## The Chow Groups

And today, we start intersection theory.  So, to establish notation a bit, we’re only going to be talking about algebraic schemes.  These are separated schemes of finite type over our base field $K$.  That is, they admit finite open affine covers such that each affine is the spectrum of a finitely generated $K$-algebra.  We’re already almost to varieties, the only thing left is to assume reduced and irreducible (equivalently, integral) to get there, but we won’t do that unless necessary.  Nor will we assume smooth if we don’t have to, so we’re hoping to get a theory that will work, at least somewhat, for singular varieties.

So first we need to introduce the basic objects of our study, a collection of groups that will eventually be turned into a graded ring.  We’re going to start with them acting somewhat more homologically, but later, they will act more cohomologically.  These are called the Chow Groups or the Cycle Groups of our scheme $X$.

Put succinctly, a $k$-cycle is a $\mathbb{Z}$-linear combination (formal, of course) of subvarieties of dimension $k$.  We’ll call this group $Z_k(X)$.  Now, we want an equivalence relation on $k$-cycles to get down to, in some sense, the fundamental ones.

For a $k+1$ dimensional subvariety $W$, and any nonzero rational function on it, we define $[\mathrm{div}(r)]=\sum \mathrm{ord}_V(r)[V]$, just as we do for divisors, except now in the larger group of cycles on $X$.  Then we say that a $k$-cycle $\alpha$ is rationally equivalent to zero if there are finitely many $W_i$ and finitely many $r_i$ rational functions such that $\alpha=\sum [\mathrm{div}(r_i)]$.  These form a subgroup of $Z_k(X)$, which we denote by $\mathrm{Rat}_k(X)$.

And now, we get to the big definition: $A_k(X)=CH_k(X)=Z_k(X)/\mathrm{Rat}_k(X)$.  We’ll denote the direct sum by $A_*(X)$ or $CH_*(X)$, and we call a general element a cycle.  We say that a cycle is positive if its coefficients are, and a class is positive if it can be represented by a positive cycle.

Example: Here’s a quick and easy example.  Look at $X=\mathbb{P}^n$.  Then, for each $k$, we get $A_k(X)=\mathbb{Z}[H_k]$, where $H_k$ is a $k$-plane.

A few other things: Chow groups don’t differentiate scheme structures on the same underlying set.  Anything has the same Chow groups as its reduction does.  Additionally, Chow groups add over disjoint union, so when studying Chow group, we can assume connected and reduced without losing any generality.  Finally, and this will be useful to reduce to the case of a variety, we have a short exact sequence: let $X_1,X_2$ be closed subschemes of $X$.  Then we have $A_k(X_1\cap X_2)\to A_kX_1\oplus A_kX_2\to A_k(X_1\cup X_2)\to 0$.

Before we talk about what we can do with cycles, let’s associate a cycle to every subscheme of $X$.  We start by defining the cycle of our ambient scheme $X$.  Now, each irreducible component $X_i$ of $X$ gives us an Artin local ring $\mathscr{O}_{X_i,X}$, and we define $m_i=\mathrm{length}_{\mathscr{O}_{X_i,X}}(\mathscr{O}_{X_i,X})$ to be the geometric multiplicity of $X_i$ in $X$.  Then, we define $[X]$ to be the cycle $\sum m_i[X_i]$ in $A_*(X)$.  To get this to work for any subscheme, we note that there is an inclusion $A_*(Y)\to A_*(X)$ for any $Y$ a subscheme of $X$, and so we just take the image of $[Y]$ in $A_*(X)$.

So now, we want to know what we can do to push these cycles around.  We’re thinking of them like homology at the moment, so our first thought should be that there should be a pushforward functor.  For this, we can’t work with just any morphism.  We’ll be needing a proper morphism.  Let $f:X\to Y$ be one.  We can define the pushforward for cycles on the subvarieties of $X$, so let $V$ be one.  We define $f_*[V]=\deg(V/W)[W]$, where $W=f(V)$, and $\deg(V/W)$ is the degree of the field extension if $V$ and $W$ are the same dimension, and zero otherwise.  It is straightforward to check that this gives a map on $A_*(X)\to A_*(Y)$, not just $Z_*(X)\to Z_*(Y)$.

In fact, we have a more precise formulation: assume also that the map is surjective.  Then, if $Y$ has dimension smaller than $X$, for every $r\in R(X)^*$, we get the pushforward of $\mathrm{div}(r)$ to be zero.  If the same dimension, we get instead the divisor of the norm of $r$, that is, the determinant of the $R(Y)$-linear map $r:R(X)\to R(X)$.

Now, given a complete scheme, that is, one proper over a point, we have a natural map $X\to\mathrm{Spec}(K)$, which we’ll denote by $\deg$ or by $\int_X$.  This actually gives a map $A_*(X)\to \mathbb{Z}=A_*(pt)$, which we call the degree of a cycle on $X$.  Degree is preserved by proper morphisms, and so we’ll often just write $\int$

So now, this does all generalize a few things we know about.  For a $n$-dimensional scheme $X$, we have $A_{n-1}(X)=\mathrm{Pic}(X)$, so for a curve, we have $A_0(X)$ is the Picard group, and the kernel of the degree map gives us back the Jacobian of the curve.

Eventually, we’re going to switch to cohomological viewpoint (essentially, we’ll reverse the grading) and introduce a product structure that encodes intersections.  Next time, we’ll talk about when we can take pullbacks, work with them for a bit, and talk about a few more things regarding general Chow groups before we start talking about divisors, line bundles, chern classes and vector bundles.