Commutative Algebra
Read this topicExplanations 3
- David Speyer's notes on Jordan-Holder for finite groups: PDF Idea is, look at a normal lattice GGG containing large normal subgroups N,N′N,N'N,N′ containing N∩N′N\cap N'N∩N′ (in…
By Victor Wang
- Flatness to-do Equivalent definitions of flatness, see e.g. Qing Liu... when is locally free a good way to think about it? Use both algebra (e.g. number theory) and geometry int…
By Victor Wang
- Spectrum of a ring: basics Essentially, go back to big picture: why the Zariski topology? Why is this useful? Why study this and not something else? Connect this with classical …
By Victor Wang
Problems 30
- Is R[x]/(x2+1)2\mathbb{R}[x]/(x^2+1)^2R[x]/(x2+1)2 isomorphic to C[y]/y2\mathbb{C}[y]/y^2C[y]/y2 as an R\mathbb{R}R-algebra? (Source: Calvin Deng)
- Select all rings below that are local rings containing coefficient fields.
- Let ϕ :A→B\phi\colon A\to Bϕ:A→B be a ring homomorphism. If ℘\wp℘ is a prime ideal of AAA, describe Spec(B℘/℘B℘)\operatorname{Spec}(B_\wp/\wp B_\wp)Spec(B℘/℘B℘), the set of …
- Let ϕ :A→B\phi\colon A\to Bϕ:A→B be a ring homomorphism. If ℘\wp℘ is a prime ideal of AAA, describe Spec(B/℘B)\operatorname{Spec}(B/\wp B)Spec(B/℘B), the set of prime ideals o…
- Let ϕ :A→B\phi\colon A\to Bϕ:A→B be a ring homomorphism. If ℘\wp℘ is a prime ideal of AAA, describe Spec(B℘)\operatorname{Spec}(B_\wp)Spec(B℘), the set of prime ideals of B℘:…
- What are the proper ideals of the localization S−1AS^{-1}AS−1A? Describe a one-to-one correspondence, using pullback by the localization map A→S−1A.A\to S^{-1}A.A→S−1A.
- Let SSS be a multiplicative subset inside a ring AAA. What are the prime ideals of the localization S−1AS^{-1}AS−1A?
- Let SSS be a multiplicative subset of a ring RRR. When is the ring homomorphism R→S−1RR\to S^{-1}RR→S−1R guaranteed to be injective? Select all that apply.
- Let I,JI,JI,J be two ideals of RRR. What are V(I)∩V(J)V(I)\cap V(J)V(I)∩V(J) and V(I)∪V(J)V(I)\cup V(J)V(I)∪V(J)?
- Must RRR have a nontrivial idempotent eee, i.e. a solution to e2=ee^2=ee2=e with e≠0,1e\neq0,1e=0,1?
- Let AAA be a ring and let q\mathfrak{q}q be a prime ideal. Let X=Spec(A)X=\operatorname{Spec}(A)X=Spec(A) be the spectrum of AAA with the Zariski topology. Which of the followi…
- Let ℘\wp℘ be a prime ideal of a ring AAA. When is A/℘→A℘/℘A℘A/\wp \to A_\wp/\wp A_\wpA/℘→A℘/℘A℘ an isomorphism?
- Let AAA be a ring. How are minimal prime ideals ℘\wp℘ related to zerodivisors sss? Select all that apply. (Assume Zorn's lemma, say.)
- Let m\mathfrak{m}m be a maximal ideal of the polynomial ring Z[T1,…,Tn]\mathbb{Z}[T_1,\dots,T_n]Z[T1,…,Tn] in n≥1n\geq1n≥1 variables. Which of the following are possibilities …
- If A→BA\to BA→B is a flat ring map, and ttt is a non-zerodivisor of AAA, must the image of ttt in BBB be a non-zerodivisor?
- Which of the following are flat ring maps?
- But first, let's try to understand why ddd induces a derivation (modulo I2I^2I2). Which identities below do d :B→Id\colon B\to Id:B→I satisfy in the AAA-algebra B⊗ABB\otimes_A …
- Let AAA be a ring and ℘\wp℘ a prime ideal of AAA. When does ℘=ker(A→A℘/℘A℘)\wp = \ker(A\to A_\wp/\wp A_\wp)℘=ker(A→A℘/℘A℘) hold?
- Let AAA be a ring and ℘\wp℘ a prime ideal of AAA. When is A/℘→A℘/℘A℘A/\wp\to A_\wp/\wp A_\wpA/℘→A℘/℘A℘ an isomorphism?
- Which of the following conditions are equivalent to ϕ\phiϕ being a local homomorphism? Select all that apply.
- Are the above two conditions on ϕ\phiϕ equivalent?
- Let RRR be a ring and ℘\wp℘ a prime. Must R℘/℘R℘R_\wp/\wp R_\wpR℘/℘R℘ be the fraction field of R/℘R/\wpR/℘?
- Which of the following objects is NOT universal (in the sense of universal properties) with respect to the given property?
- Compute the Picard group of the spectrum of the ring R[x,y]/(x2+y2−1).\mathbb{R}[x,y]/(x^2+y^2-1).R[x,y]/(x2+y2−1).
- Let I,JI,JI,J be two ideals of RRR. What are D(I)∩D(J)D(I)\cap D(J)D(I)∩D(J) and D(I)∪D(J)D(I)\cup D(J)D(I)∪D(J)?
- What can you say about this map? Select all that apply.
- Let III be a homogeneous ideal of C[s,t]\mathbb{C}[s,t]C[s,t], graded by degree. If (s−t2)2∈I(s-t^2)^2\in I(s−t2)2∈I, what is the best you can say?
- Suppose III is a homogeneous ideal of SSS contained in S+S_+S+. Let JJJ be another homogeneous ideal of SSS. Which of the following statements relating D(J),V(J),D(I),V(I),D+(I…
- Let kkk be a field and AAA a kkk-algebra. Let k′k'k′ be the integral closure of kkk in AAA. Suppose AAA is an integral domain. Must k′k'k′ be a subset of {0}∪A×\{0\}\cup A^\time…
- Let I,JI,JI,J be ideals in a (commutative, unital) ring AAA. Compute the AAA-algebra tensor product A/I⊗AA/JA/I\otimes_A A/JA/I⊗AA/J.