Motivic Cohomology I: some categorical constructions

Where we’re at, and where we’re going

In the last post, Motivic Cohomology I: presheaves with transfers and sheaves with transfers, we developed the notion of a presheaf with transfers. These were contravariant additive functors F:\mathsf{Cor}_k\rightarrow \mathsf{Ab}. We constructed a new category \mathsf{PreSh}(\mathsf{Cor}_k) whose objects were presheaves with transfers and whose morphisms were natural transformations. It was mentioned, in passing, we could extend our definitions to R-linear presheaves with transfers on the R-linear category \mathsf{Cor}_k(R). We concluded our study of presheaves with transfers by developing the notion sheaves with transfers in the appropriate Grothendieck topology. We showed objects representable by schemes in \mathsf{PreSh}(\mathsf{Cor}_k) were actually sheaves in the étale, Nisnevich, and Grothendieck topologies. Hence we constructed another faithful embedding \mathsf{Sm}_k\hookrightarrow \mathsf{Sh}_{\tau}(\mathsf{Cor}_k).

In this post, we’ll give a brief overview summarizing some general categorical constructions. In the next series of posts we’ll use these constructions to construct certain categories of motives.

Thick Subcategories

The notion of a thick subcategory makes sense in a couple of settings. We’ll give the definition in the abelian setting first. After, we give the definition for triangulated categories (since we won’t define it until later, if ever, I’ll say that a triangulated category is going to be a generalization of the notion of exactness, for categories where exactness has no meaning).

(3.1) Definition: A full subcategory T of an abelian category A is called a thick subcategory, or a Serre subcategory if, for any exact sequence

0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0

we have Y is in T if and only if X, Y are in T. That is to say, T is a full subcategory closed under extensions.

The definition for triangulated categories is similar

(3.2) Definition: A full additive subcategory T of a triangulated category A is called thick if it satisfies the two conditions: for any distinguished triangle

X\rightarrow Y\rightarrow Z\rightarrow X[1]

if two out of X,Y,Z are in T then so is the third, and, if X\oplus Y is in T then so are X,Y.

In (3.2) we could have said, alternatively but a fortiori equivalently, a full subtriangulated category is thick if it is closed under extensions. This shows the immediate generalization from definition (3.1). However this is equivalent to the definition we gave, so we’ll use the one more suitable to our use.

How we’ll apply this idea:

If T is a thick subcategory of a triangulated category T, we can consider the set of morphisms s:X\rightarrow Y in A which fit into a distinguished triangle

X\xrightarrow{s} Y\rightarrow Z\rightarrow X[1]

with Z in T. This set will be a saturated multiplicative system of morphisms (defined next section). We’ll use such sets to form the localization of a category.

In the other direction, if we start with a saturated multiplicative system of morphisms S, we can consider the full subcategory T whose objects fit into a distinguished triangle of A

X\xrightarrow{s} Y\rightarrow Z\rightarrow X[1].

In this situation T is a thick subcategory of A. ([Lev], page 425, 2.3.2) alternatively ([G-M] page 266, 4).

Localization of a Category

Let A be a category. Let S be a set (or class according to [G-M], top of page 145 – I won’t get into set theoretic details) of morphisms in A. The localization of A by S is a category which has formal inverses for any morphism in S.

(3.3) Proposition: Given A and S as above, there exists a category A[S^{-1}] and a functor F:A\rightarrow A[S^{-1}] so that, for any other category B where every s\in S has an inverse and functor G:A\rightarrow B there is a unique factorization G: A\xrightarrow{F} A[S^{-1}]\rightarrow B.

Proof. A construction is given in [G-M] also on page 145.

The objects of A[S^{-1}] are by definition the same objects as A. The proof of existence however, doesn’t give a very nice description of the morphisms in A[S^{-1}]. To remedy this we have

(3.4) Proposition: If, in the above situation, S is a saturated multiplicative system of morphisms, then we have a description of A[S^{-1}] as

1) Objects of A[S^{-1}] are the same as objects of A

2) Morphisms of A[S^{-1}] are diagrams, called roofs,

X\xleftarrow{s} X \xrightarrow{f} Y

where s\in S and f is a morphism in A modulo the equivalence relation (X\xleftarrow{s} X' \xrightarrow {f} Y) \sim (X\xleftarrow{s'} X'' \xrightarrow{g} Y) if and only if there exists an isomorphism X'\xleftarrow{t} X''' \xrightarrow{h} X'' which makes the resulting diagram commute.

Proof. [G-M] page 149, lemma 8. However, I changed their terminology from a localizing class of morphisms to a saturated multiplicative system in order to fit other sources. In the language of [K-S] we are using a right multiplicative system (or a multiplicative system in general, I’ll always use the notation above as above, but some author’s apparently don’t agree which is right and which is left). In [K-S], this is covered more formally on page 151 and onwards.

(3.5) Definition: A set of morphisms S in a category A will be called a (right) multiplicative system if it satisfies the following conditions

MS1) For any object of A, S contains the identity \text{id}_A:A\rightarrow A.

MS2) S is closed under composition of morphisms in S whenever the composition is defined in A

MS3) For any morphism f:X\rightarrow Y of A and morphism s:Z\rightarrow Y, we can find an object W of A, and morphisms g:W\rightarrow Z of A, r:W\rightarrow X in S so that we have a commutative square

\begin{matrix} W & \rightarrow & Z \\ \downarrow & & \downarrow \\ X & \rightarrow & Y \end{matrix}

MS4) Let f,g:X\rightarrow Y be two morphisms in A. If there exists s\in S with s\circ f = s\circ g then there exists a t\in S such that f\circ t = g\circ t.

Composition of roofs is defined as follows: given a roof (s,f):X\rightarrow Y, say X\xleftarrow{s} X'\xrightarrow{f} Y, and a roof (t,g):Y\rightarrow Z, say Y\xleftarrow{t} Y' \xrightarrow {g} Z, we use the condition MS3) to find a roof X'\xleftarrow{r} Z'\rightarrow{h} Y' and define the composition to be the roof (s \circ r, g\circ h): X\rightarrow Z given by X\xleftarrow{s\circ r} Z' \xrightarrow{g\circ h} Z.

Derived Categories

With localization under our belt we can now define derived categories. Let A be an abelian category.

(3.6) Definition: We define the category \text{Kom}(A) as having objects all chain complexes (A^n,d^A_n)_{n\in \mathbb{Z}} with each A_n an object of A and d_n^A a morphism of A which forms a differential. The morphisms between chain comlpexes are defined to be chain maps: morphisms A^\bullet \xrightarrow{f^\bullet} B^\bullet are a collection of maps (f_n:A_n\rightarrow B_n)_{n\in\mathbb{Z}} such that f_{n+1}\circ d^A_n = d_n^B\circ f_n.

The category \text{Kom}(A) is additive (direct sums are termwise, and the set \text{Hom}_{\text{Kom}(A)}(A^\bullet,B^\bullet) has the structure of an abelian group. There will be a bunch of indexing choices (such as the one above, where going to the right means increasing in numbers). Our choices are arbitrary, and don’t affect the outcome, as long as we remember our choices and follow them.

(3.7) Definition: The category \text{K}(A) is called the homotopy category. It’s constructed from \text{Kom}(A) as follows: the objects are the same as \text{K}(A) but, we quotient the group of morphisms by the subgroup of morphisms homotopy equivalent to 0.

The above definition can be restated. Note for any morphism of complexes, A^\bullet\xrightarrow{f^\bullet} B^\bullet, there is an induced morphism on homology of the two complexes, H^i(f^\bullet):H^i(A^\bullet)\rightarrow H^i(B^\bullet), for every index i. The null-homotopic maps, those morphisms f^\bullet\in \text{Hom}_{\text{Kom}(A)}(A^\bullet,B^\bullet) which have H^i(f^\bullet)=0 as maps, form a subgroup of \text{Hom}_{\text{Kom}(A)}(A^\bullet, B^\bullet). We quotient out by this subgroup.

To compose maps in the homotopy category, we can lift morphisms to the category of complexes and compose there; the composition then descends to the homotopy category in a well-defined way.

(3.8) Definition: Let \text{K}(A), \text{K}^+(A), \text{K}^-(A), \text{K}^b(A) be the homotopy category of A, the full subcategory of the homotopy category consisting of bounded below complexes (or bounded to the left; equivalently, for sufficiently negative indices the corresponding terms are 0), the full subcategory consisting of bounded above complexes ( i.e. terms are eventually always 0 for sufficiently positive indices), and the full subcategory of bounded complexes. We define D(A), D^+(A), D^-(A), D^b(A) to be the category obtained from the respective homotopy category by inverting the class of morphisms which induce isomorphisms on homology (such chain maps are called quasi-isomorphisms).

The reason why we must first pass to the homotopy category before localizing at quasi-isomorphisms is that they, in general, do not form a multiplicative system in the category of complexes. However, they do in the homotopy category.

(3.9) Proposition: The class of quasi-isomorphisms form a multiplicative system of morphisms in the categories \text{K}(A), \text{K}^+(A),\text{K}^-(A),\text{K}^b(A).

Proof. [G-M] Chapter 3, section 4, page 160, Theorem 4.

Using the construction of the localization of a category, this gives us a description of the unbounded derived category, the bounded below derived category, the bounded above derived category, and the bounded derived category of A.

We’ll collect some facts about derived categories here:

F1) There is a full and faithful embedding A\hookrightarrow D^*(A) where any object is sent to the complex with that object considered in degree 0 (at the 0 term) – here *=\emptyset, +, -, b.

F2) The categories D^*(A) are additive.

F3) There is an autoequivalence of categories T:D^*(A)\rightarrow D^*(A) defined by T((A^\bullet, d^A_\bullet))= (A^{\bullet+1}, -d^A_{\bullet+1}). For an object X we’ll write the nth composition of T on X as X[n]. We extend this definition to -i values by composing with the quasi-inverse of T. To reiterate, we have (A[n])^i=A^{i+n} and d^{A[n]}_i=(-1)^nd^A_i.

F4) Let X,Y be objects in A. We consider them as objects in D(A) by the embedding of F1).  We define \text{Ext}^k_A(X,Y):=\text{Hom}_{D(A)}(X,Y[k]). Again by F1), this means \text{Ext}^0_A(X,Y):=\text{Hom}_{D(A)}(X,Y)=\text{Hom}_A(X,Y). The groups \text{Ext}^k_A(X,Y) agree with the definition of the right derived functors of the \text{Hom}_A(X,-) and \text{Hom}_A(-,Y) functors when A has enough injectives and projectives respectively.

Since D^*(A) is a full subcategory of latex D(A), the ext groups don’t depend on the derived category we define it on. At the moment, I think this will be sufficient for what I want to do.

Triangulated Categories

Triangulated categories are developed in full generality in Chapter 10 of [K-S] and Chapter 4 in [G-M]. Briefly, a triangulated category C is both an additive category and an autoequivalence T:C\rightarrow C, called a shift functor,  along with a class of distinguished triangles. All of which are required to satisfy various axioms.

Triangles will be written as

X\rightarrow Y\rightarrow Z\rightarrow X[1]

where we keep the notation of X[1]=T(X). Morphisms of triangles are morphisms of each object which induces a comutative square between any two objects of the triangle.

What we’ll be interested in is:

(3.10) Proposition: the derived categories D^*(A) of an abelian category A, where *=\emptyset, +,-,b are triangulated.

and

(3.11) Proposition: Any exact sequence in \text{Kom}(A)

0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0

can be completed to a distinguished triangle in D^*(A)

X\rightarrow Y\rightarrow Z \rightarrow X[1]

by a morphism Z\rightarrow X[1], and any distinguished triangle in D^*(A) is isomorphic to one obtained in this way.

Proof of 3.10 and 3.11. These are Corollary 7 and Proposition 8 in [G-M] page 256 respectively.

In this way, our triangles will be generalizations of exact sequences in the derived category (which often won’t have a clear notion of exactness).

Tensor Triangulated Categories

The material from this section is from [MVW] chapters 8 and 8A.

Let A be an additive category with: the structure of a symmetric monoidal functor \otimes, and a triangulated structure with shift morphism T(A):=A[1].

(3.12) Definition: We call Atensor triangulated category  if it satisfies the following three conditions:

TTC1) there exist natural isomorphisms (commuting with the natural of associativity, commutativity, and unity morphisms of \otimes)

X\otimes Y[1]\xrightarrow{\sim} (X\otimes Y)[1] \xleftarrow{\sim} X[1] \otimes Y

and

TTC2) for any distinguished triangle \Delta and object X we have X\otimes \triangle and \triangle \otimes X are distinguished

and

TTC3) for any X,Y the natural diagram anti-commutes (going one way is the negative of the other way)

\begin{matrix} (X\otimes Y)[1] & \rightarrow & (X\otimes Y[1])[1]\\ \downarrow & & \downarrow \\ (X[1] \otimes Y)[1] & \rightarrow & (X\otimes Y)[2]\\ \end{matrix}

Remark: In TTC1) the isomorphisms are dependent on X,Y (and should satisfy a cocycle condition if you write out the indices). In TTC3) the morphisms in the diagram are the ones coming from TTC1. I just couldn’t find out how to write down names for the vertical maps.

(3.13) Definition: Further, A is an additive tensor triangulated category if

(\coprod_i A_i)\otimes B\cong \coprod_i (A_i\otimes B)

For example, if we have two bounded above chain complexes (X,d^X),(Y,d^Y) we can define their tensor product (X\otimes Y)^n=\oplus_{p+q=n} X^p\otimes Y^q with differential d^X\otimes \text{id} + (-1)^p\text{id}\otimes d^Y.

 We’ll end with two results which allow us to transfer the tensor triangulated category structure to another category:

(3.14) Proposition: Let A be a tensor triangulated category. Then in both of the following situations there is a way to extend the tensor triangulated structure of A:

i) if S is a saturated system of morphisms closed under sums, translations, cones and closed under tensor products with objects in A then A[S^{-1}] is a tensor triangulated structure

or

ii) if T is an object with \text{Hom}(X,Y)\xrightarrow{\sim} \text{Hom}(X\otimes T, Y\otimes T) for every X,Y in A, then the category A[T^{-1}] – whose objects are pairs (X,n) with X an object in A and n an integer, and whose morphisms (X,m)\rightarrow (Y,n) are given by elements of \varinjlim_i\text{Hom}(X\otimes T^{\otimes m+i}, Y\otimes T^{\otimes n+i})


References:

[G-M] An Introduction to Homological Algebra – Gelfand and Manin.

[K-S] Categories and Sheaves -Kashiwara and Schapira

[Lev] Mixed Motives – Marc Levine. Mathematical Surveys and Monographs volume 57. American Mathematical Society.

[MVW] Motivic Cohomology – Mazza, Voevodsky, Weibel. Link. (Try Google if the link expires).

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