Math 6150: Fall 2018

Contact

Instructor: Jonathan Wise
e-mail: jonathan.wise@colorado.edu
Office: Math 204
Office hours: calendar
Phone: 303 492 3018

Office hours

My office is Room 204 in the Math Department. My office hours sometimes change, so I maintain a calendar showing the times I will be available. I am often in my office outside of those hours, and I'll be happy to answer questions if you drop by outside of office hours, provided I am not busy with something else. I am also happy to make an appointment if my office hours are not convenient for you.

Textbook

The textbook for this course may be the best math textbook ever written:

Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra.

We will follow it as closely as we can, but we will have to make digressions into a few other topics:

Matsumura, H. Commutative Algebra, 2nd Ed.

Weibel, C. A. An Introduction to Homological Algebra

Evaluation

The requirement of the course are to submit four sets of exercises. You may choose which exercises to submit.

Syllabus

Before we begin:

Review: the definitions of commutative ring, modules, and topological spaces

Part 1: Commutative rings

Review: rings, ideals, subrings, quotient rings, polynomial rings
Read: Atiyah-MacDonald, Chapter 1 (pp. 1-10), and Chapter 3 until modules are mentioned (pp. 36-38)
Exercises: §1, #1, 2, 4 (Hint: use 1.9 and exercise 2), 5, 7, 8, 10, 12, 13, 14; Find a universal property for the group ring and prove it.

Part 2: Spectra of commutative rings

Review: topological spaces
Read: Atiyah-MacDonald, Chapter 1, Exercise 15-22, 26-28 (pp. 12-16)
Exercises: §1, #16, 22, 26, 27

Part 3: Modules

Review: modules, kernel, cokernel, direct sum, direct product
Read: Atiyah-MacDonald, Chapter 2 (pp. 17-31)
Exercises: §2, #1-26; I encourage doing some of #13-24, and especially #19 to get comfortable with direct limits

Part 4: Homological algebra

Review: exact sequence, kernel, cokernel
Read: Weibel, Chapters 2 and 3
Exercises: Prove that every projective module is a direct summand of a free module and conversely (Weibel, Prop. 2.2.1); Show that the ideal (2, 1 + x) in ℤ[x] / (x^2 + 5) is projective but not free; Prove that Ext1(M,N) is independent of whether M or N is resolved; Show that Torp(M,N) = 0 for all abelian groups M and N and all p ≥ 2 (Weibel, Prop. 3.1.2); Find an example of a ring A and A-modules M and N such that Torp(M,N) ≠ 0 for all p ≥ 0.

Part 5: Localization

Read: Atiyah-MacDonald, Chapter 3 (pp. 36-43)

Exercises: §3, #1–30 (they are all important). Prove that \( A \to S^{-1} A \) is an epimorphism.

Part 6: Integral extensions and valuation rings

Read: Atiyah-MacDonald, Chapter 5 (pp. 59-67)

Part 7: Noetherian rings

Read: Atiyah-MacDonald, Chapter 6 and Chapter 7

Part 8: Dedekind domains

Read: Atiyah-MacDonald, Chapter 9

Part 9: Dimension

Read: Atiyah-MacDonald, Chapter 11

Part 10: Nilpotent extensions, completions, and differentials

Read: Atiyah-MacDonald, Chapters 8 and 10; Matsumura, Section 26

Assignments

  1. For Friday, December 7: Read Atiyah–MacDonald, Chapter 9.
  2. For Wednesday, December 5: Do Atiyah–MacDonald, Chapter 6, #1, 3, 5, 6, 7, 9. Do Chapter 7, #1, 2, 8.
  3. For Monday, December 3: Read Atiyah–MacDonald, Chapter 6 up to p. 76, and Chapter 7, up to p. 82.
  4. For Wednesday, November 28: Construct an example of a valuation ring with more than 2 prime ideals. Do Atiyah–MacDonald, Chapter 5, #5, 27–33.
  5. For Monday, November 26: Do Atiyah–MacDonald, Chapter 5, #3, 6, 12–15. Enjoy your vacation!
  6. For Friday, November 16: Let \( A = \mathbb C[x,y] / (y^2 - x^3 - ax - b) \) and let \( M \) be the ideal \( (x,y) A \). Show that \( M \) is invertible by showing it is locally free of rank 1. What is its inverse? Finish reading Atiyah–MacDonald, Chapter 5.
  7. For Wednesday, November 14: Do Atiyah–MacDonald, Chapter 3, #24. Prove that local finite generation and local finite presentation are local properties. Warning: you need to use the definition of a local property from class; these are not local properties in the sense of Atiyah‐MacDonald (can you find a counterexample?). Read Atiyah–MacDonald, Chapter 5 (pp. 59–64).
  8. For Monday, November 12: Do Atiyah–MacDonald, Chapter 3, #16, 19 23. Let \( A \) be a ring and let \( A \to B_i \), \( i = 1, \ldots, n \) be a collection of ring homomorphisms. We say that this collection is faithfully flat if, for all sequences of \( A \)-modules \( M' \to M \to M'' \), the sequence \( M' \to M \to M'' \) is exact if and only if all of the sequences \( B_i \otimes_A M' \to B_i \otimes_A M \to B_i \otimes_A M'' \) are exact. Prove that the sequence
    \( 0 \to M \to \prod_i B_i \otimes_A M \to \prod_{i_1, i_2} B_{i_1} \otimes_A B_{i_2} \otimes_A M \to \prod_{i_1, i_2, i_3} B_{i_1} \otimes_A B_{i_2} \otimes_A B_{i_3} \otimes_A M \to \cdots \)
    is exact.
  9. For Friday, November 9: Atiyah–MacDonald, Chapter 3, #12–13
  10. For Wednesday, November 7: Suppose that \( S \) is a multiplicatively closed subset of a ring \( A \). Prove that if \( P \) is a projective \( A \)-module then \( S^{-1} P \) is a projective \( A \)-module. Prove that \( S^{-1} \operatorname{Tor}_1^A(M,N) \simeq \operatorname{Tor}_1^{S^{-1} A}(S^{-1} M, S^{-1} N) \). Is the same true for \( \operatorname{Ext}^1 \)?
  11. For Monday, November 5: Read Atiyah–MacDonald, Chapter 3. For each map \( f : N \to P \), construct a map \( \operatorname{Tor}_1(M, N) \to \operatorname{Tor}_1(M, P) \) that commutes with the maps in the long exact sequence for \( \operatorname{Tor} \). Do Atiyah–MacDonald, Chapter 2, #24–25 (you can skip the part about \( \operatorname{Tor}_n \) for \( n > 1 \). Prove that \( \operatorname{Tor}_1(N,M) \simeq \operatorname{Tor}_1(M,N) \) for all \( A \)-modules \( M \) and \( N \). Do Atiyah–MacDonald, Chapter 3, #1.
  12. For Friday, November 2: Do Atiyah–MacDonald, Chapter 2, #26.
  13. For Wednesday, October 31: Let \( A = \mathbb C[x_1, \ldots, x_n] \) and let \( M = \mathbb C \) be the \( A \)-module with all \( x_i \) acting by \( 0 \). Compute \( \operatorname{Ext}^p(M, A) \) for all integers \( p \geq 0 \).
  14. Compute \( \operatorname{Ext}^1_{\mathbb Z}(\mathbb Z/n\mathbb Z, \mathbb Z/m\mathbb Z) \) for all integers \( m, n \). Compute \( \operatorname{Ext}^1_{\mathbb C[x]}(\mathbb C[x] / f(x), \mathbb C[x] / g(x)) \) for all polynomials \( f, g \).
  15. For Friday, October 26: Complete the proof (started in class) that the Ext long exact sequences are exact.
  16. For Wednesday, October 24: Prove the Eckmann–Hilton principle: if \( G \) is a group with two multiplication operations \( m_i : G \times G \to G \) such that \( m_i \) is a homomorphism with respect to \( m_j \) for \( j \neq i \) then \( m_i = m_j \) and both are abelian. Verify that \( \operatorname{Ext}^1(-,-) \) is an additive functor of each of its variables and that the addition laws on \( \operatorname{Ext}^1(M,N) \) that these properties induce are compatible in the sense of the Eckmann–Hilton principle. Deduce that they are the same.
  17. For Monday, October 22: Prove that \( \mathbb C(x) \) is an injective module over \( \mathbb C[x] \). Let \( M_\lambda \) be the \( \mathbb C[x] \)-module \( \mathbb C \), with \( x \) acting by multiplication by \( \lambda \). Compute \( \operatorname{Ext}^1(M_\lambda, M_\mu) \) for all \( \lambda, \mu \in \mathbb C[x] \).
  18. For Friday, October 19: Let \( F : A\mathrm{-}\mathbf{Mod} \to \mathbf{Ab} \) be the functor that sends an \( A \)-module to its underlying abelian group. Define \( G, H : \mathbf{Ab} \to A\mathrm{-}\mathbf{Mod} \) by \( G(X) = \operatorname{Hom}_{\mathbf{Ab}}(A, X) \) and \( H(X) = A \otimes_{\mathbf Z} X \). Show that \( H \) is left adjoint to \( F \) and that \( G \) is right adjoint to \( F \).
  19. For Wednesday, October 17: Read Atiyah–MacDonald, pp. 36–41. Do Chapter 2, #3–5, 9, 11–13.
  20. For Friday, October 5: Finish reading Chapter 2 of Atiyah–MacDonald. Show that tensor product respects cokernels: \( M \otimes \operatorname{coker}(N' \to N) \simeq \operatorname{coker}(M \otimes N' \to M \otimes N) \) (use universal properties). Do Atiyah–MacDonald, Chapter 2, #2, 11–13.
  21. For Wednesday, October 3: Use the adjoint functor theorem to construct colimits in the categories of sets, rings, \( A \)-modules, groups, topological spaces (in all cases, the proof is the same). Let \( A \) be a ring and let \( B \) be an \( A \)-algebra (a ring with a homomorphism \( \varphi : A \to B \)). An \( A \)-derivation of \( B \) into a \( B \)-module \( M \) is a function \( \delta : B \to M \) such that \( \delta \) is a homomorphism of abelian groups, \( \delta(\varphi(A)) = 0 \), and \( \delta(fg) = f \delta(g) + g \delta(f) \) (the Leibniz rule). Prove that there is a universal \( A \)-derivation \( d : B \to \Omega_{B/A} \) using the adjoint functor theorem.
  22. For Friday, September 25: construct inverse limits in the categories of rings and \( A \)-modules; let \( X : I \to A\mathchar"2D\operatorname{Mod} \) be a diagram in the category of \( A \)-modules; show that \( \varprojlim_{i \in I} X(i) = \ker \left( \prod_{i \in I} X(i) \to \prod_{\substack{i,j \in I \\ \phi \in \operatorname{Hom}_I(i,j)}} X(j) \right) \) where the map sends a tuple \( (x_i)_{i \in I} \) to the tuple \( (x_j - \phi(x_i))_{i,j,\phi} \).
  23. For Wednesday, September 23: prove the universal properties of the direct product, the direct sum, the kernel, the cokernel, the tensor product (as constructed on p. 24 of Atiyah–MacDonald), and the module of homomorphisms all of the universal properties discussed in class; use the universal properties of kernel and cokernel to construct an isomorphism of modules \( \operatorname{coim} \varphi \to \operatorname{im} \varphi \) for any module homomorphism \( \varphi \); do Atiyah–MacDonald, Chapter 2, #1 (can you do it using the univerasl property?), 6, 7, 14–18, 21, 22 (the direct limit problems will be discussed in class)
  24. For Monday, September 21: read Atiyah–MacDonald, pp. 17–27; prove the strong Nullstellensatz from the weak Nullstellensatz. Let \( K \) be an algebraically closed field; prove from the Nullstellensatz that every maximal ideal in \( K[x_1, \ldots, x_n] \) is of the form \( (x_1 - \xi_1, \ldots, x_n - \xi_n) \) for some \( \xi_1, \ldots, \xi_n \in K \).
  25. For Wednesday, September 19: prove Chevalley's theorem
  26. For Monday, September 17: read Atiyah–MacDonald, pp. 17–22. Draw a picture of \( \operatorname{Spec} k[x,y] \) where \( k \) is a field. What does \( V(x^2 + y^2 + 1) \) look like in \operatorname{Spec} \mathbf R[x,y] \)? In class, we drew a picture of \( \operatorname{Spec} \mathbf Z[x] \). Add to that picture a depiction of \( V(x^2 + 1) \).
  27. For Friday, September 14: We will continue the discussion of the spectrum of a ring. Prove that \( D(f) = \operatorname{Spec} A[f^{-1}] \) for any \( f \in A \). If \( \varphi : A \to B \) is a ring homomorphism and \( p \) is a prime ideal of \( A \), describe a ring \( C \) such that \( \Spec C \) is the fiber of \( \operatorname{Spec} B \to \operatorname{Spec} A \) over \( p \). Continue looking at Exercises #15–22 of Chapter 1 of Atiyah–MacDonald.
  28. For Wednesday, September 12: We are still discussing the spectrum of a commutative ring. The relevant exercises are #15–22 of Chapter 1. We will also discuss Exercise #21 in class.
  29. For Monday, September 10: Please continue to work on #15–17 of Chapter 1, along with #19–22, and 27–28. You may enjoy Problem #26, but it is not essential.
  30. For Friday, September 7: prove that the prime ideals of \( S^{-1} A \) are in bijection with the prime ideals of \( A \) not meeting \( S \); prove that the prime ideals of \( A/I \) are in bijection with the prime ideals of \( A \) containing \( I \); do Atiyah–MacDonald, Chapter 1, #15–17 (make sure at least to read through all parts of these exercises, as these cover material not in the body of the textbook that will be discussed on Friday); do Atiyah–MacDonald, Chapter 1, #9, 10, 14.
  31. For Wednseday, September 5: complete the proof of the universal property of \( S^{-1} A \) (this is Proposition 3.1 of Atiyah–MacDonald); do Atiyah–MacDonald, Chapter 1, #1, 2, 5, 8; prove that \( (A/I)/(J/I) \simeq A/J \) when \( I \subset J \subset A \) are ideals of a ring \( A \) (use the universal property); prove that \( S^{-1}(A/I) \simeq (S^{-1} A) / (S^{-1} I) \) when \( S \) is a multiplicatively closed subset of \( A \) and \( I \) is an ideal of \( A \) (use the universal property); read Atiyah–MacDonald, pp. 5–10.
  32. For Friday, August 31: pick a universal property and translate it into a functor; read pp. 1–6, 36–38.

Lecture notes

  1. Lecture 37 (November 28): valuative criterion for properness
  2. Lecture 36 (November 26): valuation rings
  3. Lecture 35 (November 16): integral morphisms
  4. Lecture 34 (November 14): invertible modules
  5. Lecture 33 (November 12): projectivity is a local property
  6. Lecture 32 (November 9): flatness is a local property, modules form a sheaf
  7. Lecture 31 (November 7): local properties
  8. Lecture 30 (November 5): Nakayama's lemma
  9. Lecture 29 (November 2): Tor long exact sequence
  10. Lecture 28 (October 31): Tor
  11. Lecture 27 (October 29): existence of enough injectives
  12. Lecture 26 (October 26): computing Ext with resolutions
  13. Lecture 25 (October 24): Ext long exact sequence
  14. Lecture 24 (October 22): additivity of the extensions functor
  15. Lecture 23 (October 19): extensions
  16. Lecture 22 (October 17): injective modules
  17. Lecture 21 (October 15): projective and flat modules
  18. Lectures 17–20 (October 5–12): abelian categories
  19. Lecture 16 (October 3): tensor products of modules and algebras
  20. Lecture 15 (October 1): proof of the adjoint functor theorem
  21. Lecture 14 (September 28): the adjoint functor theorem
  22. Lecture 13 (September 26): limits
  23. Lecture 12 (September 24): module constructures
  24. Lecture 11 (September 21): Nullstellensäaut;tze
  25. Lecture 10 (September 19): Chevalley's theorem
  26. Lecture 9 (September 17): basic open sets, quasicompactness, Chinese remainder theorem
  27. Lecture 8 (September 14): preimages of open and closed subsets
  28. Lecture 7 (September 12): more first (second?) properties of Spec
  29. Lecture 6 (September 10): first properties of Spec
  30. Lecture 5 (September 7): motivation for the prime spectrum
  31. Lecture 4 (Septeber 5): units and nilpotents
  32. Lecture 3 (August 31): universal properties of ring constructions
  33. Lecture 2 (August 29): representable functors
  34. Lecture 1 (August 27): universal properties