Notes

This is a collection of theorems and lemmas in math I want to record for my future reference. I will try to make them more or less user-friendly. A lot of these are motivated by my desire to build a growing collection of examples, and I will endeavor to link to relevent examples whenever possible. Errors are to be expected; I encourage you to report them using the error tracker or (less preferred) to email me.

Warning: URLS are currently unstable, as I’m working out a scheme which will scale well and have the properties I require. Don’t rely on links remaining the same.

Eventually I plan to add some scripts to compile these into a pdf, but that’s a long term goal.

• Foundations of Math

  My (long term) goal for these notes is to build a first course in higher math, starting from the foundations of logic and set theory and building towards comfort with a lot of the general tools and techniques which are helpful to have when beginning any course in the “upper division” (analysis, algebra, topology, etc.). I’ll include problems and (perhaps) solutions along with my notes.

  • Preliminaries: Philosophy, Rigour, and Logic

    (These first few paragraphs are mostly a discussion of philosophy; the impatient reader can safely skip to Logic (A Practical Primer)

    • Problems

      Here are some problems designed to get you started thinking in terms of first order logic.
  • Introduction to Sets

    Now we’ll discuss the backbone of all modern math, set theory. Math can be thought of as, in it’s most basic form, the study of collections; a series of advanced ways of counting. A set is effectively a well-define collection of objects; however, it is actually quite difficult to specify what “well-defined” means. There was a time when a set was taken to be any collection of objects which could be described; however, this leads to paradoxes. For example, let $S$ denote the collection of all collections which do not contain themselves. Is $S$ in $S$? If it is, then $S$ contains itself, and so $S$ cannot be an element of $S$, a contradiction. But if it is not, then $S$ does not contain $S$, so $S$ must be in $S$, another contradiction. The only possibility is that “membership”, the relation that specifies if $S$ is in the collection $S$, is not well defined, and so the collection is not well-defined.
  • Relations
  • Functions
• Algebra

  These are notes on Algebra. Much of this will, at least in the beginning, be taken up by commutative algebra lemmas to support theorems in Algebraic Geometry. That being said, as I learn more algebra, I hope to improve these notes to include more non-commutative algebra.

  • Commutative Algebra

    These are my notes on commutative algebra, in part to support Algebraic Geometry, and in part because I think the subject is neat.

    • Properties of the Tensor Product

      Here are various results about the tensor product, one of my favorite mathematical constructions.
    • Basic Commutative Ring Theory

      Here we provide some crucial defininitions and lemmas for the theory of commutative rings. All rings are commutative with unity; all homomorphisms of rings take 1 to 1.
    • Graded Rings

      Here we define the projectivization of a graded ring and the projectivization of a sheaf of graded rings, and prove some important lemmas about them. We also record some important facts about graded rings. This basically follows the discussion in Hartshorne’s Algebraic Geometry, Chapter II, Section 2.
  • Group Theory

    These are my notes on group theory.
  • Notes - Nonommutative Algebra

    These are my notes on noncommutative algebra (if I ever learn any).
• Algebraic Geometry

  These are notes on Algebraic Geometry. I plan to loosely follow the structure of Hartshorne’s Algebraic Geometry, filling in some details when necessary. I’ll also suppliment with Vakil’s The Rising Sea and other texts as needed. I’m following a rather depth-first approach with these notes, so LARGE sections will remain incomplete for a long time.