Projects

This is a page for various projects I’m working on.

• Eisenbud's First Course on Commutative Algebra

  I’m working through the first course in commutative algebra recommended by Eisenbud in his book “Commutative Algebra with a view towards Algebraic Geometry”. Here are my solutions to the exercises he recommends.

  • Chapter 1: Roots of Commutative Algebra

    Exercises

    A note: I use $(f_1, ...)$ to denote the submodule generated by $f_1,...$.
• 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.

  • 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.
• Examples

  This is a collection of examples in math I find noteworthy or in some other way interesting. I will try to make them more or less user-friendly, and to include references to my notes whenever possible, but I offer you no promises.

  • Algebraic Geometry Examples

    This is a collection of examples in Algebraic Geometry I find noteworthy or in some other way interesting. I will try to make them more or less user-friendly, and to include references to my notes on AG whenever possible, but I offer you no promises.

    • Assorted Singular Plane Curves

      The Union of the Axiis is Singular

      Recall that, as a scheme, we can write the union of the axiis as $X := \spec\left[k[x, y]/(xy)\right]$. Consider the local ring $\mc O_{(x, y), X}$, a local ring with maximal ideal $(\bar x, \bar y)$. Consider $(\bar x, \bar y)/(\bar x, \bar y)^2$. Note that $(\bar x, \b y)^2$ is generated by $\b x^2, \b x\b y,$ and $\b y^2$, but $\b x\b y=0$, so $(\b x, \b y)^2 = (\b x^2, \b y^2)$. Then $\dim_k (\b x, \b y)/(\b x^2, \b y^2) = 2$. Suppose $\dim \mc O_{(x, y), X}\neq 1$. We have a nontrivial prime ideal, so the dimension must be positive; thus there must be some tower of prime ideals $0\subsetneq P \subsetneq Q \subsetneq (x, y)$. Elements of $(x, y)$ are of the form $ax + by$ for $a\in k[x]$ and $b\in k[y]$ (if $a$ had any $y$ factor it would vanish, and conversely for $b$). Consider an ideal which contains $ax+by$ for both $a$ and $b$ nonzero; quotienting by such an ideal would yield $\b{x}\b{ax} = \b{x} \b{-by} = 0$ as $xy=0$. Thus such an ideal cannot be prime. No prime ideal can thus contain $ax+by$ for $a, b$ nonzero; moreover, if an ideal contains $ax$ and $by$ it must contain $ax+by$ by closure, so each ideal must contain only multiples of $x$ or multiples of $y$. Moreover, it is clear that $a, b\in k$ for the ideal to be prime, and so any prime ideal in $(x, y)$ is of the form $(x)$ or $(y)$. These ideals don’t contain each other, and so we have found the dimension of $\mc O_{(x, y), X}$ is 1.
    • Projectivization, Homogenization, and Dehomogenezation

      Simple Dehomogenization in a Polynomial Ring

      I’ve endeavored to be as general with this example as possible so that it may generalize to the proof of Proposition (Proj is a Scheme). I also wrote this before I wrote those notes, and so the notes might be a little more well-organized. It might be good pedagogy to struggle through this example first because I included some more motivational remarks here, then read those notes, and then come back to this example with a better idea of the structure of the thing.
    • Blowup Computations

      I did my first blowup computations this semester, and found it tremendously satisfying. Those were entirely blowups of affine plane curves at the origin, using what I’ll call the “classical” blowing up procedure (take the subset of $\mb A^2 \times \mb P^1$ with coordinates $(x, y, [z:w])$ cut out by $xw=zy$, and look at where the curve pulls back to under the projection map to $\mb A^2$). I might write up those examples eventually, but this page is mostly dedicated to blowing up a sheaf of ideals.
    • Intersections on Surfaces

      The quadratic and an axis.

      (Taken from my notes on intersection theory)
    • Toric Geometry Examples

      In the words of Prof. Melody Chan, toric geometry is interesting because it yields examples entirely controlled by combinatorics; we can associate toric varieties to polyhedral fans by gluing together the prime spectra of polynomial rings given by adjoing semigroups associated to those fans to a ground field (usually $\mb C$). Rarely is such robust gluing data given in such a conscise and accessable manner.

      • Toric Surfaces from Fans

        Here are examples of 2-dimensional toric varieties with all the affine spectra worked out and the gluing data specified explicitly to the degree necessary to work out e.g. Čech cohomology by hand. I’ve used sage at times, and will include the code I’ve utilized. I’ll also include some visualizations of the cones when possible. I use $\hull(\{v_i\})$ to denote the convex hull of the $v_i$, and $\cone(\{v_i\})$ to denote the set $\{a_1v_1+a_2v_2+\cdots\ \text{s.t.}\ a_i\in \mb R_{\geq 0}\}$. If $\sigma$ is a cone, $\sigma^\nu = \{v\in\mathbb R^n \text{ s.t. } v\cdot w\geq 0\ \ \forall w\in \sigma\}$ is the dual of $\sigma$.
    • "Fibrations"

      These are various examples or counterexamples to the slogan “total space = base $\times$ fiber”.
  • Commutative Algebra Examples

    This is a collection of examples in Commutative Algebra I find noteworthy or in some other way interesting. I will try to make them more or less user-friendly, and to include references to my commalg notes whenever possible, but I offer you no promises.

    • Assorted Rings and Ideals

      Polynomial Rings as Graded rings, and some ideals.

      1. The ring $k[x, y]$ is a graded ring, with grading given by total degree. The homogeneous elements are homogeneous polynomials (polynomials where the total degree of each term is the same).
      2. The ideal $(x, y)$ is a homogeneous, prime, and maximal ideal. It is also the irrelevent ideal.
      3. The ideal $(x, y^2)$ is homogeneous.
      4. The ideal $(x + y^2)$ is not homogeneous. It cannot be generated by homogeneous elements. The same is true of $(x-a, y-b)$ for any $a, b$.