La nozione di morfismo étale

Il teorema della funzione implicita, valido nel caso di varietà differenziabili, non risulta vero se si prendono in analisi varietà algebriche affini con la topologia di Zariski. Dopo aver introdotto le nozioni di morfismo piatto e di morfismo non ramificato, si arriva ai morfismi étale, definiti proprio come quei morfismi che sono piatti e non ramificati; nella seconda parte si considerano i morfismi di varietà non singolari dimostrando che la classe dei morfismi étale coincide esattamente con quei morfismi che inducono isomorfismi sugli spazi tangenti. Si approfondisce poi la nozione di morfismo étale da un punto di vista algebrico e infine la nozione di intorno étale di un punto, che si basa su quella di morfismo étale.

When Beauty Goes to Sleep: an analysis of the symbolism behind the sleeping beauty tale

Approaching the world of the fairy tale as an adult, one soon realizes that things are not what they once seemed during story time in bed. Something that once appeared so innocent and simple can become rather complex when digging into its origin. A kiss, for example, can mean something else entirely. I can clearly remember my sister, who is ten years older than I am, telling me that the fairy tales I was told had a mysterious hidden meaning I could not understand. I was probably 9 or 10 when she told me that the story of Sleeping Beauty, which I used to love so much in Disney’s rendering, was nothing more than the story of an adolescent girl, with all the necessary steps needed to become a woman, the bleeding of menstruation and the sexual awakening - even though she did not really put it in these terms. This shocking news troubled me for a while, so much so that I haven’t watched that movie since. But in reality it was not fear that my sister had implanted in me: it was curiosity, the feeling that I was missing something terribly important behind the words and images. But it was not until last year during my semester abroad in Germany, where I had the chance to take a very interesting English literature seminar, that I fully understood what I had been looking for all these years. Thanks to what I learned from the work of Bruno Bettelheim, Jack Zipes, Vladimir Propp, and many other authors that wrote extensively about the subject, I feel I finally have the right tools to really get to know this fairy tale. But what I also know now is that the message behind fairy tales is not to be searched for behind only one version: on the contrary, since they come from oral traditions and their form was slowly shaped by centuries of recountals and retellings, the more one digs, the more complete the understanding of the tale will be. I will therefore look for Sleeping Beauty’s hidden meaning by looking for the reason why it did stick so consistently throughout time. To achieve this goal, I have organized my analysis in three chapters: in the first chapter, I will analyze the first known literary version of the tale, the French Perceforest, and then compare it with the following Italian version, Basile’s Sun, Moon, and Talia; in the second chapter, I will focus on the most famous and by now classical literary versions of Sleeping Beauty, La Belle Au Bois Dormant, written by the Frenchman, Perrault, and the German Dornröschen, recorded by the Brothers Grimm’s; finally, in the last chapter, I will analyze Almodovar’s film Talk to Her as a modern rewriting of this tale, which after a closer look, appears closely related to the earliest version of the story, Perceforest.