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I'm attending my first course in algebraic geometry, and my professor has chosen an approach which is a middle-way between the basic algebraic geometry done in $\mathbb A^n_k$ and the approach with schemes, so substantially like in Milne's notes. I have for you some simple question and I hope that the answers will not involve scheme theory:

1) Let $(X,\mathcal O_X)$ an affine variety, so it is isomorphic as ringed space to $(V,\mathcal O_V)$ where $V$ is an affine algebraic set and $\mathcal O_V$ is the sheaf of regular functions. When one says "take the covering of $X$ with standard open sets", it means that we consider the covering of $X$ done with those open sets of $X$ which are homeomorphic to the standard open sets $D(f)\subseteq V$?

2) In class we have defined a quasi-coherent sheaf on $(V,\mathcal O_V)$ (we are in $\mathbb A^n_k$) as the sheaf $\widetilde M $ uniquely associated to the assignment $\widetilde M(D(f))=M_f$ where $M$ is a $\Gamma(V,\mathcal O_V)$-module. When we talk about a quasi-coherent sheaf defined on a abstract affine variety $(X,\mathcal O_X)$ so isomorphic to $(V,\mathcal O_V)$, do we intend a sheaf $\mathcal F$ on $X$ with an isomorphism $\mathcal F\cong f_{\ast}\widetilde M$ (assuming that $f$ is the omeomorphism between $V$ and $X$)?

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(1) No. It is not just homeomorphic but identified with the corresponding standard open set in $V$ by a fixed isomorphism $X \cong V$. (2) Yes. You can check that this notion is independent of the choice of $V$ and of the choice of isomorphism $X \cong V$. –  Zhen Lin Jun 15 '12 at 10:59
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up vote 7 down vote accepted
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1) Standard open sets are defined for every locally ringed space. If $f \in \Gamma(X,\mathcal{O}_X)$, then $X_f$ (sometimes also called $D(f)$) is by definition the set of all $x \in X$ such that $f_x \notin \mathfrak{m}_x$, where $\mathfrak{m}_x$ is the maximal ideal of the local ring $\mathcal{O}_{X,x}$. Equivalently, $f(x) \neq 0$ in the residue field $k(x) = \mathcal{O}_{X,x}/\mathfrak{m}_x$. This is also the reason why often $X_f$ is called the "locus where $f$ doesn't vanish" or "where $f$ is invertible". It is an easy exercise to show that $X_f$ is in fact open, and that we have the standard identities $X_1 = X$, $X_f \cap X_g = X_{fg}$. When $X$ is an affine algebraic set, this coincides with the locus where $f$ doesn't vanish defined in the usual sense (and probably this is what you meant by $D(f) \subseteq V$).

2) Again quasi-coherent sheaves make sense for arbitrary ringed spaces. And it is a very bad idea to give definitions only for algebraic sets $\subseteq \mathbb{A}^n$ and try to extend them via chosen isomorphisms! You should work with intrinsic geometric objects instead, and (locally) ringed spaces provide a nice framework for that. So let's use this language.

A quasi-coherent module on a ringed space $X$ is just a module $M$ (i.e. what most people call a sheaf of modules) on $X$ such that locally on $X$ there is a presentation $\mathcal{O}^{\oplus I} \to \mathcal{O}^{\oplus J} \to M \to 0$. So to be more precise: There is an open covering $X = \cup_i X_i$ such that for each $i$ there is an exact sequence (which, of course, does not belong to the data) $\mathcal{O}|_{X_i}^{\oplus I} \to \mathcal{O}|_{X_i}^{\oplus J} \to M|_{X_i} \to 0$. Quasi-coherent modules constitute a (tensor) category $\mathrm{Qcoh}(X)$, which is by the way a very interesting and deep invariant of $X$, especially when $X$ is a variety.

How to construct quasi-coherent modules on a ringed space $X$? Well pick a $\Gamma(X,\mathcal{O}_X)$-module $M$. Then I claim that we can construct a quasi-coherent module $\tilde{M}$ on $X$ as follows: Choose a presentation $\Gamma(X,\mathcal{O}_X)^{\oplus I} \to \Gamma(X,\mathcal{O}_X)^{\oplus J} \to M \to 0.$ Represent the morphism on the left as a "relation matrix" consisting of elements of $\Gamma(X,\mathcal{O}_X)$. Now, every such global section corresponds to a homomorphism $\mathcal{O}_X \to \mathcal{O}_X$. Thus we can produce a matrix consisting of endomorphisms over $\mathcal{O}_X$, and thus a morphism $\mathcal{O}_X^{\oplus I} \to \mathcal{O}_X^{\oplus J}$. Define $\tilde{M}$ to be the cokernel. By definition, this is quasi-coherent! This already produces lots of examples; in fact all $X$ is an affine variety, but only few if $X$ is projective.

To give a more concise definition which does not depend on the presentation: Just define $\tilde{M}$ to be the sheaf associated to the presheaf $U \mapsto \Gamma(U,\mathcal{O}_X) \otimes_{\Gamma(X,\mathcal{O}_X)} M$. This definition easily implies a more conceptual characterization of the functor $M \to \tilde{M}$ from $\Gamma(X,\mathcal{O}_X)$-modules to quasi-coherent modules modules on $X$: It is left adjoint to the global section functor! In fact, everything you want to know about $\tilde{M}$ already follows from this adjunction. You may forget about the details of the construction, you just have to remember $\hom(\tilde{M},F) \cong \hom(M,\Gamma(X,F))$, which actually holds for every module $F$ on $X$.

So what happens when $X$ is some affine variety? Then the sets $X_f$ constitute a basis for the topology of $X$, and we have $\Gamma(X_f,\mathcal{O}_X) = \Gamma(X,\mathcal{O}_X)_f$. Namely, this is well-known if $X \subseteq \mathbb{A}^n$ and then generalizes immediately to affine varieties, which are isomorphic as ringed spaces to such concrete varieties. Let $M$ be a $\Gamma(X,\mathcal{O}_X)$-module. Now it turns out that the presheaf defined above is actually a sheaf! This comes down to the following: If $f_1,\dotsc,f_n \in \Gamma(X,\mathcal{O}_X)$ generate the unit ideal (i.e. the corresponding sets $X_{f_i}$ cover $X$), then the canonical sequence

$$0 \to M \to \prod_{i} M_{f_i} \to \prod_{i,j} M_{f_i f_j}$$

is exact. Everyone should have done this proof instead of looking it up in the standard sources. Because I think it is quite enlightening and in fact purely geometric if you think of $f_1,\dotsc,f_n$ as a partition of unity.

Anyway, so this tells us that we don't need associated sheaves in the definition of $\tilde{M}$. Thus, by definition, on the open subset $X_f$ it is given by

$$\Gamma(X_f,\tilde{M}) = \Gamma(X_f,\mathcal{O}_X) \otimes_{\Gamma(X,\mathcal{O}_X)} M = \Gamma(X,\mathcal{O}_X)_f \otimes_{\Gamma(X,\mathcal{O}_X)} M = M_f.$$

So this describes some quasi-coherent sheaves on affine varieties. In fact, one can show that every quasi-coherent sheaf on an affine variety $X$ has the form $\tilde{M}$. Namely, one shows that for every such sheaf $F$ the canonical counit morphism of the adjunction mentioned above $\tilde{\Gamma(X,F)} \to F$ is an isomorphism. Again, this is a very nice exercise. After some thought you will see that this is just another application of the exact sequence above. So this provides, for every affine variety, an equivalence of categories

$$\mathrm{Qcoh}(X) \cong \mathrm{Mod}(\Gamma(X,\mathcal{O}_X)).$$

By the way, if you define $\tilde{M}$ on an affine variety by $\Gamma(X_f,\tilde{M}) = M_f$ and extended via projective limits to arbitrary open subsets, then you probably would like to know that this is a sheaf. And again this comes down to the exact sequence above. You cannot get around it. I don't like this approach because it is somewhat clumsy, you don't get the general picture, and it doesn't produce a formula for $\tilde{M}(U)$ for arbitrary $U$. Therefore I've chosen the rather abstract but hopefully concise approach above. Of course nothing is new, you can find all that in EGA I, the Stacks Project, etc.

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can you give me a hint to prove that $X_f$ is open in $X$? –  Galoisfan Jun 26 '12 at 14:14
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Hint: $x \in X_f$ iff $f_x$ has an inverse in the stalk $\mathcal{O}_{X,x}$. –  Martin Brandenburg Jun 26 '12 at 15:08
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