Crystalline cohomology and Spider Man 2

So far so good. Each day if I understand few statements properly then I am happy. Todays gain, crystalline cohomology. There is an exposition sort of article by Luc Illusie, one of the bourbakis (or burbak), on this really interesting cohomology theory. The main theory was thought by Grothendieck and developed by P. Berthelot (whose name of adviser is unknown as far as mathematics genealogy project is concerned ). There is a nice letter of Grothendieck to J .Tate (the starting looked really gayish to me , "cher"), describing his idea of crystals and rigidity properties of this new cohomology theory.

The main idea is the following , in char 0 , we can see that d(x^p) = p.x^(p-1)dx. But in char p we get d(x^p) = 0. So on one hand we can integrate p.a^(p-1) and get back x^p in char 0,but that is not true for char p. The problem is precisely in char p we can not integrate x^(p-1) which in principle should happen to show the de rham complex is exact (remember the classical poincare lemma !). But we need to know some sort of de rham cohomology for the case of char p, since in most of the cases this turns out to be the strongest cohomology theory we ever had, as it has good properties of all the other cohomologies plus it has a mixed hodge structure (which roughly means that you cut the cohomology groups in each dimension and concentrate on something really nice). What can be a better way out ? One thing can be that given a variety X char p and f : X -> Y proper map where Y is smooth over some good ring ( in this case ring of witt vectors) we can get some cohomology theory which conincides with the de rham cohomology of Y, but then there should be some uniqueness propety related to this cohomology theory ( that is the cohomology groups are independent of the map f). So firstly we need to fix the problem of integration. Here comes Grothendiecks simple but really special observation. In char 0 , observer that if f_n(x) = x^n / n ! , then integration of f_n gives f_(n+1) and differentiation of f_(n+1) gives f_n, so bingo , this is the property which presicely gives what we need. In classical case we don't integrate over whole surface , but over some closed set, so in this case we have to get the idea of integration on some closed set. Closed sets comes from some ideal, so we have to get some ideals where we can define fuctions like f_n. Now we take the ring W(k). Why ? Well this is the special and simplest example of rings (where char k = p and k is perfect) which has an ideal where we can integrate canonically (this witt vectors has many great properties, check them out). If $k$^is $F_p$ then we have W(k) as the ring of p-adic integers. So roughly speaking what we do is for any variety X, we take all the sheaf of ideals where we can integrate, we can take all these ideals and take the schemes defined by it, this forms a Site, we can take the structure sheaf associated to this ideals, and define a sheaf on this site and calculate the cohomology. Yes for most of our cases this cohomology theory is "The theory" we were looking for. Moral of the story, ideas are always simple, and it doesnt take a genius to uncover it. You have to make your question clear ans most often the answer will be the direct consequence of the question, or if i rephrase it, if your question is really clear, then as Saurabh bhaiya says a simple Bihari arguement (Q :why ?, A : Why not.) will give you the answer.

Spider Man 2 is shit. I watched it again. Even after the second attempt I can not like it. I liked the first part, no the action scenes are not breathtaking in any of the spider man movie, but Peter Parker, not the spider man which is more lovable in this whole series. First of all, spider man is the most popular superhero till date, if its not true may lightning strike me. Reasons are very simple, he is not from any other planet, he does not want revenge or anything, his fights and weapons are simple, he can not fly, and in real life he is a simple young man, who, given an option, will always love to live an unknown life of a simple man, he falls in love, he falls from rooftop, he runs away from his flat owner, runs away from everything to find out that there are no way outs. Somehow it reminds me one of my favourite poem by Joy Goswami called Meghnad. Pardon me for the english translation, but the first few lines goes as follows

No one is fighting behind the clouds, all nonsense,
No one is winning Indra, nonsense,
everyone is grazing, chin down, in kolkata,
all ratnakar is roaming around to feed their family.

In the middle of this you can find few weak balmikis,
didn't have the courage to fight in daylight ,
there are no lights around the head of the brave,
see, there goes meghnad, hanging from the bus, Gautam Haldar.

On stage, he is a different person,
different charisma,
but everyday hanging from the bus, but every day coming back from the office,
stampling , sitting, getting up, satisfying boss,
dont we fight this stubborn war every day on the cloud top ?

Anyway Joy Goswami had his unique way of creating layer of motifs linguistically, and as english is a foreign language for me, I can not do justice to that in english. So Peter Parker is somehow this kind of a person, may be spider man is not true even in the story. May be all the things are his dream, what he want to do, how he wants to protest ? When he talks with the doctor about his dream problem i just thought may be this is the dream of all the common men. We want to be superheroes, not for only popularity but to protest against what we dont like, but we are weak, we want a normal life to. Reality and dream gets separated by our eyelid. Just a thin skin dividng two contrasting yet related world. We love our normal life, the people associated with it, so in real life we just compromise for some stability, not for us, but for all the people who define our existence. We fight with our dream of becoming a superhuman, we fight our urge to protest just to protect our near and dear ones from any trouble. So after a day full of fighting with our natural instincts, swallowing plethora of insults we give a good night kiss to our near ones, and close our eye lids, draped in our night dress and blankets, we roam around all over the world, alone, to do the job of a superhero.
As joy goswami said
"Meghnad is not alone, don't you, or that guy or me
every day fight an unequal war to find food ?
No laxman has born to kill us."
Before ending this tale of superheroes, I want to tell you an incidence in Paris. It was like every other midnight in Paris. I was as usual drunk, hungry and watching outside the trees that looked barely alive. I thought what if tonight I become a superhero like spider man or something ( i have a superhero name too, UTUMAN), and jump around . Watch all the windows and balconies of this old dead city, all the people standing their. Some of them sad, some of them trying to be happy, but everyone alone. I wrote a poem. I find this as one of the best poems I have ever written. So it is for you now
Sohortar naam hote parto everest :

ordhek thutur moto j alo chnader mukh theke berochchhe,
tate unchu unchu shikhor gulo besh bhije bhije lagchhe,
chokhe him-o pore thakte pare.
adorer time bomb fatbe ki fatbe na tar opekkhay oneke niche boshe agun shamlachchhe.
onek upor gulote hawa eshe gaye perek thukchhe
keu keu, eka eka balconytey nijer ekakityo key joy korchhe.
sohorta besh purono diner notun kotha bole
sohortar naam hote parto everest.