The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

10.69 Blow up algebras

In this section we make some elementary observations about blowing up.

Definition 10.69.1. Let $R$ be a ring. Let $I \subset R$ be an ideal.

  1. The blowup algebra, or the Rees algebra, associated to the pair $(R, I)$ is the graded $R$-algebra

    \[ \text{Bl}_ I(R) = \bigoplus \nolimits _{n \geq 0} I^ n = R \oplus I \oplus I^2 \oplus \ldots \]

    where the summand $I^ n$ is placed in degree $n$.

  2. Let $a \in I$ be an element. Denote $a^{(1)}$ the element $a$ seen as an element of degree $1$ in the Rees algebra. Then the affine blowup algebra $R[\frac{I}{a}]$ is the algebra $(\text{Bl}_ I(R))_{(a^{(1)})}$ constructed in Section 10.56.

In other words, an element of $R[\frac{I}{a}]$ is represented by an expression of the form $x/a^ n$ with $x \in I^ n$. Two representatives $x/a^ n$ and $y/a^ m$ define the same element if and only if $a^ k(a^ mx - a^ ny) = 0$ for some $k \geq 0$.

Lemma 10.69.2. Let $R$ be a ring, $I \subset R$ an ideal, and $a \in I$. Let $R' = R[\frac{I}{a}]$ be the affine blowup algebra. Then

  1. the image of $a$ in $R'$ is a nonzerodivisor,

  2. $IR' = aR'$, and

  3. $(R')_ a = R_ a$.

Proof. Immediate from the description of $R[\frac{I}{a}]$ above. $\square$

Lemma 10.69.3. Let $R \to S$ be a ring map. Let $I \subset R$ be an ideal and $a \in I$. Set $J = IS$ and let $b \in J$ be the image of $a$. Then $S[\frac{J}{b}]$ is the quotient of $S \otimes _ R R[\frac{I}{a}]$ by the ideal of elements annihilated by some power of $b$.

Proof. Let $S'$ be the quotient of $S \otimes _ R R[\frac{I}{a}]$ by its $b$-power torsion elements. The ring map

\[ S \otimes _ R R[\textstyle {\frac{I}{a}}] \longrightarrow S[\textstyle {\frac{J}{b}}] \]

is surjective and annihilates $a$-power torsion as $b$ is a nonzerodivisor in $S[\frac{J}{b}]$. Hence we obtain a surjective map $S' \to S[\frac{J}{b}]$. To see that the kernel is trivial, we construct an inverse map. Namely, let $z = y/b^ n$ be an element of $S[\frac{J}{b}]$, i.e., $y \in J^ n$. Write $y = \sum x_ is_ i$ with $x_ i \in I^ n$ and $s_ i \in S$. We map $z$ to the class of $\sum s_ i \otimes x_ i/a^ n$ in $S'$. This is well defined because an element of the kernel of the map $S \otimes _ R I^ n \to J^ n$ is annihilated by $a^ n$, hence maps to zero in $S'$. $\square$

Lemma 10.69.4. Let $R$ be a ring, $I \subset R$ an ideal, and $a \in I$. Set $R' = R[\frac{I}{a}]$. If $f \in R$ is such that $V(f) = V(I)$, then $f$ maps to a nonzerodivisor in $R'$ and $R'_ f = R'_ a = R_ a$.

Proof. We will use the results of Lemma 10.69.2 without further mention. The assumption $V(f) = V(I)$ implies $V(fR') = V(IR') = V(aR')$. Hence $a^ n = fb$ and $f^ m = ac$ for some $b, c \in R'$. The lemma follows. $\square$

Lemma 10.69.5. Let $R$ be a ring, $I \subset R$ an ideal, $a \in I$, and $f \in R$. Set $R' = R[\frac{I}{a}]$ and $R'' = R[\frac{fI}{fa}]$. Then there is a surjective $R$-algebra map $R' \to R''$ whose kernel is the set of $f$-power torsion elements of $R'$.

Proof. The map is given by sending $x/a^ n$ for $x \in I^ n$ to $f^ nx/(fa)^ n$. It is straightforward to check this map is well defined and surjective. Since $af$ is a nonzero divisor in $R''$ (Lemma 10.69.2) we see that the set of $f$-power torsion elements are mapped to zero. Conversely, if $x \in R'$ and $f^ n x \not= 0$ for all $n > 0$, then $(af)^ n x \not= 0$ for all $n$ as $a$ is a nonzero divisor in $R'$. It follows that the image of $x$ in $R''$ is not zero by the description of $R''$ following Definition 10.69.1. $\square$

Proof. Let $I \subset R$ be an ideal and $a \in I$. Suppose $x/a^ n$ with $x \in I^ n$ is a nilpotent element of $R[\frac{I}{a}]$. Then $(x/a^ n)^ m = 0$. Hence $a^ N x^ m = 0$ in $R$ for some $N \geq 0$. After increasing $N$ if necessary we may assume $N = me$ for some $e \geq 0$. Then $(a^ e x)^ m = 0$ and since $R$ is reduced we find $a^ e x = 0$. This means that $x/a^ n = 0$ in $R[\frac{I}{a}]$. $\square$

Lemma 10.69.7. Let $R$ be a domain, $I \subset R$ an ideal, and $a \in I$ a nonzero element. Then the affine blowup algebra $R[\frac{I}{a}]$ is a domain.

Proof. Suppose $x/a^ n$, $y/a^ m$ with $x \in I^ n$, $y \in I^ m$ are elements of $R[\frac{I}{a}]$ whose product is zero. Then $a^ N x y = 0$ in $R$. Since $R$ is a domain we conclude that either $x = 0$ or $y = 0$. $\square$

Lemma 10.69.8. Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $a \in I$. If $a$ is not contained in any minimal prime of $R$, then $\mathop{\mathrm{Spec}}(R[\frac{I}{a}]) \to \mathop{\mathrm{Spec}}(R)$ has dense image.

Proof. If $a^ k x = 0$ for $x \in R$, then $x$ is contained in all the minimal primes of $R$ and hence nilpotent, see Lemma 10.16.2. Thus the kernel of $R \to R[\frac{I}{a}]$ consists of nilpotent elements. Hence the result follows from Lemma 10.29.6. $\square$

Lemma 10.69.9. Let $R$ be a Noetherian ring. Let $a, a_2, \ldots , a_ r$ be a regular sequence in $R$. With $I = (a, a_2, \ldots , a_ r)$ the blowup algebra $R' = R[\frac{I}{a}]$ is isomorphic to $R'' = R[y_2, \ldots , y_ r]/(a y_ i - a_ i)$.

Proof. There is a canonical map $R'' \to R'$ sending $y_ i$ to the class of $a_ i/a$. Since every element $x$ of $I$ can be written as $ra + \sum r_ i a_ i$ we see that $x/a = r + \sum r_ i a_ i/a$ is in the image of the map. Hence our map is surjective. Suppose that $z = \sum r_ E y^ E \in R''$ maps to zero in $R'$. Here we use the multi-index notation $E = (e_2, \ldots , e_ r)$ and $y^ E = y_2^{e_2} \ldots y_ r^{e_ r}$. Let $d$ be the maximum of the degrees $|E| = \sum e_ i$ of the multi-indices which occur with a nonzero coefficient $r_ E$ in $z$. Then we see that

\[ a^ d z = \sum r_ E a^{d - |E|} a_2^{e_2} \ldots a_ r^{e_ r} \]

is zero in $R$; here we use that $a$ is a nonzerodivisor on $R$. Since a regular sequence is quasi-regular by Lemma 10.68.2 we conclude that $r_ E \in I$ for all $E$. This means that $z$ is divisible by $a$ in $R''$. Say $z = az'$. Then $z'$ is in the kernel of $R'' \to R'$ and we see that $z'$ is divisible by $a$ and so on. In other words, $z$ is an element of $\bigcap a^ n R''$. Since $R''$ is Noetherian by Krull's intersection theorem $z$ maps to zero in $R''_\mathfrak p$ for every prime ideal $\mathfrak p$ containing $aR''$, see Remark 10.50.6. On the other hand, if $\mathfrak p \subset R''$ does not contain $a$, then $R''_ a \cong R_ a \cong R'_ a$ and we find that $z$ maps to zero in $R''_\mathfrak p$ as well. We conclude that $z$ is zero by Lemma 10.22.1. $\square$

Lemma 10.69.10. Let $(R, \mathfrak m)$ be a local domain with fraction field $K$. Let $R \subset A \subset K$ be a valuation ring which dominates $R$. Then

\[ A = \mathop{\mathrm{colim}}\nolimits R[\textstyle {\frac{I}{a}}] \]

is a directed colimit of affine blowups $R \to R[\frac{I}{a}]$ with the following properties

  1. $a \in I \subset \mathfrak m$,

  2. $I$ is finitely generated, and

  3. the fibre ring of $R \to R[\frac{I}{a}]$ at $\mathfrak m$ is not zero.

Proof. Consider a finite subset $E \subset A$. Say $E = \{ e_1, \ldots , e_ n\} $. Choose a nonzero $a \in R$ such that we can write $e_ i = f_ i/a$ for all $i = 1, \ldots , n$. Set $I = (f_1, \ldots , f_ n, a)$. We claim that $R[\frac{I}{a}] \subset A$. This is clear as an element of $R[\frac{I}{a}]$ can be represented as a polynomial in the elements $e_ i$. The lemma follows immediately from this observation. $\square$


Comments (1)

Comment #1817 by Joseph Gunther on

Minor typos: I think that, in the proofs of Lemmas 10.69.6 and 10.69.9, the instances of A, A', and A'' should be R, R', and R''.


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