Introduction to the Theory of Schemes (Moscow Lectures Book 1) by Yuri I. Manin (PDF)

Description

This English edition of Yuri I. Manin’s well-received lecture notes provides a concise but extremely lucid exposition of the basics of algebraic geometry and sheaf theory. The lectures were originally held in Moscow in the late 1960s, and the corresponding preprints were widely circulated among Russian mathematicians. This book will be of interest to students majoring in algebraic geometry and theoretical physics (high energy physics, solid body, astrophysics) as well as to researchers and scholars in these areas.”This is an excellent introduction to the basics of Grothendieck’s theory of schemes; the very best first reading about the subject that I am aware of. I would heartily recommend every grad student who wants to study algebraic geometry to read it prior to reading more advanced textbooks.”- Alexander Beilinson

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Theorem 1.4.2 (pg 16-17) has a small typo. Also, in my opinion the proof could have been bit more clearer in it’s presentation. [This should not be construed as a criticism of Prof. Manin but my humble observation due to my own inadequacies]. For Example: Choose any x in V(E) then for any g in E we have g(x) = 0. Thus, for any g in the ideal generated by (E) we have g(x) = 0 for any x in V(E). So it makes sense to consider the quotient space A/(E) and consider any h in A/(E) i.e. h = f (mod (E)). Now let h(x) = 0 for any x in V(E) or f (mod (E))(x) = 0 or by definition f (mod(E)) in p_x where p_x is in spec (A/(E)) for every x in V(E). This implies f (mod(E)) is in the intersection of p_x in spec (A/(E)) or h in n(A/(E)) by Theorem 1.3.2 [this was mis-written as Theorem 1.3.1] or there exists an n in N such h^n = f^n mod (E) = 0 or f^n is in rad((E)). Q.E.D.Corollary 1.4.2a (pg 17) has a similar issue. It refers to Theorem 1.3.1 which is a typo, it is most likely Theorem 1.3.2 but it would have been more appropriate to refer to Theorem 1.4.2 because the correspondence I —–> V(I) which is a closed subset of spec (A) and thus for any x in V(I) we have f(x) = 0 for every f in I then by Theorem 1.4.2 every f belongs rad((I)) = rad(I) and thus establishing 1:1 correspondence between closed subsets of spec (A) and radical ideals of A.

⭐Exposé de la théorie des schémas qui est plus « pratique « que la présentation habituelle à la Grothendieck

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