Tags: #NT/algebraic #CA Refs: normalization Noether normalization

Integral extensions

# Integrally closed

Integral Closure

#todo

Examples

Write \(\operatorname{cl}^{\mathrm{int}} _B(A)\) for the integral closure of \(A\) in \(B\).

• \(\operatorname{cl}^{\mathrm{int}} {\mathbf{Q}}({\mathbf{Z}}) = {\mathbf{Z}}\)
• \(\operatorname{cl}^{\mathrm{int}} _K({\mathbf{Z}}) \coloneqq{\mathcal{O}}_K\) for $K\in\mathsf{Field}_{/ {{\mathbf{Q}}}} $.
• \(\operatorname{cl}^{\mathrm{int}} _{{\mathbf{Q}}[i]}({\mathbf{Z}}) = {\mathcal{O}}_{{\mathbf{Q}}[i]} = {\mathbf{Z}}[m]\) where \(m= {1\over 2}\qty{1 + \sqrt 5}\).
• \(\operatorname{cl}^{\mathrm{int}} _{{\mathbf{Q}}[\zeta_n]} = {\mathcal{O}}_{{\mathbf{Q}}[\zeta_n]} = {\mathbf{Z}}[\zeta_n]\)
• \(\operatorname{cl}^{\mathrm{int}} _{\mathbf{C}}({\mathbf{Z}}) = \operatorname{cl}^{\mathrm{alg}} ({\mathbf{Q}})\), the algebraic closure of \({\mathbf{Q}}\) or ring of algebraic integers.

Exercises

• Show that integral extensions satisfy the Cohen–Seidenberg theorems.
• Show that if \(B\) is integral over \(A\), then \(B \otimes_{A} R\) is integral over \(\mathrm{R}\) for any A-algebra R.
• Show that if A is a subring of a field K, then the integral closure of A in K is the intersection of all valuation rings of K containing A.
• Show that every UFD is integrally closed.
• For \(A \leq B\) a subring, show that \(b\in B\) is integral over \(A\) iff there exists a faithful \(A[b]{\hbox{-}}\)submodule of \(B\) which is finitely generated as an \(A{\hbox{-}}\)module. Hint: use Cramer’s rule.
• Show that an \(A{\hbox{-}}\)algebra \(B\) is finite iff it is finitely generated as an \(A{\hbox{-}}\)algebra by a generating set that is integral over \(A\).
• Show that an \(A{\hbox{-}}\)algebra \(B\) is finite iff finitely generated and integral over \(A\).
• Show that UFDs are integrally closed.
• Let \(A\) be a normal integral domain, and let \(E\) be a finite extension of the field of fractions \(F\) of \(A\). Show that element of \(E\) is integral over \(A\) iff its minimum polynomial over \(F\) has coefficients in \(A\).
• What are the integral elements of ${\mathbf{Q}}_{/ {{\mathbf{Z}}}} $?
• Show that if \(A\leq B\) is a submodule and \(x_i\in B\) are integral over \(A\), then \(A[x_1,\cdots, x_n]\) is a finitely generated \(A{\hbox{-}}\)module.
• Show that finite type and integral implies finite.
• Show that integraility is transitive. Show that this also holds for being integrally closed.
• Show that integrality is preserved by passing to quotients or the ring of fractions.
• Show that integral closure is local.
• Show that if \({\mathfrak{p}}\in \operatorname{Spec}k[x_1, \cdots, x_{n}]\) and \(A \coloneqq k[x_1, \cdots, x_{n}]/{\mathfrak{p}}\) is a 1-dimensional domain, then \(A\) is an integral extension of \(k[x]\).
• Suppose \(R\) is a Noetherian integral domain. Show that \(R\) is a UFD iff \(A\) is integrally closed in \(\operatorname{ff}(R)\) and \(\operatorname{Cl} (X) = 0\).
• Show that for \(A\subseteq k\) a subring of a field, the integral closure \({ \operatorname{cl}} _k(A)\) is the smallest valuation ring in \(k\) containing \(A\).
• Show that if \(B\) is integrally closed ensures that every prime \(\mathfrak{p}\) of \(\mathcal{O}\) has at least one prime \(\mathfrak{q}\) lying above it (this is a standard fact of commutative algebra).