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Let $A = k[x,y]/(xy,y^2)$, where $k$ is a field, and take $f \in A$. I am working on an exercise that says prove that the support of $f$ (as a global section of the structure sheaf on Spec $A$) is either empty, the origin, or the entire space. I seem to be getting answers that is different from this, and I would appreciate if someone could show me how to do it. Thank you!

user201886
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1 Answers1

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Show that $A_x \simeq k[x]_x$ (we're throwing out the nonreduced point $(x,y)$). This is a domain, so the support of a section there is all or nothing.

Hoot
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  • Could you possibly explain me why the support of a section is all or nothing when it is a domain by any chance? Thanks! – user201886 Feb 12 '15 at 15:33
  • Also, could you possibly be able to explain what is wrong here: $P = (x+1,y)$ and $Q = (x+2,y)$ are prime ideals of $A$. If I take a function $f = x+1 + I \in A$, then $f$ id $0$ at P, but not $0$ at Q... So it's neither empty, origin, or the whole space... – user201886 Feb 12 '15 at 16:14
  • @user201886 This is a common mistake when talking about supports: $f$ has value $0$ in $P$ (by this I mean it's image in the residue field is zero) but its germ there is not zero. – Hoot Feb 12 '15 at 17:05
  • The fact about domains is just because the maps $A \to A_{\mathfrak p}$ in that case are all injective. – Hoot Feb 12 '15 at 17:05
  • I see. Thank you. I am still a bit confused about the first part, so I will make it into another question. – user201886 Feb 12 '15 at 17:10
  • I think there is something I am missing (or not understanding correctly) because I am getting that it is not zero in the residue field at $Q$ either... – user201886 Feb 12 '15 at 17:16
  • In the residue field at $Q$ it should have value $-1$. – Hoot Feb 12 '15 at 18:21
  • @Hoot So the support is when the germ is not $0$? And that the germ of $x+1 + I$ is not $0$ at $P$ or at $Q$? – user211392 Feb 12 '15 at 18:41
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    @user211392 A section is supported at a point $P$ if its image in $\mathcal O_{X, P}$ is nonzero. It vanishes at $P$ if its image in the residue field $k(P) = \mathcal O_{X, P}/\mathfrak m$. These are very different. The first is saying that small neighborhoods of $P$ don't "see" any nontrivial behavior out of the function whereas the second just means that it's zero at the point but may do something interesting nearby. Think of how a smooth function can vanish at a point but still have an interesting germ or Taylor expansion there. – Hoot Feb 12 '15 at 18:48