Math 607

Homological Algebra

Tor: what is it good for?

Spring 2020

• Lecture 1: Motivation / Goals.
  video, notes, worksheet.
• Please install Macaulay2.
• Reading 1, due Thursday 4/2:
  Dummit and Foote §10.5.
• Lecture 2: Exact sequences, exactness of Hom and ⊗.
  video, notes, worksheet.
• Lecture 3: Localization is exact.
  video, notes, worksheet.
• Reading 2, due Tuesday 4/7:
  Eisenbud §2.1–2.2.
• Lecture 4: Flat and projective are local conditions.
  video, notes, continue on previous worksheet.
• Lecture 5: Continuation of previous lecture.
  video, notes, worksheet.
• Reading 3, due Tuesday 4/14:
  Dummit and Foote §17.1.
• Lecture 6: Projective means locally free.
  video, notes, worksheet.
• Lecture 7: Fibers of modules.
  video, notes, worksheet.
• Lecture 8: Projective resolutions, projective dimension.
  video, notes, worksheet.
• Lecture 9: More examples, Ext and Tor.
  video, notes, worksheet.
• Lecture 10: Ext and Tor are well-defined.
  video, notes, worksheet.
• Lecture 11: Long exact sequence; Tor and torsion.
  video, notes, worksheet.
• Lecture 12: Ext and Tor are local.
  video, notes, worksheet postponed.
• Lecture 13: Projective dimension via Tor with points.
  video, notes, worksheet.
• Lecture 14: Examples about global dimension.
  video, notes, worksheet.
• Lecture 15: The Koszul complex.
  video, notes, worksheet.
• Reading 4, due Thursday 5/7:
  Eisenbud: Chapter 17 Intro, §17.1, §17.2.
• Lecture 16: Global dimension of k[x₁, …, x_n].
  video, notes, worksheet.
• Lecture 17: Regular sequences.
  video, notes, worksheet.
• Lecture 18: Regular sequences and the Koszul complex.
  video, notes, worksheet postponed.
• Lecture 19: Depth and the Koszul complex.
  video, notes, worksheet.
• Lecture 20: Serre's theorem on finite global dimension.
  video, notes, worksheet.
• Reading 5, due Sunday 5/17:
  Eisenbud: §19.1–19.3.
• Lecture 21: Ext and extensions.
  video, notes, worksheet.
• Lecture 22: Change-of-rings spectral sequence.
  video, notes, worksheet.
• Lecture 23: Spectral sequences, continued.
  video, notes, worksheet postponed.
• Lecture 24: The spectral sequence of a double complex.
  video, notes, worksheet.
• Memorial Day: no meeting.
• Lecture 25: Last day of spectral sequences.
  video, notes, worksheet, follow-up e-mail.
• Lecture 26: Depth vs. codimension.
  video, notes, worksheet, follow-up e-mail.
• Lecture 27: Cohen–Macaulay.
  video, notes, worksheet.
• Lecture 28: Cohen–Macaulay, continued.
  video, notes, worksheet.
• Lecture 29: Smoothenss and regular sequences.
  video, notes, worksheet.