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Why does Hartshorne require that a scheme is separated for defining Weil Divisors
I flipped through Hartshorne section II.6,imitation rolex daytona steel, and nothing requires separatedness there that I could see. The following doesn't precisely address the question, but I added it because it helped me lose interest in the question :).
Fitting with our intuition for the line with doubled origin, Stacks Project Lemma 27.29.3 says that if $X$ is quasicompact then we can find a dense open subscheme $U \subseteq X$ which is separated. Then by Hartshorne Prop. 6.5 there is a surjection $Cl(X) \to Cl(U)$,imitation perpetual date rolex, which is an isomorphism when codim$_XZ > 1$. So, in many examples the class group is the same or not much different than that of something separated. Lemma 27.29.3 is also still satisfied for the easiest of non quasicompact examples, like affine space with infinitely many origins.
If you are hunting for pathologies, maybe you should look for something that fails 27.29.3 then. Still though,rolex daytona price imitation, I'm not sure what properties you would look for to fail since I think everything in Hartshorne is still true.
Relation of Function Field of a scheme to the Local Ring of its Prime DivisorCorrespondence Between Cartier Divisor and Weil Divisor (Hartshorne Proposition 6.11,imitation rolex daytona for ladies, Chapter 2)When does a principal divisor have degree 0?Effective Weil divisors on a Surface and local equations$X \times \mathbf{A}^1$ is regular in codimension 1Multiplicity of Cartier divisor on locally noetherian scheme is only non zero at generic pointCodimension 1 pointsDefinition of weil divisorsKatz Mazur chapter 1 AG questions.
I flipped through Hartshorne section II.6,imitation rolex daytona steel, and nothing requires separatedness there that I could see. The following doesn't precisely address the question, but I added it because it helped me lose interest in the question :).
Fitting with our intuition for the line with doubled origin, Stacks Project Lemma 27.29.3 says that if $X$ is quasicompact then we can find a dense open subscheme $U \subseteq X$ which is separated. Then by Hartshorne Prop. 6.5 there is a surjection $Cl(X) \to Cl(U)$,imitation perpetual date rolex, which is an isomorphism when codim$_XZ > 1$. So, in many examples the class group is the same or not much different than that of something separated. Lemma 27.29.3 is also still satisfied for the easiest of non quasicompact examples, like affine space with infinitely many origins.
If you are hunting for pathologies, maybe you should look for something that fails 27.29.3 then. Still though,rolex daytona price imitation, I'm not sure what properties you would look for to fail since I think everything in Hartshorne is still true.
Relation of Function Field of a scheme to the Local Ring of its Prime DivisorCorrespondence Between Cartier Divisor and Weil Divisor (Hartshorne Proposition 6.11,imitation rolex daytona for ladies, Chapter 2)When does a principal divisor have degree 0?Effective Weil divisors on a Surface and local equations$X \times \mathbf{A}^1$ is regular in codimension 1Multiplicity of Cartier divisor on locally noetherian scheme is only non zero at generic pointCodimension 1 pointsDefinition of weil divisorsKatz Mazur chapter 1 AG questions.
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