beginning to have a rather precise yoga for this category. Forexample, for any prime ℓ = p , there is an exact functor T ℓ from M ( k ) into the category offinite-dimensional vector spaces over Q on which the pro-group Gal( k i /k i ) i acts, where k i runs over subextensions of finite type of k and k i is the algebraic closure of k i in k ; thisfunctor is faithful but not, of course, fully faithful. . . I will not venture to make any generalconjecture on the above homomorphism; I simply hope to arrive at an actual constructionof the category of motives via this kind of heuristic considerations, and this seems to me tobe an essential part of my “long run program.” [CGC +
04, p 174-175]Grothendieck’s letter proposes several other properties of this conjetured category, and discusses his attemptsat the construction. Since then, there have been many other approximations to construct this category—foran overview see, for example, [Mil13]—but all fall short of the ideal.
Date : April 22, 2019.
Grothendieck’s approach begins with the construction of a “ K -group” of varieties. These days, this isknown as the Grothendieck ring of varieties, denoted K ( Var k ). It is generated by isomorphism classes of k -varieties, [ X ], subject to the relations that [ X ] = [ Z ] + [ X \ Z ] for closed inclusions Z ֒ → X . Kontsevich,following Drinfeld [Kon09], calls this the ring of “poor man’s motives.” He notes that any reasonable abeliancategory of motives, M k , will have a map K ( Var k ) K ( M k ). For example, in [GS96, Thm. 4], Gilletand Soule show that there is a group homomorphism K ( Var k ) K ( M ∼ ) where M is the category of(pure) motives associated to the equivalence relation ∼ . It is thus useful to understand K ( Var k ) in a deepway in order to learn more about how motives should work. It is even better to understand how it behavesin relation to abelian categories.We move toward such an understanding in this paper. Before doing so, we rephrase the question. TheGrothendieck group of an abelian category is a shadow of the much richer structure of Quillen’s higheralgebraic K -theory [Qui73]. Thus there should in fact exist a map on higher algebraic K ( Var k ) K ( M k )provided that one can define the objects in the map. It is currently far beyond the state of the art toattempt to understand the right-hand side. However, the authors separately have come up with modelsfor the left [Cam, Zak17]. Under these constructions the category of varieties behaves very similarly to anabelian category.Our goal in this paper to understand the higher algebraic K -theory of varieties, K ( Var k ) as if it werethe K -theory of an exact or abelian category. This has the benefit of putting all objects of interest on thesame footing. Such K -theories should come from an underlying categorical structure that is used to producea topological space (or spectrum) for algebraic K -theory. Thinking of sequences Z ֒ → X ← X \ Z as our“exact sequences,” and pondering the fundamental theorems of Quillen’s algebraic K -theory, we come to thefollowing desiderata for the construction of K -theories of geometric and algebraic objects:(1) The categorical machinery should somehow encompass both the category of varieties with its “exactsequences” defined above, and Quillen’s exact categories [Qui73, p.92].(2) Localization should hold: given two such categories A ⊂ B , one should be able to produce a localizedcategory B / A as one can with abelian categories. One would also like a localization sequence K ( A ) K ( B ) K ( B / A )as in [Qui73, Thm. 5].(3) Devissage should hold: Given an inclusion of categories A ⊂ B such that everying in B can be“broken up” into objects in A , there should be an equivalence K ( A ) ≃ K ( B ).In this paper we show that there is such a categorical structure, and we are able to satisfy the requirementslisted above. Although this does not get us much closer to understanding the conjectural category of motives,it does provide us with a new perspective and concrete technical tools. The perspective could be summarizeas follows: varieties, together with the exact sequences above, behave almost like abelian categories and oneshould work with this structure for as long as possible before passing to abelian categories. As will be shownbelow, this perspective is extremely fruitful when discussing algebraic K -theory, and we expect it to be moreuseful generally.The fundamental notion introduced in this paper is that of a CGW-category . It is essentially a categoryequipped with two subclasses of maps, M and E (to be thought of as analagous to admissible monomorphismsand admissible epimorphisms in exact categories), together with distinguished squares that tell us how objectsare built. In all examples we know, the horizontal and vertical morphisms need not compose in the category,and therefore we situate the classes M and E in a double category. With this minimal amount of data wedefine K -theory following the classical constructions due to Quillen (Sect. 3) or Waldhausen (Sect. 7). Weshow that the resulting K -theory spaces have the correct group of components in Thm. 3.3. CGW-categoriessatisfy requirement (1) above: they encompass varieties and exact categories.Of course, as in the case of exact categories, additional structure is required to prove these theorems. Tothis end we introduce the definition of an ACGW-category, which is meant to be a sort of “abelian” versionof a CGW-category. The category of reduced schemes of finite type is such a category, with the categoryof varieties sitting inside it as a full subcategory. Roughly, an ACGW-category is a category that formallysatisfies all of the properties that open and closed sets do (the complement of a closed set is open, you canintersect closed sets and union open sets, etc). Using this definition we prove the first main theorem of thepaper: EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 3
Theorem 1.1 (Devissage) . Let A , B be ACGW-categories with A ⊂ B satisfying certain technical conditions.Suppose every B ∈ B has a finite filtration B i such that the difference between B i and B i − lies in A . Then K ( A ) ≃ K ( B ) . Here “difference between” could mean a quotient or a complement; for the precise statement see Thm 6.1.The definition of ACGW-category has a number of requirements, but these requirements are satisfied by themotivating examples of the category of reduced schemes of finite type, polytopes [Zak17], finite sets, andabelian categories.The formal similarities between ACGW-categories and abelian categories suggest that other theorems inalgebraic K -theory can be extended to the CGW case. Quillen’s other major tool in algebraic K -theory isthe localization theorem, which relates the K-theories of two abelian categories A , B with the K -theory oftheir quotient category B / A . A very similar theorem holds for ACGW-categories: Theorem 1.2 (Localization) . Let C be an ACGW category and A a sub-ACGW-category of C satisfyingcertain technical conditions. Then there is a localization ACGW-category C\A such that K ( A ) K ( C ) K ( C\A ) is a homotopy fiber sequence. For a more precise statement of this theorem, see Theorem 8.5.An interesting observation about the proofs of these theorems is how closely they follow Quillen’s originalproofs. The category of varieties really does “behave like” an exact category, in the sense that many ofthe motions that are necessary to prove theorems have direct analogs in the category of varieties. (In fact,the category of varieties lacks only “pushouts” to behave like an abelian category; this is why switching toreduced schemes of finite type is necessary. For more detail on this, see Section 5.)We expect there to be substantial applications of the d´evissage and localization theorems. The main appli-cation that we discuss in this paper is a comparison of models for the K-theory of varieties that both authorshave constructed. Surprisingly, this theorem seems to use every bit of K-theoretic machinery the authorshave developed: assemblers, cofiber sequences in K-theory, and the d´evissage and localization theorems. Allcombine to give the following theorem.
Theorem 1.3 (Comparison) . Let K C ( Var n ) denote the K -theory of the SW-category Var n defined in [Cam] , and let K Z ( V n ) denote the K -theory of the assembler V n defined in [Zak17] . Then there is a zig-zagof weak equivalences K C ( Var n ) • K Z ( V n ) . ∼ ∼ For a more detailed statement of this theorem, see Theorem 9.1.Each of the models constructed has their own strengths, and this theorem allows us to pass between modelsto exploit these. We expect a more general theorem relating Waldhausen-style K -theory to assembler style K -theory to be true, but we leave that for future work.Whether this new perspective leads to a new theory of motives or not is unclear; however, the strikingbehavioral similarities between varieties and abelian categories was too beautiful to leave unexplored. Acknowledgements.
The authors would like to thank Pierre Deligne, Andr´e Joyal, Andrew Blumberg,and Charles Weibel for interesting conversations related to this work. They also thank Daniel Grayson andKarl Schwede for their patience with our annoying technical questions.Campbell is supported by Vanderbilt University. Zakharevich is supported by Cornell University andNSF DMS-1654522. 2.
CGW-Categories
This section contains the main definition of the paper: the definition of a CGW-category. CGW-categoriesare meant to be an abstraction of the definition of an exact category that disentangles the notion of “exactsequence” from universal properties inside underlying category. To see what we mean by this, note that thedefinition of exact category requires that if
JONATHAN A. CAMPBELL AND INNA ZAKHAREVICH
X Y Z is an exact sequence, X is the kernel of Y Z . This enforces a relationship between
X, Y and Z in theunderlying category. One then usually manipulates exact sequences by tacitly using the fact that X is thelimit of some diagram. We observe that instead of doing this, one can give the data of a collection of exactsequences and requiring that this data satisfy various properties, so that exact sequences can be formallymanipulated as they are in exact categories.It turns out that the most efficient way to encode this kind of structure is using the formalism of doublecategories. We thus begin by recalling the definition of a double category, as well as establishing somenotation for working with double categories. The notion of double categories goes back to [Ehr63]. We donot include the complete definition; for the reader interested in a more in-depth introduction, see for example[Lei, Section II.6]. Definition 2.1. A double category C is an internal category in Cat . More concretely, a double categoryconstists of a pair of categories, denoted E C and M C , which have the same objects. We denote morphismsin M C by and morphisms in E C by . This pair in endowed with a collection of squares, called distinguished squares . These are denoted A BC D (cid:3) f ′ fg ′ g . In each distinguished square, f, f ′ ∈ M C and g, g ′ ∈ E C . The squares satisfy compositional axioms, which sayin effect that gluing two squares horizontally or vertically gives another distinguished square. In addition, if f and f ′ are both isomorphisms then for any g, g ′ either both of the following squares exist, or neither does: A BC D (cid:3) ff ′ g g ′ B AD C (cid:3) f − f ′− g ′ g We sometimes write C = ( E C , M C ). When C is clear from context we omit it from the notation. Example . Let A be any category, and E and M two subcategories. We can define a double categorystructure ( E , M ) by letting the objects be the objects of C , the horizontal morphisms be given by M andthe vertical morphisms by E . We let distinguished squares be any subset of the commutative squares in A which satisfies appropriate closure conditions.In most cases of interest, the double categories we work with arise as in Example 2.2, so it is useful tointroduce language for these categories. Definition 2.3.
If a double category ( E , M ) arises from a situation as in Example 2.2 we say that A is an ambient category for ( E , M ). In such a case, the identity functor gives a natural isomorphism of categoriesiso E iso M .CGW-categories will be double categories equipped with extra data. Most of the data involves thespecification of the existence of certain distinguished squares. We define certain categories that come uprepeatedly in these specifications. Definition 2.4.
Let C = ( E , M ) be a double category. We write Ar (cid:3) E for the category whose objects aremorphisms A B in E , and whereHom Ar (cid:3) E ( A g B, A ′ g ′ B ′ ) = distinguishedsquares A A ′ B B ′ (cid:3) g g ′ EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 5
We have an analogous category Ar (cid:3) M . Note that every 2-cell in C appears uniquely as a morphism inAr (cid:3) E and Ar (cid:3) M .Now let D be any ordinary category. We write Ar △ D for the category whose objects are morphisms A B in D , and whereHom Ar △ D ( A f B, A ′ f ′ B ′ ) = commutativesquares A A ′ B B ′∼ = f ′ f . We now come to the definition of a CGW-category.
Definition 2.5. A CGW-category ( C , φ, c, k ) is a double category C = ( E , M ), an isomorphism of categories φ : iso E iso M and equivalences of categories k : Ar (cid:3) E Ar △ M and c : Ar (cid:3) M Ar △ E which satisfy:(Z) C contains an object ∅ which is initial in both E and M .(I) If f : A B is an isomorphism then all four of the following squares are distinguished:
A BB B (cid:3) f B f B A AA B (cid:3) A f A f A BA A (cid:3) f A A f − A AB A (cid:3) A f − f A . (M) Every morphism in the categories E and M is monic.(K) For every g : A B in E , k ( g : A B ) = A k/g g k B and there exists a (unique up to uniqueisomorphism) distinguished square ∅ AA k/g B (cid:3) g k g . Dually, for every f : A B in M , c ( A f B ) = A c/f f c B and there exists a (unique up to uniqueisomorphism) distinguished square ∅ A c/f A B (cid:3) f f c . (A) For any objects A and B there exist distinguished squares ∅ AB X (cid:3) and ∅ BA X (cid:3) . As isomorphisms can be considered to be “both e-morphisms and m-morphisms” we will generally drawthem as plain arrows.When it is clear from context, we write A k/B or A k instead of A k/f (and analogously for c ).The definition of a CGW-category is symmetric with respect to m-morphisms and e-morphisms. Thisduality is highly versatile and allows us to get symmetric results about e-morphisms and m-morphisms withno extra work. JONATHAN A. CAMPBELL AND INNA ZAKHAREVICH
Remark . Axiom (A) is used only to show that K ( C ) is an abelian group. Thus if in some case such aproperty is not necessary this axiom can be dropped and the rest of the analysis will still hold.Functors of CGW-categories must preserve all structure in sight. Definition 2.7. A CGW-functor of CGW-categories is a double functor F : ( E , M ) ( E ′ , M ′ ) whichcommutes with c and k . More concretely, F is a CGW-functor if the following two diagrams commute:Ar (cid:3) E Ar △ M Ar (cid:3) M Ar △ E Ar (cid:3) E ′ Ar △ M ′ Ar (cid:3) M ′ Ar △ E ′ k ck ′ c ′ Ar (cid:3) F Ar △ F Ar (cid:3) F Ar △ F The fact that c and k take distinguished squares to commutative triangles means that distinguished squaresare equifibered (the vertical arrows have equal “kernels” given by k ) and equicofibered (the horizontal arrowshave equal “cokernels” given by c ). Note that by Axiom (K), c and k are mutual inverses on arrows.When φ , c and k are clear from context we omit them from the notation. When C has an ambient category A and φ is the identity functor, we omit φ from the notation.We now prove some technical consequences of the axioms. Lemma 2.8.
For any A , the morphism f : ∅ A has f c = 1 A . Dually, the morphism f : ∅ A has f k = 1 A . The following lemma is the most important of the technical results. It states that e-morphisms andm-morphisms can be commuted past one another using distinguished squares. This is what will allow the Q -construction in Section 3 to work. Lemma 2.9.
For any diagram A f B g C there is a unique (up to unique isomorphism) distinguishedsquare A BD C (cid:3) f g . The analogous statement holds for any diagram A f B g C .Proof. As the categories M and E are symmetric in the definition of a CGW-category it suffices to checkthe first part. Given a diagram as in the statement of the lemma, we can apply c to the first morphism toobtain a diagram A c/f f c B g C. This diagram represents a morphism ( A c/f f c B ) g ( A c/f gf c C ) in Ar △ E . Applying c − to thismorphism produces a distinguished square A B ( A c/f ) k/gf c C (cid:3) f g , where we have used that c and k are inverses on objects.To check that this distinguished square is unique, suppose we are given any other such square A BD C (cid:3) ff ′ g . EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 7
Applying c to this square produces a morphism( A c/f f c B ) g ( D c/f ′ f ′ c C ) ∈ Ar △ E . Since the square is distinguished, we must have A c/f ∼ = D c/f ′ ; if we choose D c/f ′ = A c/f the codomainof the above morphism becomes A c/f gf c C . Thus any such distinguished square is mapped by c to theoriginal diagram; since c is an equivalence of categories, the square must be canonically isomorphic to thesquare produced above. (cid:3) Lemma 2.10.
Given any composition
C B A there is an induced map B c/A C c/A such that the triangle B c/A C c/A A commutes.Proof. We begin by applying the equivalence of categories given by k − from Axiom (K). Since k − = c onobjects, we have the induced diagram C c/B C c/A B A h (cid:3) We now apply the equivalence given by c to produce the diagram B c/A = ( C c/B ) c/h C c/A A. (cid:3) We conclude this section with a pair of definitions that will be useful in later sections.
Definition 2.11.
Let C = ( E , M , φ, c, k ) be a CGW-category. A CGW-subcategory is a double subcategory
A ⊆ C such that ( A , φ | A , c | A , k | A ) is also a CGW-category. Definition 2.12.
We say that a CGW-subcategory A of a CGW-category ( C , φ, c, k ) is closed under subob-jects if for any morphism B C ∈ M , if C ∈ A then B ∈ A . We say that A is closed under quotients iffor any morphism B C ∈ E , if C ∈ A then B ∈ A . We say that A is closed under extensions if for everydistinguished square A BC D (cid:3) if A , B and C are in A then so is D .3. The K -theory of a CGW-category We are now ready to define the K -theory of a CGW-category. The construction exactly follows Quillen’s Q -construction [Qui73] for exact categories. After the introduction of the definition, the rest of the sectionis taken up by noting some useful technical results and providing the standard presentation for the group K ( C ). JONATHAN A. CAMPBELL AND INNA ZAKHAREVICH
Definition 3.1.
For an CGW-category ( C , φ, c, k ) we define K ( C ) = Ω | Q C| , where Q C is the category with objects: the objects of C , morphisms: morphisms A B are equivalence classes of diagrams A f X g B, where f ∈ E and g ∈ M . Two diagrams A f X g B and A f ′ X ′ g ′ B are considered equivalent if there exists a diagram XA BX ′ ff ′ gg ′ ∼ = where the left-hand triangle commutes in E and the right-hand triangle commutes in M . composition: defined using Lemmma 2.9. More concretely, given two equivalence classes of diagramsrepresented by A f X g B and B f ′ Y g ′ C there exists a unique (up to unique isomorphism) distinguished square X ZB Y (cid:3) f ′′ f ′ g g ′′ . The composition of the two diagrams is defined to be the class of diagrams represented by A f ′′ f Z g ′ g ′′ C. The basepoint is generally taken to be ∅ . Remark . Although we have defined K -theory for CGW categories, the K -theory of a double category isdefined for any double category satisfying Lemmma 2.9.As with any definition of K -theory, the first step is to check that it gives the desired group on K . Theorem 3.3. K ( C ) is the free abelian group generated by objects of C , modulo the relation that for anydistinguished square A BD C (cid:3) f g we have [ D ] + [ B ] = [ A ] + [ C ] .Proof. There are two ways to proceed. One could prove this by showing that K ( C ) is equivalent to somevariant of the S • construction, and proceeding from there, or one could mimic Quillen’s original proof that π ( BQ C ) = K ( C ) for exact categories. We opt for the latter, again to emphasize the analogy with exactcategories.We follow a more modern version of the proof (see, e.g. [Wei13, Proposition IV.6.2]). EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 9
In what follows we let [
A X B ] denote the equivalence class of a morphism
A B in π ( BQ C ).The notation [ A B ] corresponds to the morphism [ A B = B ] and similarly [ A B ] correspondsto [
A A B ]. For shorthand, we write [ A ] = [ ∅ A ].The morphisms ∅ A form a maximal tree in BQ C . By [Wei13, Lemma IV.3.4], the fundamental group π ( BQ C ) is generated by the morphisms of BQ C , modulo the relations [ ∅ A ] = 1 and [ f ] · [ g ] = [ f ◦ g ]for composable morphisms in Q C . We proceed by a series of reductions to get the set of generators andrelations in the theorem.From the definition of Q we have [ A B ][ B C ] = [
A C ]. In particular, since [ ∅ X ] = 1 in π ( BQ C ) for all objects X , [ A B ] = 1 for all such morphisms.We begin by noting that by definition[
D A ][ A B ] = [
D A B ] . Now consider [
A B ][ B C ]. By Lemmma 2.9 there exists a distinguished square
A BD C (cid:3) which implies the relation [
A B ][ B C ] = [
A D ][ D C ]via the composition relation. Note that each distinguished square produces such a relation. Since allmorphisms in M are equal to the identity, this reduces to the equation[ B C ] = [
A D ]for all distinguished squares. We have now shown that π ( BQ C ) has as generators the morphisms of E , withrelations induced by composition and distinguished squares.Since(3.4) [ A ][ A A ] = [ A ] .π ( BQ C ) is generated by the elements [ A ]. This expression also eliminates the composition relation. We cansubstitute for both sides in the relations induced by the distinguished squares to get[ B ] − [ C ] = [ A ] − [ D ] . This gives the desired presentation of K ( C ).It remains to check that K ( C ) is abelian; in other words, that [ A ][ B ] = [ B ][ A ]. The relations imposed bythe squares in Axiom (A) state that [ A ][ B ] = [ X ] = [ B ][ A ] , as desired. (cid:3) The rest of this section is devoted to some technical lemmas exploring the properties of this Q -construction.The first identifies the isomorphisms in Q C via their components. Lemma 3.5. If α : A B is an isomorphism inside Q C for a CGW-category C represented by A f X g B then both f and g are isomorphisms in C .Proof. Suppose that the inverse of α is represented by B f ′ Y g ′ A. Then the composition is represented by a diagram
A X BZ YA f gf ′′ f ′ g ′′ g ′ (cid:3) Since this is equivalent to 1 A , f ′′ f is an isomoprhism. Since f ′′ is monic and f is its right inverse, it mustbe an isomorphism; thus f is an isomorphism. Doing the composition in reverse, we see that g has a rightinverse and thus must also be an isomorphism. (cid:3) The next lemma illustrates that we can think of a morphism in Q C as a set of “layers” inside M . Thisallows us to think about the Q -construction in CGW-categories analogously to the way that Quillen originallythought about exact categories in [Qui73]. Lemma 3.6.
For any CGW-category B and any B ∈ B , the category Q B /B is equivalent to the category L B B with objects: diagrams B B B in C , morphisms: commutative diagrams B B BB ′ B ′ B In particular, Q B /B is a preorder for any B .Proof. It suffices to prove the first part of the lemma; the second follows from the definition of L B B andaxiom (M).We define a functor κ : Q B /B L B B . An object of Q B /B is a diagram B g B B . We send thisto the diagram B k g k B B . Seeing that this extends to a functor is a bit more complicated. Supposethat B g B f B and B ′ g ′ B ′ f ′ B are two objects of Q B /B , and suppose that we are given a morphism between them. This morphism consistsof an object C ∈ B and a diagram B C B ′ B B ′ B g h ′ h f g ′ f ′ (cid:3) Applying c − to the upper-left triangle, this diagram corresponds to a unique diagram EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 11 B k/C C B ′ B k/B B B ′ B h ′′ h f g ′ f ′ (cid:3) (cid:3) Applying k , this time to the two distinguished squares on the top, gives us a unique diagram B ′ k/B ′ = ( B k/B ) k/h ′′ B k/B B B ′ B f f ′ This can be rearranged into a diagram B k/B B BB ′ k/B ′ B ′ B ff ′ as desired.The inverse equivalence is given by sending a diagram B B B to B c/B B B . ByAxiom (K) these two functors give inverse equivalences. (cid:3) Examples
In this section we give several motivating examples of CGW-categories. All double categories in ourexamples have ambient categories, so we omit mention of φ . Example . Let A be an exact category. Let ( C , c, k ) be given by E = { admissible epimorphisms } op and M = { admissible monomorphisms } ;The distinguished squares are stable squares: those squares that are both pushouts and pullbacks in A . Theequivalence k is given by mapping every admissible epimorphism to its kernel; the equivalence c is given bytaking every admissible monomorphism to its cokernel.We check the axioms explicitly.(Z) The zero object is initial in M and terminal in E , so it is initial in both M and E .(I) This follows directly from the definition.(M) This holds by definition.(K) k and c give the correct equivalences, since distinguished squares are both equifibered (since theyare pullbacks) and equicofibered (since they are pushouts).e(A) This holds with X = A ⊕ B .With this definition, BQ C = BQ A , so K ( C ) = K ( A ).Thus an exact category gives rise to a CGW-category with the same K -theory. However, there areexamples of CGW-categories which are not exact. Example . Consider the category
FinSet ∗ of based finite sets. We define a CGW-category ( C , c, k ) bysetting M = { injections } and E = n f : A B (cid:12)(cid:12)(cid:12) f | f − ( B \{∗} ) is a bijection o op . The distinguished squares are the pushout squares. We define k by taking f : A B to f − ( ∗ ) A .We define c by taking g : A B to B B \ g ( A ), with the elements not in the image of g mapping tothemselves, and everything else mapping to the basepoint.That axioms (Z), (I), (M), and (A) are satisfied is direct from the definition. To see that (K) is sat-isfied, note that the distinguished squares are pullback squares in the underlying category. In particular,in a distinguished square the preimages of the basepoint of the two vertical maps are isomorphic. Dually,the complements of the two injections horizontally are also isomorphic, since g is injective away from thebasepoint.We have K ( C ) ≃ S . To see this, note that Ω BQ C is a two-fold subdivision of the S • -construction for theWaldhausen category FinSet ∗ with injections as the cofibrations (see Section 7). Thus K ( C ): = Ω BQ FinSet ∗ ≃ K Wald ( FinSet + ) ≃ S where the last equivalence is by Barrat-Priddy-Quillen [BP72].One of the advantages of CGW-categories is the observation that the contravariance in the E -direction isnot necessary. Example . Consider the category
FinSet . We define a CGW-category ( C , c, k ) by setting E = M = { injections } . The distinguished squares are the pushout squares; note that since all morphisms are injections, they arealso pullback squares. The equivalences c and k are given by taking any injection A B to the inclusion B \ A B .That axioms (Z), (I), (M), and (A) are satisfied is direct from the definition. To see that (K) is satisfied,note that since distinguished squares are pushouts, the complements of the images in the horizontal mapsare isomorphic; the same holds dually for the vertical maps.In this case we also have K ( C ) ≃ S . To see this, note that there is an equivalence of CGW-categoriesbetween ( FinSet , c, k ) and (
FinSet ∗ , c, k ) from Example 4.2 given as follows. An injection [ i ] [ j ]considered as an element of E ⊂
FinSet corresponds to an injection [ i ] + [ j ] + in FinSet ∗ . An injection u : [ i ] [ j ] considered as an element of M ⊂
FinSet corresponds to a surjection [ j ] + [ i ] + by taking m ∈ [ j ] to u − ( m ) and the rest of [ j ] to the distinguished basepoint.We can also improve the intuition from the finite sets example to get a CGW-category structure on thecategory of varieties. Example . Let C = Var E = { open immersions } and M = { closed immersions } . We let both c and k take a morphism to the inclusion of the complement. The distinguished squares A BC D (cid:3) g f are the pullback squares in which im f ∪ im g = D . Axiom (Z) is satisfied by the empty variety. Axiom (I)holds by definition. Axiom (M) is verified by noting that open and closed immersions satisfy base change inthe category of varieties. Axiom (A) holds by setting X = A ∐ B . To see that Axiom (K) holds, consider adistinguished square A BC D (cid:3) . By definition, D r C ∼ = B r A , since the image of B in D contains the complement of the image of C . Thedual statement for e-morphisms holds as well.Then K ( Var ) is equivalent to the K -theory of varieties defined in [Cam]; for a more detailed discussion,see Section 7. EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 13
The CGW-category of varieties includes into the larger category of reduced schemes of finite type via aCGW-functor:
Example . Let
Sch rf be the category of reduced schemes of finite type, with morphisms the compositionsof open and closed immersions. We define the E -morphisms to be the open immersions and the M -morphismsto be the closed immersions.We can also restrict attention just to smooth varieties. Example . The category
Var sm/k of smooth varieties can be given a CGW-structure. We set the m-morphisms to be closed immersions with smooth complements, and the e-morphisms to be open immersionswith smooth complements. Thus
Var sm/k is a sub-CGW-category (but not a full sub-CGW-category) of
Var /k . 5. ACGW-Categories
A CGW-category behaves like an exact category. In order to create categories that are analogous toabelian categories (with the goal of proving Quillen’s d´evissage and localization) we need to assume someextra conditions. The extra conditions amount to the requirement that certain “pushout-like” objects existand are compatible with c and k ; in geometric settings this corresponds to certain gluings of objects. Definition 5.1. An enhanced double category is a double category C with two notions of 2-cell, called the distinguished and commutative squares. Forgetting either of the sets of squares produces a double category,and all distinguished squares are commutative. We denote distinguished squares with (cid:3) and commutativesquares with (cid:9) .We write Ar (cid:9) M for the category whose objects are morphisms in M and whose morphisms are com-mutative squares in C . We write Ar × M for the category whose objects are morphisms in M and whosemorphisms are pullback squares in M . Note that Ar (cid:3) M is a subcategory of Ar (cid:9) M and Ar △ M is asubcategory of Ar × M (since all morphisms in M are monic). Definition 5.2. A pre-ACGW-category ( C , φ, c, k ) is an enhanced double category C which is a CGW-category when the commutative squares are forgotten, and in which the following extra axioms are satisfied:(P) M and E are closed under pullbacks.(U) The functors c and k extend to equivalences of categories c : Ar (cid:9) M Ar × E and k : Ar (cid:9) E Ar × M . These are compatible in the sense that for any diagram
A C B there exists a uniqueisomorphism ϕ : ( A c × C B ) k/pr B ( A × C B k ) c/pr A such that the square ( A × C B k ) c ( A c × C B ) k BA C ϕ (cid:9) is a commutative square.We write A ⊘ C B def = ( A c × C B ) k/pr B ∼ = ( A × C B k ) c/pr A , so that we have a “mixed pullbacksquare” A ⊘ C B BA C (cid:9) (S) Suppose that we are given a pullback square A × C B AB C ℓ in M . The object X = A ∪ A × C B B exists. The induced square X c/C B c/C A c/C ( A × C B ) c/C is a pushout square.The dual of this statement also holds. Definition 5.3. An ACGW-category is a pre-ACGW-category (
C, φ, c, k ) such that the following conditionholds:(PP) For every diagram
C A B there exists a unique (up to unique isomorphism) choice of square
A BC B ⋆ A C f g ′ g f ′ which is a pullback square, and whose image under k − induces an isomorphism A c/B ∼ = C c/B⋆ A C .This is functorial in the sense that given a diagram C A f B f B ′ we have ( f f ) ′ = f ′ f ′ . In addition, this is compatible with distinguished squares in the sense thatgiven a diagram C A BC ′ A ′ B ′ (cid:3) (cid:3) there is an induced map B ⋆ A C B ′ ⋆ A ′ C ′ . These maps are compatible with compositions ofdistinguished squares.The dual statement for e-morphisms holds as well. Example . Let A be an abelian category. Then A defines an ACGW-category for which M is the categoryof monomorphisms, E is the opposite category of the epimorphisms, distinguished squares are stable squaresand commutative squares are commutative squares. Here, the “mixed pullback” of a diagram A B C is the factorizarion of the morphism
A C into an epic followed by a monic.
Example . The category
Var is a pre-ACGW-category. Here we define the commutative squares to bethe pullback squares.We check the axioms in turn. Axiom (P) holds because varieties are closed under pullbacks. In order tocheck Axiom (U) it suffices to check that given a variety X and an open subvariety U and a closed subvariety Z , we have Z \ ( Z ∩ ( X \ U )) ∼ = U ∩ (( X \ Z ) ∩ U ) . This is true because it is true in the underlying topological spaces, where each one is simply Z × X U . Axiom(S) holds because it holds in the underlying topological spaces. EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 15
Counterexample . The CGW-category
Var sm/k is not a pre-ACGW-category, since it is possible that theintersection of smooth subvarieties is not smooth. This means that the m-morphisms are not closed underpullbacks. Example . The category
Sch rf is an ACGW-category, with the commutative squares being pullbacksquares. With this definition we can consider Var a pre-ACGW-subcategory of the ACGW-category
Sch rf .That Axioms (P), (U), and (S) hold follows identically as for the case of varieties.Thus it remains to check Axiom (PP). For both open and closed embeddings, we define ⋆ to be thepushout in the category of schemes. The pushout of schemes along open embeddings produces a square ofopen embeddings by the definition of a scheme; the pushout of schemes along closed embedding producesa square of closed embeddings of schemes by [Sch05, Corollary 3.9]. Note that these are not pushouts inthe categories of closed/open embeddings; these are pushouts in the entire category of schemes. That thissatisfies the conditions of (PP) follows from the universal property of pushouts.We finish this section with a couple of technical lemmas which will be useful later. Lemma 5.8.
Let C be a pre-ACGW category. Given a diagram C B AC ′ B ′ A ′ (cid:9) where C ∼ = C ′ × B ′ B there exists a cube C DB AC ′ D ′ B ′ A ′ where the top and bottom squares are distinguished, the front and back squares and the left and right squaresare pullback squares.The statement with the roles of e-morphisms and m-morphisms swapped also holds.Proof. Let D = ( C k/B ) c/A and D ′ = (( C ′ ) k/B ′ ) c/A ′ Applying c − to the right-hand square produces a diagram C k/B B A ( C ′ ) k/B ′ B ′ A ′ (cid:9)(cid:9) which corresponds, under c , to the pullback square on the right of the cube. Lemma 2.9 shows that thesquares on the top and bottom of the cube must be distinguished. To finish the proof of hte lemma it remainsto check that the back face of the cube is distinguished. To prove this it suffices to check that, after applying c to the m-morphisms in the diagram, it corresponds to a pullback square. This is a straightforward diagramchase using the fact that all morphisms are monic. (cid:3) Lemma 5.9.
Let C be a pre-ACGW category. In any commutative square A BC D (cid:9) ff ′ if f ′ is an isomorphism, so is f .Proof. Apply · k vertically. This produces a pullback square A k B k C D ( f ′ ) k f Since f ′ is an isomorphism, ( f ′ ) k must be, as well. Thus the commutative square is mapped to an iso-morphism inside Ar × M ; in particular, both horizontal morphisms in the commutative square must beisomorphisms. Thus f ′ is an isomorphism, as desired. (cid:3) D´evissage
We can now prove a direct analog to Quillen’s devissage [Qui73, Theorem 5.4].
Theorem 6.1.
Let A be a full pre-ACGW-subcategory of the pre-ACGW-category ( B , φ, c, k ) , closed undersubobjects and quotients, such that the inclusion A ∩ E E creates pushouts. Suppose that for all objects B ∈ B there is a sequence ∅ = B B · · · B n = B such that B c/B i i − is in A for all i = 1 , . . . n . Then K ( A ) ≃ K ( B ) Proof.
The proof proceeds exactly as in [Qui73]. Let ι : A B be the inclusion of A into B . We would like ι to give a homotopy equivalence BQ A BQι BQ B . By Quillen’s Theorem A it is enough to show that Qι /B is contractible for any B ∈ B . Note that since A isclosed under subobjects, Qι /B is the full subcategory of Q B /B of those objects A B B where A ∈ A . By Lemma 3.6, Q B /B is a preorder, and thus Qι /B is also a preorder.By the hypothesis of the theorem, there exists a sequence ∅ = B B · · · B n = B with B c/B i i − ∈ A for all i = 1 , . . . , n . We prove that Qi /B n is contractible by induction on n .We have B ∈ A ; in this case Qι /B is contractible, since it has the terminal object B B B .To prove the inductive step it suffices to show that for any h : B B ′ with B c ∈ A the map Qι /B Qι /B ′ induced by postcomposition is a homotopy equivalence. Let L A B B be the full subcategory of L B B containingthose objects B B B where B c/B ∈ A . By Lemma 3.6 it suffices to check that the functor ι : L A B B L A B ′ B induced by postcomposition with h is a homotopy equivalence.Let B B B ′ be any object of L A B ′ B . We have the diagram EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 17 B × B ′ B B × B ′ B BB B B ′ g ′ h where both squares are pullback squares. We define functors r : L A B ′ B L A B B r ( B B B ′ ) = B × B ′ B g ′ B × B ′ B B.s : L A B ′ B L A B ′ B s ( B B B ′ ) = B × B ′ B B B ′ . Note that if s is well-defined (so ( B × B ′ B ) c/B ∈ A ) then so is r , because ( B × B ′ B ) c/g ′ is a subobject of( B × B ′ B ) c/B . Thus we just need to check that s is well-defined.First, we note that by Axiom (U) there exists a map ( B × B ′ B ) c/B B c/B ′ ; since B c/B ′ ∈ A , it followsthat ( B × B ′ B ) c/B must be, as well. Now by Axiom (S), ( B × B ′ B ) c/B ∼ = B c/B ∪ Y ( B × B ′ B ) c/B , where Y is the pushout constructed in Axiom (S). By assumption B c/B ∈ A and by the above ( B × B ′ B ) c/B ∈ A ,so Y is also in A . Thus ( B × B ′ B ) c/B ∈ A , and s is well-defined, as desired.Redrawing the above diagram, we have the following diagram: B B B ′ L A B ′ B B × B ′ B B B ′ sB × B ′ B B × B ′ B B ′ rι The upper row of squares gives a natural transformation 1 L A B ′ B s ; the lower row gives a natural trans-formation rι s . Since natural transformations realize to homotopies, we see that rι is homotopic to theidentity on L A B ′ B . On the other hand, ιr is equal to the identity on L A B B , so these produce a homotopyequivalence of spaces, as desired. (cid:3) We can now apply this theorem to compare the K -theory of varieties to the K -theory of reduced schemesof finite type. Example . We use the dual of Theorem 6.1 to prove that K ( Var ) ≃ K ( Sch rf ). Var is a sucategory of
Sch rf closed under subobjects and quotients; the inclusion Var ∩M M createspushouts since the pushout of varieties along closed embeddings is a variety [Sch05, Cor. 3.9]. To apply thetheorem we must show that for every reduced scheme of finite type X there exists a filtration X X · · · X n = X such that X i r X i − is a variety for all i . Since X is of finite type there exists a finite cover of X by affineopens U , . . . , U n ; each of these is reduced since X is and separated because each is affine. We then define X i = i [ j =1 U i . This gives a finite open filtration of X ; it remains to show that X i r X i − is a variety for all i . Note that X i r X i − = U i r S i − j =1 ( U j ∩ U i ). This is reduced, separated and of finite type, and is thus a variety, asdesired. Relationship with the S • -construction In this section we relate our Q -construction to the S • -construction of Waldhausen [Wal85]. We will showthat the Q -construction is equivalent to the construction defined for Var /k in [Cam]. In addition, in thesequel we will need fiber sequences in K -theory similar to Waldhausen’s [Wal85, Thm. 1.5.5]. While wecould develop these using the Q -construction, it is expedient not to, and instead we appeal to [Cam]. Thesefiber sequences will be critical in Section 9.In order to compare our Q -construction with a version of the S • -construction, we need to briefly reviewsimplicial subdivision. The version of simplicial subdivision that we need is described in [Wal85, Sec. 1.9]or in [Seg73, App. 1] Definition 7.1.
Let [ n ] op ∗ [ n ] ∼ = [2 n + 1] be the ordered set n < n − < · · · < < < < < · · · < n − < n Then there is a functor sd : ∆ ∆ given on objects by [ n ] [ n ] op ∗ [ n ] ∼ = [2 n + 1] and on maps by f f op ∗ f . Definition 7.2. [Seg73, App. 1] We define the edgewise subdivision of the simpicial set X • to be thesimplicial set sd ∗ X • . Theorem 7.3 ([Seg73]) . For any simplicial set X • , | X • | ∼ = | sd ∗ X • | Remark . There are other versions of edgewise simplicial subdivision, see for example [BHM93, Sec.1],but Segal’s is the one with the most convenient variance properties.
Definition 7.5.
Let ( C , M , E ) be a CGW-category. Consider the category Fun([1] , [ n ]); for an object C ∈ Fun([1] , [ n ]) we write C ij for C ( i j ).. We define a simplicial category S • C to have as n -simplices thefull subcategory of objects C such that(1) C jj = ∅ for all j , and(2) Every subdiagram C ik C ℓk C ij C ℓj (cid:3) for i ≤ ℓ and j ≤ k is a distinguished square.The face and degeneracies are defined as in the usual S • -construction: the i th face map is deleting the i throw and i th column, and the degenercies are given by reptition. The 0-th face is given by applying c inthe appropriate direction. (For more on the traditional S • -construction, see [Wal85, Section 1.3]; for a moreexplicit description of how this plays out in the case of varieties, see the e S • -construction in [Cam, Definition3.31].) Example . When C = Var , then K S ( Var ) is exactly the e S • construction of [Cam]. Definition 7.7.
Given a CGW category ( C , M , E ) define K S ( C ): = Ω | ob S • C| Remark . In [Cam], the author introduced the e S • construction, which is a version of the Waldhausenconstruction that works on SW-categories [Cam, Defn 3.23]. These categories are meant to encode cuttingand pasting, just as CGW categories do. In fact, in that paper there are three notions of such categoriesthat appear: 1. pre-subtractive category 2. subtractive categories and 3. SW-categories. Pre-subtractiveare closely related to CGW-categories; they are categories where one can define a higher geometric objectthat encodes cutting and pasting. Subtractive categories correspond to ACGW-categories: certain pushoutsand pullbacks are required to exist. Finally, SW-categories, like Waldhausen categories, are allowed to haveweak equivalences other than isomorphisms. Subtractive categories satisfy the axioms for ACGW-categories,and in this case the corresponding S • constructions are equivalent and, in fact, equal. An ACGW-categorywhere the distinguished squares are cartesian in the underlying category A , is an SW-category, and we EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 19 may use the full machinery of SW-categories. This is true, for example, for
Var /k and FinSet ∗ . However,CGW-categories are more general. Theorem 7.9.
Let ( C , M , E ) be a CGW category. Then there is a weak equivalence of topological spaces K S ( C ) K ( C ) induced by a map of simplicial sets S • C Q C . The equivalence above is one of topological spaces, not of infinite loop space or spectra. While in manycases the equivalence are equivalences of infinite loop spaces, that is not true in this generality (for example,smooth varieties cannot be delooped in the way described in [Cam] since it relies on the existence of pushouts).We hope to address deloopings in future work.asdf
Proof.
The definitions are designed to work exactly as in Waldhausen [Wal85, Sect 1.9]. Let iQ C be the adouble category where vertical morphisms are isomorphisms in Q C and horizontal morphisms are morphismsin Q C . Taking the nerve in the horiztonal direction, we get a simplicial category iQ • C : the rows in thediagram below are elements of the category and vertical arrows are the morphisms: Q Q Q · · · Q ′ Q ′ Q ′ · · · ∼ = ∼ = ∼ = There is an equivalence | iQ • C| ≃ −→ | Q ǫ C| given by Waldhausen’s Swallowing Lemma [Wal85, Lem. 1.6.5].We also note that the composition of n morphisms in iQ C are equivalences classes of diagrams of the shapeFun([1] , [ n ] op ) C where all full squares in the diagram are distinguished.Let sd iS • C be the simplicial category we obtain from edgewise dividing the S • -construction. There isnow a map sd iS • C iQ • C defined as follows. The simplicial set sd iS n C is a functor from Fun([1] , [ n ] op ∗ [ n ]) to C satisfiying theconditions in the definition of S • C . From this, we obtain a diagram in C of shape Fun([1] , [ n ] op ) via restrictingto the subcategory of Fun([1] , [ n ] op ∗ [ n ]) where 0 may only land in [ n ] op and 1 may only land in [ n ], andif 0 is mapped to j and 1 is mapped to i , then j ≥ i . We then consider equivalences classes of diagrams ofthis type. These are exactly elements in iQ • C — the relevant squares are distinguished. This functor is anequivalence of categories, and so an equivalence upon realization. Altogether we have | iS • C| ∼ = −→ | sd iS • C| ≃ −→ | iQ • C| ≃ ←− | Q C| . Finally, we have the commutative diagram S • C sd S • C Q C iS • C sd iS • C iQ C ≃≃ ≃ ≃ ≃ ≃ where we know that all of the indicated arrows are weak equivalences, and so the remaining arrow is a weakequivalence. The composite across the top S • C Q C is thus a weak equivalence. Upon realization andtaking loop spaces, this gives the statement of the theorem. (cid:3) We now go on to prove a version of Waldhausen’s cofiber theorem [Wal85, Prop. 1.5.5] using a mix of the S • -construction, and the Q -construction. Remark . As pointed out above, this fiber sequence could be proved internally to the Q -construction.However, that would require proving an additivity theorem for the Q -construction. While this is not difficult,it is faster to proceed as below. Definition 7.11.
Let A be an ACGW-category. We define a CGW-structure on S n A . We give S n A distinguished families of M and E morphisms as follows. First, recall a diagram F ∈ S n A is given bya functor Fun([1] , [ n ]) A , and accordingly, a morphism F G in S n A is given by a functor [1] × Fun([1] , [ n ]) A . M -morphisms: A map
F G in S n A is in M if each restriction to [1] A is in M and eachrestriction [1] × [1] A is in Ar × ME -morphisms: A map
F G in S n A is in E if each restriction [1] × [1] A is in Ar (cid:9) M Distinguished squares:
These are functors [1] × × Fun([1] , [ n ]) A whose restrictions to [1] × aredistinguished squares in A .With the definitions above, the following is tedious, but straightforward. Lemma 7.12. S n A , with the structure from Definition 7.11, is an ACGW-category. Using this we can define the relative S • -construction. Definition 7.13.
Let B be an ACGW-category and A ⊂ B be a sub-ACGW-category. Define S n ( B , A ) viathe pullback S n ( B , A ) S n +1 B S n A S n B The category S n ( B , A ) inherits the structure of an ACGW-category. We now invoke the additivity theoremfor the S • -construction to deduce the additivity theorem for the Q -construction. Lemma 7.14.
Let B be an ACGW-category and A ⊂ B be a sub ACGW-category. Then we have the weakequivalence QS n ( B , A ) Q B × QS n A ∼ Proof.
We have the following commutative diagram QS n ( B , A ) Q B × QS n A S • S n ( B , A ) S • B × S • S n A ∼ ∼∼ The vertical arrows are weak equivalences by Theorem 7.9 and the bottom arrow is a weak equivalence by[Cam, Prop. 5.5]. Thus, the top map is a weak equivalence. (cid:3)
Finally, we obtain the desired fiber sequence.
Proposition 7.15.
Let B be an ACGW-category and A a sub-ACGW-category of B . Then the following isa homotopy fiber sequence: S • B QS • ( B , A ) QS • A . Proof.
By [Wal78, Prop. 5.1] since S n B QS n ( B , A ) QS n A S n B Q B × QS n ( B , A ) QS n A ∼ EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 21 is a fiber homotopy fiber sequence (in fact a trivial fiber sequence), S n B QS n A is constant, and QS n A is connected, the geometric realization of S n B QS n ( B , A ) QS n A is a homotopy fiber sequence. (cid:3) Localization of ACGW-categories
In this section we state the new definition necessary to state the localization theorem. The goal of alocalization theorem is to identify the homotopy cofiber of the map K ( A ) K ( C ) induced by the inclusionof a sub-CGW-category. In order to prove the cleanest version of the theorem it is necessary to make extraassumptions about the structure of A and C , and thus passage to ACGW-categories is necessary. In addition,in order to ensure that objects in A can be worked with easily, we assume some nice closure properties on A (similar to the closure properties assumed by Quillen).Let C = ( E , M ) be an ACGW-category, and let A be a full ACGW-subcategory closed under subob-jects, quotients and extensions, as defined in Definition 2.12. The first step towards stating localization isidentifying the CGW-category whose K -theory we hope to be the cofiber. Definition 8.1.
Let
A B be a morphism in M . We write if A c ∈ A . We define analogouslyLet C\A be the double category with objects: the objects of C , m-morphisms: A morphism
A B is an equivalence classes of diagrams in C A A ′ X B ′ B. If there exists a diagram in C X B ′ A ′ A BA ′′ X ′ B ′′ (cid:9) (cid:9)(cid:9) (cid:9) ∼ = then the two formal compositions around the outside are considered equivalent. Note that theright-most square with the isomorphism in the middle is the same square that determines when twomorphisms in Q C are equivalent.Composition is defined via a similar type of diagram, commuting the different types of morphismspast one another. e-morphisms: A morphism
A B is an equivalence class of diagrams in C A A ′ X B ′ B. The equivalence relation between these is defined to be the dual condition to the condition on m-morphisms. distinguished squares:
The distinguished squares are generated by the distinguished squares in C and axiom (I). For a more detailed description, see Appendix A. In this section we will often be working with morphisms in
C\A as represented by diagrams in C . As thesecategories have the same objects this can get confusing. To help with this, we denote morphisms in C byarrows with straight shafts, and morphisms in C\A by morphisms with wavy shafts. We can thus say thatan m-morphism
A B in C\A is represented by a diagram
A A ′ X B ′ B in C .We define c : Ar (cid:3) M Ar △ E by c ( A B ) = c C ( B ′ B ), and k : Ar (cid:3) E Ar △ M by k ( A B ) = k C ( B ′ B ).There is a functor of double categories s : C C\A which takes each object to itself and takes everymorphism to itself.
Remark . As currently defined,
C\A does not have the structure of a CGW-category, as we do not have adefinition of how to extend c and k to morphisms. Proving that such a structure exists appears to require adevelopment of a theory of a left calculus of fractions for a double category. As this is far beyond the scopeof this paper, we state as a condition of the localization theorem that C\A extends to a CGW-category ina fashion compatible with the CGW-structure on C and the functor s : C C\A and show that this worksfor our relevant examples. In future work we hope to simplify these conditions.If
C\A is a CGW-category then by definition the functor s is a CGW-functor.Before we state the main theorem, we need some auxillary definitions. Definition 8.3.
Let V be an object in C\A . The category I mV has as its objects pairs ( N, φ ), where N ∈ C and φ : sN V is an isomorphism in C\A . A morphism (
N, φ ) ( N ′ , φ ′ ) is an equivalence class ofdiagrams g : N g e Y g m N ′ (where diagrams are allowed to differ by an isomorphic choice of Y ) suchthat φ ′ s ( g ) = φ . Here, s ( g ) is considered as an isomorphism in C\A . Composition is defined using mixedpullbacks.The category I eV is defined analogously with the roles of m-morphisms and e-morphisms swapped.If I mV is filtered we say that A is m-well-represented in C . Dually, if I eV is filtered we say that A is e-well-represented in C .We think of I mV as the category of representatives inside C of an isomorphism class of objects in C\A .When this category is filtered it means that representatives of V can always be chosen compatibly, at leastin the m-morphism direction. Definition 8.4.
Suppose that for every diagram
A B C in C there exists a commutative square A ′ BA ′ C (cid:9) such that A ′ B factors through A B . Then we say that A is m-negligible in C . If the same statementholds with the m-morphisms and e-morphisms swapped, we say that A is e-negligible in C .Negligibility is a “dual” notion to well-representability. Whereas well-representability states that repre-sentatives can always be compatibly combined, negligibility says that certain representatives can be ignored.If A is m-negligible in C this means that we never have to think about e-components of morphisms inside Q C ; all such morphisms can be represented (up to commutative square) purely as an m-morphism.We are now ready to state the CGW version of localization. Theorem 8.5.
Suppose that C is an ACGW-category and A is a sub-ACGW-category satisfying the followingconditions: (W) A is m-well-represented or m-negligible in C and A is e-well-represented or e-negligible in C . EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 23 (CGW)
C\A is a CGW-category. (E)
For two diagrams
A X B and
A X ′ B which represent the same morphism in C\A there exists a diagram
B C and an isomorphism α : X ⊘ B C X ′ ⊘ B C such that the induceddiagram A X ⊘ B CX ′ ⊘ B C C α commutes. The same statement holds with e-morphisms and m-morphisms swapped.Then the sequence K ( A ) K ( C ) K ( C\A ) is a homotopy fiber sequence. We postpone the proof of Theorem 8.5 until Section 10; in this section we focus on two applications ofthe theorem.The first application is a sanity check, showing that in the case of an abelian category the theorem is thesame as Quillen’s localization [Qui73, Theorem 5.5].
Example . Let C be an abelian category, considered as an ACGW-category. Then C\A is exactly theabelian category C / A , considered as an ACGW-category. This can be seen by noting that a morphism in C whose kernel and cokernel are in A is monic in C / A exactly when it can be represented as a diagram as inthe description above, and similarly for epics. Since C / A is abelian it immediately follows that C\A mustbe a CGW-category.We must now check condition (W); we will show that A is both m- and e-well-represented in C . Notethat by symmetry it suffices to check that I mV is filtered. An object ( N, φ ) ∈ I mV is an object N ∈ C togetherwith a mod- A -isomorphism N V ; a morphism (
N, φ ) ( N ′ , φ ′ ) is a morphism g : N N ′ in C suchthat φ ′ s ( g ) = φ . Suppose that we are given two morphisms g, g ′ : ( N, φ ) ( N ′ , φ ′ ). Then the morphism N ′ N ′ / im( g − g ′ ) is a mod- A -isomorphism which equalizes g and g ′ ; thus I mV has coequalizers. Nowsuppose that we are given two objects ( N, φ ) and ( N ′ , φ ′ ) in I mV . Choosing representatives appropriately,these give a diagram in C e N V ′ f N ′ N N ⊕ e N V ′ V N ′ ⊕ f N ′ V ′ N ′ ( N ⊕ e N V ′ ) ⊕ V ′ ( N ′ ⊕ e N ′ V ′ ) ψ where the bulleted arrows represented mod- A -isomorphisms. The object (( N ⊕ e N V ′ ) ⊕ V ′ ( N ′ ⊕ e N ′ V ′ ) , ψ )then represents an object under both ( N, φ ) and ( N ′ , φ ′ ). Thus I mV is filtered, as desired.The second example is the case of reduced schemes of finite type of bounded dimension; we will be usingthis example in Section 9 to compare different models of the K -theory of varieties. Example . Let
Sch drf be the category of reduced schemes of finite type over k which are at most d -dimensional. As mentioned in Example 6.2, Sch rf is an ACGW-category; since morphisms can only increasethe dimension of a scheme it follows directly that Sch drf is also an ACGW-category.We claim that Theorem 8.5 applies for
Sch d − rf ⊆ Sch drf . We check the conditions in turn.First, consider condition (W). We claim that
Sch d − rf is m-well-represented and e-negligible in Sch drd .Here, an isomorphism in
Sch drf \ Sch d − rf is (the germ of) an isomorphism between open subsets whose com-plements are at most d − components of dimension less than d . In addition, we can assume that all d -dimensional components aresmooth and consider isomorphisms to be birational isomorphisms. To check that Sch drf is m-well-representedit suffices to check that for any two representatives of a birational isomorphism there exists a common denseopen subset on which they are defined. This is clearly true.To check that
Sch d − rf is e-negligible in Sch drf we note that for any diagram
A B C if we take thenonsingular locus of the d -dimensional irreducible components of C and intersect it with the image of A weget exactly the desired subset, as all that the inclusion B C can add is either (a) disjoint components ofdimension less than d or (b) components of dimension less than d that intersect d -dimensional components.In case (b) the intersections are singular in C , so when we remove them we produce exactly the desiredmorphism.We now check condition (CGW). In Appendix A we show that in order for C\A to be a CGW-categorywe are only required to show that c and k are well-defined equivalences of categories; the other axioms followdirectly from the definitions. In Sch drf \ Sch d − rf all objects are canonically isomorphic to the disjoint union oftheir d -dimensional connected components, so it suffices to consider these examples. By definition, both thee-morphisms and m-morphisms in Sch drf \ Sch d − rf are birational isomorphisms of the domain with a subsetof the components of the codomain. Both c and k simply take the components not hit by the morphism.Consider taking each object to its set of connected components; from the definition of the distinguishedsquares (see Appendix A) a square in Sch drf \ Sch d − rf is distinguished if and only if the produced square inthe category of finite sets is distinguished. The fact that c and k are equivalences of categories thus followsfrom the fact that they are induced from c and k on the category FinSet .It remains to check condition (E). Since ⊘ in Sch drf is simply intersection of schemes the condition as statedfollows by the same argument as the negligibility condition above. To check the condition with m-morphismsand e-morphisms reversed, let A d be the d -dimensional irreducible components of A . Then A d X × A X ′ exists, and the maps A d X B and A d X ′ B are equal inside the (ordinary) categoryof schemes (since they must be equal on a dense open subset, as they are equivalent in Sch drf \ Sch d − rf ).Factoring this morphism as A d C B gives the desired object C .We now observe that, by the Barratt–Priddy–Quillen theorem, K ( Sch drf \ Sch d − rf ) ≃ M α ∈ B n Ω ∞ Σ ∞ B Aut( α ) . Here, B n is the set of birational automorphism classes of schemes of dimension d , and Aut( α ) is the groupof birational automorphisms of a representative of the class.9. A comparison of models
In this section we compare both authors’ models for K ( Var /k ). For the time being let K C ( Var /k ) denotethe model which appears above and let K Z ( Var /k ) denote the model in [Zak17]. We then have the followingcomparison theorem. Theorem 9.1. K C ( Var /k ) is weakly equivalent to K Z ( Var /k ) . The rest of this section focuses on the proof of the theorem. For conciseness we fix the base field k andomit it from the notation. To prove the theorem we construct an auxillary SW -category Var w and showthat there are weak equivalences K C ( Var ) ∼ K C ( Var w ) ∼ K Z ( Var ) . Definition 9.2.
We define a new SW -category Var w . Its underlying category is Var . We define thestructure maps by setting cofibrations: the closed immersions, and complement maps: the open immersions, and weak equivalences: those morphisms f : X Y such that there exists a stratification ∅ = Y Y · · · Y n = Y of Y such that for all i , the induced map f i : X × Y ( Y i \ Y i − ) Y i \ Y i − is an isomorphism. EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 25
Remark . This is equivalent to the statement that there is a corresponding filtration X i on X such that f i : X i \ X i − Y i \ Y i − is an isomorphism. We sometimes use the condition in this form.As the proof of Theorem 9.1 has many parts, we begin by presenting the basic outline. This will reducethe proof to showing that certain morphisms are equivalences on K -theory, and the rest of the section willfocus on each of those maps in turn. Outline of proof for Theorem 9.1.
The category of reduced schemes of finite type comes equipped with afiltration by dimension. This filtration is inherited by
Var and
Var w , and the inclusion Var Var w iscompatible with this filtration.Proposition 9.7 constructs a map K Z ( Var n ) K C ( Var nw ) which is an equivalence for all n . Note that K C ( Var ) = hocolim n K C ( Var n ) , and similarly for K Z ( Var ) and K C ( Var w ).Our proof proceeds by induction on n . When n = 0, Var = Var w , so the K -theories of these are equal.We now assume that the natural inclusion K C ( Var n − ) K C ( Var n − w ) is an equivalence. Consider thefollowing diagram:(9.4) K C ( Var n − ) K C ( Var n − w ) K Z ( V n − ) K C ( Var n ) K C ( Var nw ) K Z ( V n ) K C ( Var n , Var n − ) K C ( Var nw , Var n − w ) K Z (( V n /i ) • ) i ∼ i ′ ∼ ig ∼ g ′ f The columns in this diagram are homotopy fiber sequences. The column on the right is produced by [Zak17,Theorem C], the other two columns are produced by [Cam, Prop. 5.5]. The maps between the columns aregiven below. Since the columns are homotopy fiber sequences of loop spaces, f must be a weak equivalenceby the five lemma. Note that g is a weak equivalence if and only if g ′ is, so we focus on proving that g ′ is aweak equivalence.Let D be the category with objects finite disjoint unions of smooth n -dimensional varieties. A morphism ` s ∈ S X s ` t ∈ T Y t is an injective map f : S T together with birational isomorphisms X s Y f ( t ) .This is a CGW-category which is equivalent (as CGW-categories) to M α ∈ B n Aut( α ) ≀ FinSet , where B n is the set of birational isomorphism classes of varieties of dimension n . There exists a map K C ( Var nw , Var n − w ) K ( D ) induced by the functor Var nw D taking each variety X to the disjointunion of the nonsingular subvarieties of X ’s irreducible components of dimension n .Restricting our attention to the bottom row of (9.4), consider the following diagram: K C ( Var n , Var n − ) K C ( Var nw , Var n − w ) K Z (( V n /i ) • ) K C ( D ) g ′ f ∼ ρβλ The map ρ is an equivalence by [Zak17, Theorem D]. Since f is a weak equivalence, β must also be a weakequivalence. Thus we see that g ′ is an equivalence if and only if λ is; that λ is an equivalence is exactly theconclusion of Proposition 9.8. (cid:3) We now turn our attention to filling in the details of the proof above. We begin by checking that
Var w is a category. Lemma 9.5.
Let
X, Y, Z ∈ Var w and suppose X Y and
Y Z are weak equivalences. Then
X Z is a weak equivalence.
Proof.
Recall that
X Y being a weak equivalence is the statement that there is a stratification ∅ = Y Y · · · Y n such that X × Y ( Y i \ Y i − ) ∼ = −→ Y i \ Y i − . Similarly for Y Z . We must produce a new stratification of Z ,call it Z ′ i , such that X × Z ( Z ′ i \ Z ′ i − ) ∼ = −→ ( Z ′ i \ Z ′ i − ). We do this by stratifying each ( Z i \ Z i − ) in turn,using the stratification of Y , and gluing these together.The problem thus reduces to the following. Given Y Y and Z Z with an isomorphism ϕ : Y \ Y Z \ Z , and a further stratification Y , · · · Y ,n = Y , produce a correspondingstratification for Z Z . To do this, define Z ,i = Z \ ϕ ( Y \ Y ,i ). One checks that Z ,i \ Z ,i − = ( Z \ ϕ ( Y \ Y ,i )) \ ( Z \ ( Y ,i − )) = ϕ ( Y \ Y ,i − ) \ ϕ ( Y \ Y ,i ) ∼ = ϕ ( Y i \ Y i − ) (cid:3) Lemma 9.6. Var w is an SW -category.Proof. For this we only need to check the axioms of SW-categories that apply to weak equivalences [Cam,Defn. 3.24], which are wholly analagous to [Wal85, p.326]. First, the isomorphisms are certainly containedin w . Second, we must check that subtraction respects weak equivalences. That is, if we have a commutativesquare with sides as indicated: X X ′ Y Y ′ ∼ ∼ then there is a weak equivalence X ′ \ X Y ′ \ Y making the induced square commute. Thus, we needa stratification on Y ′ \ Y . Since we are subtracting off Y , the stratification of Y will not come into play.Define the stratification to be ∅ = ( Y ′ \ Y ) × Y ′ Y ′ ( Y ′ \ Y ) × Y ′ Y ′ · · · Y ′ \ Y. Finally, we must check that in a diagram as below, where all the horizontal maps are cofibrations and thesquares are pullbacks, the induced map between pushouts is a weak equivalence: X ′′ X X ′ Y ′′ Y Y ′ ∼ ∼ ∼ Before we continue, note that since X ′ Y ′ is a weak equivalence, X Y is trivially so: a stratification Y ′ i Y ′ pulls back to one on Y , Y × Y ′ Y ′ i Y . A similar statement obviously holds for X ′′ Y ′′ .It suffices to consider the case where both X ′ Y ′ and X ′′ Y ′′ are given by two step stratifications.Let these be Y ′ Y ′ and Y ′′ Y ′′ . Denote the two induced stratifications on Y by Y (1)1 Y and Y (2)1 Y so that Y (1)1 = Y × Y ′ Y ′ and Y (2)1 = Y × Y ′′ Y ′′ . We now consider the three-step stratification Y ′′ ∐ Y (2)1 × Y (1)1 Y ′ Y ′′ ∐ Y (1)1 Y ′ Y ′′ ∐ Y Y ′ One verifies that ( Y ′′ ∐ Y (1)1 Y ′ ) \ ( Y ′′ ∐ Y (2)1 × Y (1)1 Y ′ ) ∼ = ( Y ′′ \ Y ′′ )( Y ′′ ∐ Y Y ′ ) \ ( Y ′′ ∐ Y (1)1 Y ′ ) ∼ = ( Y ′ \ Y ′ ) (cid:3) EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 27
The main work of this section goes into proving Propositions 9.7 and 9.8 which together immediatelyimply Theorem 9.1.
Proposition 9.7.
For n ≤ ∞ , K Z ( V n ) ≃ K C ( Var nw ) , induced by taking each tuple of varieties in V n to their disjoint union.Proof. For conciseness of notation, we give the proof for the case n = ∞ and omit the n from the notation.The proof works identically for all finite n . Throughout this proof we freely use the notation and definitionsof [Zak17].We construct a functor of simplicial categories F • : W ( V ∨• ) wS • Var w which has a levelwise rightadjoint. Thus the functor is levelwise a homotopy equivalence, and we get an equivalence on the geometricrealizations of the simplicial categories. This equivalence produces an equivalence K Z ( Var ) K C ( Var ) ,and (since these are both Ω-spectra above level 1) an equivalence of K -theories.We construct the functor in the following manner. W ( Var ∨ m ) is the full subcategory of W ( Var ) m consisting of those objects with disjoint indexing sets. We will thus refer to objects of W ( Var ∨ m ) as tuples( { A i } i ∈ I , . . . , { A mi } i ∈ I m ) in W ( Var ) m and simply ensure that at all stages the indexing sets are disjoint.Let F m ( A , . . . , A m ) be the functor X : f Ar[ m ] Var given by X i,j = j a k = i +1 a ℓ ∈ I k A kℓ , with morphisms given by the natural inclusions into the coproduct. A morphism of tuples gives a naturaltransformation of functors, each component of which is a weak equivalence in Var w , so F m is well-defined.The simplicial maps in W ( Var ∨• ) are induced by maps on the indexing sets, so these commute with thesimplicial structure maps in wS • Var . Thus F • is a simplicial functor.It remains to check that F m has a right adjoint. Given a diagram X : f Ar[ m ] Var , we define G m ( X )to have as its i -th component { X i \ X i − } { i } .We define the unit of the adjuction by taking each { A ji } i ∈ I j to { ` i ∈ I j A ji } { j } ; note that this is a validmorphism in W ( Var ), so gives a valid morphism in W ( Var ) m , with the indexing set disjoint by definition.Now consider F m ◦ G m . This takes a functor X : f Ar[ m ] Var to the functor X ′ : f Ar[ m ] Var , where X ′ ij = j a k = i +1 X ij \ X i ( j − . There is a natural weak equivalence X ′ X by simply mapping each component to itself. This gives thecounit of the adjunction and completes the proof of the proposition. (cid:3) We now turn our attention to the map λ . Proposition 9.8.
The map K C ( Var n , Var n − ) λ K (cid:18) M α ∈ B n Aut( α ) ≀ FinSet (cid:19) is a weak equivalence.Proof.
Consider the following diagram: Ω | Q Sch n − rf | Ω | Q Var n − | Ω | iS • Var n − | Ω | Q Sch nrf | Ω | Q Var n | Ω | iS • Var n | Ω | Q ( Sch nrf , Sch n − rf ) | Ω | iQ ( Var n , Var n − ) | Ω | QS • ( Var n , Var n − ) | Ω | Q D| Ω | Q D| Ω | iS • D| ∼∼∼∼ iλ ∼ ∼ ∼ λ ′ The right-hand side of the diagram is the definition of λ . The three top right-hand horizontal maps are givenby the natural transformation described in the proof of Theorem 7.9 for the comparison between the Q -construction and the S • -construction. The three top left-hand horizontal maps are induced by the inclusion Var n Sch nrf ; these are weak equivalences by Example 6.2. All of the maps between the third and fourthrow are induced by the map
Sch nrf D taking each variety to the disjoint union of its n -dimensionalirreducible components. Thus λ is a weak equivalence if and only if λ ′ is.Consider λ ′ . This map is induced by the CGW-functor Sch nrf
Sch nrf \ Sch n − rf D . By Theorem 8.5we know that Ω | Q ( Sch nrf , Sch n − rf ) | ≃ Ω | Q ( Sch nrf \ Sch n − rf ) | ; in particular, we see that λ ′ is a weak equiva-lence if and only if the functor Sch nrf \ Sch n − rf D induces an equivalence on K -theory. However, theseare equivalent categories, so this follows. (cid:3) Proof of Theorem 8.5
The goal of this section is to prove Theorem 8.5. The idea of the proof is to use Quillen’s Theorem B[Qui73, Theorem B] applied to the functor Qs . There are therefore two steps to the proof: proving that thetheorem applies to Qs , and proving that the fiber agrees with K ( A ).Let i : A C be the inclusion functor. Then Qi factors as Q A Qs ∅ / Q C M ( M, ∅ ) ( N, u ) N Theorem B implies that the fiber of Qs is Qs ∅ / . Thus to show that the fiber agrees with K ( A ) it suffices tocheck that the left-hand map in this factorization is a weak equivalence. We see that the theorem is thus adirect consequence of the following two propositions: Proposition 10.1.
The inclusion Q A Qs ∅ / is a homotopy equivalence. Proposition 10.2.
Quillen’s Theorem B applies to the functor Qs . More concretely, for any u : V V ′ in Q ( C\A ) , the induced functor u ∗ : Qs V ′ / Qs V/ is a homotopy equivalence. The rest of this section is taken up with the proof of these two propositions. We begin by analyzing howmorphisms in
C\A and Q ( C\A ) work.
Lemma 10.3. M A and E A satisfies 1-of-3, in the sense that M A and E A are subcategories of M and E ,respecively, and given any composable morphisms f, g ∈ M (resp. E ), if gf ∈ M A (resp. E A ) then so are f and g .Proof. We prove this for M A ; the result for E A follows by duality.Suppose that we are given f : A B and g : B C in C . This corresponds to a diagram EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 29 ∅ B c/g ∅ A c/f A c/gf A B C (cid:3) (cid:3)(cid:3)
The lower-left square exists by the definition of A c/f ; the lower-right square exists by applying k − to thebottom row; the upper square exists because ( A c/f ) c/A c/gf ∼ = B c/g by the definition of c . Consider the uppersquare, we note that since A is closed under subobjects, quotients and extensions, A c/gf is in A if and onlyif A c/f and B c/g are. Thus, if f and g are in M A so is gf (showing that M A is a subcategory) and if gf isin M A then f and g must be, as well. (cid:3) Lemma 10.4.
The categories E A and M A satisfy the following properties: (a) The subcategories E A and M A are preserved under pullbacks and mixed pullbacks along morphismsin E and M . (b) Pullback squares and mixed pullback satisfy satisfy 3-of-4: if three of the morphisms in a square arein M A or E A , the fourth must be as well.Proof. We first prove (a). Suppose that we have a square
A BC D (cid:9) f ′ f . We want to show that if f is in M A , so is f ′ . Applying c to this diagram produces a pullback square A c/f BC c/f ′ D By definition, C c/f ′ ∈ A ; thus, since A is closed under quotients, A c/f ∈ A , as desired. The other proofs ofclosure under pullbacks follow analogously.We turn our attention to (b). To check 3-of-4, consider a square as above where we know that A B is in M A and B D is in E A . Because E A is closed under pullbacks, it follows that A c/f C c/f ′ is alsoin E A . Thus we have a distinguished square ∅ A c/f ( A c/f ) k/c C (cid:3) in which we know everything but C is in A . Since A is closed under extensions, C ∈ A as well. The otherforms of 3-of-4 follow analogously. (cid:3) This proposition implies that we can identify the isomorphisms in
C\A in the following manner:
Lemma 10.5.
An m-morphism in
C\A represented by a diagram
A A ′ X B ′ B is an isomorphism if and only if B ′ B is in M A ; the dual statement holds for e-morphisms. Any morphism u : A B in Q ( C\A ) can be represented by a diagram A A ′ X B ′ B in C . Such a diagram represents an isomorphism if and only if X B ′ is in E A and B ′ B is in M A .Proof. If B ′ B is in M A then the given diagram represents an isomorphism by definition (by reversingthe composition for the inverse). Conversely, if an m-morphism has an inverse then tracing through thedefinition of composition and using Lemmas 10.3 and 10.4 gives that B ′ B must be in M A .A morphism A B in Q ( C\A ) is represented by a composition of an e-morphism
A C and anm-morphism
C B . We can represent these by the top and right side of the following diagram:
A A ′ X C ′ CC ′′ A ′′ Z YB ′ B ′′ B (cid:9) (cid:9)(cid:3)(cid:3)(cid:3) (cid:3) The rest of the diagram shows that the composition around the bottom is an equivalent representation ofthis morphism.Since morphisms in Q ( C\A ) are isomorphisms exactly when both components are isomorphisms (byLemma 3.5), the composition is an isomorphism if and only if the morphisms
C C ′ and B ′ B areisomorphisms, meaning that they are in E A and M A , respectively. If this is the case then Z B ′′ and B ′′ B are in E A and M A , respectively, and this represents an isomorphism. Conversely, if this is anisomorphism then we must have Z B ′′ and B ′′ B in E A and M A ; tracing through and using that E A and M A satisfy 1-of-3 we obtain the converse. (cid:3) We turn our attention to proving Proposition 10.1.
Definition 10.6.
Let V ∈ Q ( C\A ), and let F V be the full subcategory of Qs V/ of those objects ( M, u : V sM )in which u is an isomorphism.Proposition 10.1 is the V = ∅ case of the following: Proposition 10.7.
The inclusion ι V : F V Qs V/ is a homotopy equivalence for all V ∈ Q ( C\A ) .Proof. By [Qui73, Theorem A], it suffices to check that for all (
M, u ) ∈ Qs V/ , the category ι V / ( M, u ) iscontractible for all (
M, u ). By the dual of [Qui73, Proposition 3, Corollary 2] it suffices to check that it is acofiltered category. By Lemma 10.5, u can be represented by a diagram V u mD V ′ u eD X u e Y u m sM. An object of ι V / ( M, u ) is an isomorphism u ′ : V sM ′ in F V together with a morphism f : M ′ M in Q C such that s ( f ) u ′ = u . A morphism ( u ′ , M ′ , f ) ( u ′′ , M ′′ , f ′ ) is a morphism g : M ′ M ′′ in Q C suchthat f ′ g = f . Thus by Lemma 3.6 ι V / ( M, u ) is a preorder. All it remains to check is that it is nonemptyand that any two objects have a common object above them.To see that ι V / ( M, u ) is nonempty, consider the following diagram in C : EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 31
V V ′ X Y sMV ′ X Y u mD u eD u e u m u mD u eD u e u m uu ′ f This represents an object of ι V / ( M, u ) as desired.Now suppose that we are given two different objects of ι V / ( M, u ); we want to show that there is an objectmapping to both of them. Suppose that the two objects are given by ( u ′ : V sM ′ , f : M ′ M ) and( u ′′ : V sM ′′ , f ′ : M ′′ M ). Writing these in terms of their representations we get the outside of thefollowing diagram; it is possible to complete the outside to the diagam on the inside because s ( f ) u ′ = s ( f ′ ) u ′′ . V X ′ Y ′ sM ′ W T sZX ′′ T ′ A BY ′′ sM ′′ sZ ′ sM u ′ u ′′ s ( f ′ ) s ( f ) (cid:3)(cid:3) (cid:9)(cid:9) (cid:3) (cid:3) ∼ = ∃ Consider the object represented by (
A W V, A B M ) . This is a well-defined morphism of ι V / ( M, u ). This comes with a morphism to ( u ′ , f ) given by the formalcomposition A T X ′ Y ′ M ′ and an analogous morphism to ( u ′′ , f ′ ). Thus ι V / ( M, u ) is cofiltered, as desired. (cid:3)
We now turn our attention to Proposition 10.2; this proof is quite complicated and will take the rest ofthis section. First, note that in order to prove that u ∗ is a homotopy equivalence for all u it suffices to showthat it is true for the morphisms ∅ V and ∅ V . Since all of the conditions of the theorem aresymmetric in m-morphisms and e-morphisms, it suffices to prove this for ∅ V ; we focus on this case forthe rest of this proof. The key idea of the proof is to construct a category H N with sN ∼ = V and functors P ( N,φ ) : H N F V and k N : H N Q A such that the diagram(10.8) H N F V Qs V/ Q A F ∅ Qs ∅ /P ( N,φ ) ∼ = k N u ∗ commutes up to homotopy. We will then show that k N and P ( N,φ ) are both homotopy equivalence. Fromthis Proposition 10.2 follows by 2-of-3.We thus turn our attention to constructing H N , k N and P ( N,φ ) . Definition 10.9.
The category H N has as objects equivalence classes of diagrams M h e X h m N, where two diagrams are allowed to differ by an isomorphic choice of X . A morphism( M h e X h m N ) ( M ′ h ′ e X ′ h ′ m N )is a diagram M j M i M ′ such that there exists a map e h m : X X ′ such that the triangle on the leftcommutes and the square on the right X X ′ N e h m h m h ′ m X X ′ M M ′ (cid:9) e h m ijh e h ′ e is a commutative square. Composition works via composition in Q C ; using the following diagram we seethat it is well-defined: X X ′ X ′′ M M ′ • M M ′′ e h m e h ′ m i i ′ jh e h ′ e h ′′ e j ′ (cid:9) (cid:9)(cid:3) The functor k N : H N Q A takes M h e X N to X k/h e . A morphism is taken to the representation X k/h e X k/jh e e h m X ′ , where the first map is obtained by applying c − . Definition 10.10.
Let (
N, φ ) be an object of I mV . We define P ( N,φ ) : H N F V by letting it take everyobject M X N to the composition V φ − sN sX sM in F V ⊆ Qs V/ . Lemma 10.11. P ( N,φ ) is a well-defined functor.Proof. Checking that P ( N,φ ) is well-defined on morphisms is straightforward from the definition. Supposethat we are given a morphism in H N as defined in Definition 10.9. We must show that this produces awell-defined morphism in F V ; from the definition the produced morphism in Qs V/ is an isomorphism, so it EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 33 suffices to show that a morphism in H N gives a well-defined morphism in Qs V/ . For this to be true it sufficesto check that the morphisms represented by N X M M M ′ and N X ′ M ′ are equivalent in Q ( C\A ). This is true because the are equivalent isomorphisms inside the m-morphisms of
C\A via the following diagram:
N X M M ′ X ′ M ′ (cid:9) where the marked square is commutative from the definition of a morphism in H N . That P ( N,φ ) respectscomposition follows directly from the definition, since composition in both Qs V/ and H N is defined usingcomposition in Q C . (cid:3) We begin our analysis by showing that (10.8) commutes up to homotopy.
Lemma 10.12.
In (10.8) the composition around the top and the composition around the bottom are ho-motopic.Proof.
Consider an object M h e X N in H N . Under the composition around the top it is mapped to ∅ V φ − sN sX sM ;this is equivalent to the representation ∅ sM. Around the bottom this is mapped to ∅ X k/h e . There is a natural map h ke : X k/h e M which inducesa morphism between these in Qs ∅ / , so we just need to check that this gives a natural transformation. Tosee that this transformation is natural, suppose that we are given a morphism( M h e X h m N ) ( M ′ h ′ e X ′ h ′ m N )represented by M j M i M ′ . Consider the following diagram in C : X k/h e X k/jh e ( X ′ ) k/h ′ e M M M ′ h ke ( jh e ) k h ′ ke (cid:3) The left-hand square exists and is distinguished by the definition of k . The right-hand square exists andcommutes by the condition on morphisms in H N ; this is exactly k applied to the commutative square. Afterapplying s to the diagram and considering the outer corners as objects under ∅ , we see that this diagramexactly corresponds to a naturality square for functors H N Qs ∅ / , as desired. (cid:3) It remains to show that k N and P ( N,φ ) are homotopy equivalences. We begin with k N ; however, beforewe can prove that k N is a homotopy equivalence we must develop some theory. Definition 10.13.
Let J N be the full subcategory of M /N containing those morphisms A N such that A c ∈ A . Note that J N has a terminal object: 1 N . Definition 10.14.
Let H ′ N be the full subcategory of H N containing those objects where h m = 1 M . Forany m-morphism i : B A we define the functor ρ i : H ′ A H ′ B by sending the e-morphism A f X tothe e-morphism B X ′ , where B X ′ is determined by the following distinguished square: B AX ′ X (cid:3) i f . Lemma 10.15.
Let i : ( J N ) (
I N ) be a morphism in J N . Then the diagram H ′ I H ′ J Q A ρ i k ′ I k ′ J commutes up to natural isomorphism.Proof. Consider the object I h M in H ′ I . Its image under k ′ I is I k/h . For the other composition, weconsider the distinguished square J IM ′ M (cid:3) ih ′ h .h is mapped to h ′ , and then to J k/h ′ . Since these are related by a natural distinguished square, they arenaturally isomorphic, as desired. (cid:3) Consider the functor F : H N J N sending M X N to X N . Lemma 10.16. H N is fibered over J N .Proof. Note that for any i : I N ∈ J N , F − ( i ), the fiber over i , is isomorphic to H ′ I . The category F i/ has as its objects the solid part of the diagram M ′ I NM X i (cid:3) The functor taking such a diagram to
I M ′ is the right adjoint to the inclusion H ′ I = F − ( i ) F i/ .Thus H N is prefibered over J N . To check that it is fibered it suffices to check that this right adjoint iscompatible with composition in the following sense. For any j : ( I i N ) ( I ′ i ′ N ) in J N we get aninduced functor j ∗ : F − ( i ′ ) F − ( i ) defined by the composition (cid:18) M I ′ N i ′ (cid:19) M I NM ′ I ′ (cid:3) j i i ′ (cid:18) M ′ I N i (cid:19) . We must show that for any composable j and k , ( kj ) ∗ is naturally isomorphic to j ∗ k ∗ . This is true becausecompleting a formal composition to a distinguished square is unique up to unique isomorphism. As both j ∗ k ∗ and ( kj ) ∗ are obtained by completing a formal composition M I ′′ k I ′ j I EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 35 to a distinguished square, they are naturally isomorphic. (cid:3)
We are now ready to prove that k N is a homotopy equivalence. Lemma 10.17. k N is a homotopy equivalence.Proof. We begin by checking that k ′ N def = k N | H ′ N is a homotopy equivalence. Let T be an object in Q A ; it suf-fices to check that k ′ N /T is contractible for all T . An object of k ′ N /T is a triple ( M, h e : N M, u : N k T )with u ∈ Q A . Let C ′ be the full subcategory of k ′ N /T consisting of those morphisms u which can be repre-sented purely by an e-morphism.Represent u as X k i Y j T , and consider the following diagram: N k M NY M ⋆ N k Y NT h ke h e i ( h ke ) ′ k − (( h ke ) ′ ) (cid:9) ju Here, the upper-left square is produced by condition (PP). We claim that the map taking (
M, h e , u ) to( M ⋆ N k Y, h ′ e , j ) is a functor which produces a retraction from k ′ N /T to C ′ . To check that this is functorial,consider a morphism in k ′ N /T . This is represented by a diagram T Y N k/h e MY ′ N k/jh e M NN k/h ′ e M ′ N h e h ′ e = (cid:9)(cid:3) ij i ′ j ′ (cid:3) where the morphism is considered to go from the object represented by the diagram around the top tothe object represented by the diagram around the bottom. This diagram produces a map M ⋆ N k/he Y M ⋆ N k/jhe Y ′ M ′ ⋆ N k/h ′ e Y ′ by the functoriality conditions in (PP). This is compatible withcomposition by Lemma 5.8 and the definition of morphism composition in Q A . This functor also comes witha natural transformation from the identity produced by the map M M ⋆ N k Y . Thus k ′ N /T is homotopyequivalent to C ′ . Note that C ′ has an initial object ( N, N , ∅ T ), so it is contractible. Thus k ′ N /T iscontractible for all T , and so k ′ N is a homotopy equivalence.We have now shown that k ′ N is a homotopy equivalence. By 2-of-3, in order to show that k N is a homotopyequivalence it suffices to check that the inclusion H ′ N H N is a homotopy equivalence.Since k ′ n is a homotopy equivalence, by Lemma 10.15 we see that ρ i is a homotopy equivalence for all i ∈ J N . Thus, since H N is fibered over J N , by [Qui73, Theorem B, Cor], for all I N , H ′ I is homotopyequivalent to the homotopy fiber of F . However, since J N is contractible it follows that the inclusion H ′ I H N is a homotopy equivalence. In particular, taking the m-morphism to be the identity on N givesthe desired result. (cid:3) We now turn our attention to P ( N,φ ) .We will need two different proofs for this functor, depending on whether A is m-negligible or m-well-represented in C . Lemma 10.18. If A is m-negligible in C then P ( N,φ ) is a homotopy equivalence. Proof.
We prove this using Theorem A. An object of F V is an isomorphism V ψ sA . We will show that( P ( N,φ ) ) A/ is contractible. We can fix representatives for φ and ψ such that an object of ( P ( N,φ ) ) /A isrepresented by a diagram(10.19) V V ′ Z N ′ NA ′ XA M ′ M φ − ψ where the dashed arrows commute inside Q ( C\A ). V ′ , Z, N ′ , A ′ are all fixed by our choice of representatives;the only part of the diagram that is allowed to change are the bottom and rightmost rows. A representativeof an object is well-defined up to unique isomorphism, since both the right-hand column (an object in H N )and the bottom row are well-defined up to unique isomorphism. The maps M M ′ and M ′ A mustalso be in M A and E A , respectively, since M A and E A are closed under 2-of-3 by Lemma 10.3. (This followsby computing a representative of the composition and noting that since its components are in M A (resp. E A ) the two maps across the bottom are.)A morphism ( M/A ) ( M ′ /A ) of ( P ( N,φ ) ) /A is a diagram V V ′ Z N ′ N NA ′ X b XA M ′ M M c M where the morphism c M A in Q C is given by the composition across the bottom.Let D be the full subcategory of ( P ( N,φ ) ) /A of those objects which can be represented by a diagramwhere the morphism X M is the identity. Note that given any object represented by (10.19) there is awell-defined morphism given by
V V ′ Z N ′ N NA ′ X XA M ′ M M X which is natural in our object (since the choice of X is unique up to unique isomorphism). This show that D is a retractive subcategory of ( P ( N,φ ) ) /A , and is thus homotopy equivalent to it.A morphism inside D is represented by a diagram EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 37
V V ′ Z N ′ N NA ′ A M ′ M c M Note that the only important information here is the lower-right-hand side. Thus we will think of morphismsin D as represented by diagrams N M M ′ A which are equivalent inside C\A . Since all morphisms in M are monic, such a morphism (if it exists) isunique; thus D is a preorder. To show that D is contractible we will show that it is nonempty and cofiltered.Given two objects N M M ′ A and N f M f M ′ A we know that they are equivalent inside C\A if there exists a diagram
X Y N such that precompo-sition by this diagram sends these to diagrams which are equivalent in C . However, since A is m-negligiblein C we see that such a diagram exists if and only if such a diagram exists with the e-component equal tothe identity. Picking such a morphism Y N we see that the object represented by
N Y × N M M ′ A maps to both of these objects. Thus D is cofiltered.To see that D is nonempty, consdier the diagram Z N ′ N given by the chosen representative for φ . Since A is m-negligible in C there exists an m-morphism M N such that M ⊘ N N ′ ∼ = M and M N ′ factors through Z N ′ . Then the diagram N M A ′ A gives a well-defined object of D . Thus D is nonempty and cofiltered, and therefore contractible. (cid:3) If A is m-negligible in C we are now done. Thus we can now assume that A is m-well-represented in C .Consider a diagram N g e X g m N ′ which we denote g . We define the functor g ∗ : H N H N ′ by M Y N Y ⊘ N X X N ′ M Y N g e g m (cid:9) M Y ⊘ N X N ′ . This is functorial because commutative squares compose.
Lemma 10.20.
There is a natural transformation k N k N ′ g ∗ .Proof. We have k N ( M h e Y N ) = Y k/M . On the other hand, k N ′ g ∗ ( M h e Y N ) = ( Y ⊘ N X ) k/M . The map Y ⊘ N X M factors through
Y M , so (by Lemma 2.10) there is a functorially induced map Y k/M ( Y ⊘ N X ) k/M . This map gives the natural transformation. To check that this is actually natural,consider a map ( M Y N ) ( M ′ Y ′ N ) represented by M M M ′ . We must showthat the square Y k/M Y k/M ( Y ′ ) k/M ′ ( Y ⊘ N X ) k/M ( Y ⊘ N X ) k/M ( Y ′ ⊘ N X ) k/M ′ commutes in Q A . To show this it suffices to show that there exists a map Y k/M ( Y ⊘ N X ) k/M such thatthe left-hand square is distinguished and the right-hand square commutes. The map exists and makes theright-hand square commute by Lemma 2.10. To check that the left-hand square is distinguished it sufficesto check that given any diagram A B C D the square B k/C A k/C B k/D A k/D is distinguished. This follows directly from the definition of c and k . (cid:3) Since k N and k N ′ are both homotopy equivalences, we get the following corollary: Corollary 10.21. g ∗ is a homotopy equivalence. Consider the functor H : I mV Cat sending (
N, φ ) to H N and g : ( N, φ ) (
N, φ ′ ) to g ∗ . Lemma 10.22.
There is an isomorphism of categories e H : colim I mV H F V induced by P ( N,φ ) : H N F V .Proof. We first check that e H is well-defined. To prove this it suffices to check that for g : ( N, φ ) ( N ′ , φ ′ ), P ( N ′ ,φ ′ ) g ∗ = P ( N,φ ) . First, note that since morphisms in H N are defined to be morphisms in Q C satisfying extra conditions, andsince both P ( N,φ ) and g ∗ do not change any of the representation data in the morphism, if the two sides agreeon objects they must also agree on morphisms. P ( N,φ ) maps an object ( M X N ) to the composition V φ − sN sX sM, while P ( N ′ ,φ ′ ) g ∗ maps it to the composition V φ ′− sN ′ sY sN sX sM. However, since φ ′ s ( g ) = φ , these two compositions represent equivalent diagrams (since after being consideredinside C\A , all g ∗ does is compose with g ) and thus the left and right sides agree on objects. Therefore thefunctors P ( N,φ ) produce a valid cone under H and e H is well-defined.It now remains to show that it is, in fact, an isomorphism of categories.First we show that e H is surjective on objects; in other words, that for every ( M, u : V ∼ = sM ) in F V there exists an ( N, φ ) and an object ( M ′ , h ) in H N such that P ( N,φ ) ( M ′ , h ) = ( M, u ). To do this, let(
N, φ ) = (
M, u − ) and let ( M ′ , h ) = ( M, M M M ).. Thus e H is surjective on objects.Now consider injectivity. Since I mV is filtered, it suffices to check that each individual P ( N,φ ) is injectiveon objects. Suppose that P ( N,φ ) ( M, h ) = P ( N,φ ) ( M ′ , h ′ ) . EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 39
We must show that there exists g : ( N, φ ) ( N ′ , φ ′ ) in I mV such that g ∗ ( M, h ) = g ∗ ( M ′ , h ′ ). Note, that bydefinition in order for this to hold we must have M = M ′ and s ( h ) = s ( h ′ ). The fact that such a g exists isimplied by condition (E); in fact, this g will be represented by a morphism where the m-component is theidentity. Thus e H is injective on objects.We now consider morphisms. As before, we consider surjectivity first. Consider a morphism g : ( M, u ) ( M ′ , u ′ )in F V . This is given by a morphism g : M M ′ in Q C such that s ( g ) u = u ′ in Q ( C\A ). Since both u and u ′ are isomorphisms, s ( g ) must be as well; thus it is represented by a diagram M X M ′ . Considerthe distinguished square M X ′ X M h m g m g e h e (cid:3) where the composition around the bottom is given by the components of g . Since all distinguished squaresare commutative, this defines a morphism( M, M h m X ′ ) f ( M ′ , M ′ h e X ′ )in H X ′ . Note that s ( g e ) u = s ( h − e ) u ′ . Thus P ( X,s ( g e ) u ) ( f ) = g , as desired.Now consider injectivity. As before, it suffices to consider a single P ( N,φ ) and show that it is faithful.Suppose that P ( N,φ ) ( g ) = P ( N,φ ) ( g ′ ). By definition, g, g ′ : ( M X N ) ( M ′ X ′ N )are given by morphisms e g, e g ′ : M M ′ in Q C satisfying the diagram in Definition 10.9. For P ( N,φ ) ( g ) = P ( N,φ ) ( g ′ ) we must have e g = e g ′ ; however, in this case we must have g and g ′ equal as well. Thus e H isinjective on morphisms, and we are done. (cid:3) We are now ready to finish:
Proposition 10.23. If A is m-well-represented in C then P ( N,φ ) is a homotopy equivalence.Proof. [Qui73, Proposition 3, Corollary 1] states the following: given any filtered category C and a functor F : C Cat such that for all f : A B ∈ C , F ( f ) is a homotopy equivalence. Then the induced map F ( A ) colim C F is a homotopy equivalence for all A ∈ C .Applying this to the functor H , we get that the map H ( N, φ ) colim I mV H ∼ = F V is a homotopyequivalence for all ( N, φ ) ∈ I mV . By definition this is exactly P ( N,φ ) : H N F V , and we are done. (cid:3) Appendix A. Checking that
C\A is a CGW-category
In this section we check as much as possible that the definition of
C\A gives a well-defined CGW-category.For this to work, we must check that the m-morphisms and e-morphisms give well-defined categories, thatthe distinguished squares compose correctly, that φ exists, that c and k are equivalences of categories, andthat axioms (Z), (I), (M), (K), and (A) hold. For this to hold we must make the following extra assumptions:(Ex) The definitions of c and k extend to equivalences of categories.Note that as the definition of C\A is symmetric with respect to e-morphisms and m-morphisms it sufficesto focus on proving only half of each statement; the other half will follow by symmetry.We first begin with a somewhat more explicit definition of the distinguihsed squares in
C\A . These aregenerated by the following types of squares:
A BC D (cid:3)
A BC D (cid:9)
A BC D (cid:9)
A BC D A BC DA BC D (cid:9)
A BC D (cid:9)
A BC D (cid:9)
A BC D A BC DA BC D (cid:9)
A BC D (cid:9)
A BC D A BC D A BC D (cid:9)
A BC D (cid:9)
A BC D A BC D A BC D (cid:9)
A BC D (cid:9)
We now prove a series of lemmas about how different types of squares in C interact. The commonconsequence of all of these lemmas is that the given squares fit into a cube with opposite sides of the same“type” (be that commutative squares, distinguished squares, or simply squares that commute inside E or M ). We do not worry about which arrows have c or k in A ; the properties of A ensure that whenever suchan arrow is “pulled back”, the pullback also has c or k in A . Lemma A.1.
Given two diagrams in C A B A ′ X C ′ C D C ′ C D (cid:9) (cid:9) we can assemble these into a cube X ′ XA CA ′ C ′ B D in which all faces with mixed morphisms are commutative. If
ABCD was originally distinguished, then X ′ A ′ XC ′ will be, as well.An analogous statement with the roles of e-morphisms and m-morphisms swapped also holds.Proof. Apply c to the left-hand diagram. This turns both of the squares into pullback squares in E (bydefinition). We can then form the following diagram: EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 41 A c × C X XA c C ( A ′ ) c C ′ B c D To prove the main statement of the lemma it suffices to show that a morphism A c × C X ( A ′ ) c existsand makes the back face into a pullback. To show the last stement it suffices to show that if A c B c isan isomorphism then this morphism is also an isomorphism. This is a straightforward diagram chase usingthe fact that all solid faces in the above diagram are pullbacks and all morphisms in E are monic. (cid:3) As a corollary we can see that assembling distinguished squares and pullbacks commutes:
Corollary A.2.
Suppose that we are given a diagram
A B C D.
The two possible different compositions of this diagram fit into a cube
A BX CY ZW D in which the top and bottom face are distinguished squares, the front and the back face are commutativesquares, and the right and left face are commutative in E with the right-hand face a pullback. We now prove a “complement” to Lemma 5.8: instead of assuming that a commutative square in E isattached to the back of a commutative square, we assume that it is attached to the front: Lemma A.3.
Suppose that we are given a diagram
A B B ′ C D D ′ (cid:9) Then this diagram assembles into a cube
A BA ′ B ′ C DC ′ D ′ where the front, back, and top faces are commutative and the bottom face is distinguished. If the right-handsquare is a pullback then the top face will also be distinguished.The dual statement also holds.Proof. Define C ′ so that the bottom face of the cube is a distinguished square. Define A ′ = ( B ′ × D ′ ( C ′ ) c ) k/B ′ . By definition this produces a diagram where the front face is commutative and the bottom faceis distinguished. It therefore suffices to check that there exists a morphism A A ′ such that the leftface commutes in E and the top face is commutative. To prove this it suffices to check that there exists amorphism A c/B B ′ × D ′ ( C ′ ) c such that in the diagram C c/D A c/B B ( C ′ ) c ( C ′ ) c × D ′ B ′ B ′ the left-hand square commutes and the right-hand square is a pullback. This follows directly from thedefinitions. (cid:3) We are now ready to turn our attention to proving that
C\A is a CGW-category.
The m-morphisms form a well-defined category
The m-morphisms in
C\A are defined to be equivalence classes of diagrams
A A ′ X B ′ B. The equivalence relation is generated by the following types of diagrams (up to isomorphism), where the reddiagram is declared to be equivalent to the blue diagram:(A.4) BB ′′ B ′ A ′′ X ′ X ′ ⊘ B ′′ B ′ A A ′ X ′ ⊘ A ′′ A ′ X (cid:9)(cid:9) (cid:9) Note that the relation defined between m-morphisms is a formal composition of two such relations, oneinverse to another. Thus to show that the relation is well-defined we must check that if we are given twosuch relations built on top of one another, then either they compose to a single one, or that we can “pullback” two such relations.
EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 43
Let us consider the first such case. Suppose that we are given two such diagrams, one relating
A A ′ X B ′ B to A A ′′ X ′ B ′′ B , and one relating A A ′′ X ′ B ′′ B to A A ′′′ X ′′ B ′′′ B .We can rearrange this data into the following diagram, where the first formal composition is in red, the secondis in blue, and the third is in green: BB ′′′ B ′′ B ′ A ′′′ X ′′ B ′′ ⊘ B ′′′ X ′′ B ′ ⊘ B ′′′ X ′′ A ′′ X ′′′ ⊘ A ′′′ A ′′ X ′ B ′ ⊘ B ′′ X ′ A A ′ X ′′ ⊘ A ′′′ A ′ X ′ ⊘ A ′′ A ′ X (cid:9) (cid:9)(cid:9) (cid:9) (cid:9)(cid:9) (cid:9) (cid:9) By regrouping the commutative squares, we see that the red composition is equivalent to the green compo-sition, as desired.To show the second case, consider the following diagram, which shows that red and blue are both equivalentto green: XA ′ X ′′ ⊘ A ′′′ A ′ B ′ ⊘ B ′′′ X ′′ B ′ A A ′′′ X ′′ B ′′′ BA ′′ X ′′ ⊘ A ′′′ A ′′ B ′′ ⊘ B ′′′ X ′′ B ′′ X (cid:9)(cid:9)(cid:9)(cid:9) (cid:9)(cid:9) Then the composition
A A ′ × A ′′′ A ′′ (( B ′ × B ′′′ B ′′ ) ⊘ B ′′′ X ′′ ) ⊘ X ′′ ( X ′′ ⊘ A ′′′ ( A ′ × A ′′′ A ′′ )) B ′ × B ′′′ B ′′ B is equivalent to both the red and the blue, completing the desired picture. Putting these together showsthat the relation defined on m-morphisms is an equivalence relation, as desired.Now we can work with the definition of the m-morphisms directly. Given two morphisms A B and
B C their composition is defined to be represented by the diagonal in the following square:
A A ′ X B ′ BA ′′ X × ( B ′ ⊘ B B ′′ ) B ′ ⊘ B B ′′ B ′′ Z ( B ′ ⊘ B B ′′ ) × Y YC ′′ C ′ C (cid:9)(cid:3) (cid:9) (cid:3) Here, Z = ( X × ( B ′ ⊘ B B ′′ )) ⊘ B ′ ⊘ B B ′′ (( B ′ ⊘ B B ′′ ) × Y ) and A ′′ and C ′′ are uniquely determined by thedistinguished squares they are in.To check that this is well-defined, it suffices to check that given a diagram as in (A.4) and a morphismrepresented as one of , , or the composition (resp. precomposition) with the redmorphism and the composition (resp. precomposition) with the blue morphism are equivalent. We checkthe case of composing with a morphism represented by ; all of the other cases are analogous. This is astraightforward diagram chase, using Lemma A.1 to push the diagram showing the equivalence of the tworepresentations along the composition; the only nontrivial part is ensured by Lemma 5.8.We need to check that composition is associative. As a morphism is a formal composition of four arrows,it suffices to check that compositions of those component arrows is associative. Note that we do not needto worry about which morphisms have kernel/cokernel in A , since that is preserved by the definition ofcomposition; all we are checking is associativity. Thus our definition of morphism is symmetric in e-morphismand m-morphism. In addition, since both E and M are closed under pullbacks, by standard argumentsabout span categories we know that when all three morphisms are e-morphisms or all three morphisms arem-morphisms composition is associative. Thus it remains to consider the case of 2 m-morphisms and 1e-morphism or 1 m-morphism and 2 e-morphisms. By symmetry again it suffices to consider this secondcase, and, in fact, it suffices to consider the case when the m-morphism is directed covariantly with thecomposition.Now there are 12 cases left (three positions for the m-morphism and four directions in which the e-morphisms can point). Most of these have only a single composition, so associativity holds automaticallyfor these. The remaining three cases are , and . Thefirst and second of these give associative compositions because distinguished and commutative squares workcorrectly with respect to composition. Thus the last case is the only one of interest, which directly followsfrom Corollary A.2. The fact that the two different compositions assemble into a cube implies that they areequivalent in C\A . Distinguished squares compose correctly
This is true by definition.
There exists a φ We must show that the subcategory of m-isomorphisms is isomorphic to the category ofe-isomorphisms by a functor which takes objects to themselves. To construct this functor, use Lemma 2.9to change a representation of an m-isomorphism as
A A ′ X B ′ B to A A ′′ X B ′′ B, which gives a representation of an e-isomorphism. Note that since distinguished squares are unique up tounique isomorphism, this is an isomorphism of categories. Axiom (Z)
We must check that ∅ is initial in M .There exists a morphism ∅ B for any B by simply taking the representation where all but the lastmorphism are the identity. We must now check that this morphism is unique. Suppose that we are given EVISSAGE AND LOCALIZATION FOR THE GROTHENDIECK SPECTRUM OF VARIETIES 45 any diagram ∅ ∅ ∅ B ′ B. We must have B ′ ∈ A for this diagram to be valid. The diagram BB ′ ∅∅ ∅ ∅∅ ∅ ∅ ∅ (cid:9)(cid:9) (cid:9) shows that the two are equivalent. Thus ∅ is horizontally initial. Axiom (I)
Note that the m-morphisms which are isomorphisms are exactly those morphisms of the form
A B.
Using this description and the listing of different kinds of distinguished squares we can construct each of therequired squares by hand.
Axiom (M)
It suffices to check this for the m-morphisms of
C\A ; the statement for the e-moprhisms willfollow by symmetry. Thus we want to check that if we are given two morphisms f, g : A B and a morphism h : B C in C\A then if hf = hg then f = g . Note that all morphisms in M are equal, up to isomorphism,to ones represented by diagrams • • . Thus it suffices to assume that h is of this form. This means thatthe compositions hf and hg are computed simply by composing the last m-moprhism components.The fact that hf = hg implies that for any choice of representatives for f and g , the following diagramexists: • B ′ • • C ′ A • • C B • • C ′′ • B ′′ (cid:9) (cid:9)(cid:9) (cid:9) ∼ = hfg To show that f = g it suffices to check that there exist maps C ′ B and C ′′ B such that the triangle C ′ BC ′′∼ = commutes. Setting these maps to be the evident ones generated by the above diagram, we see that the giventriangle must commute, as it commutes after postcomposition with h and h is monic. Axiom (K)
As before, we prove this only for c ; the result for k follows by symmetry.Let f : A B be a morphism. Given a representative
A A ′ X B ′ B of f , we can conclude that c ( f ) ∼ = ( B ′ ) c B . Thus if we can show that a distinguished square as desiredexists for this representative, we will be done. The following diagram shows that this is the case ∅ ∅ ∅ ∅ ( B ′ ) c A A ′ X B ′ B (cid:3)(cid:9) as it is a composition of squares which are distinguished in C\A . Axiom (A)
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