I'm having problem comprehending the Picard group of a ring, and figured that perhaps it would be easier if I had a few examples of actual computations of Picard groups to look at.
Thus, my question to you good people is, given the ring $\mathbb{C}[x,y,z]/(x^2+y-z)$, how does one go about finding the Picard group?
EDIT 1: Seeing this question has already gotten 3 votes for closing for not following proper community guidelines, let me say I hear your concerns. Just give me another two hours or so to get home and elaborate a little, and I hope I shouls be able to rectify that!
EDIT 2: Okay, so, the complaint raised against this question has been that it does not follow certain standards for questions on this wonderful forum, specifically the following:
Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc.
And, I say, fair enough. It was rather out asked without context, I shall now try to remedy this.
The closest analogy that I can think of to finding the Picard group of a ring is finding the representation ring of a finite group, something that is a standard homework question in undergraduate courses in group and representation theory. Usually, the groups being the object of interest in these homeworks are, well, not too sophisticated, but neither are they too trivial. Stuff like, say, the tetrahedral symmetry group or the octahedral symmetry group, etc.
Nevertheless, when it comes to finding these representation rings (the question is often phrased rather as find the reduced character table, which is of course equivalent), the student has many tools available to them which are explained in intricate detail during their lectures. They are given character theory with its class functions, its inner products, and the Grand Orthogonality Theorem. They are given Schur's lemma and Maschke's theorem, from which they are given the result that they can decompose the regular representation into the irreducible representations (up to isomorphism) in a rather neat form. And they are given the result that there exists a bijection between the conjugacy classes of a group and the irreducible representations of the same.
As a consequence, though it might prove tedious, they have all the ingredients for actually computing the reduced character table of the group.
And so, I suppose that my real question here is, not specifically, what is the Picard group of $\mathbb{C}[x,y,z]/(x^2+y-z)$, but rather, how does one in general go about finding the Picard group of a ring, in the same manner as one would go about finding the representation ring of a finite group? What is the "analogy" for Schur's lemma, Maschke's theorem, the inner product of characters, etc. What is the process?
I hope that should satisfy you and provide you with the context you requested.
Please show some more of your own effort in the future. What did you try? What are your thoughts?
The first step here would maybe be to simplify your ring. We have an isomorphism $$\mathbb{C}[x,y,z]/(x^2+y-z) \rightarrow \mathbb{C}[x,z]$$ of $\mathbb{C}$-algebras given by identifying $y$ with $z-x^2$. Therefore it suffices to understand the Picard group of a polynomial ring, which is known to be trivial (this is a nice exercise in case that you do not know that yet: The Picard group of any UFD is trivial). Hence we have $$\text{Pic}(\mathbb{C}[x,y,z]/(x^2+y-z)) = 0.$$