Let \(I\) and \(J\) be ideals in \(k[x_1,\ldots,x_n]\) where \(k\) is a field.
Prove \(I\cap J=(tI+(1-t)J)\cap k[x_1,\ldots,x_n]\).
Prove that \(V(I\cap J)=V(I)\cup V(J)\) where for any set \(X\subseteq k[x_1,\ldots,x_n]\), \(V(X)\) is the affine variety defined by \(X\).
Prove that \(\sqrt{I\cap J}=\sqrt{I}\cap\sqrt{J}\).
Let \(I=\langle x^3y\rangle\) and \(J=\langle xy^3+xy\rangle\) be ideals in \(k[x,y]\). Find a Gröbner basis for \(I\cap J\) relative to lexicographic order where \(x>y\).
Solution
Part (i).
1. (\(\subseteq\)) If \(f \in I \cap J\), then \(f = tf + (1-t)f \in tI + (1-t)J\), and \(f \in k[x_1,\ldots,x_n]\).
\(f \in I\) gives \(tf \in tI\), and \(f \in J\) gives \((1-t)f \in (1-t)J\).
2. (\(\supseteq\)) If \(f \in (tI + (1-t)J) \cap k[x_1,\ldots,x_n]\), then \(f = tg + (1-t)h\) with \(g \in I\), \(h \in J\).
definition of \(tI + (1-t)J\).
3. Setting \(t = 0\) gives \(f = h \in J\); setting \(t = 1\) gives \(f = g \in I\).
substitute \(t = 0\) and \(t = 1\) (valid since \(f\) does not involve \(t\)).
4. Hence \(f \in I \cap J\).
3.
5. Q.E.D. (i).
1 and 4.
Part (ii).
1. \(V(I \cap J) \supseteq V(I) \cup V(J)\).
\(I \cap J \subseteq I\) and \(I \cap J \subseteq J\), so \(V(I) \subseteq V(I \cap J)\) and \(V(J) \subseteq V(I \cap J)\).
2. \(V(I \cap J) \subseteq V(I) \cup V(J)\).
2.1. Let \(p \in V(I \cap J)\).
take a point vanishing on \(I \cap J\).
2.2. If \(p \notin V(I)\), then there is \(f \in I\) with \(f(p) \neq 0\).
definition of \(V(I)\).
2.3. For any \(g \in J\), \(fg \in I \cap J\), so \((fg)(p) = f(p)g(p) = 0\); since \(f(p) \neq 0\), \(g(p) = 0\).
\(fg \in I \cap J\) and \(p \in V(I \cap J)\).
2.4. Hence \(p \in V(J)\).
3 shows every \(g \in J\) vanishes at \(p\).
2.5. Therefore \(p \in V(I) \cup V(J)\).
<2>2–<2>4.
3. Q.E.D. (ii).
1 and 2.
Part (iii).
1. \(\sqrt{I \cap J} = \sqrt{I} \cap \sqrt{J}\).
1.1. (\(\subseteq\)) If \(f \in \sqrt{I \cap J}\), then \(f^m \in I \cap J\) for some \(m\), so \(f^m \in I\) and \(f^m \in J\), hence \(f \in \sqrt I\) and \(f \in \sqrt J\).
definition of the radical.
1.2. (\(\supseteq\)) If \(f \in \sqrt I \cap \sqrt J\), then \(f^m \in I\) and \(f^n \in J\) for some \(m, n\), so \(f^{m+n} \in I \cap J\), hence \(f \in \sqrt{I \cap J}\).
definition of the radical.
4. Q.E.D. (iii).
1.
Part (iv).
1. \(I \cap J = \langle x^3 y^3 + x^3 y \rangle\).
by part (i), \(I \cap J = (t\langle x^3 y\rangle + (1-t)\langle xy^3 + xy\rangle) \cap k[x,y]\); computing the elimination ideal (Gröbner basis with \(t > x > y\) and intersecting with \(k[x,y]\)) gives the single generator \(x^3 y^3 + x^3 y\).
2. A Gröbner basis for \(I \cap J\) (lex, \(x > y\)) is \(\{x^3 y^3 + x^3 y\}\).
a single polynomial is its own Gröbner basis.
3. Q.E.D. (iv).
2.