# Sections of non-reduced schemes

Let $$X$$ be an affine, irreducible (complex), generically reduced, scheme containing an embedded point, say at $$x \in X$$. Suppose further dimension of $$X$$ is strictly positive (can assume to be one dimensional). Assume that the associated reduced scheme $$X_{\mbox{red}}$$ is non-singular. Note that, there is a natural closed immersion $$X_{\mbox{red}} \to X$$. Then,

1) Are there examples of $$X$$ as above such that there exists a morphism $$X \to X_{\mbox{red}}$$ such that the composition $$X_{\mbox{red}} \to X \to X_{\mbox{red}}$$ is the identity? I know of such examples in zero dimension. I am interested to see if such a phenomenon can happen in the setup of embedded points in schemes of positive dimension.

2) If such a map $$X \to X_{\mbox{red}}$$ exists, then does the structure sheaf of $$X$$ decompose (as an $$\mathcal{O}_{X_{\mbox{red}}}$$-module) as $$\mathcal{O}_{X_{\mbox{red}}} \oplus I\mathcal{O}_{X_{\mbox{red}}}$$ for some nilpotent ideal $$I$$?

• There's plenty of examples. Let $k = \mathbf{C}$. Take $R = k[x_1,...,x_n]$, an artinian local $k$-algebra $A$ with residue field $k$, and consider the fibre product $S = R \times_k A$, where the map $A \to k$ is the obvious one and $R \to k$ sets all $x_i = 0$. Geometrically, you've tacked on a copy of $\mathrm{Spec}(A)$ at the origin in $\mathbf{A}^n_k$. So the projection $S \to R$ is a nilpotent thickening with kernel $\mathfrak{m}_A$ (and there's a section). For (2), in what category do you want the splitting? There's such a splitting after pushing down to $X_{red}$ for trivial reasons... – Anonymous May 26 at 17:00
• @Anonymous Is the scheme in your example really generically reduced? It seems it is generically non-reduced. – Jana May 26 at 17:44
• @Jana fibre product, not tensor product. If $A = k[\varepsilon]/(\varepsilon^2)$, then $S \cong R[y]/(x_1y, \ldots, x_ny, y^2)$, which has an embedded point at the origin and the reduction is $S[y]/(y) = R$. Anonymous's construction is just a generalisation of that example. – R. van Dobben de Bruyn May 26 at 19:44
• @R.vanDobbendeBruyn Thanks. I was thinking of tensor product. – Jana May 26 at 19:50