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Mnemonic: $\quad M=IM \Rightarrow m=im$ The version of Nakayama described: If $I$ is an arbitrary ideal of an arbitrary commutative ring $A$ and if a finitely generated module $M$ satisfies $M=IM$, …

Dear Victoria: here is a summary of the main comparison results I know of between Grothendieck cohomology (which is usually just called cohomology and written $\newcommand{\F}{\mathcal F}H^i(X,\F)$ …

Wrong! Here is Bourbaki document on algebraic geometry, taken from the now available Master's Archives: click on Autres rédactions, then on Chap.I Théorie globale élémentaire (91 p.) This prelimina …

If $X$ is a nonsingular algebraic (or analytic) variety over $\mathbb C$ or $\mathbb R$ then it is certainly $C^\infty$ over the reals. The converse is false for a silly reason : in the real or comp …

a) Conceptually an algebraic topologist should be interested in étale cohomology, because it answers a very naïve question: given an algebraic variety over $\mathbb C$, how do I calculate algebraicall …

I have been intrigued for a long time by the formal similarity of results from different areas of mathematics. Here are some examples. Set theory Given a map $f:X\to Y$ and subsets $X' \subset X, Y'\ …

If $I,J \subset A$ are comaximal ideals in a commutative ring $A$, i.e. $I+J=A$, then for all $n,m \in \mathbb N$ the ideals $I^n$ and $J^m$ are also comaximal. Proof: $\emptyset= V(A)=V(I+J)=V(I)\ …

Certainly Bourbaki never wrote an introduction to algebraic geometry: we would have heard about it, right?

Let $M$ be a differentiable manifold, $A=C^\infty (M)$ its ring of global differentiable functions and $\Omega^1 (M)$ the A-module of global differential forms of class $C^\infty$. The A-module of …

Instead of trying to say what flatness in analytic geometry means I'll give you some street-fighting tricks for recognizing whether a morphism of analytic spaces ( not necessarily reduced) $f:X\to Y $ …

Dear Ravi, maybe the simplest example is one by Serre: the holomorphic Stein surface $\mathbb C^\ast\times \mathbb C^\ast $ underlies two non-isomorphic smooth complex algebraic varieties. 1) $\math …

Since in 2007-2008 you evoked [ Class 24, §1.8, The problem with locally free sheaves] the equivalence between locally free sheaves and vector bundles on a scheme, the following point, potentially co …

This is false. Consider a characteristic zero field $k$ and the cusp $C\subset \mathbb A^2_k$ with equation $y^2=x^3$ . Its normalization $n: \mathbb A^1_k \to C: t\mapsto (t^2, t^3)$ is biject …

Thinking of arbitrary tensor products of rings, $A=\otimes_i A_i$ ($i\in I$, an arbitrary index set), I have recently realized that $Spec(A)$ should be the product of the schemes $Spec(A_i)$, a prior …

Every (?) algebraic geometer knows that concepts like homotopy groups or singular homology groups are irrelevant for schemes in their Zariski topology. Yet, I am curious about the following. Let's st …