Tags: #study-guides ProblemsUndergrad Show that prime implies irreducible, and the converse only holds for UFDs. Show that maximal implies prime but not conversely, so \(\operatorname{mSpec}R \subseteq \operatorname{Spec}R\). 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 \(k[x_1, \cdots, x_{n}]\) is a UFD. Show that nonzero proper principal ideals in \(k[x_1, \cdots, x_{n}]\) are generated by irreducible polynomials. Describe \(\operatorname{Spec}A\) for \(A \coloneqq R \left[ { \scriptstyle { { ({{\mathfrak{p}}}^c) }^{-1}} } \right]\), and show that \(A\) is local with \({\mathfrak{m}}= {\mathfrak{p}}_e\). Show that \(M\otimes_R N\) may equal zero when \(M\neq 0, N\neq 0\). Find a polynomial \(f\in {\mathbf{C}}[x]\) such that \(f({\mathbf{Z}}) \subseteq {\mathbf{Z}}\) but \(f \not\in {\mathbf{Z}}[x]\). 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 \(R{\hbox{-}}\)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 \(I\in \operatorname{Id}(R)\) is prime iff \(R\setminus I\) is a submonoid of the multiplicative monoid of \(R\), i.e. \(R\setminus I\) 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 \(\operatorname{Spec}, \operatorname{mSpec},\) and radical ideals. Show that if \(a, b\) are radical ideals then \(a\cap b\) is radical but \(a+b\) need not be. Show that for a local ring \((R, {\mathfrak{m}})\), \({J ({R}) } = {\mathfrak{m}}\). 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 \({\mathfrak{m}}\) are \({\mathfrak{m}}{\hbox{-}}\)primary. Show that if \(A\leq B\) is a subring and \({\mathfrak{p}}\in \operatorname{Spec}A\) then there exists \({\mathfrak{q}}\in \operatorname{Spec}B\) such that \({\mathfrak{q}}\cap A = {\mathfrak{p}}\). Show that an ideal \(I{~\trianglelefteq~}k[x_1, \cdots, x_{n}]\) is homogeneous iff it is graded, i.e. \(I = \bigoplus I_d\) where \(I_d \coloneqq I \cap k[x_1, \cdots, x_{n}]_d\), the homogeneous degree \(d\) part of the graded ring \(k[x_1, \cdots, x_{n}]\). Let \(k[V]\) be the coordinate ring of a variety, and show that every maximal ideal \({\mathfrak{m}}\in \operatorname{mSpec}k[V]\) is of the form \({\mathfrak{m}}_p \coloneqq\left\{{f\in k[V] {~\mathrel{\Big\vert}~}f(p) = 0}\right\}\) for some point \(p\in V\). Show that every \(I\in \operatorname{Id}(R)\) is a projective \(R{\hbox{-}}\)module. Local Rings Given \((A, {\mathfrak{m}}_A)\), show that \(A \left[ { \scriptstyle { {{\mathfrak{m}}}^{-1}} } \right]\) is local. What is its maximal ideal? Show that for a fixed \(S \subseteq R\), there is an exact functor $S^{-1}: {}{R}{\mathsf{Mod}} \to {}{S^{-1}R}{\mathsf{Mod}} $. Show that the following are local properties: Being zero, i.e. \(M= 0\) if \(M \left[ { \scriptstyle { {{\mathfrak{m}}}^{-1}} } \right] = 0\) for all \({\mathfrak{m}}\in \operatorname{mSpec}R\) for ${}_{R}{\mathsf{Mod}} $. Injectivity of module morphisms, i.e. \(M\to N\) is injective iff \(M \left[ { \scriptstyle { {{\mathfrak{m}}}^{-1}} } \right] \to N \left[ { \scriptstyle { {{\mathfrak{m}}}^{-1}} } \right]\) is injective for all \({\mathfrak{m}}\in \operatorname{mSpec}R\). Being reduced: \(R\) is reduced iff \(R \left[ { \scriptstyle { {{\mathfrak{m}}}^{-1}} } \right]\) is reduced for all \({\mathfrak{m}}\in \operatorname{mSpec}R\). Flatness Exactness, i.e. \(A\to B\to C\) is exact iff \(A \left[ { \scriptstyle { {{\mathfrak{m}}}^{-1}} } \right] \to B \left[ { \scriptstyle { {{\mathfrak{m}}}^{-1}} } \right] \to C \left[ { \scriptstyle { {{\mathfrak{m}}}^{-1}} } \right]\) is exact for all \({\mathfrak{m}}\in \operatorname{mSpec}R\) Being integrally closed. Being coprime ideals, i.e. \(I+J = R \iff I_{\mathfrak{p}}+ J_{\mathfrak{p}}= R_{\mathfrak{p}}\) for all \({\mathfrak{p}}\in \operatorname{Spec}R\). When is a Unsorted/localization of rings 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 \((A/I) \left[ { \scriptstyle { {S}^{-1}} } \right] \cong M/IM\) for \(M\coloneqq A \left[ { \scriptstyle { {S}^{-1}} } \right]\) Noetherian Rings Modules and Algebras Show that a morphism \(A\to B\) is the same as giving \(B\) an \(A{\hbox{-}}\)algebra structure. Show that Nakayama’s lemma may fail if \(M\) is not finitely generated. Show that if $A\in \mathsf{Alg} _{/ {k}} $ is finite over \(k\) (and an integral domain), then \(A\) is a field. Show that any $M\in \mathsf{Alg} _{/ {R}} $ satisfies \(M = \colim M_\alpha\) where \(\left\{{M_\alpha \leq M}\right\}\) are all of the finitely generated subalgebras of \(M\). Show that if $M\in {}_{R}{\mathsf{Mod}} $ with \((R, {\mathfrak{m}})\) a local ring, then the action \(R\curvearrowright M/{\mathfrak{m}}M\) factors through the residue field \(\kappa(R)\), and \(\left\{{g_i}\right\}\subseteq M\) generate the quotient as an \(R{\hbox{-}}\)module iff \(\left\{{a_i + {\mathfrak{m}}M}\right\}\) generate the quotient as a \(\kappa(R){\hbox{-}}\)module. Show that if $M\in {}_{R}{\mathsf{Mod}} $ then \(M {}^{ \vee } {}^{ \vee }\) 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 \(\dim A = \dim A{ {}_{ \widehat{{\mathfrak{m}}} } }\) Let \(R = k[x_1, \cdots, x_{n}]\) and \(I = \left\langle{x_1,\cdots, x_n}\right\rangle\). Show that \(\dim R \left[ { \scriptstyle { {I}^{-1}} } \right] = n\). 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 \({\mathcal{O}}_K\) 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 \(\left\{{0}\right\}\) is closed. Show that for \(R \coloneqq k[x_1, \cdots ,x_n]\) and \(I\coloneqq\left\langle{x_1, \cdots, x_n}\right\rangle\), the \(I{\hbox{-}}\)adic completion is $R{ {}_{ \widehat{I} } } = k {\left[\left[ x_1, \cdots, x_n \right]\right] } $. Show that A PID is a Dedekind domain. For \(L/{\mathbf{Q}}\) a finite field extension, show that the ring of algebraic integers \({\mathcal{O}}_L\) 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 Unsorted/localization of rings 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 \({\sharp}\operatorname{Spec}R < \infty\), 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 \(M_k \twoheadrightarrow M_{k+1}\) for all \(k\). Show that direct limits commute with tensor products. Show that \(k[x_1, \cdots, x_{n}]\otimes_k k[x_{n+1}, \cdots, x_m] \cong k[x_1,\cdots, x_m]\) Show that \(M\otimes_R {-}\) preserves direct sums. Prove that \({-}\otimes_R M\) is not generally exact for all $M\in {}_{R}{\mathsf{Mod}} $. Show that localization is exact. Show that \(M\) is flat iff \(\operatorname{Tor}^1(M, R/I)=0\) for all finitely generated ideal \(I{~\trianglelefteq~}R\). Show that \(M\) is flat if \({-}\otimes_R M \to M\) is a flat morphism, where it suffices to check on all ideals \(I\). Show that if \(R\) is Noetherian and local and $M\in {}_{R}{\mathsf{Mod}} $, then \(M\) is free iff \(\operatorname{Tor}^1(\kappa(R), M) = 0\) for \(\kappa(R)\) the residue field. Show that over \(R=k[x,y]\), the ideals \(I\in \operatorname{Id}(R)\) which are projective are precisely the principal ideals. Similarly, free ideals in/over a commutative integral domain must be principal. Show that integral domains have no proper direct summands, and conclude that \(R/I\) is never a projective \(R{\hbox{-}}\)module. Show that \(\left\langle{x, y}\right\rangle {~\trianglelefteq~}R\coloneqq k[x, y]\) is not a free \(R{\hbox{-}}\)module. Show the following: a finitely generated module over a noetherian domain is locally free if and only if the dimensions of the fibres over maximal ideals is constant Show that \(\operatorname{Tor}_i^R(k, k) = k\) for all \(i\) for \(R = k[x]/x^2\). Hint: resolve by \(1\mapsto x\). Flatness 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 \mathsf{Alg} _{/ {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 \(\ker(A^m \xrightarrow[]{f} { \mathrel{\mkern-16mu}\rightarrow }\, M)\) 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, \(\mathop{\mathrm{Hom}}(P, P') \cong P {}^{ \vee }\otimes P'\). Show that the going up theorem fails for \({\mathbf{Z}}\subseteq {\mathbf{Z}}[x]\). 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 \begin{align*} \bigcap_{n \geq 1} m^{n}=(0) \text {. } \end{align*} 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.
