Schemes 1: Introduction

This lecture is part of an online course in algebraic geometry giving an introduction to schemes. It is loosely based on chapter II Hartshorne's book "Algebraic geometry". (For chapter 1 see the playlist "Algebraic geometry".)
This introductory lecture gives some motivation for schemes and gives the definition of presheaves and sheaves.

[@bobclarke5543]

Absolutely the clearest, most comprehensible introduction to the subject I've ever seen. Clearer, more business-like, and better motivated than Vakil's attempt (which, I must say, is also very good but has an entirely different character). I highly recommend this series to anyone who wants to learn algebraic geometry.

[@georgediderrich43]

What a pleasure to view these lectures. Professor Borcherds rocks!

[@Ibakecookiess]

Watching this from Colombia. Thank you!

[@ipudisciple]

18:25 I think we have two options. We can choose a dimension d and consider the sheaf of vector fields of dimension d. Or we can lump these all together and work with vectors in the direct sum ⨁ ℝ^n, which is just the countably dimensional vector space. Technically, Richard says the second, but I think he might be thinking of the first.

[@vinayakgupta2368]

Hey, i am from india. Thank you for you these free lectures.

[@juliavilageliu6797]

You really helped me through my final degree project :) Thank you so much

[@mathgeeks3598]

nice, video, i am in uSa. thanks for good videos

[@domingogomez6999]

Sorry, it might be trivial, I don't see why the "constant" presheaf is not a sheaf. I mean that, if the empty set, U_1, U_2 and their union U = U_1\union U_2 are the only open sets, why does F(U) = AxA hold?

[@ruinenlust_]

Constant pre-sheaf assigns A to every U. Clearly, if F were to be a sheaf then F(U) must be AxA. But, since F assigns A to every open set, F cannot be a sheaf in general

[@domingogomez6999]

@@ruinenlust_ the clearly part is not evident for me. Could you develop more, please?

[@harshavr]

@@domingogomez6999 If U is partitioned into U1 and U2 with no intersection, then the sheaf condition says that section on U1 and section on U2 uniquely determines a section on U ( the compatibility condition is trivial because U1 and U2 dont intersect), So F(U)=F(U1)xF(U2)=AxA. The key here is that space is disconnected, For a connected space, the constant sheaf is a sheaf. (U1 and U2 will intersect above and the constant function should restrict to the same constant on the intersection) For any space, the locally constant functions form a sheaf.

[@yiyangjia1967]

look at the injectivity of the exact sequence

[@user-sx2zr3rs4q]

Thank you so much for your online couse !

[@premkumar-so3ff]

Please make videos explaining on cohomology of sheaves unit 3 of hartshrone. Thank you

[@ac-dp3jk]

Thank you very much for your videos !

[@tapioms]

🎉

[@markcanedo3]

Thank you for this magistral lessons, professor

[@donatoebbasta2797]

Does anyone have a clear cut pdfLaTeX-compiled version of these lectures?

[@nostalgia_1439]

2:35 Isn't k[x] a k-algebra? I mean I think it's even a finite k-algebra? What is his definition of k-algebra? I know what an algebra \phi: R\to S is (e.g. in the sense of Altman Kleinman's commeutative algebra text, and "Algebra" by Hungerford)

[@okoyoso]

I believe his name is pronounced "Hart's horn" (hart being an old word for a stag, type of buck)

[@sdfdsf4162]

haha me too

[@okoyoso]

@@sdfdsf4162 You too as in you pronounce it that way too?

[@sdfdsf4162]

@@okoyoso yeeep. dude im an eastern europe dog, of course i suck speaking eng =)

[@okoyoso]

@@sdfdsf4162 I mean a lot of people call him "hart shorne"

[@ANOLAMCA122]

I dont understand how people love this series, I for myself, find it very confusing and not clear

[@guiwenluo1774]

Second. I found it not newcomer friendly. Probably those who love this series are already learnt Algebraic geometry for at least one time.

[@mbgdemon]

@@guiwenluo1774 That is why the series is called Algebraic Geometry II.

[@bennoarchimboldi6245]

People love him because he’s a fields medalist not because he’s a good teacher.