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Chapter 3: Problem 7

Let \(X=\operatorname{Spec} A\) be an affine noetherian scheme. Let \(M, N\) be \(A\) -modules, with \(M\) finitely generated. Then $$\operatorname{Ext}_{X}^{i}(\tilde{M}, \tilde{N}) \cong \operatorname{Ext}_{A}^{i}(M, N)$$ and $$\mathscr{E} x t_{X}^{i}(\tilde{M}, \tilde{N}) \cong \operatorname{Ext}_{A}^{i}(M \cdot N)^{\sim}$$

Short Answer

Expert verified
The proofs of the two isomorphisms are based on the use of category theory, specifically the properties of derived categories and functors, and sheaf theory. We used the equivalence between the categories of modules over a ring and sheaves over a spectrum, applied it to the derived categories, and used a spectral sequence argument to show the equality of global Ext functions. The key concept applied here is that the derived functors of naturally equivalent functors are naturally equivalent.

Step by step solution

01

Proof of \(\operatorname{Ext}_{X}^{i}(\tilde{M}, \tilde{N}) \cong \operatorname{Ext}_{A}^{i}(M, N)\)

To show the first isomorphism, we follow these steps: A. Remember that for an affine scheme \(X = Spec(A)\), the global sections functor \(\Gamma(X,-) : Mod(O_X) \to Mod(A)\) is an equivalence of categories with quasi-inverse \(A \mapsto \tilde{A}\). B. It follows that the derived functors of \(\Gamma(X,-)\) and \(\tilde{(-)}\) are also quasi-inverse equivalences \(D(Mod(O_X)) \simeq D(Mod(A))\), \(D^+(Mod(O_X)) \simeq D^+(Mod(A))\) and \(D^-(Mod(O_X)) \simeq D^-(Mod(A))\). C. As a result, for any \(O_X\)-module \(F\) and any \(A\)-module \(M\), \(R^n\Gamma(X, F) = H^n(Gamma(X, F))\) and \(R^n\tilde{M} = H^n(\tilde{M})\). So, \(R\Gamma(X, F) \simeq \Gamma(X, F)\) and \(R\tilde{M} \simeq \tilde{M}\). D. Thus, \(RHom_{O_X}(R\tilde{M}, F) \simeq RHom_A(M, R\Gamma(X, F))\). E. Since \(M\) is finitely presented, \(R\tilde{M} \simeq \tilde{M}\). Hence, \(RHom_{O_X}(\tilde{M}, F) \simeq RHom_A(M, R\Gamma(X, F))\). F. Lastly, using \(R\Gamma(X, F) \simeq \Gamma(X, F)\), we get \(RHom_{O_X}(\tilde{M}, F) \simeq RHom_A(M, \Gamma(X, F))\). This completes the proof of the first isomorphism.
02

Proof of \(\mathscr{E} x t_{X}^{i}(\tilde{M}, \tilde{N}) \cong \operatorname{Ext}_{A}^{i}(M \cdot N)^{\sim}\)

This proof involves understanding of the sheafified Ext functor and the sheaf associated to a module. A. The sheafified Ext functor on a scheme is defined by taking the presheaf Ext functor and applying sheafification. B. The sheaf associated to a module \(M\) is defined by assigning to every open set the module of sections over that set. C. We know from the equivalence \(Mod(A) \simeq Mod(O_X)\) that the local-to-global Ext spectral sequence degenerates, so we only need to check that the two sides agree after taking global sections. D. By the proof of the first statement, we have \(H^i(X, Ext^j_{O_X}(\tilde{M}, \tilde{N})) \cong Ext^{i+j}_A(M, N)\). E. In particular, for \(j = 0\), we have \(H^i(X, Hom_{O_X}(\tilde{M}, \tilde{N})) \cong Ext^i_A(M, N)\), which gives the result upon sheafifying the right-hand side.

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Key Concepts

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

Affine Noetherian Scheme
Understanding Affine Noetherian Schemes is crucial in algebraic geometry, as they build a bridge between algebra and geometry. An Affine Noetherian Scheme, labeled as \(X = \text{Spec} A\), is a space that represents the spectrum of ring theoretic properties brought to a geometric setting.

In the case of algebraic geometry, the ring \(A\) must be a Noetherian ring, which implies that it satisfies the ascending chain condition on ideals 鈥� essentially, this means that any increasing sequence of ideals will eventually stabilize. This property has significant consequences on the structure of the ring, ensuring that it is well-behaved in a sense that parallels how finite sets are in basic arithmetic.

Schemes that are affine and Noetherian are particularly nice to work with because they are the geometric counterparts to these well-behaved rings. Moreover, the scheme \(X\) allows us to consider modules \(M\) and \(N\) over the ring \(A\) in a geometric context, which can be quite powerful for understanding the relationships between these modules.
Ext Functor
The Ext functor, denoted as \(\text{Ext}\), is an essential construction in homological algebra, which is one of the powerful tools in both algebra and algebraic geometry. It serves a purpose much like that of a detective: it investigates the extent to which a sequence fails to be exact. As a reminder, an exact sequence is a sequence of module homomorphisms wherein the image of each homomorphism is equal to the kernel of the next.

The Ext functor actually comprises a family of functors \(\text{Ext}^i\), indexed by integers \(i\). For \(i = 0\), \(\text{Ext}^0\) is just \(\text{Hom}\), the set of homomorphisms between two modules, which reflects direct relationships. Meanwhile, for \(i > 0\), \(\text{Ext}^i\) measures the failure to extend a module to an exact sequence, hence its name 鈥淓xt鈥�.

It comes into play most directly when dealing with extensions of modules and derives its name from this very application. The calculations of \(\text{Ext}^i\) can be quite complex, often requiring long exact sequences and dimension-shifting techniques to simplify the computation. These functors help in understanding the deeper interactions between modules that aren't immediately obvious, and they play a pivotal role in many theorems and proofs within the realm of algebraic geometry.
Derived Functors
Derived functors are among the heavy-lifters in homological algebra and come in many flavors including the Ext and Tor functors. They generalize the idea of taking homology in a different context - they are constructed from a given functor by iterating this functor in a homological sense.

The clear-cut advantage of derived functors is their ability to capture global information about the functor across all dimensions, rather than just the picture at a fixed level. For instance, in our problem, the derived functors give us the correspondence between the Ext groups computed in the category of modules over \(A\) and the category of quasi-coherent sheaves on \(X\).

The symbol \(R\) before a functor, such as \(R\text{Hom}\) or \(R\Gamma\), stands for the right derived functor, indicating a series of functors obtained by looking at right resolutions of objects. These derived functors account for deeper properties, such as extensions and obstructions, which can be critical in various contexts, such as complex geometry, representation theory, and even in string theory. Understanding these functors provides a robust toolkit for peeling back the layers of complexity in algebraic structures.

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Most popular questions from this chapter

Let \(Y\) be an integral scheme of finite type over an algebraically closed field \(k\) Let \(\delta\) be a locally free sheaf on \(Y\), and let \(X=\mathbf{P}(\delta)-\) see (II, \(\$ 7\) ). Then show that Pic \(X \cong(\text { Pic } Y) \times \mathbf{Z} .\) This strengthens (II, Ex. 7.9)

A Nonprojectire Scheme. We show the result of \((\mathrm{Ex} .5 .8)\) is false in dimension 2 Let \(k\) be an algebraically closed field of characteristic \(0,\) and let \(X=\mathbf{P}_{k}^{2}\). Let \(w\) be the sheaf of differential 2 -forms (II, se). Define an infinitesimal extension \(X\) of \(X\) by \((\cdot)\) by giving the element \(\xi \in H^{1}(X,(\cdot) \otimes . \mathscr{J})\) defined as follows (Ex. 4.10 ). Let \(x_{0}, x_{1}, x_{2}\) be the homogeneous coordinates of \(X\). let \(U_{0} . \ell_{1}, \ell_{2}\), be the standard open covering. and let \(\check{\xi}_{1,1}=\left(x_{1}, x_{1}, d / x_{i}, x_{1}\right) .\) This gives a Cech 1 -cocycle with values in \(\Omega_{x}^{1}\). and since dim \(X=2\), we have \((\cdot) \otimes . \bar{J} \cong \Omega^{1}(\mathrm{II}, \mathrm{E} \times .5 .16 \mathrm{b}) .\) Now use the exact sequence $$\ldots \rightarrow H^{1}\left(X,_{(\cdot))} \rightarrow \operatorname{Pic} X^{\prime} \rightarrow \operatorname{Pic} X \stackrel{\bullet}{\rightarrow} H^{2}(X,(\cdot)) \rightarrow \ldots\right.$$ of \((\mathrm{Ex} .4 .6)\) and show \(\delta\) is injective. We have ()\(\cong\left(x^{\prime}-3\right)\) by \((11,8.20 .1) .\) so \(H^{2}(X, \omega) \cong k .\) since char \(h=0,\) you need only show that \(\delta(C(1)) \neq 0,\) which can be done by calculating in Cech cohomology. since \(H^{1}(X, \omega)=0,\) we see that Pic \(X^{\prime}=0 .\) In particular, \(X^{\prime}\) has no ample invertible sheaves, so it is not projective. Note. In fact, this result can be generalized to show that for any nonsingular projective surface \(X\) over an algebraically closed field \(k\) of characteristic 0 , there is an infinitesimal extension \(X^{\prime}\) of \(X\) by \(\omega,\) such that \(X^{\prime}\) is not projective over \(k\) Indeed, let \(D\) be an ample divisor on \(X\). Then \(D\) determines an element \(c_{1}(D) \in\) \(H^{1}\left(X, \Omega^{1}\right)\) which we use to define \(X^{\prime},\) as above. Then for any divisor \(E\) on \(X\) one can show that \(\delta(\mathscr{P}(E))=(D . E) .\) where \((D . E)\) is the intersection number (Chapter \(\mathrm{V}\) ), considered as an element of \(k .\) Hence if \(E\) is ample, \(\delta\left(\mathscr{L}^{\prime}(E)\right) \neq 0 .\) Therefore \(X^{\prime}\) has no ample divisors. On the other hand, over a field of characteristic \(p>0,\) a proper scheme \(X\) is projective if and only if \(X_{\text {rad }}\) is!

(a) Let \(X=\mathbf{A}_{k}^{1}\) be the affine line over an infinite field \(k\). Let \(P . Q\) be distinct closed points of \(X,\) and let \(U=X-\left\\{P . Q ; \text { Show that } H^{1}\left(X . \mathbf{Z}_{l}\right) \neq 0\right.\) (b) More generally, let \(Y \subseteq X=\mathbf{A}_{k}^{n}\) be the union of \(n+1\) hyperplanes in suitably general position, and let \(U=X-Y .\) Show that \(H^{n}\left(X, \mathbf{Z}_{U}\right) \neq 0 .\) Thus the result of (2.7) is the best possible.

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 \(Y\)

A scheme \(X_{0}\) over a field \(k\) is rigid if it has no infinitesimal deformations. (a) Show that \(P_{k}^{1}\) is rigid, using \((9.13 .2)\) (b) One might think that if \(X_{0}\) is rigid over \(k\), then every global deformation of \(X_{0}\) is locally tris ial. Show that this is not so. by constructing a proper, flat morphism \(f: X \rightarrow \mathbf{A}^{2}\) over \(k\) algebraically closed, such that \(X_{0} \cong \mathbf{P}_{k}^{1},\) but there is no open neighborhood \(U\) of 0 in \(\mathbf{A}^{2}\) for which \(f^{-1}(U) \cong L \times \mathbf{P}^{1}\) (c) Show, however, that one can trivialize a global deformation of \(\mathbf{P}^{1}\) after a flat base extension, in the following sense: let \(f: X \rightarrow T\) be a flat projective morphism, where \(T\) is a nonsingular curve over \(k\) algebraically closed. Assume there is a closed point \(t \in T\) such that \(X_{t} \cong P_{k}^{1} .\) Then there exists a nonsingular curve \(T^{\prime},\) and a flat morphism \(g: T^{\prime} \rightarrow T,\) whose image contains \(t,\) such that if \(X^{\prime}=X \times_{T} T^{\prime}\) is the base extension, then the new family \(f^{\prime}: X^{\prime} \rightarrow T^{\prime}\) is isomorphic to \(\mathbf{P}_{T}^{1} \rightarrow T^{\prime}\)
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