Derived functors are powerful tools for studying modules and sheaves. They extend familiar concepts like Hom and tensor product to capture more subtle relationships between objects. This section explores how to compute and apply these functors.

We'll dive into Ext and Tor groups, which measure extensions and tensor product failures. We'll also look at group and sheaf cohomology, along with advanced tools like spectral sequences and derived categories. These concepts are key to understanding homological algebra.

Ext and Tor Groups

Computing Ext and Tor

• Ext groups $Ext_R^n(M,N)$ measure extensions between $R$-modules $M$ and $N$
  • $Ext_R^0(M,N)$ is the set of module homomorphisms $Hom_R(M,N)$
  • Higher Ext groups correspond to equivalence classes of long exact sequences
• Tor groups $Tor_n^R(M,N)$ measure the extent to which the tensor product of $R$-modules $M$ and $N$ fails to be exact
  • $Tor_0^R(M,N)$ is the tensor product $M \otimes_R N$
  • Higher Tor groups arise from the failure of the tensor product to be left-exact

Computational Techniques

• Dimension shifting allows computing higher Ext or Tor groups from lower ones
  • For Ext: $Ext_R^{n+1}(M,N) \cong Ext_R^n(M,\Omega N)$ where $\Omega N$ is the first syzygy of $N$
  • For Tor: $Tor_{n+1}^R(M,N) \cong Tor_n^R(\Omega M, N)$ where $\Omega M$ is the first syzygy of $M$
• Horseshoe lemma relates the projective resolutions of $M$, $N$, and $M \oplus N$
  • Allows constructing a projective resolution of $M \oplus N$ from those of $M$ and $N$
  • Useful for computing Ext and Tor groups involving direct sums

Cohomology Theories

Group Cohomology

• Group cohomology $H^n(G,A)$ measures the cohomology of a group $G$ with coefficients in a $G$-module $A$
  • $H^0(G,A)$ is the set of $G$-invariant elements of $A$
  • $H^1(G,A)$ classifies crossed homomorphisms modulo principal crossed homomorphisms
  • Higher cohomology groups have interpretations in terms of extensions and obstructions
• Computed using the bar resolution or projective resolutions of the trivial $G$-module $\mathbb{Z}$

Sheaf Cohomology

• Sheaf cohomology $H^n(X,\mathcal{F})$ measures the cohomology of a sheaf $\mathcal{F}$ on a topological space $X$
  • $H^0(X,\mathcal{F})$ is the set of global sections of $\mathcal{F}$
  • Higher cohomology groups measure obstructions to extending local sections to global sections
• Computed using Čech cohomology or derived functors of the global sections functor
• Künneth formula relates the cohomology of a product space to the cohomology of its factors
  • For sheaves $\mathcal{F}$ on $X$ and $\mathcal{G}$ on $Y$, there is a short exact sequence involving $H^n(X \times Y, \mathcal{F} \boxtimes \mathcal{G})$ and $\bigoplus_{p+q=n} H^p(X,\mathcal{F}) \otimes H^q(Y,\mathcal{G})$

Advanced Tools

Spectral Sequences

• Spectral sequences are algebraic tools for computing homology or cohomology groups
  • Consist of a sequence of pages $E_r^{p,q}$ with differentials $d_r: E_r^{p,q} \to E_r^{p+r,q-r+1}$
  • Each page is the homology of the previous page with respect to its differential
  • The sequence converges to the desired homology or cohomology groups
• Examples include the Leray spectral sequence and the Grothendieck spectral sequence

Derived Categories

• The derived category $D(R)$ of an abelian category $R$ is obtained by formally inverting quasi-isomorphisms
  • Objects are chain complexes, morphisms are chain maps modulo homotopy
  • Quasi-isomorphisms (chain maps inducing isomorphisms on homology) become isomorphisms
• Derived functors can be defined as functors on the derived category
  • Examples: derived tensor product $\otimes^L$, derived Hom functor $RHom$
• Provide a unified framework for homological algebra constructions