20

I think the comments may be (inadvertently) giving the impression that this is more complicated than it really is. So let's just say how everything fits together: Given a vector bundle, its sheaf of sections is locally free. Conversely, if we have a locally free sheaf then it's the sheaf of sections of a vector bundle which we can build by taking sheaf ...


18

All Serre needed to use was quasicoherent sheaves. For these it's a fundamental theorem that Cech cohomology is the "correct" cohomology. That means that it agrees with the cohomology defined abstractly, via derived functors. Since derived functors give rise to long exact sequences, the fundamental theorem implies that short exact sequences of (quasi)...


18

Serre didn't mention quasi-coherent sheaves nor derived functor cohomology (=Grothendieck Tohoku cohomology, introduced in 1957) when he finished writing FAC in October 1954, because nobody in the world knew about these at that date. He didn't know about abelian categories either for the same reason. He introduced coherent sheaves in algebraic geometry in ...


13

Simply put, the evolution of the foundational definitions in Algebraic Geometry was driven not by the search of new objects, but of a better understanding of classical objects. In fact, the crowning achievement of Grothendieck's school was precisely to rigorously state and prove a myriad of classical problems. The exemplary case are the Weil conjectures. ...


11

I spent a lot of time during my masters (and still now) trying to resolve such questions which frequently appear : "Should I apply the sheaf or its dual here? Which directions should my arrows go?" and the reason why it's so confusing is because this question has not been settled once and for all ; people keep arguing about it and there is no widely accepted ...


10

There's an exact sequence $$0\to \pi^*\Omega^1_X\to \Omega^1_{\widetilde{X}}\to i_*\Omega^1_{E/Y}\to 0$$ where $i:E\to \widetilde{X}$ is the inclusion of the exceptional divisor. This corresponds to the fact that pulling back differential forms gives you differential forms which are constant along the fibers. Checking that this is exact may be done from ...


8

Suppose that $M $ is generated by $ b_{1}, \ldots, b_{k} $ over $ A $, and $ M_{\mathfrak{p}} $ has a basis given by $ \beta_{1}, \ldots, \beta _{ n} \in M_{ \mathfrak{p}} $. Let $ \beta_{i} = m_{i} / s_{i} $ for $ m_{i} \in M $, $ s_{i} \in A \setminus \mathfrak{p} $. There exist $ a_{ij} \in A $, $ t_{ij} \in A \setminus \mathfrak{p} $ for $ 1 \leq i \...


7

It's been a while since this was posted, but I'd like to note that the reason it is hard to show is that it is false. If $I$ is the ideal sheaf of a codimension $1$ subvariety (say $X$ is projective), then the determinant is not trivial: $I$ is reflexive, and $\det I = I$. If $I$ is the ideal sheaf of a subvariety of codimension $\geq 2$, then the result ...


7

I actually think something like the quasi-compactness of $Z$ is necessary (or at least, I haven't been clever enough to figure out how to do it otherwise), but not really for the reasons why Justin asked the question. But the hypotheses on $f,g$ seem unnecessary… Stephen's answer is completely right if we assume the morphisms $f \colon X \to Y$ and $g \...


7

My favorite way to do this is to work with the conormal bundle instead, seeing very concretely what its transition functions are. Here is a bit to get you started. We have $$0\to N^*_{C/\Bbb P^3}\to T^*\Bbb P^3\big|_C \overset{\phi}\to T^*C \to 0.$$ In the "main" chart $[1,x,y,z]$, where $C$ is parametrized by $(t,t^2,t^3)$, we have \begin{align*}\phi(dx)&...


7

The Algebra of Coherent Algebraic Sheaves with an Explicated Translation of Serre's "Faisceaux Algébriques Cohérents" is now available at my personal website https://andymclennan.droppages.com/Books/ or more directly from https://andymclennan.droppages.com/fac_trans.pdf (thanks to Martin for pointing this out) Thanks for your interest!


6

Chapter II, §5.2 in: N. Bourbaki, Elements of mathematics. Commutative algebra. Hermann, Paris, 1972. Translated from the French. Alternatively, you can always consult the stacks project for these basics. In this case, it's Tag 00NV.


6

To simplify greatly, one is interested in some sort of a "parameter space" for vector bundles, where it makes sense to say one bundle is the "limit" of sequence of other bundles, or to have a "curve" of bundles, and so on. Here you can compare it to simplest parameter spaces, like the projective space or a Grassmanian (in the ...


6

You are correct that $\mathscr F_{\eta}$ is generated over $k(t)$ by $\frac{1}{1}$. However, this doesn't tell you the rank is $1$; it only tells you the rank is at most $1$, since the set $\{\frac{1}{1}\}$ may be linearly dependent in $\mathscr F_{\eta}$. This can only happen if $\frac{1}{1}=0$, but that is in fact exactly what happens. Indeed, remember ...


5

Because $Z$ and ${\bf Z}$ look too much alike, I'm going to work instead with projective morphisms $f:X \rightarrow Y$ and $g:Y \rightarrow W$. Also, the little I have to say about the difference between EGA and Hartshorne I will put at the end of this answer (briefly, Hartshorne is not wrong and I suspect the hypotheses in EGA are necessary); from now on I'...


5

Yes, you can construct such a comparison transformation in any context of adjoint functors: Suppose $$(\ast)\quad\quad\quad\begin{array}{ccc} {\mathscr C} & \xrightarrow{\alpha_{\ast}} & {\mathscr D} \\ {\scriptsize\beta_{\ast}}\downarrow & &\downarrow{\scriptsize\gamma_{\ast}} \\\ {\mathscr E} & \xrightarrow{\delta_{\ast}} & {\...


5

A holomorphic vector bundle $E$ is globally generated if there exist holomorphic sections $s_1, \dots, s_r$ such that for all $x \in X$, $s_1(x), \dots, s_r(x)$ span $E_x$. In terms of the sheaf-theoretic notion you came across, a holomorphic vector bundle $E$ is globally generated if $\mathcal{O}(E)$, its sheaf of holomorphic sections, is globally ...


5

This would simply be a morphism $\phi$ of sheaves of sets (which in general is almost certainly not $\mathscr{O}_X$-linear, nor even a morphism of sheaves of abelian groups) such that for any $U \subseteq X$ open, $\phi_U : \mathscr{F}(U) \times \mathscr{G}(U) \to \mathscr{H}(U)$ is $\mathscr{O}_X(U)$-bilinear, i.e. for any $x_1, x_2 \in \mathscr{F}(U)$, $...


5

The question is local, so you may replace $F$ by a module $M$ over a local ring $A$, and ask about $Hom(M,k)$. Note that if $\mathfrak{m} \subset A$ is the maximal ideal, then $M \ne \mathfrak{m}M$ by Nakayama Lemma, hence $M/\mathfrak{m}M$ is a non-zero vector space, so you can choose a non-zero linear function on it and define a morphism $$ M \to M/\...


5

There are two things to keep in mind. First, we always need to fix an ample line bundle to speak about stability. Secondly, Intersection Theory is made a way that you can work (at least if $X$ is smooth) on the Grothendieck group $K_0(X)$, so one might consider locally free resolution. Fix an ample line bundle $H$. To define the degree of any coherent sheaf ...


4

Let $$\mathcal{F}_\bullet=\dots\stackrel{d^\mathcal{F}_3}{\to}\mathcal{F}_2 \stackrel{d^\mathcal{F}_2}{\to}\mathcal{F}_1 \stackrel{d^\mathcal{F}_1}{\to}\mathcal{F}_0 \stackrel{d^\mathcal{F}_0}{\to}0\to\dots$$ be a bounded complex of coherent sheaves (assuming without loss of generality that $\mathcal{F}_i=0$ for $i<0$). Then you can construct a locally ...


4

This is indeed quite an interesting and subtle question! The point is that when $\mathcal{L}(-P)\subset \mathcal{L}$, every section $s$ of $\mathcal{L}(-P)$ is a section of $\mathcal{L}$ but its order of vanishing at $P$ seen in $\mathcal{L}(-P)$ is one less than its order of vanishing seen in $\mathcal{L}$. Proof: Since the problem is local, we may ...


4

Torsion means torsion over $\mathcal O_X$. For a coherent sheaf on a curve, it is equivalent to having support a finite number of closed points.


4

Look in Hartshorne, Ch. II, Section 7 (I think), especially the test for very ampleness in terms of separating points and tangent vectors. Applying this test should be pretty straightforward. If we write $\mathcal O(m,n)$ for your tensor product of pull-backs, then e.g. $\mathcal O(1,1)$ gives the embedding of the product into $\mathbb P^3$ as a quadric, ...


4

Hartshorne often assumes schemes to be noetherian so that his definition of "coherence" is correct. The general and correct definition of coherence is a bit complicated; what Hartshorne defines as coherence is actually called "locally of finite type", which agrees with "locally of finite presentation" over noetherian schemes. It turns out that, over ...


4

Why not read Serre's crystal-clear article which introduced this equivalence? As Gauss said: "Read the masters".


4

Let $S$ be a base scheme and $f : X \to X'$ and $g : Y \to Y'$ morphisms of $S$-schemes. If $F,G \in \mathsf{Qcoh}(X),\mathsf{Qcoh}(Y)$, we have the external tensor product $F \boxtimes G \in \mathsf{Qcoh}(X \times_S Y)$. We have a canonical homomorphism (induced by the usual adjunctions) $$f_* F \boxtimes g_* G \to (f \times g)_* (F \boxtimes G)$$ in $\...


4

There must be some more elegant arguments, but here is one. The hypercohomology of such complex should be zero. Since $\mathcal O(-1)$ have no cohomology, the usual spectral sequence implies that $\mathcal O^{\oplus b}=H^0(F)\otimes \mathcal O$ and $\mathcal O^{\oplus d}=H^1(F)\otimes \mathcal O$. Moreover, the map $$ H^0(F)\otimes \mathcal O\to F $$ has ...


4

To find the basis explicitly you can use the Poincare residue map $$ \text{res}: H^0(\mathbb{P}^2, \Omega^2_{\mathbb{P}^2}(C)) \to H^0(C, \Omega^1_C), $$ which in this case is an isomorphism. Let $g(x_1, x_2) = f(1,x_1,x_2)$, any 2-form $\omega$ on $\mathbb{P}^2$ with a single pole along $C$ can be written locally in coordinates $x_1$, $x_2$ as $$ \omega = ...


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