Ext Functor and Sheaf Cohomology

Oct 13, 2025 by magicalceylontours.com 31 views

In case of Proj S: Isomorphism between Ext^i(M~,N^~) and Ext^i(M,N)

In the realm of algebraic geometry, sheaf cohomology plays a pivotal role in understanding the properties of coherent and quasicoherent sheaves on a scheme. The Ext functor, a fundamental tool in homological algebra, is used to compute the cohomology groups of sheaves. In this article, we will explore the isomorphism between Ext^i(M~,N^~) and Ext^i(M,N) for all i>0, where M and N are finitely generated torsion-free modules on a projective scheme Proj S.

Let X=Proj(S) be a projective scheme, where S is a graded ring. We denote the category of graded S-modules by Mod_S. For a graded S-module M, we define the sheafification of M, denoted by \tilde{M}, as the sheaf on X associated to the presheaf U \mapsto M_U, where U is an open subset of X. We denote the category of sheaves on X by Shv(X).

Finitely Generated Torsion-Free Modules

A graded S-module M is said to be finitely generated torsion-free if it is generated by a finite set of elements and has no torsion elements. In other words, M is a finitely generated module over the ring S_0, where S_0 is the degree 0 part of S.

Ext Functor and Sheaf Cohomology

The Ext functor is a fundamental tool in homological algebra, used to compute the cohomology groups of sheaves. For two sheaves F and G on X, the Ext functor is defined as:

Ext^i(F,G) = H^i(X, \mathcal{H}om(F,G))

where \mathcal{H}om(F,G) is the sheaf of homomorphisms from F to G.

Isomorphism between Ext^i(M~,N^~) and Ext^i(M,N)

We are now ready to prove the isomorphism between Ext^i(M~,N^~) and Ext^i(M,N) for all i>0, where M and N are finitely generated torsion-free modules on Proj S.

Let M and N be two finitely generated torsion-free graded modules on Proj S. Then, for all i>0, we have:

Ext^i(\tilde{M},\tilde{N}) \cong Ext^i(M,N)

We will prove the theorem by induction on i. For i=1, we have:

Ext^1(\tilde{M},\tilde{N}) = H^1(X, \mathcal{H}om(\tilde{M},\tilde{N}))

Ext^1(M,N) = H^1(X, \mathcal{H}om(M,N))

Using the fact that \mathcal{H}om(\tilde{M},\tilde{N}) \cong \tilde{\mathcal{H}om(M,N)}, we get:

Ext^1(\tilde{M},\tilde{N}) \cong H^1(X, \tilde{\mathcal{H}om(M,N)})

\cong H^1(X, \mathcal{H}om(M,N))

= Ext^1(M,N)

This proves the base case of the induction.

For the inductive step, assume that the theorem holds for i=k. We need to show that it holds for i=k+1. We have:

Ext^{k+1}(\tilde{M},\tilde{N}) = H^{k+1}(X, \mathcal{H}om(\tilde{M},\tilde{N}))

Ext^{k+1}(M,N) = H^{k+1}(X, \mathcal{H}om(M,N))

Using the fact that \mathcal{H}om(\tilde{M},\tilde{N}) \cong \tilde{\mathcal{H}om(M,N)}, we get:

Ext^{k+1}(\tilde{M},\tilde{N}) \cong H^{k+1}(X, \tilde{\mathcal{H}om(M,N)})

\cong H^{k+1}(X, \mathcal{H}om(M,N))

= Ext^{k+1}(M,N)

This completes the inductive step and proves the theorem.

Q: What is the significance of the isomorphism between Ext^i(M~,N^~) and Ext^i(M,N)?

A: The isomorphism between Ext^i(M~,N^~) and Ext^i(M,N) has important implications for the study of sheaf cohomology and Ext functors on projective schemes. It allows us to compute the cohomology groups of sheaves on a projective scheme using the Ext functor, which is a fundamental tool in homological algebra.

Q: What are the assumptions on M and N in the theorem?

A: The theorem assumes that M and N are finitely generated torsion-free graded modules on a projective scheme Proj S.

Q: What is the relationship between \mathcal{H}om(\tilde{M},\tilde{N}) and \tilde{\mathcal{H}om(M,N)}?

A: We have the isomorphism \mathcal{H}om(\tilde{M},\tilde{N}) \cong \tilde{\mathcal{H}om(M,N)}, which is used to prove the theorem.

Q: How does the theorem generalize to higher dimensions?

A: The theorem can be generalized to higher dimensions by using the same proof technique. However, the assumptions on M and N may need to be modified to ensure that the sheafification of M and N is well-defined.

Q: What are some applications of the theorem?

A: The theorem has important applications in algebraic geometry, particularly in the study of sheaf cohomology and Ext functors on projective schemes. It can be used to compute the cohomology groups of sheaves on a projective scheme, which is a fundamental problem in algebraic geometry.

Q: Can the theorem be generalized to other types of modules?

A: The theorem can be generalized to other types of modules, such as coherent sheaves or quasicoherent sheaves, by modifying the assumptions on M and N. However, the proof technique may need to be modified accordingly.

Q: What are some open problems related to the theorem?

A: Some open problems related to the theorem include:

Generalizing the theorem to higher dimensions
Modifying the assumptions on M and N to ensure that the sheafification of M and N is well-defined
Generalizing the theorem to other types of modules, such as coherent sheaves or quasicoherent sheaves

Q: How does the theorem relate to other results in algebraic geometry?

A: The theorem is related to other results in algebraic geometry, such as the Serre duality theorem and the Grothendieck duality theorem. It can be used to prove these theorems and has important implications for the study of sheaf cohomology and Ext functors on projective schemes.

Q: What are some future directions for research related to the theorem?

A: Some future directions for research related to the theorem include:

Generalizing the theorem to higher dimensions
Modifying the assumptions on M and N to ensure that the sheafification of M and N is well-defined
Generalizing the theorem to other types of modules, such as coherent sheaves or quasicoherent sheaves
Studying the applications of the theorem in algebraic geometry and other areas of mathematics.