When one thinks of the ring of integers in an algebraic number field as defining a one-dimensional scheme, what would the ideal class group correspond to geometrically please? Is it maybe the group of invertible sheaves under tensor product? Does it correspond to equivalence classes of line bundles perhaps?
I don't ask for a very detailed answer. A brief outline with perhaps a few references will do.
Edit: Lord Shark the Unknown's comment seems to confirm my suspicion. For a curve, the Jacobian and Chow group coincide as notions. A line bundle defines a linear equivalence of divisors. This should correspond to an equivalence class of a non-zero ideal. It remains only to prove that this equivalence relation is the right one. Ok got it! Thank you!
Read my edit first. Two nonzero ideals in the ring of integers are equivalent if and only if they differ by a nonzero rational factor. This corresponds to linear equivalence of divisors on a curve. Thus the ideal class group is isomorphic to the (abelian) additive group of divisors up to linear equivalence of the corresponding curve. It is the Weil divisor class group $Cl(X)$ of the corresponding curve $X$.