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*}

10.14 Miscellany

The proofs in this section should not refer to any results except those from the section on basic notions, Section 10.3.

Lemma 10.14.1. Let $R$ be a ring, $I$ and $J$ two ideals and $\mathfrak p$ a prime ideal containing the product $IJ$. Then $\mathfrak {p}$ contains $I$ or $J$.

Proof. Assume the contrary and take $x \in I \setminus \mathfrak p$ and $y \in J \setminus \mathfrak p$. Their product is an element of $IJ \subset \mathfrak p$, which contradicts the assumption that $\mathfrak p$ was prime. $\square$

slogan

Lemma 10.14.2 (Prime avoidance). Let $R$ be a ring. Let $I_ i \subset R$, $i = 1, \ldots , r$, and $J \subset R$ be ideals. Assume

  1. $J \not\subset I_ i$ for $i = 1, \ldots , r$, and

  2. all but two of $I_ i$ are prime ideals.

Then there exists an $x \in J$, $x\not\in I_ i$ for all $i$.

Proof. The result is true for $r = 1$. If $r = 2$, then let $x, y \in J$ with $x \not\in I_1$ and $y \not\in I_2$. We are done unless $x \in I_2$ and $y \in I_1$. Then the element $x + y$ cannot be in $I_1$ (since that would mean $x + y - y \in I_1$) and it also cannot be in $I_2$.

For $r \geq 3$, assume the result holds for $r - 1$. After renumbering we may assume that $I_ r$ is prime. We may also assume there are no inclusions among the $I_ i$. Pick $x \in J$, $x \not\in I_ i$ for all $i = 1, \ldots , r - 1$. If $x \not\in I_ r$ we are done. So assume $x \in I_ r$. If $J I_1 \ldots I_{r - 1} \subset I_ r$ then $J \subset I_ r$ (by Lemma 10.14.1) a contradiction. Pick $y \in J I_1 \ldots I_{r - 1}$, $y \not\in I_ r$. Then $x + y$ works. $\square$

Lemma 10.14.3. Let $R$ be a ring. Let $x \in R$, $I \subset R$ an ideal, and $\mathfrak p_ i$, $i = 1, \ldots , r$ be prime ideals. Suppose that $x + I \not\subset \mathfrak p_ i$ for $i = 1, \ldots , r$. Then there exists an $y \in I$ such that $x + y \not\in \mathfrak p_ i$ for all $i$.

Proof. We may assume there are no inclusions among the $\mathfrak p_ i$. After reordering we may assume $x \not\in \mathfrak p_ i$ for $i < s$ and $x \in \mathfrak p_ i$ for $i \geq s$. If $s = r + 1$ then we are done. If not, then we can find $y \in I$ with $y \not\in \mathfrak p_ s$. Choose $f \in \bigcap _{i < s} \mathfrak p_ i$ with $f \not\in \mathfrak p_ s$. Then $x + fy$ is not contained in $\mathfrak p_1, \ldots , \mathfrak p_ s$. Thus we win by induction on $s$. $\square$

Lemma 10.14.4 (Chinese remainder). Let $R$ be a ring.

  1. If $I_1, \ldots , I_ r$ are ideals such that $I_ a + I_ b = R$ when $a \not= b$, then $I_1 \cap \ldots \cap I_ r = I_1I_2\ldots I_ r$ and $R/(I_1I_2\ldots I_ r) \cong R/I_1 \times \ldots \times R/I_ r$.

  2. If $\mathfrak m_1, \ldots , \mathfrak m_ r$ are pairwise distinct maximal ideals then $\mathfrak m_ a + \mathfrak m_ b = R$ for $a \not= b$ and the above applies.

Proof. Let us first prove $I_1 \cap \ldots \cap I_ r = I_1 \ldots I_ r$ as this will also imply the injectivity of the induced ring homomorphism $R/(I_1 \ldots I_ r) \rightarrow R/I_1 \times \ldots \times R/I_ r$. The inclusion $I_1 \cap \ldots \cap I_ r \supset I_1 \ldots I_ r$ is always fulfilled since ideals are closed under multiplication with arbitrary ring elements. To prove the other inclusion, we claim that the ideals

\[ I_1 \ldots \hat I_ i \ldots I_ r,\quad i = 1, \ldots , r \]

generate the ring $R$. We prove this by induction on $r$. It holds when $r = 2$. If $r > 2$, then we see that $R$ is the sum of the ideals $I_1 \ldots \hat I_ i \ldots I_{r - 1}$, $i = 1, \ldots , r - 1$. Hence $I_ r$ is the sum of the ideals $I_1 \ldots \hat I_ i \ldots I_ r$, $i = 1, \ldots , r - 1$. Applying the same argument with the reverse ordering on the ideals we see that $I_1$ is the sum of the ideals $I_1 \ldots \hat I_ i \ldots I_ r$, $i = 2, \ldots , r$. Since $R = I_1 + I_ r$ by assumption we see that $R$ is the sum of the ideals displayed above. Therefore we can find elements $a_ i \in I_1 \ldots \hat I_ i \ldots I_ r$ such that their sum is one. Multiplying this equation by an element of $I_1 \cap \ldots \cap I_ r$ gives the other inclusion. It remains to show that the canonical map $R/(I_1 \ldots I_ r) \rightarrow R/I_1 \times \ldots \times R/I_ r$ is surjective. For this, consider its action on the equation $1 = \sum _{i=1}^ r a_ i$ we derived above. On the one hand, a ring morphism sends 1 to 1 and on the other hand, the image of any $a_ i$ is zero in $R/I_ j$ for $j \neq i$. Therefore, the image of $a_ i$ in $R/I_ i$ is the identity. So given any element $(\bar{b_1}, \ldots , \bar{b_ r}) \in R/I_1 \times \ldots \times R/I_ r$, the element $\sum _{i=1}^ r a_ i \cdot b_ i$ is an inverse image in $R$.

To see (2), by the very definition of being distinct maximal ideals, we have $\mathfrak {m}_ a + \mathfrak {m}_ b = R$ for $a \neq b$ and so the above applies. $\square$

Lemma 10.14.5. Let $R$ be a ring. Let $n \geq m$. Let $A$ be an $n \times m$ matrix with coefficients in $R$. Let $J \subset R$ be the ideal generated by the $m \times m$ minors of $A$.

  1. For any $f \in J$ there exists a $m \times n$ matrix $B$ such that $BA = f 1_{m \times m}$.

  2. If $f \in R$ and $BA = f 1_{m \times m}$ for some $m \times n$ matrix $B$, then $f^ m \in J$.

Proof. For $I \subset \{ 1, \ldots , n\} $ with $|I| = m$, we denote by $E_ I$ the $m \times n$ matrix of the projection

\[ R^{\oplus n} = \bigoplus \nolimits _{i \in \{ 1, \ldots , n\} } R \longrightarrow \bigoplus \nolimits _{i \in I} R \]

and set $A_ I = E_ I A$, i.e., $A_ I$ is the $m \times m$ matrix whose rows are the rows of $A$ with indices in $I$. Let $B_ I$ be the adjugate (transpose of cofactor) matrix to $A_ I$, i.e., such that $A_ I B_ I = B_ I A_ I = \det (A_ I) 1_{m \times m}$. The $m \times m$ minors of $A$ are the determinants $\det A_ I$ for all the $I \subset \{ 1, \ldots , n\} $ with $|I| = m$. If $f \in J$ then we can write $f = \sum c_ I \det (A_ I)$ for some $c_ I \in R$. Set $B = \sum c_ I B_ I E_ I$ to see that (1) holds.

If $f 1_{m \times m} = BA$ then by the Cauchy-Binet formula (72) we have $f^ m = \sum b_ I \det (A_ I)$ where $b_ I$ is the determinant of the $m \times m$ matrix whose columns are the columns of $B$ with indices in $I$. $\square$

Lemma 10.14.6. Let $R$ be a ring. Let $n \geq m$. Let $A = (a_{ij})$ be an $n \times m$ matrix with coefficients in $R$, written in block form as

\[ A = \left( \begin{matrix} A_1 \\ A_2 \end{matrix} \right) \]

where $A_1$ has size $m \times m$. Let $B$ be the adjugate (transpose of cofactor) matrix to $A_1$. Then

\[ AB = \left( \begin{matrix} f 1_{m \times m} \\ C \end{matrix} \right) \]

where $f = \det (A_1)$ and $c_{ij}$ is (up to sign) the determinant of the $m \times m$ minor of $A$ corresponding to the rows $1, \ldots , \hat j, \ldots , m, i$.

Proof. Since the adjugate has the property $A_1B = B A_1 = f$ the first block of the expression for $AB$ is correct. Note that

\[ c_{ij} = \sum \nolimits _ k a_{ik}b_{kj} = \sum (-1)^{j + k}a_{ik} \det (A_1^{jk}) \]

where $A_1^{ij}$ means $A_1$ with the $j$th row and $k$th column removed. This last expression is the row expansion of the determinant of the matrix in the statement of the lemma. $\square$


Comments (5)

Comment #1124 by OlgaD on

In the proof of Lemma 10.14.1

Comment #3419 by Peter Fleischmann on

In Lemma 07DQ (2) change m x m to m x n,

Comment #3424 by Fan on

Second Peter Fleischmann: "for some matrix " should be "for some matrix ".


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