Commutative Algebra Problems

Undergrad

• Show that prime implies irreducible, and the converse only holds for UFDs.
• Show that maximal implies prime but not conversely, so \(\mspec R \subseteq \spec R\).
• Give an example of a non-principal ideal.
• Show that if \(R\) is a UFD then \(R[x]\) is a UFD.
• Prove Gauss’ lemma.
• Show that if a ring \(R\) has factorization into irreducibles and irreducibles are prime, then \(R\) is a UFD.
• Show that \(\kxn\) is a UFD.
• Show that nonzero proper principal ideals in \(\kxn\) are generated by irreducible polynomials.
• Describe \(\spec A\) for \(A \da R\plocalize{\mfp}\), and show that \(A\) is local with \(\mfm = \mfp_e\).
• Show that \(M\tensor_R N\) may equal zero when \(M\neq 0, N\neq 0\).
• Find a polynomial \(f\in \CC[x]\) such that \(f(\ZZ) \subseteq \ZZ\) but \(f \not\in \ZZ[x]\).
• Show that primes are principal in a UFD.
• Show that UFD and PID are equivalent for Dedekind domains.
• Show that PID implies UFD.
• Give a necessary and sufficient condition for an irreducible polynomial to be inseparable.
• Show that in characteristic zero, algebraic implies separable for field extensions.
• Show that a finitely generated torsionfree \(R\dash\)module need not be free if \(R\) is not a PID.
• Show that the dual of a finitely generated module need not be finitely generated.
• Give examples of non-Noetherian rings.
• Show that \(I\in \Id(R)\) is prime iff \(R\sm I\) is a submonoid of the multiplicative monoid of \(R\), i.e. \(R\sm I\) is multiplicatively closed and contains \(1_R\).

Ring Basics

• Show that every integral domain is reduced.
• Give an example of a ring that is not reduced.
• Show that \(I\) is a radical ideal iff \(A/I\) is reduced.
• Show that \(R\) Noetherian implies \(R[[x]]\) Noetherian.
• Show that \(\sqrt{I}\) is an ideal if \(I\) is an ideal.
• Find a ring satisfying the ACC but not the DCC, and vice-versa.
  • Find a ring that satisfies neither the ACC nor the DCC.
• Show that the ideal correspondence preserves \(\spec, \mspec,\) and radical ideals.
• Show that if \(a, b\) are radical ideals then \(a\intersect b\) is radical but \(a+b\) need not be.
• Show that for a local ring \((R, \mfm)\), \(\jacobsonrad{R} = \mfm\).
• Show that in a UFD, prime ideals of height 1 are principal.
• Show that regular rings are Cohen-Macaulay.
• (Standard exercise) Let \(K\) be a field. A commutative \(K\)-algebra of finite dimension is semisimple if and only if it is reduced.

Ideals

• Show that \(R\) is a UFD iff every height 1 prime ideal is principal.
• Show that powers of a maximal ideal \(\mfm\) are \(\mfm\dash\)primary.
• Show that if \(A\leq B\) is a subring and \(\mfp \in \spec A\) then there exists \(\mfq \in \spec B\) such that \(\mfq \intersect A = \mfp\).
• Show that an ideal \(I\normal \kxn\) is homogeneous iff it is graded, i.e. \(I = \bigoplus I_d\) where \(I_d \da I \intersect \kxn_d\), the homogeneous degree \(d\) part of the graded ring \(\kxn\).
• Let \(k[V]\) be the coordinate ring of a variety, and show that every maximal ideal \(\mfm\in \mspec k[V]\) is of the form \(\mfm_p \da \ts{f\in k[V] \st f(p) = 0}\) for some point \(p\in V\).
• Show that every \(I\in \Id(R)\) is a projective \(R\dash\)module.

Local Rings

• Given \((A, \mfm_A)\), show that \(A\localize{\mfm}\) is local. What is its maximal ideal?
• Show that for a fixed \(S \subseteq R\), there is an exact functor \(S\inv: \mods{R} \to \mods{S\inv R}\).
• Show that the following are local properties:
  • Being zero, i.e. \(M= 0\) if \(M\localize{\mfm} = 0\) for all \(\mfm\in \mspec R\) for \(\rmod\).
  • Injectivity of module morphisms, i.e. \(M\to N\) is injective iff \(M\localize{\mfm} \to N\localize{\mfm}\) is injective for all \(\mfm \in \mspec R\).
  • Being reduced: \(R\) is reduced iff \(R\localize{\mfm}\) is reduced for all \(\mfm\in \mspec R\).
  • Flatness
  • Exactness, i.e. \(A\to B\to C\) is exact iff \(A\localize{\mfm} \to B \localize{\mfm} \to C\localize{\mfm}\) is exact for all \(\mfm \in \mspec R\)
  • Being integrally closed.
  • Being coprime ideals, i.e. \(I+J = R \iff I_\mfp + J_\mfp = R_\mfp\) for all \(\mfp\in \spec R\).
• When is a localization a subring of the fraction field?
• Show that if \(A \rightarrow B\) is a ring homomorphism and \(M\) is a flat \(A\)-module, then \(M_{B}=B \otimes_{A} M\) is a flat B-module. (Use the canonical isomorphisms (2.14), (2.15).)
• Show that \((A/I)\localize{S} \cong M/IM\) for \(M\da A\localize{S}\)

Noetherian Rings

Modules and Algebras

• Show that a morphism \(A\to B\) is the same as giving \(B\) an \(A\dash\)algebra structure.
• Show that Nakayama's lemma may fail if \(M\) is not finitely generated.
• Show that if \(A\in \Alg\slice k\) is finite over \(k\) (and an integral domain), then \(A\) is a field.
• Show that any \(M\in\Alg\slice R\) satisfies \(M = \colim M_\alpha\) where \(\ts{M_\alpha \leq M}\) are all of the finitely generated subalgebras of \(M\).
• Show that if \(M\in \mods{R}\) with \((R, \mfm)\) a local ring, then the action \(R\actson M/\mfm M\) factors through the residue field \(\kappa(R)\), and \(\ts{g_i}\subseteq M\) generate the quotient as an \(R\dash\)module iff \(\ts{a_i + \mfm M}\) generate the quotient as a \(\kappa(R)\dash\)module.
• Show that if \(M\in \rmod\) then \(M\dual\dual\) is torsionfree, and conclude that not every module is reflexive.

Integrality

Dimension

• Show that a 0-dimensional domain is a field.
• Show that a PID is dimension 1 unless it is a field.
• Show that a 1-dimensional regular local ring is a DVR.
• For \(A\) a complete local ring, show that \(\dim A = \dim A\complete{\mfm}\)
• Let \(R = \kxn\) and \(I = \gens{x_1,\cdots, x_n}\). Show that \(\dim R\localize{I} = n\).
• Find an infinite-dimensional Noetherian domain.
• Show that a regular local ring is integrally closed, but that there are integrally closed local domains of dimension \(d\geq 2\) which are not regular.
• Show that \(R\) is a DVR iff \(R\) is a regular local ring of dimension 1.

Number Theory

• Show that ideals satisfy unique factorization in a Dedekind domain.
• Show that a local Dedekind domain is a PID.
• Show that PID implies Dedekind, and all of its localizations are PIDs, DVRs, and thus also Dedekind domains.
• Show that \(\OO_K\) for \(K\) an algebraic number field is Dedekind.
• Show that valuation rings are local.
• Prove Hensel's lemma.
• Show that a topological group is Hausdorff if \(\ts{0}\) is closed.
• Show that for \(R \da k[x_1, \cdots ,x_n]\) and \(I\da \gens{x_1, \cdots, x_n}\), the \(I\dash\)adic completion is \(R\complete{I} = k\formalpowerseries{x_1, \cdots, x_n}\).
• Show that
  • A PID is a Dedekind domain.
  • For \(L/\QQ\) a finite field extension, show that the ring of algebraic integers \(\OO_L\) is a Dedekind domain.
  • If \(A\) is a Dedekind domain with field of fractions \(K\) and if \(K \subset L\) is a finite separable field extension, then the integral closure, \(B\), of \(A\) in \(L\) is a Dedekind domain.
  • A localization of a Dedekind domain is also a Dedekind domain.
• Show that if \(R\) is a Noetherian local integral domain whose maximal ideal is principal, then \(R\) is a PID and thus a DVR.

  Hint: look for a uniformizer.

• Show that in a Dedekind domain, all fractional ideals are invertible ideals.
• Show that if \(R\) is a Dedekind domain with \(\size \spec R < \infty\), then \(R\) is a PID.
• Show that every ideal in a Dedekind domain can be generated by two elements.

Homological Algebra

• Show that \(\cocolim^1 M_k = 0\) if \(M_k \surjects M_{k+1}\) for all \(k\).
• Show that direct limits commute with tensor products.
• Show that \(\kxn \tensor_k k[x_{n+1}, \cdots, x_m] \cong k[x_1,\cdots, x_m]\)
• Show that \(M\tensor_R \wait\) preserves direct sums.
• Prove that \(\wait \tensor_R M\) is not generally exact for all \(M\in \mods{R}\).
• Show that localization is exact.
• Show that \(M\) is flat iff \(\Tor^1(M, R/I)=0\) for all finitely generated ideal \(I\normal R\).
• Show that \(M\) is flat if \(\wait\tensor_R M \to M\) is a flat morphism, where it suffices to check on all ideals \(I\).
• Show that if \(R\) is Noetherian and local and \(M\in\mods{R}\), then \(M\) is free iff \(\Tor^1(\kappa(R), M) = 0\) for \(\kappa(R)\) the residue field.

Flatness

• Show that a nonzero finitely generated flat module over a local ring is faithfully flat.
• Show that \(R\to R[x]\) is always a faithfully flat morphism.
• Show that the category of flat \(R\dash\)modules is closed under direct sums and summands.
• Show that free implies flat.
• Show that nonzero vector spaces are faithfully flat.
• Show that projections \(A\times B\to A\) are flat.
  • Show that if \(A\) is Noetherian, then every flat quotient map is a projection of this form.
• Show that if \(A\to B\) is a faithfully flat morphism of rings, then for \(M\in\mods{A}\), \(M\) is (faithfully) flat iff \(B\tensor_A M\) is (faithfully) flat.
• Show that flat = locally free for finitely generated modules.
• Find a flat but not faithfully flat \(\ZZ\dash\)algebra.
• Show that a flat morphism of local rings is faithfully flat.
• If \(f:R\to S\) is flat, show that \(R\plocalize{ (f\inv(\mfp)) } \to S\plocalize{\mfp}\) is faithfully flat.
• For \(R\to S\) faithfully flat, show that \(S\) Noetherian implies \(R\) is Noetherian, but the converse is not true.
• Show that if \(f:R\to S\) is faithfully flat then \(f^*: \spec S\to \spec R\) is surjective.
  • Conversely, show that if \(f\) is flat and \(f^*\) is surjective, then \(f\) is faithfully flat.
  • Show that if \(f\) is faithfully flat, then \(f^*\) is a quotient map of topological spaces.
• Show that being fiathfully flat is preserved under base change.

Numerical Invariants

• Prove rationality of the Poincare series of any Poincare series of an additive function on modules.
• What is the Poincare series of \(A = R[x_0,\cdots, x_n]\) for \(R\) an Artin ring?

Unsorted

• Show that for, \(M\in \Alg\slice R\) finitely presented implies finitely generated.
  • Show that the converse doesn’t generally hold, unless \(R\) is Noetherian.
  • Show that if \(M\) is finitely presented, then for all \(f\) and \(m\), the module of relations \(\ker(A^m \surjectsvia{f} M)\) is finitely generated.
  • Show that finitely generated projective implies finitely presented.
• Show that any ideal Dedekind domain is finitely generated and projective but not free unless they are principal.
• Show that for \(P\) or \(P'\) finitely generated projective, \(\Hom(P, P') \cong P\dual \tensor P'\).
• Show that the going up theorem fails for \(\ZZ \subseteq \ZZ[x]\).
• Show that the completions of local rings at non-singular points of a variety over \(k\) are all isomorphic.
• Give an example of a local ring with zero divisors.

Nakayama

• Show the following: if \(A\) is a Noetherian local ring with maximal ideal \(m \subset A\) and if \(m^{n+1}=m^{n}\) then \(m^{n}=(0)\). If \(A\) is a Noetherian integral domain and \(P \subset A\) is a prime ideal then the powers \(\left\{P^{n}\right\}, n \geq 1\), are distinct.
• Show the following: if \(A\) is a Noetherian local ring with maximal ideal \(m \subset A\), then [ \bigcap_{n \geq 1} m^{n}=(0) \text {. } ] If \(A\) is a Noetherian integral domain and \(P \subset A\) is a prime ideal, then \(\bigcap P^{n}=\) (0).
• Show that a finitely generated projective module \(E\) over a local ring \(A\) is free.