Tag Info

11

I have three suggestions: Mac Lane, S., and Moerdijk, I., "Sheaves in Geometry and Logic: A First Introduction to Topos Theory" Kashiwara, M., and Schapira, P., "Categories and Sheaves" The first is my favorite. The latter is more advanced, and doesn't really start talking about sheaves until late in the book. It's a quality text nonetheless. Finally, ...

8

I suppose what you are looking for is an introduction to classical topos theory. A topos is a category which can be thought of as a category of sheaves. One thing to keep in mind when reading about topos is that there are three equivalent definitions. The first is a left exact localization of a presheaf category. The second constructs sheaves from a ...

8

I imagine one could write books on the subject, but probably the first moment one encounters homological algebra in algebraic geometry is the following. If $f: X\to Y$ is a proper map of varieties, then given a coherent sheaf $\mathcal{F}$ on $Y$ one can form the pull-back $f^{*} \mathcal{F}$. Now, the pull-back functor $f^{*}: \mathcal{C}oh(Y) \rightarrow ... 6 Neither is true. In the first your mistake is in this step: $$(\pi^{-1}F)(\pi^{-1}V) = \lim_{W \supset V}F(W)$$ It should be $$(\pi^{-1}F)(\pi^{-1}V) = \lim_{W \supset \pi\pi^{-1}V}F(W)$$ because in general we don't have$\pi(\pi^{-1}(V)) = V$. Take, for example,$\pi$to be the inclusion map of a point. For the second assume$X$and$Y$are affine ... 6 Take$X=\coprod_n \mathrm{Spec}(\mathbb{Z}/p^n)$for a prime$p$. Then$s:=(p)_n \in \prod_n \mathbb{Z}/p^n = \mathcal{O}_X(X)$is not nilpotent, but the restriction to each$\mathrm{Spec}(\mathbb{Z}/p^n)$is nilpotent. Hence, we have$[s] \neq [0]$in$\mathcal{O}_X(X)_{red}$, although we have$[s]=[0]$in each ... 5 Here is some intuition for choosing a linearization as far as I understand. I'm also just learning the subject so take this with a grain of salt but here is the picture I've gotten. Let us first consider a finite group$G$acting nicely on an affine variety$X = \operatorname{Spec}A$. Here we want a (presumably affine) quotient$X/G$to be a space whose ... 5 Instead of giving a vague a-priori reason or the usual proof, let me give a quite unknown direct proof, which even works in the larger category of locally ringed spaces: If$X \to S$and$Y \to S$are morphisms of locally ringed spaces, then you can construct their fiber product$X \times_S Y$explicitly, see here for a summary. The set consists of all ... 5 Regarding #2, here is a related question. The idea is rather simple: the existence of nontrivial automorphisms implies existence of nontrivial isotrivial families (i.e. with all fibres isomorphic/equivalent). To see why this messes up representability, suppose we are given a nontrivial isotrivial family$\mathcal X\to B$. There exists a unique map$B\to P$, ... 4 What have you tried with this problem? I will help you out with (1) but you must show effort with (2). Here's how (1) goes. For every$x \in \ker \varphi(U)$we can consider it as an element of$\mathcal{F}(U)$since$\ker \varphi(U)$is a subgroup of$\mathcal{F}(U)$. Since$\mathcal{F}$is a sheaf the fact that$x|_{V_i} = 0$implies immediately that$x = ...

4

$\newcommand{\spec}{\operatorname{Spec}}$I just want to expand on Pece's answer a bit to try to make it more explicit. Consider some affine open subset $U = \operatorname{Spec} B \subseteq X$ intersecting the fiber $X_y$ nontrivially and some affine open $V = \operatorname{Spec} R \subseteq Y$ containing $y$. By possibly shrinking $\operatorname{Spec} B$ we ...

4

I'm not sure exactly how Eisenbud intends to argue, because I don't know what he's assuming known at this point. But here is some kind of explanation, which perhaps you can adapt to what you know: The inclusion $(Q_1,Q_2) \subset I$ shows that $V(Q_1,Q_2) \supset V(I) = C$. Now $V(Q_1,Q_2)$ is a curve of degree $4$ (by Bezout), as is $C$ (by assumption) ...

3

It is not true that a hyperplane corresponds to the invertible sheaf $\mathcal O(1)$. What is true, is that the class of a hyperplane corresponds to the invertible sheaf $\mathcal O(1)$. Here's (a sketch of) the correspondence. The sheaf $\mathcal O(1)$ is generated by global sections, and the sections correspond to hyperplanes in $X$. Recall that the sheaf ...

3

Yes. In fact, even when we are defining a hypersurface of degree $r$ in the naive way, we are saying that it is the zero set of a homogeneous polynomial $f$ of degree $r$. But homogeneous polynomials of degree $r$ are not functions on projective space. They are sections of the line bundle $\mathcal{O}(r)$. Thus, even under the usual naive definition, a ...

3

Passing to underlying reduced subschemes is a functor on schemes, which doesn't change the underlying topological spaces. Both the properties of closedness and of being closed under specialization are purely topological properties, so there is no problem with passing to underlying reduced subschemes.

3

Your question Given a variety $V$ over a field $k$ and an extension field $k\to K$, the variety $V$ is smooth over $k$ if and only of the variety $V_K$ is smooth over $K$. Vastly more general result Smoothness is compatible with base change in all generality: if an arbitrary morphism of schemes $X\to S$ is smooth, it will remain smooth after an ...

3

What does "canonical" mean? It's not actually a technical term. If you mean something like I mean in this paper, then under the hypotheses given there, the answer is yes just because the main theorem implies that the stalk maps are in fact induced by the global map. Note that your hypotheses on the sheaves are lacking: the induced maps on fibers being ...

3

It's looking like the answer is "yes." That said, in case it helps anyone who hits this via Google, the relevant definitions are actually written down in: EGA O, 3.5.1 - 3.5.2, 4.1.1 (composition of morphisms of ringed spaces) EGA O, 4.1.2 (the inclusion morphism and restriction of morphisms)

3

This question is addressed in Appendix I of Topological Methods in Algebraic Geometry. Let $V_n^{d_1, \ldots, d_r}$ denote the complete intersection of $r$ generic hypersurfaces of degrees $d_1, \ldots d_r$ in $\mathbb{P}^{n + r}$. Let $$\chi_y(V_n^{d_1,\ldots, d_r}) = \sum_{p,q \geq 0} (-1)^qh^{p,q}(V_n^{d_1,\ldots, d_r})y^p = \sum_{p \geq 0} ... 3 This is indeed true. As Keenan's answer stated, the geometric version of the notion you're talking about is geometric irreducibility and reducedness. Consider the scheme X cut out by I. Then X is irreducible if and only if \sqrt{I} is prime. Then X is geometrically irreducible if it stays irreducible after any base extension. Similarly, X is ... 3 Don't work with elements, work with morphisms. This general lesson of category theory is also very useful for algebraic geometry. Let g := p_1 h = p_2 h : T \to X. Then h is equal to T \xrightarrow{g} X \xrightarrow{\Delta} X \times_Y X, since p_i \Delta g = g = p_i h. Hence, h factors through \Delta. By the way, you are right with your guess ... 3 This works for arbitrary morphisms of ringed spaces (or in fact ringed topoi). A global section of F is a morphism \mathcal{O}_Y \to F. Since pullbacks are functorial, we get a morphism f^* \mathcal{O}_Y \to f^* F. Since f^* \mathcal{O}_Y = \mathcal{O}_X, this is a global section of f^* F. 3 First of all you must suppose p_1\neq0, else the projection clearly does not make sense (you would always project onto p !). A parametric description of your line is L_x(t)=p+t(x-p). It will hit the hyperplane H=V(x_1) for t_0=-\frac{p_1}{x_1-p_1} and the hitting point is thus the required projection \pi(x)=L_x(t_0)=p-\frac{p_1}{x_1-p_1}(x-p). ... 2 I think the tangent bundle you should consider is spanned by the Lie algebra generated by L_i, not just L_{ij}. In particular, if the Lie algebra is commutative then L_{ij}=0, which is not what you are looking for. My favourite reference on these things is "Mathematical Methods of Classical Mechanics" by V.I. Arnold. In the chapter on Hamiltonian ... 2 It is not totally clear to me what your question is, but let me show the following statement. Let S = K[x_0,\dots, x_n] and S_+ = (x_0,\dots, x_n). Let F_1,\dots, F_m be forms of degree d and m \le n. If (F_1,\dots, F_n) \neq S, then S_+ \nsubseteq \sqrt{(F_1,\dots, F_m)}. In particular, S_+^l \nsubseteq (F_1,\dots, F_n) for all l. ... 2 I don't think this is true. In fact if we denote a rigidified line bundle as (L,\sigma) where we make explicit the isomorphism \sigma:\varepsilon^*L \to \mathcal{O}_S, then the set of rigidified line bundles forms a group under$$ (L,\sigma_L) \otimes (M, \sigma_M) = (L\otimes M, \sigma_L \otimes \sigma_M).  We can verify $\sigma_L \otimes \sigma_M$ ...

2

A morphism of schemes is not determined by the underlying map on topological spaces! There is also a map of sheaves to consider. For example, take $C = X=\mathbb{C}$, and let $\sigma: \mathbb{C}\to\mathbb{C}$ be complex conjugation. Then $\sigma$ gives rise to a morphism of schemes $\operatorname{Spec}\mathbb{C}\to\operatorname{Spec}\mathbb{C}$ which is ...

2

For the first question: Since $p$ maps to the generic point of $X$, we have $p\cap A = 0$. Thus, in the ring $A[t]_p$, all nonzero elements of $A$ are inverted. So if $S$ denotes the multiplicatively closed set $A[t]-p$, the natural inclusion $A[t]_p \rightarrow S^{-1}(K[t])$ is an isomorphism. Since $K[t]$ is a UFD, it follows that $A[t]_p$ is a UFD (cf. ...

2

An indirect answer is that Hartshorne's sheaf constructions are often a little lacking in the elegance department. This is one example; the definition of the structure sheaf is another. If you are looking for a better way to understand the reduced scheme associated with an arbitrary scheme, think of it this way: it's a closed subscheme and thus given by a ...

2

The second formula you quote is true in the following case: $Z \overset{\pi}\to X$ is a closed immersion with $Z\,$ equal to the support of a coherent sheaf $\mathcal{F}$ on $X$. Here is a proof of this fact. We may assume that $X$ is affine (equal to $\operatorname{Spec} A$) with $\mathcal{F} = \widetilde{M}$ for some finite type $A$-module $M$. ...

2

Let $X=\{a,b\}\subset \mathbb P^n\; (n\geq 1)$ be a two-point subset of $n$-dimensional projective space and $\mathcal F=\mathcal I_X$ the ideal sheaf of functions vanishing at $a$ and $b$. This means that for an open subset $U\subset \mathbb P^n$ a section $s\in F(U)=\mathcal I_X(U)\subset \mathcal O(U)$ is a section in $\mathcal O(U)$ vanishing on \$U\cap ...

Only top voted, non community-wiki answers of a minimum length are eligible