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Let $A$ be a noetherian ring and $X=\operatorname {Spec}A$ the corresponding affine scheme. There are three rings which might reasonably be called the ring of rational functions on $X$. a) The total r …

The ring $\mathcal K(X)$ is called the ring of meromorphic functions on $X$ in EGA, the Stacks project or by Kleiman and many others. The terminology is disastrous if one considers schemes over $\math …

Consider a family of flabby (= flasque) sheaves $(\mathcal F_i)_{i\in I}$ of abelian groups on the topological space $X$. My question : is their direct sum sheaf $\mathcal F=\oplus _{i\in I} \mathcal …

The answer is Yes for complex manifolds of dimension one. Indeed for any open subset $U\subset X$ the long exact cohomology sequence associated to the exponential sequence you mention yields the fr …

Grothendieck and Dieudonné prove in $EGA_{II}$ (Proposition 5.5.5.(ii), page 105) that if $f:X\to Y, g:Y\to Z$ are projective morphisms of schemes and if $Z$ is separated and quasi-compact, or if t …

Let $R$ be a domain and $\tilde R$ its integral closure in its fraction field: $R\subset \tilde R\subset Frac(R)$. Is it true that a prime ideal $ \tilde {\mathfrak p} \subset \tilde R$ and its trac …

A very explicit example is given by the cusp $X=\operatorname {Spec}(A)$ where $A=\frac {\mathbb C[\xi,\eta]}{(3\eta^2-2\xi^3)}=\mathbb C[x,y]$. Since the set of closed points $X(\mathbb C)$ in its cl …

In 1963 Grothendieck introduced the algebraic de Rham cohomolog in a letter to Atiyah, later published in the Publications Mathématiques de l'IHES, N°29. If $X$ is an algebraic scheme over $\mathbb C …

Given a projective variety $X$, each of its embeddings $i:X\hookrightarrow \mathbb P^N$ gives rise to an integer valued Hilbert polynomial $P_{X,i}(t)\in \mathbb Q[t]$. These polynomials depend howev …

Here are some examples illustrating the genuine necessity of noetherian assumptions: 1) Every scheme with just one point is the spectrum of a local artinian ring? This is true for every noetheria …

1) It is sometimes stated that a finitely generated module $M$ over a Noetherian commutative ring $R$ is projective if for all maximal ideals $\mathfrak m\subset R$ the localized module $M_\mathfrak m …

Given an affine scheme $S=\operatorname {Spec (R)} $ and a closed subscheme $T=V(J)\subset S$, the restriction mapping $$\mathcal O(S)=R\to \mathcal O(T)=R/J$$ is obviously surjective. Applying this t …

Claim: The complement $U=\mathbb C^h\setminus \{F=0\}$ is path-connected and thus connected. Proof: Given $a,b\in U$ consider the affine complex line $L_{a,b}=L$ joining $a$ to $b$. The polynomial $ …

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 …

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