Zariski Topologies

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.

More Posts from Mildlyramified and Others

2 months ago

ADHD is so debilitating and it isn’t talked about enough. Imagine your body doesn’t produce enough of the most essential neurotransmitter. You are constantly seeking this neurotransmitter through any way possible, and it’s why you get addicted to doing things or focusing so heavily on something you forget to meet every single basic need.

You sit there and question what the fuck is wrong with you because it was so easy to study yet you just didn’t do it. It was so easy to do the things you stopped doing but you literally can’t do them.

Like wtf do you fucking mean I was born with a chemical imbalance that makes me incapable of getting up??? Wtf do you mean I have to take stimulants to counteract crippling ADHD symptoms, and then those stimulants actually just make me like everyone else????

Dude. What the fuck.

6 months ago

REBLOG THIS POST IF YOU ARE A MATH ENJOYER

No it is not optional, I desperately need to follow y’all so that there is more math on my dashboard.

2 months ago

*heart eyes* I love you too /p

@mybeanalgebra I SEE YOU ONLINE 🫵 HELLO

2 months ago

Every once in a while where decapitation seems like a much more tolerable idea than having a head on top of my neck

Ugh… Headache. Head Splitting Migraine, Even.

Ugh… headache. Head splitting migraine, even.

1 month ago

KASHMIR MASTERLIST

Background

History of Kashmir from 250 BC to 1947 [to understand Kashmir's multi religious history and how we got to 1947]

Broad timeline of events from 1947 to the abrogation of Article 370 of the Indian Constitution in 2019 (BBC) [yes, BBC. hang on just this once]

Human Rights Watch report based on a visit to Indian controlled Kashmir in 1998 [has a summary, background, human rights abuses and recommendations]

Another concise summary of the issue

Sites to check out

Kashmir Action - news and readings

The Kashmiriyat - independent news site about ongoings in Kashmir

FreePressKashmir - same thing as previous

Kashmir Law and Justice Project - analysis of international law as it applies to Kashmir

Stand with Kashmir - awareness, run by diaspora Kashmiris [both Pandit and Muslim]

These two for more readings and resources on Kashmir: note that the petitions and donation links are from 2019 and also have explainers on the background (x) (x)

To read

Do You Remember Kunan Poshpora? - about women in the Kashmiri resistance movement and the 1991 mass rape of Kashmiri women in the twin villages of Kunan and Poshpora by Indian armed forces

Until My Freedom Has Come: The New Intifada in Kashmir - a compliation of writings about the lives of Kashmiris under Indian domination [available on libgen]

Colonizing Kashmir: State Building under Indian Occupation - how Kashmir was made "integral" to the Indian state and examines state-building policies [excerpt]

Resisting Occupation in Kashmir - about the social and legal dimensions of India's occupation [available on libgen]

Of Occupation and Resistance - another collation of stories of Kashmiris living under state repression

On India's scapegoating of Kashmiri Pandits, both by Kashmiri Pandits (x) (x)

Of Gardens and Graves - translations of Kashmiri poems

Social media

kashiirkoor

museumofkashmir

kashmirpopart

posh_baahar

readingkashmir

standwithkashmir and their backup account standwithkashmir2 [their main account is banned in India. I wonder why!]

kashmirlawjustice

kashmirawareness

kashmirarchive

jammugenocide [awareness about the 1947 genocide abetted by Maharaja Hari Singh and the RSS]

To watch

Jashn-e-Azadi: How We Celebrate Freedom parts 1 and 2 - a documentary about the Kashmiri freedom struggle [filmed by a Kashmiri Pandit]

Paradise Lost - BBC documentary about how India and Pakistan's dispute over the valley has affected the people

Kashmir - Valley of Tears - the exhaustion with the conflict in the post nineties

In the Shade of Fallen Chinar - art as a form of Kashmiri resistance

Human rights violations (x) (x) (x) (x) (x)

Land theft and dispossession (x) (x) (x) (x) (x) (x)

A note: The list of readings is not exhaustive. It is only an introduction to the history of the occupation. I know annoying "Desis" are going to see this and bitch and moan about how Kashmir is actually integral to their country out of a sense of colonial entitlement. Kashmir belongs to Kashmiris, the natives, no matter what religion they belong to. Neither Pakistan nor India get to decide the matter of Kashmiri sovereignty. The reasons given by both parties as to why Kashmir should be a part of either nation are bullshit. The United Nations itself recognises Kashmir as a disputed region, so I will entertain neither dumbfuckery nor whataboutism. I highly encourage fellow Indians especially to take the time to go through and properly understand the violence the state enacts on Kashmiris. I've also included links to learn more about Kashmiri culture because really, what do the rest of us know about it? Culturally & linguistically Kashmir differs so much from the rest of India and Pakistan (also the way Kashmiri women are fetishised... yikes). It's not just a bilateral issue between the two nations over land, it actually affects the people of Kashmir

7 months ago

Mathematics is taught very rigidly. When I'm independently working and studying math, it feels like art - like I'm making something and it tickles the creative side of my brain. In class it feels like the structured STEM course I initially signed up for.

It's a world of rules and structures people have carefully built over the millennium and you can add to it (if you can) or just walk around and observe and learn.

Analogously, learning mathematics, especially higher mathematics and even more so Algebra and Category Theory also feels like learning a new language. Working with it feels like writing poetry. Mathematics literature has a lot of the characteristic features of literature. There are many rules, but if you can break them, you are a mad genius!

I was talking to a professor and he told me about realising that he could read mathematics, granted it's not the same as picking up a story book, but there is this entire new world out there when you start reading mathematics. He also pulled up the linguistics definition of a language and said that perhaps mathematics is the only language with no exceptions in class once.


Tags
1 week ago
These Are So Fun To Make
These Are So Fun To Make

These are so fun to make

2 months ago

guess who

2 months ago

Awww

@mybeanalgebra I SEE YOU ONLINE 🫵 HELLO

4 months ago

Sleepy, confused puffins and dodo birds

A Puffin Rests While Overlooking The Ocean Near Látrebjarg, Iceland.

A puffin rests while overlooking the ocean near Látrebjarg, Iceland.

Go World Travel Magazine


Tags
  • algebraic-dualist
    algebraic-dualist liked this · 1 week ago
  • veganchickennuggie
    veganchickennuggie liked this · 1 month ago
  • vastlyestranged
    vastlyestranged liked this · 1 month ago
  • phoenixdiedaweekago
    phoenixdiedaweekago liked this · 1 month ago
  • kfunc-vmlinux-utf8nlookup
    kfunc-vmlinux-utf8nlookup liked this · 1 month ago
  • bubbliterally
    bubbliterally liked this · 3 months ago
  • stargirll111
    stargirll111 liked this · 6 months ago
  • dorothytheexplorothy
    dorothytheexplorothy liked this · 7 months ago
  • dofo-humming
    dofo-humming liked this · 7 months ago
  • frankiekafka
    frankiekafka reblogged this · 7 months ago
  • frankiekafka
    frankiekafka liked this · 7 months ago
  • mybeanalgebra
    mybeanalgebra reblogged this · 7 months ago
  • mybeanalgebra
    mybeanalgebra liked this · 7 months ago
  • mildlyramified
    mildlyramified reblogged this · 7 months ago
  • xjnegen
    xjnegen liked this · 7 months ago
  • notarealwelder
    notarealwelder liked this · 7 months ago
  • changing-notation
    changing-notation liked this · 7 months ago
  • locally-normal
    locally-normal liked this · 7 months ago
  • adjoint-law
    adjoint-law liked this · 7 months ago
  • ilovelollipops
    ilovelollipops liked this · 7 months ago
  • bubbloquacious
    bubbloquacious reblogged this · 7 months ago
mildlyramified - Abstract Nonsense
Abstract Nonsense

They/Them/She/Her | I Math

89 posts

Explore Tumblr Blog
Search Through Tumblr Tags