Pour des ensembles de cardinalité finie, on a le principe d’inclusion-exclusion:

En comparaison, si @X@ et @Y@ sont deux parties fermées d’un espace topologique ayant chacune un nombre fini de composantes irréductibles¹, et qu’on note @n(-)@ le nombre de composantes irréductibles, alors

On déduit la première équation de celle-ci en donnant aux ensembles @A@ et @B@ la topologie discrète.

Jusqu’à maintenant, je me suis presque¹ toujours servi de la disposition canadienne multilingue standard. Elle permet d’écrire en anglais et en français sans trop de mal. Un objectif principal de cette disposition est de minimiser les touches mortes. Par exemple, on peut entrer les caractères “à”, “ç” et “è” directement, ce qui est assez pratique. Cependant! Mes doigts partent en incartade quand je leur demande une paire de crochets ou des accolades. Ma main fait des frasques si j’ai le malheur de lui quémander un diacritique tout nu. Je fige si je dois fermer une balise HTML ou ouvrir une commande TeX.

Learning Clojure is a lot of fun. For one, the community is great.¹ On top of that, there’s many exercises available; that’s excellent for people like me, who learn by doing. For instance, rich4clojure is a great resource. It works out-of-the-box with Visual Studio Code, which is nice when you just want to start working on problems asap and you’ll just “setup the tools later”.

Tonight, I’m solving problem 84 from that set. I thought, maybe you’d like to follow along with me. Here it goes:

Here I talked about computing a direct limit on some cofinal subset. In this post, I want to prove what my assertions.

Note that direct limits are a special case of colimits, and what I’ll talk about can easily be generalized. In particular, colimit may also be computed on cofinal subcategories of their indexing category, and yield equal (“isomorphic up to a unique canonical isomorphism”) colimits. However, in this post I’ll stay in the particular case of direct limits, and I’ll try to define things first in terms of elements, to stay as concrete as possible.

Let @X@ be a topological space, let @p@ be a point of @X@, and let @S@ be any set. In the notation of Vakil, define the skyscraper sheaf supported at @p@ by the formula

Here @1@ is the singleton set. We can also define such a sheaf in other categories (abelian groups, rings, etc.), replacing the singleton set by the appropriate terminal object.

TODO: I need to fix the commutative diagrams in this post because they just don’t show up right now.

I’m planning on writing a series of posts that explore the theory and applications of characteristic classes in algebraic topology, following the book from Milnor and Stasheff. Here I start with the appendix A, where homology is discussed and basic theorems are laid out. This post will also serve me well as a quick reminder, since I keep forgetting small details and ideas regarding (singular) (co)homology. Many results and explanations can be found in Hatcher’s book.

TODO: Fix Young diagrams as they just don’t show up right now!

Recall that a partition @\lambda@ of some fixed natural number @n \geq 0@ is a list of nonnegative integers @(\lambda_1,\lambda_2,\dots)@ such that @\lambda_1 \geq \lambda_2 \geq \dots \geq 0@ and such that @\lambda_1+\lambda_2+\dots = n@. For the purposes of this post, the actual number of parts in a partition is irrelevant, and since one can always extend a partition with zeroes we may suppose that all partitions have the same fixed number of parts, let’s say @\ell@.

The goal of this short post is to convince myself that the ideal generated by @\operatorname{Sym}^+@ inside of @\mathbb{C}[x,y]@ can be more simply described as the ideal generated by @xy@ and @x+y@, i.e.

As a shorthand, set @A = \mathbb{C}[x,y]@. The ring @A@ is graded:

where @A^{(d)}@ is the @\mathbb{C}@-module consisting of the homogeneous polynomials of degree @d@:

In general, a polynomial is said to be symmetric when it is invariant under any permutation of the variables. In our case, a polynomial @p(x,y) \in A@ is symmetric when @p(x,y) = p(y,x)@. For instance, @x^3 + y^3 + 2xy@ is symmetric while @x+y^2@ is not. The product and difference of two symmetric polynomials is also a symmetric polynomial. Also, @1@ is a trivial example of a symmetric polynomial. Hence the set of all symmetric polynomials is a subring of @A@, which we denote by @\operatorname{Sym}@. This subring is naturally graded:

Let @X@ and @Y@ be two partially ordered sets (posets). Two functions @f : X \to Y@ and @g : Y \to X@ are said to form a Galois connection when the following is true for all @a \in X@ and all @b \in Y@: \begin{equation*}f(a) \leq b \iff a \leq g(b).\tag{C}\end{equation*} We say that @f@ is the lower (or left) adjoint, and @g@ is the upper (or right) adjoint. Oftentimes the lower adjoint is marked with a lower-star @f_*@ and the upper adjoint with an upper-star @f^*@. We will see for instance that the direct and inverse image form such a connection, and this hopefully explains the usual notation for these concepts. The shorthand I’m going to use for saying two functions are in Galois connection is @f_* \dashv f^*@.

A fact I’ve known for a very long time, but never bothered to prove for myself, is the unicity of the determinant for square matrices. More precisely, let @A@ be any commutative ring with identity; then the determinant is the only @A@-valued function on matrices that: (i) is linear on each column, (ii) is such that switching two columns changes the sign, and (iii) sends the identity matrix to 1.