Re: geometric multiplicity of schemes

On Aug 5, 4:33 am, Hagen <k...@xxxxxxxxxxx> wrote:
Hi

consider the affine scheme X=spec k[x,y]/(xy), k a
field. If one
speaks of the "irreducible components" of X then one
means the
irreducible components of topological space of X
endued with the
reduced subscheme structure, right? Therefore one
gets the irreducible
components of X as closed subschemes. The irreducible
components of X
are X1=spec k[x,y]/(x)=spec k[y] and X2=spec k[x]. I
wonder what the
geometric multiplicity of X1 and X2 is. The geometric
multiplicity of
X1 and X2 is (perhaps) defined as the length of the
local ring of X at
the generic point of X1 and X2 respectively. I am not
able to localize
k[x,y]/(xy) at the generic point of X1 or X2 but I
suppose the
geometric multiplicity is one in both cases. How can
I calculate that?

The intuition suggests that the geometric
multiplicity of the y-axis
in X=spec k[x,y]/(x^2y) is two but if my
understanding of an
irreducible component as mentioned above is correct,
this makes no
difference to X. Am I dumb?

The notion of an irreducible component is a purely topological
one and from a top. viewpoint the schemes defined by the
ideals (xy) and (x^2y) are identical.

Well... experience with polynomials in one
variable shows that it is nice that roots
be counted with multiplicity, don't you think?
Do you really want to both irreducible components
of the scheme

Spec CC[x] / (x^2(x-1))

to be indistinguishable?

-- m

.

Relevant Pages

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