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Topologically the scheme $\rm{Spec}(\mathcal{O}_{X,x})$ is exactly the intersection of all neighbourhoods of $x$ and algebraically it contains every infinitesimal neighbourhood of $X$. Although techni …

Dear Jérôme, I doubt that Grothendieck ever said that. However, in an analogous vein, Jean Leray, a brilliant French mathematician, was taken prisoner by the Germans in 1940 and sent to Oflag XVIIA …

Dear Brian, it seems that algebraic geometers who adopt your favoured approach are essentially specialists in algebraic groups. My favourite example would be Jantzen's Representations of Algebraic g …

You should look at Atiyah's aricle "On the Krull-Schmidt theorem with applications to sheaves", freely available here. He gives fairly general conditions on an exact category for the Krull-Remak-Sc …

Dear Shenghao, I) I am happy to report that according to Ueno's Classification of Algebraic Varieties and Compact Complex Spaces, Springer LNM 439, 1975, Moishezon's papers have been translated into …

Dear Jankir, 1) The book by Sergey V. Matveev Lectures on Algebraic Topology, Sergey V. Matveev (EMS Series of Lectures in Mathematics, 2006) contains about 10 pages of hints and solutions to i …

Is every algebraic curve in $\mathbb P^3$ the set-theoretic intersection of two algebraic surfaces ? Not known!

Dear Anton, my ex-colleague Michel Granger wrote his thesis on the punctual Hilbert scheme you are asking about. It is rather technical and I haven't read it. It has been published since as a "Mémoire …

If $X$ is a noetherian separated scheme and $X_{red}$ its reduction , we have $K_0(X)=K_o(X_{red})$: in other words $K_o$ doesn't see nilpotents . Much more generally and profoundly, Quillen has pro …

Dear Klim, when you say "the" book, i suppose you mean Intersection Theory published by Springer . However Fulton has written a much more elementary overview called Introduction to Intersection Theory …

Here are three excellent books. Tensor Spaces and Exterior Algebra by Takeo Yokonuma. Translations of Mathematical Monographs, volume 108, AMS 1992 You can browse it in Google books here Laurent Sc …

Actually a more general result is true: If $X$ is a complex space (maybe not smooth) and if $H^1(X,\mathcal I)$ is zero for all coherent sheaves of ideals $\mathcal I\subset \mathcal O_X$, then $X$ is …

1) There is a great book From Holomorphic Functions to Complex Manifolds by Fritzsche-Grauert. It is very geometric and gives you the fundamentals on complex manifolds, including specialized topics, f …

I'll complement the list of well known books on the subject by some freely available documents, which I find user-friendly. Here are great lecture notes , from a course that de Jong (of Stacks Proj …

Dear Tim, on page 31 they consider a ring $A$ and two $A$- algebras defined by their structural ring morphisms $f:A\to B$ and $g:A\to C$. They then define the tensor product as a ring $D=B\otimes _A C …