What does the blow up along a locally principal look like?

Question

Let $\mathcal{I}$ be a locally principal ideal sheaf of a scheme $X$ (i.e. for all $x\in X$ exists an open neighborhood $U$ of $x$ and $f_U\in \mathcal{O}_X(U)$ with $\mathcal{I}|_U=(f_U)^\tilde{}$).

Is the blow up $X'$ of $X$ with center $\mathcal{I}$ a closed subscheme of $X$? And if it is the case, which is the ideal sheaf defining $X'$ as a closed subscheme of $X$?

I've founded positive answers for the affine case with $\mathcal{I}$ globally principal:

Suppose $X=\mathrm{Spec}(A)$ and $\mathcal{I}$ being the sheaf associated to the ideal $(f)\subseteq A$ for a given $f\in A$.

The ideal $J$, defining the closed subscheme $X'$ of $X$, is the extension and contraction of the zero ideal by the homomorphism $A\to A_f$. If the zero ideal of $A$ has a minimal primary decomposition $q_1\cap\dots\cap q_l$ (e.g. $A$ Noetherian), the ideal $J$ is the intersection of the $q_i$ such that $\{1,f,f^2,\dots\}\cap \sqrt{q_i}=\emptyset$, or, in any case, $J$ is $\bigcup_{n\ge 1} (0:f^n)$.

The closed embedding $\mathrm{Spec}(A/J)\to \mathrm{Spec}(A)$ satisfies the required universal property: If $f$ is nilpotent then $V(f)=A$, $J=A$ and the blow up is the empty set. If it is not then the class of $f$ in $A/J$ is diferent from zero an not a zero divisor, so the pull back of $\mathcal{I}$ to $X'$ is a Cartier divisor. And given a morphism $W\to X$ with this last property, affine locally we have a ring homomorphism $\varphi:A\to B$ such that $\varphi(f)$ is not a zero divisor and diferent from zero, so given $j\in J$, exists $n$ such that $jf^n=0$, hence $\varphi(j)\varphi(f)^n=0$ and $\varphi(j)=0$ (i.e. $J\subseteq \ker(\varphi)$).

Morally, for one side, we are erasing the irreducible components of $X$ inside the closed scheme $Z$ determined by $\mathcal{I}$ and, the other side, we are lifting the non-reduced structure of the embedded components of $X$ inside $Z$.

Answer 1

We can, a bit more conceptually, reformulate the OP's observation as follows:

The blow-up of a scheme $X$ along a locally principal sub-scheme $Z\subset X$ is the scheme-theoretic closure of the complement $U\ = X-Z$ in $X$.

We first give a description of the ideal of the scheme-theoretic closure. Let $i\colon U\to X$ be the inclusion of the complement $U = X-Z$. By definition, the scheme-theoretic closure of $U$ is the scheme-theoretic image of $i$. I claim that $i$ is an affine morphism. In fact, if $V = \mathrm{Spec}(A)\subset X$ is an affine open such that $Z\cap V = \mathrm{Spec}(A/f)$, then $U\cap V = \mathrm{Spec}(A_f)$. Thus, $i_*\mathcal{O}_U$ and consequently $\mathcal{J}:=\ker(\mathcal{O}_X\to i_*\mathcal{O}_U)$ are quasi-coherent $\mathcal{O}_X$-modules and so $\mathcal{J}$ is the ideal defining the scheme-theoretic closure of $U$ (c.f. Stacks Project tag 01R8 for example).

Denoting the scheme-theoretic closure of $U$ by $Y := \mathrm{Spec}(\mathcal{O}_X/\mathcal{J})$ it remains to show that this inclusion morphism $X'\to X$ is the blow-up of $X$ along $Z$. Since the blow-up can be computed on an open cover, we can assume that $X = \mathrm{Spec}(A)$ is affine and such that $Z = \mathrm{Spec}(A/f)$. Then, as before, $U = \mathrm{Spec}(A_f)$ and $\mathcal{J}$, the ideal sheaf defining $X'$, is the sheaf associated with the kernel of the natural homomorphism $A\to A_f$. This is exactly the ideal $J\subset A$ considered by the OP and so $X' = \mathrm{Spec}(A/J)\subset X$ is the blow-up in $Z$, as claimed.

In addition, here are two further incarnations of this ideal.

First alternative: If $\mathcal{I}\subset\mathcal{O}_X$ is a locally principal ideal sheaf, then it is invertible if and only if the dual section $s\colon \mathcal{O}_X\to\mathcal{I}^\vee = \mathcal{H}\mathit{om}(\mathcal{I},\mathcal{O}_X)$ is injective. The kernel is $\mathrm{ann}(\mathcal{I})$, but possibly, the analogous map when restricted to $\mathrm{Spec}(\mathcal{O}_X/\ker(s))$ is not injective, yet again; then its kernel is induced from $\mathrm{ann}(\mathcal{I}^2)$, and so on. So we're led to consider $\mathrm{colim}_n\mathrm{ann}(\mathcal{I}^n)$, the colimit being taken in the category of quasi-coherent $\mathcal{O}_X$-modules so that on an affine open $V\subset X$ it equals $\sum \mathrm{ann}_{\mathcal{O}_X(V)}(\mathcal{I}(V)^n)$. In particular, $\mathrm{colim}_n\mathrm{ann}(\mathcal{I}^n) = \mathcal{J}$.

Second alternative (at least locally): If $Z = \mathrm{Spec}(A/I)\subset X = \mathrm{Spec}(A)$ where $I = (f)$, then the blow-up is given by the morphism $\mathrm{Proj}(\bigoplus_nI^nt^n)\to \mathrm{Proj}(A[t]) \cong X$ corresponding to the homogeneous $A$-algebra homomorphism $A[t]\to\bigoplus_nI^nt^n$ mapping $t$ to $ft$. It is surjective with kernel $J = \bigoplus_n\mathrm{ann}(I^n)t^n$ and so the blow up is given by $\mathrm{Proj}(A[t]/J)\subset \mathrm{Proj}(A[t])$ and via the canonical isomorphism $\mathrm{Proj}(A[t]) \cong \mathrm{Spec}(A) = X$, the closed sub-scheme defined by the ideal $J$ becomes the closed sub-scheme defined by $\sum_nJ_n = \sum_n\mathrm{ann}(I^n)$.

Answer 2

If $X$ is locally Noetherian, there is an other analog construction with a smaller closed sub-set $T$ of $X$ which maybe can enlighten what is going on (so, from now on we assume $X$ locally Noetherian).

Define $T:=\bigcup_{p\in \mathrm{Ass}(X)\cap Z} \overline{\{p\}}$ which is closed (you can use Stacks Project Tag 05AF).

The blow up of a scheme $X$ along a locally principal sub-scheme $Z\subset X$ is (also) the scheme-theoretic closure of $U':=X\setminus T$.

Since $U\subseteq U'$, if a morphism $W\to X$ factors through $\overline{U}$ then also factors through $\overline{U'}$. So, by @Ben's answer, we just need to check that the pullback of $Z$ to $\overline{U'}$ is a Cartier divisor. It follows straightforward from the two following general facts:

1.- A locally principal closed sub-scheme $S\subset X$ is an effective Cartier divisor if and only if $\mathrm{Ass}(X)\cap S=\emptyset$.

For a Noetherian ring the set of zero divisors is equal to the union of its associated prime ideals (see Atiyah & Macdonald, prop.4.7).

2.- If $s:S\to X$ is a locally closed embedding, then the associated points of the scheme-theoretic closure are (naturally in bijection with) the associated points of S. (See Vakil's notes February 7, 2017 version; exercise 8.3.D, p.239. I've not found a more formal reference).

This few more lines are just an observation that relates this answer with Ben's. By Stacks Project Tag 01OX, the embedding $j:U'\to X$ is quasi-compact. Hence, the schematic closure of $U'$ is the closed scheme defined by the quasi-coherent sheaf $\ker(\mathscr{O}_X\to j_*\mathscr{O}_{U'})$, like $U$ and $\ker(\mathscr{O}_X\to i_*\mathscr{O}_U)$ in Ben's answer. So, since they define the same closed sub-scheme of $X$ (the blow up of $X$ along $Z$) they have to be the same sheaf of ideals of $\mathscr{O}_X$ (see Ulrich | Torsten, Algebraic Geometry I, corollary 7.33, p.192).