--- date: 2022-12-25 03:34 modification date: Sunday 25th December 2022 03:34:35 title: "Commutative Algebra" flashcard: "Quals::Commutative Algebra" created: 2023-07-26T10:17 updated: 2023-07-26T10:18 --- # Commutative Algebra - [x] What is a regular ring? ✅ 2023-01-1 - [ ] A local ring $(R, \mfm_R)$ is **regular** iff $\mfm_R$ can be generated by $\dim R$ elements. - [ ] The minimal number of generators of $\mfm_x$ equals its Krull dimension. - [ ] An arbitrary ring $R$ is **regular** iff $R_p$ is regular for every prime. - [x] What is $\jacobsonrad{R}$? How is it interpreted geometrically? ✅ 2023-01-14 - [ ] $$\jacobsonrad{R} \da \Intersect_{\mfm\in \mspec R} \mfm = \Intersect_{M\in\rmod \text{ simple}} \Ann_R(M)$$ - [ ] If $R\in \kmod^{\fg}$, so $R\cong \kxn/I$, the $\nilrad{R} = \jacobsonrad{R}$, so $\jacobsonrad{R}$ measures how far away $I$ is from defining the ring of functions on some variety. - [x] What is a quasiregular element in a ring? ✅ 2023-01-14 - [ ] $a\in R$ with $1-a\in R\units$; $\jacobsonrad{R}$ is the largest ideal with every element quasiregular. - [x] What is the Artin-Rees lemma? ✅ 2023-01-14 - [ ] $I\normal R, N\leq M\in \rmod \implies \exists C\in \ZZ$ such that $$I^nM = I^{n-C}(I^CM \intersect N)$$ - [x] What is Krull's intersection theorem? ✅ 2023-01-14 - [ ] For $I\normal R$ Noetherian with $I \subseteq \jacobsonrad{R}$, $$\Intersect_{n\geq 0} I^n = 0$$ - [ ] Works for $R\in \Loc\CRing$ with $I\da \mfm_R$. - [x] What is the geometric interpretation of Krull's intersection theorem? ✅ 2023-01-14 - [ ] Define $\Ord_{p} f$ as the smallest $n$ such that $f\in \mfm_p^n$ but $f\not\in \mfp^{n+1}$; then $\Ord_p(f) = \infty \iff f=0$. - [ ] There is no hypersurface passing though $p$ with arbitrarily high order of vanishing. - [ ] One can detect whether $n$ functions vanishing at $0\in \AA^n$ generate a local coordinate system (so generate $\mfm_0$) by knowing their differentials (residues in $\T_0 \AA^n = \mfm_0/\mfm_0^2$ are a vector space basis), like an implicit function theorem. - [x] What is a reduced ring? ✅ 2023-01-14 - [ ] No nonzero nilpotents. - [x] What is Nakayama's lemma? ✅ 2023-01-24 - [ ] $M = IM \implies m= im$, i.e. if $I\normal R$ and $M\in\rmod^\fg$ with $M=IM$, then $\exists i\in I$ such that $m=im$ for all $m\in M$. So if $I$ stabilizes $M$, then an individual $i\in I$ stabilizes all of $M$. - [ ] For $R$ local, $M\to N$ is surjective iff $M/\mfm_RM \to N/\mfm_R N$ is surjective as a map of vector spaces. - [x] Give a categorical/homological corollary of Nakayama's lemma. ✅ 2023-01-24 - [ ] For $R$ Noetherian, $M\in \rmod$ flat, then $M$ is free: - [ ] Take a presentation $K\injects R^n \surjects M$, reduce to $k$ to get $K'\injects k^n \surjects M'$ where $K'=0 \implies K=0$. - [ ] Thus flat coherent sheaves over a Noetherian scheme are vector bundles. - [x] What is Nakayama's lemma for a local ring? ✅ 2023-01-24 - [ ] For $(R, \mfm_R)$ a loca ring, minimal generators of $M\in \rmod$ biject with $k\mod$ bases for $M/\mfm_R M$ where $k \da R/\mfm_R$. - [ ] Any two such sets of minimal generators for $M$ of size $n$ are related by a matrix in $\GL_n(R)$. - [x] What is the geometric interpretation of Nakayama's lemma? ✅ 2023-01-14 - [ ] For any $\mcf\in \oxmods$, regard stalks $\mcf_p \in \mods{\OO_{X, p}}$, then the fibers $\mcf(p) \da \mcf_p/\mfm_p \mcf_p$ are vector spaces and any basis for the fiber $\mcf(p)$ lifts to a minimal generating set for the stalk $\mcf_p$. - [ ] If $\mcf \leadsto E$ representes a vector bundle, then a basis for $\ro E p$ can be lifted to a basis of global sections in some neighborhood of $p$. - [ ] Give a geometric application of Nakayama's lemma. - [ ] If $f:X\to Y$ is a projective morphism between quasiprojective varieties, then $f$ is an isomorphism iff $df_p$ is injective for all $p\in X$. - [x] What is the going up theorem? ✅ 2023-01-14 - [ ] If $R\injects S\in \zalg$ is an integral extension, then $\spec S\to \spec R$, and for every $p\in \spec R$ there is a $\tilde p\in \spec S$ such that $\tilde p \intersect R = p$. - [x] What is Noether normalization? ✅ 2023-01-06 - [ ] ![](attachments/2022-02-06_19-48-53.png) - [x] What is Krull's theorem? ✅ 2023-01-14 - [ ] Every nonzero ring has a maximal ideal. - [ ] Stronger version: every proper ideal is contained in a maximal ideal (can apply to $I=\gens{0}$). - [x] What is Krull's Hauptidealsatz? ✅ 2023-01-14 - [ ] Let $R$ be a Noetherian ring and $a$ an element of $R$ which is neither a zero divisor nor a unit. Then every minimal prime ideal $P$ containing $a$ has height 1. - [ ] What is Zariski's lemma? - [ ] What is a regular ring? - [ ] What is a Dedekind ring? - [ ] What is an idempotent? ## Rings - [ ] What is the characteristic of a ring? - [ ] What is Zorn's lemma? - [ ] What is an irreducible element of a ring? - [ ] What is a prime element of a ring? - [ ] What is a prime ideal in a ring (defined without using elements)? - [ ] What is Gauss' lemma? - [ ] What are the quotient characterizations of prime and maximal ideals? ## Modules - [x] What is the one-step submodule test? ✅ 2023-01-06 - [ ] If $r\in R,\, m,n \in M \implies rm + n \in M$ then $M$ is a submodule. - [x] For matrices, does $A^n=B^n\implies A=B$? ✅ 2023-01-14 - [ ] No: $$A=\left[\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right] \Longrightarrow A^2=\left[\begin{array}{ll}1 & 0 \\0 & 1\end{array}\right]$$ - [ ] Characterize when a matrix over a field $\FF$ is diagonalizable. - [ ] What is a free module? - [ ] What is a projective module? - [ ] What is a $p\dash$primary ring? - [ ] What is a Noetherian module? An Artinian module? - [ ] What is the annihilator of an element? Of an $R\dash$module? - [ ] What is an irreducible module? An indecomposable module? A simple module? - [ ] Prove that free $\implies$ projective. - [ ] Give a Euclidean domain that is not a field. - [ ] Give a PID that is not an ED. - [ ] Give a UFD that is not a PID. - [ ] Give a domain that is not a UFD. ## Groups - [ ] State the Sylow theorems - [ ] State the recognizing direct products theorem - [ ] What is a solvable group? - [ ] Define $A_n$. - [ ] Define the sign of a permutation. - [ ] Given a permutation $\sigma\in S_n$ in cycle notation, how does one compute its sign? - [ ] What is Burnside's formula? - [ ] What is the centralizer? - [ ] What is the stabilizer? - [ ] What is orbit-stabilizer? - [ ] What is the class equation? - [ ] What are the fixed points of a group action? - [ ] What is a $p\dash$group? - [ ] What is Euler's theorem? - [ ] What is Cauchy's theorem? - [ ] How does one compute $\phi(n)$ in general? - [ ] Classify groups of order: 5. - [ ] Classify groups of order: 6. - [ ] Classify groups of order: 7. - [ ] Classify groups of order: 8. - [ ] Classify groups of order: 9. - [ ] What is the smallest nonabelian group? - [ ] What is the one-step subgroup test? - [ ] What are elementary divisors? - [ ] What are invariant factors? - [ ] What is the normalizer? - [ ] What is the normal core? - [ ] What is a characteristic subgroup? ## Fields - [ ] What is Eisenstein's criterion? - [ ] What is an algebraic extension? - [ ] What is a normal extension? - [ ] What is a separable extension? - [ ] Describe $K(x)$ - [ ] Describe $K((x))$ - [ ] What is a cyclotomic polynomial? - [ ] Define $K[\alpha]$ and $K(\alpha)$. - [ ] When is an extension Galois? - [ ] What is the fundamental theorem of Galois theory? - [ ] What is the splitting field of a polynomial? - [ ] What is a perfect field?