These Are So Fun To Make

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.

One of the most frustrating things about being ADHD is the way hyperfixations and skill levels work.

So I, an ADHD person, will get obsessed with A Thing. I will research, I will practice. I'll check out library books, watch YouTube videos, seek out podcasts, all to learn everything I can about The Thing.

Thiat Thing is often a skill or hobby. Painting, writing, candlemaking, baking, mixology, tea blending.

But the thing with ADHD is that you'll be obsessed with it only to a certain skill level. Something where all the mystery is gone. It's not as fun once the learning part is over and it's just boring practice to get better.

Then abruptly, you'll lose interest and move to another fixation.

That skill level you've earned may be higher than your average person with a passing interest. But it's also lower than someone who specializes in said thing, who has put in those hard hours of practice and work.

So you start just forming this miscellaneous collection of things that you're good enough at to earn some praise, but still leave you feeling like you're just never *quite* good enough at anything because you can't just choose anything.

And you want to pick a Thing. To find Your Thing. The thing that fits, that you can finally excel at. But you just can't seem to.

SabrangIndia

Jyotiba Phule’s Trenchant Critique of Caste: Gulamgiri | SabrangIndia

First Published on: 11 Apr 2016 On his 189th Birth Anniversary, April 11, we bring to you excerpts from Jyotiba Phule’s path breaking work,

Jyotiba Phule was born on April 11, 1827 If a Bhat happened to pass by a river where a Shudra as washing his clothes, the Shudra had to collect all his clothes and proceed to a far distant spot, lest some drops of the (contaminated) water should be sprayed on the Bhat. Even then, if a drop of water were to touch the body of the Bhat from there, or even if the Bhat so imagined it, the Bhat did not hesitate to fling his utensil angrily at the head of the Shudra who would collapse to the ground, his head bleeding profusely. On recovering from the swoon the Shudra would collect his blood- stained clothes and wend his way home silently. He could not complain to the Government Officials, as the administration was dominated by the Bhats. More often than not he would be punished stringently for complaining against the Bhats. This was the height of injustice! It was difficult for the Shudras to move about freely in the streets for their daily routine, most of all in the mornings when persons and things cast long shadows about them. If a `Bhat Saheb’ were to come along from the opposite direction, the Shudra had to stop by the road until such time as the `Bhat Saheb’ passed by – for fear of casting his polluting shadow on him. He was free to proceed further only after the `Bhat Saheb’ had passed by him. Should a Shudra be unlucky enough to cast his polluting shadow on a Bhat inadvertently, the Bhat used to belabour him mercilessly and would go to bathe at the river to wash off the pollution. The Shudras were forbidden even to spit in the streets. Should he happen to pass through a Brahmin (Bhat) locality he had to carry an earthen-pot slung about his neck to collect his spittle. (Should a Bhat Officer find a spittle from a Shudra’s mouth on the road, woe betide the Shudra!)……. [[The Shudra suffered many such indignities and disabilities and were looking forward to their release from their persecutors as prisoners fondly do. The all-merciful Providence took pity on the Shudras and brought about the British raj to India by its divine dispensation which emancipated the Shudras from the physical (bodily) thraldom (slavery). We are much beholden to the British rulers. We shall never forget their kindness to us. It was the British rulers who freed us from the centuries-old oppression of the Bhat and assured a hopeful future for our children. Had the British not come on the scene (in India) (as our rulers) the Bhat would surely have crushed us in no time (long ago.)]]

Some may well wonder as to how the Bhats managed to crush the depressed and down-trodden people here even though they (the Shudras) outnumbered them tenfold. It was well-known that one clever person can master ten ignorant persons (e.g. a shepherd and his flock). Should the ten ignorant men be united (be of one mind), they would surely prevail over that clever one. But if the ten are disunited they would easily be duped by that clever one. The Bhats have invented a very cunning method to sow seeds of dissension among the Shudras. The Bhats were naturally apprehensive of the growing numbers of the depressed and down- trodden people. They knew that keeping them disunited alone ensured their (the Bhats’) continued mastery ever them. It was the only way of keeping them as abject slaves indefinitely, and only thus would they be able to indulge in a life of gross indulgence and luxury ensured by the `sweat of the Shudras’ brows. To that end in view, the Bhats invented the pernicious fiction of the caste-system, compiled (learned) treatises to serve their own self-interest and indoctrinated the pliable minds of the ignorant Shudras (masses) accordingly. Some of the Shudras put up a gallant fight against this blatant injustice. They were segregated into a separate category (class). In order to wreak vengeance on them (for their temerity) the Bhats persuaded those whom we today term as Malis (gardeners), Kunbis (tillers, peasants) etc. not to stigmatise them as untouchables. Being deprived of their means of livelihood, they were driven to the extremity of eating the flesh of dead animals. Some of the members of the Shudras community today proudly call themselves as Malis (gardeners), Kunbis (peasants), gold-smiths, tailors, iron smiths, carpenters etc, on the basis of the avocation (trade) they pursued (practised), Little do they know that our ancestors and those of the so¬called untouchables (Mahars, Mangs etc.) were blood-brothers (traced their lineage to the same family stock). Their ancestors fought bravely in defence of their motherland against the invading usurpers (the Bhats) and hence, the wily Bhats reduced them to penury and misery. It is a thousand pities that being unmindful of this state of affairs, the Shudras began to hate their own kith and kin. The Bhats invented an elaborate system of caste-distinction based on the way the other Shudras behaved towards them, condemning some to the lowest rung and some to a slightly higher rung. Thus they permanently made them into their proteges and by means of the powerful weapon of the `iniquitous caste system,’ drove a permanent wedge among the Shudras. It was a classic case of the cats who went to law! The Bhats created dissensions among the depressed and the down- trodden masses and are battening on the differences (are leading luxurious lives thereby). The depressed and down­trodden masses in India were freed from the physical bodily) slavery of the Bhats as a result of the advent of the British raj here. But we are sorry to state that the benevolent British Government have not addressed themselves to the important task of providing education to the said masses. That is why the Shudras continue to be ignorant, and hence, their ‘mental slavery’ regarding the spurious religious tracts of the Bhats continues unabated. They cannot even appeal to the Government for the redressal of their wrongs. The Government is not yet aware of the way the Bhats exploit the masses in their day to day problems as also in the administrative machinery. We pray to the Almighty to enable the Government to kindly pay attention to this urgent task and to free the masses from their mental slavery to the machinations of the Bhats.

Read the full text in the original Marathi, in Hindi, in English.

also parents: *showing the same exact signs and thinks everyone is like them*

Isomorphism

everything IS like everything else

Laziness Does Not Exist

Psychological research is clear: when people procrastinate, there's usually a good reason

good read for teachers.

she abstract my nonsense till my diagram commutes

she graph my theory til i form an edge set

Someone put the beach beneath my feet and the salty humid air

forget about touching grass, i need to touch THE SEA I NEED TO GO INTO THE WATER I NEED TO DIVE INTO THE SEA!!!!!!!!!!!!

date someone who is interested in you. i don’t mean someone who thinks you’re cute or funny. i mean someone who wants to know every insignificant detail about you. someone who wants to read every word you write. someone who wants to hear every note of your favourite song, or watch every scene of your favourite movie. someone who wants to find every scar on your body, and learn where they came from. someone who wants to know your favourite brand of toothpaste, and which quotes resonate deep inside your bones when you hear them. there is a difference between attraction and interest. find the person who wants to learn every aspect of who you are.

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 I'll be like "nothing has ever been good and I shall die ;__;" but as soon as I get a little sip of water and it's sunny outside I'm like nvm I'm thriving I love life :)

But if my houseplants do that exact same thing, I'll call evert single one of them an overdramatic bitch.