Sitemap

A map of the full site. Click on the triangle pointing towards the title of a page to preview the page.

• 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 $a, b$.
• Sitemap

  A map of the full site. Click on the triangle pointing towards the title of a page to preview the page.
• Musings
  • Blowups and Fiber Products

    This past week, I’ve been working on computing blowups. Classically, the blowup of $\mb A^n$ is the algebraic subset of $\mb A^n\times \mb P^{n-1}$ given by $\{(a_1, ..., a_n, [p_1, ..., p_n]) | a_ip_i=a_jp_j\}$; now, we blowup along a sheaf of ideals $\ms I$ by forming the sheaf of graded rings $\ms B = \mc O_X\oplus \ms I\oplus \ms I^2\oplus \cdots$. In the affine case this is called the blowup algebra. We then take relative $\proj$ of this sheaf of graded rings, which yields the blowup. In particular, this week, I was trying to compute the blowup of the affine cone $x^2-y^2-z^2$ at the origin (to resolve that singularity). I found this shockingly difficult, mostly due to my inability to work with the graded ring and the proj construction.
• Papers

  Here you can find papers, preprents, and notes. All may contain errors; please email me at skyler@skylermarks.com with any thoughts you may have.

  • Intersection Theory on Surfaces

    Intersection theory is a cornerstone of modern algebraic geometry, and the case of surfaces is the simplest and most classical case thereof. This paper develops the intersection pairing and intersection multiplicity, following Hartshorne’s “Algebraic Geometry”, and presents some examples.
  • A Categorical Development of Right Derived Functors

    Category theory is the language of homological algebra, allowing us to state broadly applicable theorems and results without needing to specify the details for every instance of analogous objects. However, authors often stray from the realm of pure abstract category theory in their development of the field, leveraging the Freyd-Mitchell embedding theorem or similar results, or otherwise using set-theoretic language to augment a general categorical discussion. This paper seeks to demonstrate that - while it is not necessary for most mathematicians’ purposes - a development of homological concepts can be contrived from purely categorical notions. We begin by outlining the categories we will work within, namely Abelian categories (building off additive categories). We continue to develop cohomology groups of sequences, eventually culminating in a development of right derived functors. This paper is designed to be a minimalist construction, supplying no examples or motivation beyond what is necessary to develop the ideas presented.
  • An Invitation to Algebraic Geometry

    In high school algebra 1 and 2, we study the theory of single variable polynomial equations. In linear algebra, we study systems of linear equations, or polynomial equations with no exponents greater than 1. Algebraic Geometry combines these disciplines to study polynomial equations (in particular, their solutions) in many variables. This theory is useful as it is specific enough that we can compute with it, yet general enough that it applies to many problems we care about. The promised invitation will be extended by way of plane conics and cubics. After some definitions and preliminaries, we will review a family of classical results regarding plane conics (quadratic polynomials in two variables.
  • Elliptic Curve Cryptography

    The digital world is kept secure by cryptography. The idea behind most modern cryptographic systems is that if I have a public number (some sort of seed) and a secret number (usually called a private key) I can perform some operation with them to generate a third number (which usually called a public key). The public key is easy to generate by combining a seed and a private key, but hard to generate any other way; this allows us to verify the authenticity of a private key very easily. One such operation involves finding collinear points on an elliptic curve, giving rise to elliptic curve cryptography. This talk introduced elliptic curves over finite fields, explained how we can use their geometry to define an appropriate operation, and touched on why this operation is appropriate for cryptography. Slides for the talk can be found here.
  • The Kodaira Embedding Theorem

    Chow’s Theorem and GAGA are renowned results demonstrating the algebraic nature of projective manifolds and, more broadly, projective analytic varieties. However, determining if a particular manifold is projective is not, generally, a simple task. The Kodaira Embedding Theorem provides an intrinsic characterization of projective varieties in terms of line bundles; in particular, it states that a manifold is projective if and only if it admits a positive line bundle. We prove only the ‘if’ implication in this paper, giving a sufficient condition for a manifold bundle to be embedded in projective space. Along the way, we prove several other interesting results. Of particular note is the Kodaira-Nakano Vanishing Theorem, a crucial tool for eliminating higher cohomology of complex manifolds, as well as Lemmas 6.2 and 6.1, which provide important relationships between divisors, line bundles, and blowups. Although this treatment is relatively self-contained, we omit a rigorous development of Hodge theory, some basic complex analysis results, and some theorems regarding Cech cohomology (including Leray’s Theorem).