I'm A Huge Hypocrite, If We're Being Honest. If I Haven't Had A Beverage And It's Been Dark For Too Long

Zariski topologies

So if you take kⁿ, the n-dimensional coordinate space over some field k, the Zariski topology on kⁿ is the topology whose closed sets are of the form

Z(S) = { x ∈ kⁿ : f(x) = 0 for all f ∈ S }

for some subset S ⊆ k[x₁,...,xₙ]. That is, the closed sets are the common zero loci of some set of polynomials over k in n variables, i.e. they are the solution sets for some system of algebraic equations. Such sets are called algebraic sets. If I is the ideal generated by S, then Z(S) = Z(I), so we can restrict ourselves to ideals.

Now if you take a commutative unital ring R, we let Spec R denote its prime spectrum, the set of prime ideals of R. We let Max R ⊆ Spec R be the subset consisting of the maximal ideals, the maximal spectrum. The Zariski topology on Spec R is the topology whose closed sets are of the form

Z(S) = { P ∈ Spec R : P ⊇ S }

for some subset S ⊆ R. A prime ideal P contains S if and only if it contains the ideal generated by S, so again we can restrict to ideals. What's the common idea here? Classically, if k is algebraically closed, then Hilbert's Nullstellensatz (meaning Zero Locus Theorem) allows us to identify the points of kⁿ with those of Max k[x₁,...,xₙ], by mapping a point (a₁,...,aₙ) to the maximal ideal (x₁ - a₁,...,xₙ - aₙ), and the Zariski topologies will agree along this identification. There's nothing very special about these algebraic sets though.

Let X be any (pre-)ordered set with at least one bottom element. For a subset Y ⊆ X, define the lower and upper sets associated to Y as

L(Y) = { x ∈ X : x ≤ y for all y ∈ Y }, U(Y) = { x ∈ X : x ≥ y for all y ∈ Y }.

We call a lower [upper] set principal if it is of the form L(x)= L({x}) [U(x) = U({x})] for some x ∈ X. If X is complete (any subset has at least one least upper bound and greatest lower bound), then any lower or upper set is principal. Note that ⋂ᵢ L(Yᵢ) = L(⋃ᵢ Yᵢ), so lower sets are closed under arbitrary intersections; they provide what's called a closure system on the power set of X. The lower closure of a set Y is the intersection of all lower sets containing Y. We have that Y ⊆ L(x) if and only if x ∈ U(Y), so the lower closure of Y is given by L(U(Y)). If the lower sets were furthermore closed under finite unions (including empty unions), then they would form the closed sets of a topology on X.

This is not generally true; first of all, note that any lower set contains the bottom elements of X, of which there is at least one, so the empty set is not a lower set. As for binary unions, generally we have L(Y₁) ∪ L(Y₂) ⊆ L(Y₁ ∩ Y₂), but this inclusion might be strict. This is something we can fix by restricting to a subset of X.

We say that p ∈ X is prime if p is not a bottom element and for all x, y such that for all z such that x ≤ z and y ≤ z we have p ≤ z, we have that p ≤ x or p ≤ y. That is, if p is smaller than every upper bound of x and y, then p is smaller than x or y. Furthermore, we say that p is a prime atom if it is a minimal prime element. Let P(X) and A(X) denote the sets of primes and prime atoms of X, respectively. For a subset Y ⊆ X, let the Zariski closed set associated to Y be given by

Z(Y) = L(Y) ∩ P(X) = { p ∈ P(X) : p ≤ y for all y ∈ Y }.

We again have ⋂ᵢ Z(Yᵢ) = Z(⋃ᵢ Yᵢ), so the Zariski closed sets are closed under arbitrary intersections. Note also that Z(X) = ∅, so the empty set is closed. Now let Y₁, Y₂ be subsets of X. We find that Z(Y₁) ∪ Z(Y₂) = Z(U(Y₁ ∪ Y₂)). Clearly if p is smaller than all of the elements of one Yᵢ, then it is smaller than every upper bound; the interesting part is the other containment.

Assume that p ∈ Z(U(Y₁ ∪ Y₂)), so p is smaller than every upper bound of Y₁ ∪ Y₂. If p is smaller than every element of Y₁ then we are done, so assume that there is some y ∈ Y₁ with p ≰ y. For every y' ∈ Y₂ we have that p is smaller than every upper bound of y and y', so because p is prime we get that it is smaller than y or y'. It is not smaller than y, so p ≤ y'. We conclude that p ∈ Z(Y₂), and we're done.

As before, the Zariski closure of a set of primes Q ⊆ P(X) is given by Z(U(Q)). Note however that for a point x ∈ X we have L(x) = L(U(x)), so the Zariski closure of a prime p is Z(p). It follows that A(X) is exactly the subspace of closed points of P(X).

So we have defined the Zariski topology on P(X). How can we recover the classical examples?

If X is the collection of algebraic subsets of kⁿ ordered by inclusion, then P(X) consists of the irreducible algebraic subsets, and we can identify kⁿ itself with A(X). Our Zariski topology coincides with the standard definition.

If X = R is a unital commutative ring, ordered by divisibility, then being prime for the ordering coincides with being either prime for the ring structure, or being equal to 0 if R is an integral domain. Note that this ordering is not generally antisymmetric; consider 1 and -1 in a ring of characteristic not equal to 2.

A more well-behaved version of the previous example has X = { ideals I ⊴ R }, ordered by reverse inclusion. Note that for principal ideals (r), (s) we have (r) ⊇ (s) if and only if r divides s. We have P(X) = Spec R and A(X) = Max R, and our Zariski topology coincides with the standard definition.

You can play the same game if X is the lattice of subobjects of any structure H. If H is a set (or a topological space) and X is its power set, then the primes and prime atoms are the same; the points. The Zariski topology is the discrete topology on H. If H is a vector space, then P(X) is empty, because any non-zero subspace V can be contained in the span of two subspaces that don't contain V. It seems that the sweet spot for 'interesting' Zariski topologies is somewhere in between the rigidity of vector spaces and the flexibility of sets.

If H is an affine space, then again the prime elements are exactly the points. The resulting Zariski topology has as closed sets the finite unions of affine subspaces of H.

An interesting one is if X is the set of closed sets of some topological space S (generalizing the first example). The prime elements are the irreducible closed sets, and if S is T1 (meaning all points are closed), then the points of A(X) can be identified with those of S. Then the Zariski topology on A(X) is the same as the topology on S, and the Zariski closure of an irreducible closed set is the set of all irreducible closed sets contained in it.

Actually this isn't how we count.

These are hand symbols (Hasta or Mudra) used in various different Indian dance forms. They have different names.

👍 is shikaram. ✌️ Is actually slightly different, the middle finger comes forward and the index goes backwards almost like you are trying to twist them. It is called Kartharimukam(scissors).

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So, my spouse has been exploring his gender lately; he also just built himself a new laptop. Today he told me that he in an attempt to process some genderfeels through metaphor, he made a post on a trans forum along the lines of: "I'm a lifelong Windows user and I think I'm pretty good at it. I want to find out what Linux has to offer but I'm afraid I wouldn't be any good at it. And how do you choose the right Linux distro, anyway? Do you have to try them all?"

The responses, he said, were a mix of useful advice about feeling out your gender and useful advice about choosing a Linux distro.

I love trans people so much

I need to stop thinking about my work days as "productive" days and my days off as "unproductive" days that I waste if I haven't built something or deep cleaned my house. What the fuck am I accomplishing at work. My job doesn't wash my dishes

If I were writing a math textbook I’d have a section in the beginning called “A Note on Notation” where I introduce all the notation Im using. And everyone would want to fuck me sooooo bad

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Just learnt about this dude Galois, absolutely crazy. so basically, he:

- tries to get into "Ecole Polytechnique", the French school for science at the time, and fails

-publishes several papers on polynomials

-his father dies

-tries again (and fails) to get into the Ecole

-publishes several papers basically founding group theory

-attends a lesser school, gets kicked out and eventually arrested for political beliefs

-died in a duel shortly after he got out of prison

AT THE AGE OF 20

CanpeoplestopputtingmathhateonthemathtagCanpeoplestopputtingmathhateonthemathtagCanpeoplestopputtingmathhateonthemathtagCanpeoplestopputtingmathhateonthemathtag

Can people stop putting math hate on the math tag ?????

pleaseeee like I just want to see some fun math stuff, a bunch of theorems and memes

Thanks! Yall keep switching to a language I don't speak every 20 seconds and I can't bother to be involved anymore

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