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Let $X$ be a noetherian scheme. So in particular $X$ is a finite, locally finite, union of its irreducible components $X = \bigcup^n_i K_i$. Non-reduced points in $X$ fall into 2 categories:

  1. Fat point - $Nil(\mathcal{O}_{X,x})$ is a non-zero prime ideal. A generic point of some irreducible subscheme contained in a unique irreducible component of $X$. i.e. $\overline{\{ x \}} = Z$ for some irreducible subscheme $Z\subset K_j$ for a unique integer $j\le n$. . Special cases of these are:

    • Non-reduced Associated point. A generic point of an irreducible component of $X$ whose local ring is non-reduced.
    • Closed embedded points. These are generic points of themselves. i.e. closed non-reduced points $x \in X$ which lie in a unique irreducible component of $X$.
  2. Intersection point - $Nil(\mathcal{O}_{X,x})$ is a (non-zero) non-prime ideal. A generic point of some irreducible subscheme contained in more than one irreducible component. i.e. those for which $\overline{\{ \eta\}} = Z \subset K_i \cap K_j$ for some irreducible subscheme $Z$ and integers $i,j \le n$. Special case:

    • Closed intersection points - Closed points which lie at an intersection of irreducible components. Equivalently those closed points $x\in X$ for which $Nil (\mathcal{O}_{X,x})$ is a non-zero non-prime ideal.

So there are essentially two ways in which an irreducible component $K \subset X$ may fail to be an integral subscheme.

  • $K$ contains a fat point. i.e. a non-reduced irreducible subscheme. (What does that even mean? after all $K$ can be always be given the reduced induced subscheme structure).
  • $K$ Intersects another irreducible component of $X$.

When I try to make sense of this distinction I get confused. Is the above charactrization even well defined?

How should I view the non-reduced information of the scheme?

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The irreducible components of $X = \operatorname{Spec} A$ correspond to the minimal primes $\mathfrak p \subseteq A$; thus they are always reduced by convention. The scheme structure is the reduced induced one, i.e. $A/\mathfrak p$. A better name might have been integral components...

If two components $A/\mathfrak p_1, A/\mathfrak p_2$ meet, then the scheme structure on their intersection is $A/(\mathfrak p_1 + \mathfrak p_2)$. This may or may not be reduced, even though both $A/\mathfrak p_1$ and $A/\mathfrak p_2$ are reduced at the points of intersection (since they are reduced everywhere).

Example. Consider the ring $A = k[x,y]/(y^2-x^2y)$. This is the union of the $x$-axis and the parabola $y=x^2$. The minimal primes are $(y)$ and $(y-x^2)$. Both components are reduced (because components always are), but their intersection is $k[x,y]/(y,y-x^2) = k[x,y]/(y,x^2) \cong k[x]/(x^2)$, which is a thick point.

Remark. The subscheme $K$ does not see the other components. Thus, the intersection criterion you gave doesn't make sense. Maybe you're confusing $\mathcal O_{X,P}$ with $\mathcal O_{K,P}$ (for $P$ an intersection point).

Remark. Let $\eta$ be the generic point of $K$. Then $\mathcal O_{X, \eta} = \mathcal O_{K, \eta}$ if and only if $X$ is reduced at $K$. Indeed, for $\mathfrak p \subseteq \operatorname{Spec} A$ a minimal prime, we have a natural surjection $$\mathcal O_{X, \eta} = A_\mathfrak p \twoheadrightarrow A_\mathfrak p/\mathfrak p = (A/\mathfrak p)_\mathfrak p = \mathcal O_{K, \eta},$$ which is an isomorphism iff $\mathfrak p A_\mathfrak p = 0$. Since $\mathfrak p A_\mathfrak p$ is the only prime ideal in $A_\mathfrak p$, it equals the nilradical.

One might hope that there is some nice scheme structure on components for which $\mathcal O_{K, \eta} = \mathcal O_{X, \eta}$, but I think that there are too many choices for this. This probably has something to do with the non-uniqueness of primary decomposition.

Remark. For a Noetherian affine scheme at least, one has the following neat criterion: $A$ is reduced if and only if $A_\mathfrak p$ is reduced for all $\mathfrak p \subseteq A$ minimal, and $A$ has no embedded primes. (See Eisenbud, Exercise 11.10.)

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  • $\begingroup$ Thanks, this is pretty much what I had before asking albeit phrased very elegantly. I've realized that finding interpretation for reduced points without knowing any intersection theory is a bit premature. $\endgroup$ Commented Jan 11, 2016 at 21:52

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