# Picard group for dummies

a picard group is the set of isomorphism classes of invertible R-modules. I just read that phrase in the CRing project notes without further explanations:

Here are my questions:

1-under which law (I am guessing it's a restriction of a tensor product) how could we restrict it to classes? 3-why an abelian group?

I am not familiar with any algebraic geometry. Thus shall be grateful to you for using a basic vocabulary :)

Many thanks

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1. tensor product, 2. tensor product respects isomorphisms by the universal property, 3. tensor product respects switching of the factors by the universal property. –  Qiaochu Yuan May 20 '11 at 11:28
@Yuan thanks for the hint –  El Moro May 20 '11 at 12:00
If you are familiar with Number theory, the class group for a number field is in fact the Picard group for the ring of integers. The point is that every invertible module is isomorphic to a fractional ideal, and tensor product is isomorphic to products when you work with fractional ideals. Multiplication by a principal ideal gives you a fractional ideal that's isomorphic to the previous one as $R$-modules. See Theorem 11.6 of Eisenbud's "commutative algebra with a view toward algebraic geometry" for more details. –  Jiangwei Xue May 20 '11 at 12:05
Dear El Moro, The reason considering invertible modules is so important is that, in algebraic geometry, it's important to know what the line bundles on a scheme are, and on an affine these correspond precisely to the invertible modules. In topology, much mathematics has focused on classifying arbitrary vector bundles -- perhaps since line bundles alone on a space X can be classified by the easily computable (usually) $H^2(X,Z)$. –  Akhil Mathew May 20 '11 at 12:35
In algebraic geometry, the analogous characterization (the line bundles are classified by $H^1(X,O_X^*)$) seems much less easy to compute with, since it is the cohomology of a rather complicated (and non-quasicoherent) sheaf. Moreover, on a normal affine scheme, the Picard group measures the deviation from factoriality. –  Akhil Mathew May 20 '11 at 12:35