Divisorial Jungle

I'd like to compile a short list of definitions of Weil and Cartier Divisors, Line Bundles and Invertible Sheaves, Class Groups and Picard Groups, Cohomology, (higher) Chow Groups and K-theory for algebraic schemes and their relations. I intentionally omit proofs, but there are some ideas. I couldn't resist to jot down some properties of the objects which are important to me (homotopy invariance, existence of pullbacks and pushforwards).

Let $X$ be an algebraic scheme over a field $k$, i.e. a scheme with structural morphism $X \to Spec(k)$ of finite type. Whenever any coefficients appear, I chose $\mathbb{Z}$, but it might be much more convenient to take $\mathbb{Q}$ in applications. It might also be quite convenient to restrict attention to varieties which are smooth over $k$ and assume $k$ perfect (since these conditions imply regular and normal), but I try to make these assumptions only when necessary. In fact, after writing this, I wish I would have stuck to rational coefficients and smooth schemes.

Definitions

Cycles and divisors

A $k$-dimensional cycle of $X$ is a $\mathbb{Z}$-linear combination of $k$-dimensional closed subvarieties, i.e. closed immersions from integral algebraic schemes of dimension $k$. The group of all $k$-dimensional cycles is denoted $Z_k(X)$. If $X$ is equidimensional, the group of all codimension $k$ cycles is denoted $Z^k(X) := Z_{dim(X)-k}(X)$. I will discuss rational equivalence and Chow groups below.

For $X$ equidimensional and regular in codimension $1$, a Weil divisor is a $\mathbb{Z}$-linear combination of codimension $1$ subvarieties (which are also called prime divisors). A Weil divisor is called effective if the coefficients of the prime divisors are non-negative. To a non-zero rational function $f$ one can associate a Weil divisor $(f) := \sum_Y v_Y(f)[Y]$, where the sum runs over all prime divisors. Weil Divisors of the form $(f)$ are called principal divisors. The group $Cl(X) := Div(X) / Princ(X)$ of divisors modulo principal divisors is called the divisor class group. I will discuss the relation with Chow groups below.

A Cartier Divisor is a global section of $\mathcal{K}_X^\times / \mathcal{O}_X^\times$, where $\mathcal{K}_X$ is the sheaf of rational functions on $X$. A Cartier divisor is called principal divisor if it is in the image of the quotient map from $\mathcal{K}_X^\times$, i.e. if it can be represented by a global non-zero rational function. The group $CaCl(X) := CaDiv(X) / Princ(X)$ of Cartier divisors modulo principal divisors is called Cartier class group. I will discuss the relation with cohomology below.

Invertible sheaves and bundles

An invertible sheaf is a coherent $\mathcal{O}_X$-module sheaf which is locally free of rank $1$, i.e. an invertible sheaf $L$ is locally isomorphic to $\mathcal{O}_X$. For an invertible sheaf $L$ the dual $\mathcal{O}_X$-module $L^\vee := \mathcal{H}om(L,\mathcal{O}_X)$ is a $\otimes$-inverse via the evaluation isomorphism $L \otimes_{\mathcal{O}_X} \mathcal{H}om(L,\mathcal{O}_X) \to \mathcal{O}_X$.

An algebraic line bundle is an algebraic vector bundle of rank $1$, i.e. a morphism of schemes $E \to X$ with zero section $X \to E$, scalar multiplication morphism $Spec(k) \times_X E \to E$ and vector addition morphism $E \times_X E \to E$ that turn each fiber into a $k$-vector space of dimension $1$, and there exists a Zariski-covering of $X$ and a local trivialization of $E$ such that the corresponding glueing morphisms are $k$-linear in each fiber. From a general argument identifying locally free $\mathcal{O}_X$-modules as sheaves of sections of algebraic vector bundles (which gives an equivalence of categories), algebraic line bundles are equivalent to invertible sheaves. The group of all line bundles modulo isomorphism (with group structure from $\otimes$) is called Picard group $Pic(X)$.

Definitions of some fancy objects

The $0$-th K-group $K_0(A)$ of a ring $A$ is the Grothendieck ring of projective modules, i.e. the group of isomorphism classes of projective modules modulo the relation identifying a module $M$ with $M' \oplus M''$ if there exists a short exact sequence $M' \to M \to M''$ (which doesn't necessarily split). The $0$-th K-group $K_0(X)$ is the Grothendieck ring of algebraic vector bundles. Since projective modules over a ring are, as category, equivalent to vector bundles over its spectrum, $K_0(Spec(A)) \simeq K_0(A)$. There are various isomorphic definitions of higher algebraic K-theory of a scheme, but I won't write any of these down. For a ring $A$ one can take $K_n(A) := \pi_n(BGL_\infty(A)^+)$. There is a filtration on $K_n(X)$ called the Adams filtration, or $\gamma$-filtration (coming from the $\lambda$-ring structure), with weight-$1$-subspace of $K_0(X)$ just $K_0(X)^{(1)} = Pic(X)$.

For $X$ equidimensional and smooth over $k$, write $z^i(X,m)$ for $\mathbb{Z}$-linear combinations of codimension $i$ closed subvarieties of $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ (for $j < m$) properly, where $\Delta^m := Spec(k[t_0,\dots,t_m]/\sum t_i-1)$ is the affine standard simplex. These fit together to form a simplicial abelian group $z^i(X,\bullet)$ (with respect to intersection product) and the homotopy groups $CH^i(X,m) := \pi_m(z^i(X,\bullet))$ are called higher Chow groups. Obviously, $CH^i(X,0) = \pi_0(z^i(X,\bullet)) = CH^i(X)$.

Motivic cohomology are certain Ext-groups in Voevodsky's category of mixed motives (one can take that as definition): $H^{p,q}(X) = Hom_{DM^{eff}_{Nis}}(\mathbb{Z}_{tr}(X),\mathbb{Z}_{tr}(p)[q])$, where $\mathbb{Z}_{tr}(X)$ is the presheaf with transfers associated to $X$.

Some properties

After looking at the comparison theorems below, the properties noted here become a lot nicer on smooth schemes, by using the comparisons to get more pullback/pushforward homomorphisms.

Pullbacks

• Cycles can be pulled back along flat morphisms of constant relative dimension, which preserves codimension, rational equivalence and is compatible with the intersection product. The same holds for Weil divisors. There is also flat pullback for higher Chow groups.
• Cartier divisors can be pulled back along any map whose image is not contained in the support of the divisor. By taking a linearly equivalent Cartier divisor, if necessary, one may always pull back the class of a Cartier divisor (on an integral scheme).
• Bundles can be pulled back to bundles along arbitrary morphisms and isomorphic bundles pull back to isomorphic bundles, hence there is a pullback on the Picard group. This generalizes to algebraic K-Theory.
• On sheaf cohomology one has arbitrary pullbacks (sheaf cohomology is a contravariant functor, after all) and the same is true for motivic cohomology.

Pushforwards

• Cycles can be pushed forward along proper morphisms, which preserves dimension and rational equivalence. Since Weil divisors are codimension 1 cycles, they admit proper pushforward only along maps of relative dimension $0$. There is the degree map, which is just pushforward along the structural morphism $X \to Spec(k)$. I don't know whether higher Chow groups also have proper pushforward... do you?
• I don't know if there is any kind of general pushforward for Cartier divisors or classes thereof.
  Bundles can be pushed forward (in the sense of pushforward of sheaves that yields a bundle again) along finite flat morphisms, and there are more morphisms whose pushforward is a vector bundle again. I don't know if there is a nice criterion for which kind of morphism allows pushforward in K-Theory.
• For cohomology, pushforwards of sheaves exist in general, but you don't get a pushforward morphism on the cohomology in general. If you take a proper map which is in some sense oriented, then one gets pushforwards on ordinary (singular) cohomology, which look like integration over the fiber in the de Rham picture. I don't know what the general statement for motivic cohomology looks like... do you?

Homotopy invariance

• Chow groups have homotopy invariance, i.e. $CH^\bullet(X\times \mathbb{A}^1) \simeq CH^\bullet(X)$, where the isomorphism is induced by pullback along the projection $X \times \mathbb{A}^1 \to X$. In particular, the Weil divisor class group has homotopy invariance. Even the higher Chow groups are homotopy invariant, i.e. $CH^i(X\times \mathbb{A}^1,m) \simeq CH^i(X,m)$.
• The Cartier class group is not homotopy invariant in general, but I don't know good examples.
• The Picard group (and the K-Theory) are homotopy invariant on regular schemes (but I don't know counter-examples in general).
• Sheaf cohomology is not homotopy-invariant in general, but if you take a locally constant sheaf as coefficients, it is. However, in this article, coefficients $\mathcal{O}_X^\times$ are most relevant, and there we don't have homotopy invariance. Motivic cohomology, on the other hand, is homotopy invariant (by construction).

Comparisons

As far as I can see, all these comparisons are compatible with pullbacks (as far as they exist). For pushforwards, the precise relationship between the Chern character and pushforwards is called Grothendieck-Riemann-Roch theorem.

Weil divisor class group and Chow group of codimension 1 cycles

The usual definition of rational equivalence (that shows up in the definition of the Chow group as in the book of Y.André on motives) looks rather different from the usual definition of linear equivalence (that shows up in the definition of the Weil divisor class group). Linear equivalence is just equality modulo adding principal divisors.
Rational equivalence of two cycles $\alpha,\beta$ of codimension $1$ in $X$ is the existence of a cycle $\gamma$ of codimension $1$ in $X \times \mathbb{P}^1$ such that the projection $p_{\mathbb{P}^1}^{X\times\mathbb{P}^1}$ to $\mathbb{P}^1$ is dominant, and $(p_X^{X\times\mathbb{P}^1})_\ast\left(\gamma \cdot (X \times \{0\} - X \times \{\infty\})\right) = \alpha-\beta$.
These two equivalence relations on codimension-1 cycles agree. From linear equivalence you can easily cook up a $\mathbb{P}^1$ joining the two divisors, but the other direction is tricky. One has to relate the pushforward in the latter definition to the former. The crucial technical result (Proposition 1.4b in Fulton's book on intersection theory) states that for $p : X \to Y$ a proper surjective morphism of varieties of the same dimension and $f \in \mathcal{K}(X)^\times$ a non-zero rational function, $p_\ast[div(f)] = [div(N(f))]$, where $N$ is the norm of the field extension $K(X)/K(Y)$. This is applied in the situation where we have $\gamma$ as in the definition of rational equivalence, but $\gamma = V$ consists of just a subvariety $V$ (the general case is done by extending linearly), so there is a morphism $f : V \to \mathbb{P}^1$ which is induced by the projection $p_{\mathbb{P}^1}^{X\times\mathbb{P}^1}$ (so, $f$ determines a rational function $f \in K(X)$) and the other projection $p_X^{X\times\mathbb{P}^1}$ induces morphisms $p : f^{-1}(P) \to X$ from the scheme-theoretic fiber $f^{-1}(P)$ for any rational point $P \in \mathbb{P}^1$, which are isomorphisms onto some subscheme we want to call $V(P)$. Then one has $[f^{-1}(0)] - [f^{-1}(1)] = [div(f)]$, hence $[V(0)] - [V(\infty)] = p_\ast[div(f)] = [N(div(f))]$, which is linearly equivalent to $0$ (since it's a principal divisor).

Cartier class group and cohomology:

The short exact sequence of sheaves $1 \to \mathcal{O}_X^\times \to \mathcal{K}_X^\times \to \mathcal{K}_X^\times / \mathcal{O}_X^\times \to 1$ yields a long exact cohomology sequence which reads $1 \to \Gamma(\mathcal{O}_X^\times) \to Princ(X) \to CaDiv(X) \to H^1(X,\mathcal{O}_X^\times) \to 1$ and identifies $H^1(X,\mathcal{O}_X^\times) \simeq CaCl(X)$.

Line bundles and cohomology:

Given a line bundle $L$ one can take any local trivialization $\phi_i : L|_{U_i} \simeq \mathcal{O}_{U_i}$ over an open cover $\mathcal{U} = \{U_i \to X\}$ with patching data $\phi_{ij} := \phi_i \circ \phi_j^{-1} \in GL_1(\mathcal{O}_{U_i \times_X U_j})$ and these patching data $\phi_{ij}$ form a Cech 1-cocycle over $\mathcal{U}$ with values in $\mathcal{O}_X^\times$, thus a class in Cech cohomology $H^1_{Cech}(\mathcal{U},\mathcal{O}_X^\times)$. In the limit, one gets a class in $H^1_{Cech}(X,\mathcal{O}_X^\times)$, thus from general nonsense a class in the sheaf cohomology group $H^1(X,\mathcal{O}_X^\times)$. Different trivializations yield cohomologous cocycles, hence the same Cech classes.
It also works the other way around: Take such a cohomology class, represent it by a Cech 1-cocycle over some open cover $\mathcal{U}$ (which has to be fine enough) and write down the bundle patched together from trivial bundles via the patching data. Taking a different 1-cocycle in the same class (or a different open cover) yields an isomorphic bundle, hence $H^1(X,\mathcal{O}_X^\times) \simeq Pic(X)$.

Cartier divisors and line bundles:

For each Cartier divisor $D \in CaDiv(X) = \Gamma(\mathcal{K}_X^\times / \mathcal{O}_X^\times)$ we can choose an open affine cover $\{U_i \to X\}$ such that $D$ is represented by sections $f_i$ of $\mathcal{K}_X^\times$ over $U_i$. Then we define a line bundle $L(D)$ as the sub-$\mathcal{O}_X$-module of $\mathcal{K}_X$ generated by the $f_i^{-1}$ over $U_i$. This gives a monomorphism $CaCl(X) \to Pic(X)$.
To any line bundle $L$ with embedding $L \to \mathcal{K}_X$ we can associate a Cartier divisor $D$ such that $L=L(D)$, by taking $f_i \in \mathcal{K}_X(U_i)$ to be the inverse of a local generator of $L$ over $U_i$ (where the $U_i$ have to be a trivializing cover). This is obviously an inverse to the other construction.
If $X$ is integral, $\mathcal{K}_X = \underline{K(X)}$, a constant sheaf, and then we can embed every line bundle in $\mathcal{K}_X$ by $L \to L\otimes \mathcal{K}_X$, since $L \otimes \mathcal{K}_X$ is locally constant, hence constant. Therefore, on $X$ integral, $CaCl(X) \simeq Pic(X)$.

Compatibility of the isomorphisms so far:

The isomorphism class of line bundles we attached to a Cartier class turns out to be the same isomorphism class of line bundles specified by the cohomological data given by the Cartier class, which one can see explicitly by taking a Cartier divisor $D = (f_i)_i$ and assign the isomorphism class of line bundles specified by the patching data $f_i/f_j$.

Cartier divisors and Weil divisors:

On $X$ integral, separated, noetherian and regular in codimension $1$, Cartier divisors can be mapped to locally principal Weil divisors, by representing the Cartier divisor over a cover $U_i$ as some $f_i \in K(X)$ and for each prime divisor $Y$ taking some index $i$ such that $Y \cap U_i \neq \emptyset$, then $n_Y := v_Y(f_i)$ is well-defined (doesn't depend on $i$) and $\sum v_Y Y$ is a locally principal Weil divisor.
If $X$ is furthermore normal, each locally principal Weil divisor $D$ comes from a Cartier divisor, since for each point $x \in X$ we can take the restriction $D_x$, a divisor on $Spec( \mathcal{O}_{X,x})$, and $\mathcal{O}_{X,x}$ is a UFD, so $D_x$ is a principal divisor, $D_x = (f_x)$ and $(f_x)$ defines a divisor on $X$ which restricts to $D_x$ as well, so agrees on an open neighborhood $U_x$ with $D$; the various $f_x$ give a Cartier divisor, since $X$ is normal, so on any open $U$ with $f,g$ inducing the same Weil principal divisor, $f/g$ is a section of $\mathcal{O}_X$.
If $X$ is furthermore locally factorial, every Weil divisor is locally principal, so $Div(X) \simeq CaDiv(X)$. Since principal divisors agree, $Cl(X) \simeq CaCl(X)$. In particular this holds for smooth $X$.

The map from an isomorphism class of line bundles to a linear equivalence class of Cartier divisors to a rational equivalence class of Weil divisors is also called $c_1$, the first Chern class. Note how one has such a map $Pic(X) \to CH^1(X)$ even if $Pic(X) \to CaCl(X)$ and $CaCl(X) \to Cl(X)=CH^1(X)$ are only monomorphisms. This map is called $c_1$, the first Chern class of a line bundle.

I want to summarize, for $X$ smooth over $k$, we have
$H^{1}(X,\mathcal{O}_X^\times) \simeq Pic(X) \simeq CH^1(X)$

Comparison of fancy objects

Comparison of Chow groups and K-Theory in general, for $X$ smooth over $k$:
$K_0(X)^{(q)} \otimes \mathbb{Q} \simeq CH^q(X) \otimes \mathbb{Q}$
where the $(q)$ indicates the $\gamma$-filtration.
This is the Chern character, which you can build by identifying $K_0(X)$ with the $K$-group of coherent sheaves on $X$ (there smoothness is used), and then one can take a resolution of a coherent sheaf by vector bundles (on a stratification) and define ordinary chern classes for vector bundles via a splitting principle and $c_1$.

This generalizes to higher Chow groups and higher K-Theory (but I don't know who proved that):
$gr_\gamma^q\left( K_n(X)\otimes \mathbb{Q} \right) \simeq CH^q(X,n) \otimes \mathbb{Q}.$
Voevodsky proved that these are strongly related to motivic cohomology:
$CH^q(X,2q-p) \simeq H^{p,q}(X)$ for $X$ smooth over $k$ and $k$ a perfect field, where the map is the so-called motivic cycle class map.

We recover the comparison of divisors with line bundles and cohomology (but now with rational coefficients):
$H^{2,1}(X)_{\mathbb{Q}} \simeq K_0(X)^{(1)}_{\mathbb{Q}} \simeq CH^1(X,0)_{\mathbb{Q}}$
where $H^{2,1}(X)$ is just $H^1(X,\mathcal{O}_X^\times)$ and $K_0(X)^{(1)} = Pic(X)$ and $CH^1(X,0) = CH^1(X)$.

I don't know how to prove that stuff :-) but I hope I'll learn that some day.