Derived Functors - Applications

Derived functors have wide-ranging applications across algebraic geometry, topology, and representation theory.

Theorem4.16Čech-to-Derived Spectral Sequence

For a sheaf $\mathcal{F}$ on a topological space $X$ with an open cover $\mathcal{U} = \{U_i\}$ , there is a spectral sequence: $E_2^{p,q} = \check{H}^p(\mathcal{U}, \mathcal{H}^q(\mathcal{F})) \Rightarrow H^{p+q}(X, \mathcal{F})$

where $\check{H}^p$ is Čech cohomology and $H^{p+q}$ is sheaf cohomology.

Remark

This spectral sequence is fundamental in algebraic geometry for computing sheaf cohomology using Čech techniques, which are often more accessible for explicit calculations.

Theorem4.17Local-to-Global Ext

For a Noetherian scheme $X$ and coherent sheaves $\mathcal{F}, \mathcal{G}$ on $X$ : $\mathcal{E}xt^i(\mathcal{F}, \mathcal{G}) = \text{sheafification of } (U \mapsto \text{Ext}^i_{\mathcal{O}_X(U)}(\mathcal{F}|_U, \mathcal{G}|_U))$

These sheaves measure local extension problems and relate to global Ext via spectral sequences.

ExampleSerre Duality

For a smooth projective variety $X$ of dimension $n$ over a field $k$ , there is a functorial isomorphism: $H^i(X, \mathcal{F})^\vee \cong H^{n-i}(X, \mathcal{F}^\vee \otimes \omega_X)$

where $\omega_X$ is the canonical sheaf. This uses derived functors extensively in its proof.

Theorem4.18Leray Spectral Sequence

For a continuous map $f: X \to Y$ and sheaf $\mathcal{F}$ on $X$ , there is a spectral sequence: $E_2^{p,q} = H^p(Y, R^qf_*\mathcal{F}) \Rightarrow H^{p+q}(X, \mathcal{F})$

where $R^qf_*$ is the right derived functor of the direct image $f_*$ .

Remark

The Leray spectral sequence is essential for computing cohomology of fibrations and understanding how cohomology behaves under continuous maps.

ExampleHochschild Cohomology

For an associative $k$ -algebra $A$ , the Hochschild cohomology is: $HH^n(A, M) = \text{Ext}^n_{A^e}(A, M)$

where $A^e = A \otimes_k A^{op}$ is the enveloping algebra and $M$ is an $A$ -bimodule. This measures the deformation theory of $A$ .

Theorem4.19Künneth Formula for Ext

Over a field $k$ , for finite-dimensional vector spaces: $\text{Ext}^n_{k[x]}(M, N) \cong \bigoplus_{i+j=n} \text{Ext}^i_{k[x]}(M, k) \otimes_k \text{Ext}^j_{k[x]}(k, N)$

This generalizes to more general rings with appropriate flatness conditions.