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Chapter 3: Problem 3

Let \(X=\mathbf{P}_{k}^{n} .\) Show that \(H^{q}\left(X, \Omega_{X}^{p}\right)=0\) for \(p \neq q . k\) for \(p=q, 0 \leqslant p, q \leqslant n\)

Short Answer

Expert verified
Indeed, it can be verified that \(H^{q}\left(X, \Omega_{X}^{p}\right)=0\) if \(p \neq q\) and that \(H^q(X, \Omega_X^p) \neq 0\) if \(p = q\) with \(0 \leqslant p, q \leqslant n\). This was achieved by considering the properties of sheaf cohomology, differential forms, and the topology of projective spaces.

Step by step solution

01

Understanding the Notations

The notation \(H^{q}\left(X, \Omega_{X}^{p}\right)\) refers to the \(q\)-th sheaf cohomology group of the space \(X = \mathbf{P}_{k}^{n}\) with coefficients in the sheaf \(\Omega_{X}^{p}\) of \(p\)-forms on \(X\). This is a fundamental concept in algebraic geometry that measures the 'failure' of global sections to generate the entire sheaf.
02

Applying the Properties of Sheaf Cohomology and Differential Forms

One should know that for projective space \(\mathbf{P}_{k}^{n}\), there are no non-trivial differential forms of positive degree. This is because global differential forms on \(\mathbf{P}_{k}^{n}\) can be viewed as global sections of the sheaf \(\Omega_{X}^{p}\), and we know there are no global sections of this sheaf for \(p > 0\). Therefore, \(H^{q}\left(X, \Omega_{X}^{p}\right) = 0\) for all \(p > 0\), regardless of the value of \(q\). For \(p=0\), \(\Omega_{X}^{p}\) is just the structure sheaf \(\mathcal{O}_{X}\) of regular functions, and it's a fact that \(H^{q}(X,\mathcal{O}_{X})=0\) for \(q>0\). Hence we conclude that \(H^{q}\left(X, \Omega_{X}^{p}\right) = 0\) if \(p \neq q\).
03

Analyzing the Case When \(p=q\)

When \(p=q\), we know that \(\Omega_{X}^{p}\) is the \(p\)-th wedge power of the cotangent bundle \(T^*X\). Furthermore, a key property in algebraic topology states that the \(q\)-th cohomology of a \(p\)-dimensional projective space \(X\) is nonzero only when \(q = p\). So, when we restrict \(p = q\), we indeed find \(H^q(X, \Omega_X^p) \neq 0\) for \(0 \leqslant p = q \leqslant n\).

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Key Concepts

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

Projective Space
Projective space, denoted as \(\mathbf{P}_k^n\), is a fundamental concept in algebraic geometry. It can be thought of as the set of lines passing through the origin in \(\mathbb{A}^{n+1}\), the \(n+1\)-dimensional affine space.
  • This space is essential in studying geometric objects, as it allows for a more complete understanding by including points at infinity.
  • Projective space is compact, meaning it is closed and bounded, which lends it some distinctive properties in geometry.
  • It plays a key role in simplifying many algebraic geometry problems.
Understanding projective space is crucial when dealing with sheaf cohomology. Sheaf cohomology on projective spaces helps in analyzing the behavior of certain functions and forms globally over the space. With \(X = \mathbf{P}_k^n\), we examine the global sections or the lack thereof, which will connect us to deeper insights into cohomology and differential forms.
Differential Forms
Differential forms are a central tool in both algebraic and differential geometry. They are used to generalize the concepts of gradients, divergences, and curls in calculus, extending them to more complex spaces and manifolds.
  • In algebraic geometry, they often appear as sections of the cotangent bundle, which gives insight into the geometric structure of varieties and projective spaces.
  • The sheaf of differential forms on a projective space, \(\Omega^p_X\), consists of \(p\)-forms, which are typically smooth, continuous functions that can be integrated over \(p\)-dimensional subspaces.
  • For projective spaces, interestingly, there aren't any non-trivial global sections for positive-degree differential forms. This is a particular characteristic that greatly influences cohomological computations.
In our exercise, understanding \(\Omega_X^p\) was vital to see why \(H^q(X, \Omega_X^p) = 0\) for \(p > 0\) and \(p eq q\), as these forms do not exist globally when their degree is positive.
Algebraic Geometry
Algebraic geometry is a branch of mathematics that studies solutions to algebraic equations and their properties. It combines techniques from algebra, particularly commutative algebra, with the geometric intuition of spaces of solutions.
  • The foundational elements include varieties, which are the solution sets of systems of polynomial equations. Projective space is a key structure in this context.
  • Sheaf cohomology is a tool in algebraic geometry that measures the extent to which local solutions can be extended globally. This is crucial, as it indicates potential obstructions in finding solutions globally when certain conditions are met locally.
  • In the context of our exercise, \(H^q(X, \Omega_X^p)\)'s calculation focuses on identifying cases where certain sheaves do not support global sections, hence showing said cohomology vanishies.
Understanding algebraic geometry's principles helps establish why specific results, such as our exercise's conclusion regarding sheaf cohomology on projective space, are true. It provides a frame to explore deeply how geometric spaces behave and interact with algebraic structures.

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Most popular questions from this chapter

For any ringed space \(\left(X, \mathcal{O}_{X}\right),\) let Pic \(X\) be the group of isomorphism classes of invertible sheaves (II, \(\S 6\) ). Show that Pic \(X \cong H^{1}\left(X, O_{X}^{*}\right),\) where \(O_{X}^{*}\) denotes the sheaf whose sections over an open set \(U\) are the units in the ring \(\Gamma\left(U, \mathcal{O}_{X}\right),\) with multiplication as the group operation. [Hint: For any invertible sheaf \(\mathscr{L}\) on \(X\) cover \(X\) by open sets \(U_{i}\) on which \(\mathscr{L}\) is free, and fix isomorphisms \(\varphi_{i}:\left.\mathcal{O}_{v_{i}} \simeq \mathscr{L}\right|_{U_{i}}\) Then on \(U_{i} \cap U_{j},\) we get an isomorphism \(\varphi_{i}^{-1} \circ \varphi_{j}\) of \(\mathscr{O}_{U_{i} \cap U_{j}}\) with itself. These isomorphisms give an element of \(\check{H}^{1}\left(\mathfrak{U}, \mathscr{O}_{\boldsymbol{X}}^{*}\right) .\) Now use (Ex. 4.4).]

Let \(X\) be a projective scheme over a noetherian ring \(A,\) and let \(\mathscr{J}^{1} \rightarrow \mathscr{F}^{2} \rightarrow \ldots \rightarrow\) If be an exact sequence of coherent sheaves on \(X\). Show that there is an integer \(n_{0},\) such that for all \(n \geqslant n_{0},\) the sequence of global sections $$\Gamma\left(X, \overline{\mathscr{F}}^{1}(n)\right) \rightarrow \Gamma\left(X, \overline{\mathscr{H}}^{2}(n)\right) \rightarrow \ldots \rightarrow \Gamma\left(X, \mathscr{F}^{\prime}(n)\right)$$ is exact.

The Cohomology Class of a Subrariety. Let \(X\) be a nonsingular projective variety of dimension \(n\) over an algebraically closed field \(k\). Let \(Y\) be a nonsingular subvariety of codimension \(p\) (hence dimension \(n-p\) ). From the natural map \(\Omega_{x} \otimes\) \(\varphi_{Y} \rightarrow \Omega_{Y}\) of \((\mathrm{II}, 8.12)\) we deduce a \(\operatorname{map} \Omega_{X}^{n-p} \rightarrow \Omega_{Y}^{n-p} .\) This induces a map on cohomology \(H^{n-p}\left(X, \Omega_{X}^{n-p}\right) \rightarrow H^{n-p}\left(Y, \Omega_{Y}^{n-p}\right) .\) Now \(\Omega_{Y}^{n-p}=\omega_{Y}\) is a dualizing sheaf for \(Y,\) so we have the trace map \(t_{Y}: H^{n-p}\left(Y, \Omega_{Y}^{n-p}\right) \rightarrow k .\) Composing, we obtain a linear map \(H^{n-p}\left(X, \Omega_{X}^{n-p}\right) \rightarrow k .\) By (7.13) this corresponds to an element \(\eta(Y) \in\) \(H^{p}\left(X, \Omega_{X}^{p}\right),\) which we call the cohomology class of \(Y\) (a) If \(P \in X\) is a closed point, show that \(t_{X}(\eta(P))=1,\) where \(\eta(P) \in H^{n}\left(X, \Omega^{n}\right)\) and \(t_{X}\) is the trace map. (b) If \(X=\mathbf{P}^{n},\) identify \(H^{p}\left(X, \Omega^{p}\right)\) with \(k\) by \((\mathrm{Ex} .7 .3),\) and show that \(\eta(Y)=(\operatorname{deg} Y) \cdot 1\) where deg \(Y\) is its degree as a projective variety \((\mathrm{I}, \mathrm{s} 7) .[\text { Hint}:\) Cut with a hyperplane \(H \subseteq X,\) and use Bertini's theorem (II, 8.18 ) to reduce to the case \(Y\) is a finite set of points.] (c) For any scheme \(X\) of finite type over \(k\), we define a homomorphism of sheaves of abelian groups \(d \log :\left(\stackrel{*}{X} \rightarrow \Omega_{X} \text { by } d \log (f)=f^{-1} d f . \text { Here }C*\text { is a group }\right.\) under multiplication, and \(\Omega_{X}\) is a group under addition. This induces a map on cohomology Pic \(X=H^{1}\left(X, C_{X}^{*}\right) \rightarrow H^{1}\left(X, \Omega_{X}\right)\) which we denote by \(c-\) see (Ex. 4.5 ). (d) Returning to the hypotheses above, suppose \(p=1 .\) Show that \(\eta(Y)=c(\mathscr{Q}(Y))\) where \(\mathscr{L}(Y)\) is the invertible sheaf corresponding to the divisor \(Y\)

Duality for a Finite Flat Morphism. (a) Let \(f: X \rightarrow Y\) be a finite morphism of noetherian schemes. For any quasicoherent \(\mathscr{O}_{Y}\) -module \(\mathscr{G}, \mathscr{H}_{O} m_{Y}\left(f_{*} \mathscr{O}_{X}, \mathscr{L}\right)\) is a quasi-coherent \(f_{*} \mathscr{C}_{X}\) -module, hence corresponds to a quasi-coherent \(\mathscr{O}_{X}\) -module, which we call \(f^{\prime} \mathscr{G}\) (II, Ex. \(5.17 \mathrm{e}\) ). (b) Show that for any coherent \(\mathscr{F}\) on \(X\) and any quasi-coherent \(\mathscr{G}\) on \(Y\), there is a natural isomorphism $$f_{*} \mathscr{H o m}_{X}\left(\mathscr{F}, f^{\prime: g}\right) \simeq \mathscr{H}_{O M_{Y}}\left(f_{*} \mathscr{F}, \mathscr{L}\right)$$ (c) For each \(i \geqslant 0\), there is a natural map $$\varphi_{i}: \operatorname{Ext}_{x}^{i}\left(\mathscr{F}, f^{\prime} \mathscr{G}\right) \rightarrow \operatorname{Ext}_{Y}^{i}\left(f_{*} \mathscr{F}, \mathscr{G}\right)$$ [Hint: First construct a map \\[\operatorname{Ext}_{x}^{i}\left(\mathscr{F}, f^{\prime} \mathscr{G}\right) \rightarrow \operatorname{Ext}_{\mathrm{Y}}^{i}\left(f_{*} \mathscr{F}, f_{*} f^{\prime} \mathscr{G}\right)\\] \([\text {Hint}:\) First construct a map $$\operatorname{Ext}_{x}^{i}\left(\mathscr{F}, f^{\prime} \mathscr{G}\right) \rightarrow \operatorname{Ext}^{i}\left(f_{*} \mathscr{F}_{1} f_{*} f^{\prime} \mathscr{G}\right)$$ Then compose with a suitable map from \(\left.f_{*} f^{\prime} \mathscr{G} \text { to } \mathscr{G} .\right]\) (d) Now assume that \(X\) and \(Y\) are separated, \(\operatorname{Cob}(X)\) has enough locally frees, and assume that \(f_{*} \mathscr{O}_{X}\) is locally free on \(Y\) (this is equivalent to saying \(f\) flat- see 89.). Show that \(\varphi_{i}\) is an isomorphism for all \(i\), all \(\mathscr{F}\) coherent on \(X\), and all \(\mathscr{G}\) quasi-coherent on \(Y\). [Hints: First do \(i=0 .\) Then do \(\mathscr{F}=\mathscr{O}_{X},\) using (Ex. 4.1 ). Then do \(\mathscr{F}\) locally free. Do the general case by induction on \(i\), writing \(\mathscr{F}\) as a quotient of a locally free sheaf.

A morphism \(f: X \rightarrow Y\) of schemes of finite type over \(k\) is 茅tale if it is smooth of relative dimension 0. It is unramified if for every \(x \in X\), letting \(y=f(x)\), we have \(\mathrm{m}_{\mathrm{y}} \cdot \mathscr{C}_{x}=\mathrm{m}_{\mathrm{x}},\) and \(k(x)\) is a separable algebraic extension of \(k\left(y^{\prime}\right) .\) Show that the following conditions are equivalent: (i) \(f\) is 茅tale (ii) \(f\) is flat, and \(\Omega_{X, Y}=0\) (iii) \(f\) is flat and unramified.
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