7.3 Computation techniques for Tor and Ext

Tor and Ext are crucial functors in homological algebra, and knowing how to compute them is key. This section covers various techniques, from resolutions to spectral sequences, that help us calculate these functors efficiently.

Understanding these computational methods not only makes working with Tor and Ext easier but also deepens our grasp of their properties. We'll explore how these tools connect and build upon each other to solve complex problems.

Resolution Techniques

Projective and Injective Resolutions

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• Projective resolution constructs a resolution of a module using projective modules
  • Allows for the computation of left derived functors (Tor)
  • Starts with the module and builds a sequence of projective modules with maps between them
  • The sequence is exact and the maps are surjective
• Injective resolution constructs a resolution of a module using injective modules
  • Allows for the computation of right derived functors (Ext)
  • Starts with the module and builds a sequence of injective modules with maps between them
  • The sequence is exact and the maps are injective
• Resolutions are used to replace a module with a more structured sequence of modules
  • This allows for the application of functors and the computation of derived functors

Horseshoe Lemma

• Horseshoe lemma is a tool for constructing resolutions and comparing them
• Given a short exact sequence of modules and a projective (or injective) resolution of one of the modules, the lemma allows for the construction of resolutions for the other two modules
• The resulting diagram has the shape of a horseshoe, hence the name
• Useful for proving the long exact sequence of Tor (or Ext) associated with a short exact sequence of modules
• Allows for the comparison of resolutions and the transfer of information between them

Computational Tools

Spectral Sequences

• Spectral sequences are a powerful computational tool in homological algebra
• They are a systematic way of computing homology (or cohomology) groups by successive approximations
• Consist of a sequence of pages, each containing a grid of groups (or modules) with differentials between them
• The differentials on each page satisfy certain properties (e.g., $d^2 = 0$)
• The homology (or cohomology) of one page forms the groups (or modules) on the next page
• The process continues until the differentials become zero, yielding the final result
• Spectral sequences can be used to compute Tor and Ext groups in various settings (e.g., group homology, sheaf cohomology)

Künneth Formula

• Künneth formula is a tool for computing the homology (or cohomology) of a tensor product of chain complexes (or modules)
• Relates the homology (or cohomology) of the tensor product to the homology (or cohomology) of the individual complexes (or modules)
• Involves a short exact sequence that splits, leading to a direct sum decomposition
• Can be used to compute Tor (or Ext) of a tensor product of modules
• Useful in algebraic topology for computing the homology (or cohomology) of product spaces

Dimension Shifting

• Dimension shifting is a technique for relating Tor (or Ext) groups in different dimensions
• Allows for the computation of higher Tor (or Ext) groups in terms of lower ones
• Involves the use of short exact sequences and the long exact sequence of Tor (or Ext)
• Can simplify computations by reducing the problem to a lower dimension
• Useful in conjunction with other computational tools (e.g., spectral sequences, Künneth formula)

Ring-Theoretic Considerations

Change of Rings

• Change of rings is a technique for relating Tor (or Ext) groups over different rings
• Given a ring homomorphism $f: R \to S$ and an $S$-module $M$, one can compute $\text{Tor}_*^R(M, N)$ in terms of $\text{Tor}_*^S(M, S \otimes_R N)$ for an $R$-module $N$
• Similarly, for Ext, one can compute $\text{Ext}_R^*(M, N)$ in terms of $\text{Ext}_S^*(M, \text{Hom}_R(S, N))$
• Useful when the Tor (or Ext) groups are easier to compute over one ring than the other
• Can be used to transfer results between different rings (e.g., from a ring to its quotient, or from a ring to its localization)
• Involves the use of spectral sequences (e.g., the Grothendieck spectral sequence) to relate the Tor (or Ext) groups over the different rings