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Chapter 3: Problem 4

Let \(X\) be a noetherian scheme, and suppose that every coherent sheaf on \(X\) is a quotient of a locally free sheaf. In this case we say \(\mathfrak{C}\) ob \((X)\) has enough locally frees. Then for any \(\mathscr{G} \in \mathbb{V}\) iod \((X),\) show that the \(\delta\) -functor \(\left(\mathscr{E} x t^{i}(\cdot, \mathscr{G})\right),\) from \(\operatorname{Cob}(X)\) to \(\operatorname{Mod}(X)\), is a contravariant universal \(\delta\) -functor. [Hint: Show \(\delta x t^{i}(\cdot, \mathscr{L})\) is coeffaceable (\$1) for \(i>0 .]\)

Short Answer

Expert verified
The \(\mathscr{E}xt^{i}(\cdot, \mathscr{G})\) functor is indeed a contravariant universal \(\delta\)-functor. This is shown by first positing the functor as a \(\delta\)-functor, then proving its contravariance, and finally demonstrating its universality by considering its effaceability on locally free sheaves.

Step by step solution

01

Posit the \(\delta\)-functor

We first posit that \(\mathscr{E}xt^{i}(\cdot, \mathscr{G})\) is a \(\delta\)-functor. This follows by definition, as the Ext functors are generally known to be \(\delta\)-functors.
02

Prove contravariance of the \(\delta\)-functor

Next we need to prove that \(\mathscr{E}xt^{i}(\cdot, \mathscr{G})\) is contravariant in its first argument. This involves recalling the definition of a contravariant functor, which reverses the direction of morphisms. Considering a morphism \(f: \mathscr{F}_1 \rightarrow \mathscr{F}_2\) in \(\mathfrak{C}\) ob\((X)\), the contravariance of the functor gives us a morphism \(f^*: \mathscr{E}xt^{i}(\mathscr{F}_2, \mathscr{G}) \rightarrow \mathscr{E}xt^{i}(\mathscr{F}_1, \mathscr{G})\).
03

Show the \(\delta\)-functor is universal

To show universality, we have to prove the effaceability condition for the \(\delta\)-functor. This means that for any sheaf \(\mathscr{F}\) in \(\mathfrak{C}\) ob\((X)\) and all \(i > 0\), there exists a subobject \(\mathscr{F}'\) of \(\mathscr{F}\) such that \(\mathscr{E}xt^{i}(\mathscr{F}', \mathscr{G}) = 0\). From the hint provided in the exercise, we will focus on the case where \(\mathscr{F} = \mathscr{L}\) is locally free. We know that every coherent sheaf is a quotient of a locally free sheaf by assumptions, and local freeness of \(\mathscr{L}\) guarantees effaceability and therefore universality.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coherent Sheaves
Coherent sheaves play an essential role in algebraic geometry, serving as a bridge between algebra and geometry. These sheaves can be thought of as a generalization of vector bundles and have powerful applications in both theory and practical problem-solving. Coherent sheaves are defined over a ringed space and satisfy certain finiteness conditions, ensuring a kind of "reasonable" behavior.

They are particularly important when dealing with Noetherian schemes, as they allow the use of algebraic techniques within geometric contexts. Coherent sheaves:
  • Contain finitely generated modules over the structure sheaf of the scheme
  • Preserve the property of finite presentation when considering them over open subschemes
  • Are useful in constructing various sheaf cohomology theories
For students, understanding coherent sheaves is vital as they provide a framework for results and methods in both local and global settings. They enable the application of powerful mathematical tools connecting sheaf theory and algebraic geometry.
Locally Free Sheaves
Locally free sheaves are a special class of sheaves that resemble vector bundles in many ways. On a Noetherian scheme, these sheaves are characterized by their capacity to be described as free modules locally. This allows for easier manipulation and understanding of otherwise complex sheaf structures. Locally free sheaves:

  • Are defined as sheaves that can locally be expressed as direct sums of copies of the structure sheaf
  • Allow for the use of linear algebra tools because locally, they resemble free modules
  • Are instrumental in proving properties about other sheaves through the use of exact sequences
In particular, the exercise highlights locally free sheaves' utility in proving universality of the ext{Ext} functor, given their effaceability. This property makes locally free sheaves incredibly valuable in constructing surjective morphisms within sheaf theory.
Ext Functor
The ext{Ext} functor is a powerful tool in homological algebra, used to measure the extent to which a module fails to be projective. In the context of sheaves, it helps to explore the relationships between different sheaf structures. The ext{Ext} functor:

  • Takes two modules (or sheaves) and returns a higher-dimensional module (or sheaf), enabling us to capture extension information
  • Plays a crucial role in defining ext{delta}-functors, which are sequences of functors sharing a long exact sequence property
  • Is key in defining cohomology groups and understanding the deeper structure of sheaves
For Noetherian schemes and coherent sheaves, the ext{Ext} functor can go further by providing insight into local and global properties. Understanding it involves exploring long exact sequences and homological dimensions, providing a deep look into structural aspects of sheaves.
Contravariant Functors
Contravariant functors have a distinct feature: they reverse the direction of morphisms in a category, providing a contrasting perspective to that of covariant functors. This characteristic makes contravariant functors crucial for many constructions in algebraic geometry and related fields.

With a contravariant functor:
  • A morphism from an object A to B in the original category gets transformed into a morphism from B to A upon applying the functor
  • This reversal facilitates the study of duality theories and important transformations in mathematical spaces
  • For Noetherian schemes, contravariant functors are particularly valuable in promoting deeper understanding of sheaves and their interactions
By converting information and morphism directions, students get to appreciate fundamental dualities and how these are employed to solve complex problems. The exercise's requirement to show contravariance of the Ext functor showcases this concept's utility in understanding mathematical structures effectively.

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Most popular questions from this chapter

Arithmetic Genus. Let \(X\) be a projective scheme of dimension \(r\) over a field \(k .\) We define the arithmetic genus \(p_{a}\) of \(X\) by $$p_{a}(X)=(-1)^{r}\left(\chi\left(C_{x}\right)-1\right)$$ Note that it depends only on \(X,\) not on any projective embedding. (a) If \(X\) is integral, and \(k\) algebraically closed, show that \(H^{0}\left(X, \mathscr{O}_{X}\right) \cong k,\) so that $$p_{a}(X)=\sum_{i=0}^{r-1}(-1)^{i} \operatorname{dim}_{k} H^{r-i}\left(X, \mathcal{O}_{X}\right)$$ In particular, if \(X\) is a curve, we have $$p_{a}(X)=\operatorname{dim}_{k} H^{1}\left(X, C_{X}\right)$$ (b) If \(X\) is a closed subvariety of \(\mathbf{P}_{k}^{r},\) show that this \(p_{a}(X)\) coincides with the one defined in (I, Ex. 7.2), which apparently depended on the projective embedding. (c) If \(X\) is a nonsingular projective curve over an algebraically closed field \(k\), show that \(p_{a}(X)\) is in fact a birational invariant. Conclude that a nonsingular plane curve of degree \(d \geqslant 3\) is not rational. (This gives another proof of \((\mathrm{II}, 8.20 .3)\) where we used the geometric genus.

Let \(A\) be a regular local ring, and let \(M\) be a finitely generated \(A\) -module. In this case, strengthen the result (6.10A) as follows. (a) \(M\) is projective if and only if \(\operatorname{Ext}^{i}(M, A)=0\) for all \(i>0 .[\text { Hint: Use }(6.11 \mathrm{A})\) and descending induction on \(i\) to show that \(\operatorname{Ext}^{i}(M, N)=0\) for all \(i>0\) and all finitely generated \(A\) -modules \(N .\) Then show \(M\) is a direct summand of a \(\text { free }A \text { -module (Matsumura }[2, \mathrm{p} .129]) .]\) (b) Use (a) to show that for any \(n,\) pd \(M \leqslant n\) if and only if \(\operatorname{Ext}^{i}(M, A)=0\) for all \(i>n\)

Let \(Y\) be a scheme of finite type over an algebraically closed field \(k\). Show that the function \\[ \varphi(y)=\operatorname{dim}_{k}\left(m_{y} / m_{y}^{2}\right) \\] is upper semicontinuous on the set of closed points of \(Y\).

Show that a morphism \(f: X \rightarrow Y\) of schemes of finite type over \(k\) is 茅tale if and only if the following condition is satisfied: for each \(x \in X\), let \(y=f(x)\). Let \(\hat{C}_{\text {. }}\) and \(\hat{C}_{y}\) be the completions of the local rings at \(x\) and \(y\). Choose fields of representatives \((\mathrm{II}, 8.25 \mathrm{A}) \mathrm{k}(\mathrm{x}) \subseteq \hat{\mathrm{C}}_{\mathrm{A}}\) and \(k(\mathrm{y}) \subseteq \hat{\mathrm{C}}_{\mathrm{y}}\) so that \(k(\mathrm{y}) \subseteq k(\mathrm{x})\) via the natural map \(\hat{C}_{y} \rightarrow \hat{C}_{x} .\) Then our condition is that for every \(x \in X, k(x)\) is a separable algebraic extension of \(k(y),\) and the natural map \\[ \hat{C}_{y} \otimes_{h(y)} k(x) \rightarrow \hat{C}_{x} \\] is an isomorphism.

Let \(X\) be a noetherian scheme, and let \(\mathscr{F}, \mathscr{G} \in \mathfrak{M}_{0} \mathfrak{d}(X)\) (a) If \(\mathscr{F}, \mathscr{G}\) are both coherent, then \(\mathscr{E} x t^{i}(\mathscr{F}, \mathscr{S})\) is coherent, for all \(i \geqslant 0\) (b) If \(\mathscr{F}\) is coherent and \(\mathscr{G}\) is quasi-coherent, then \(\mathscr{E} x t^{i}(\mathscr{F}, \mathscr{G})\) is quasi-coherent for all \(i \geqslant 0\)
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