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?


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.)

  • $\begingroup$ 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. $\endgroup$ Jan 11 '16 at 21:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.