1.3 Motivation and historical context of homological algebra

Homological algebra emerged from the study of topology, connecting algebraic structures to geometric spaces. It provides powerful tools for understanding complex mathematical objects, using concepts like cohomology and derived functors to extract meaningful information.

Cohomology theories assign algebraic objects to topological spaces, capturing essential properties. Derived functors extend this approach, measuring how functors deviate from exactness. These tools have wide-ranging applications in mathematics, from algebraic geometry to representation theory.

Cohomology and Topological Invariants

Cohomology Theories and Eilenberg-Steenrod Axioms

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• Cohomology theories assign algebraic objects (abelian groups or vector spaces) to topological spaces in a functorial way
• Cohomology theories capture important topological information about spaces
• Eilenberg-Steenrod axioms provide a set of properties that characterize a cohomology theory
  • Homotopy invariance: Homotopy equivalent spaces have isomorphic cohomology groups
  • Exactness: A short exact sequence of spaces induces a long exact sequence of cohomology groups
  • Excision: Cohomology of a space can be computed using an excised subspace and its complement
  • Dimension: The cohomology of a point is trivial except in dimension 0, where it is the coefficient group
• Examples of cohomology theories satisfying the Eilenberg-Steenrod axioms include singular cohomology and de Rham cohomology

Topological Invariants and Algebraic Topology Connection

• Topological invariants are properties of topological spaces that remain unchanged under homeomorphisms or homotopy equivalences
• Cohomology groups serve as topological invariants, providing a way to distinguish non-homeomorphic or non-homotopy equivalent spaces
  • Betti numbers: The ranks of the cohomology groups are topological invariants (the first Betti number counts the number of connected components, the second counts the number of holes, etc.)
  • Cup product structure: The cup product gives the cohomology ring additional structure, which is a finer invariant than just the cohomology groups
• Cohomology plays a central role in algebraic topology, which studies topological spaces using algebraic tools
  • Algebraic topology uses functors to translate topological problems into algebraic ones, which are often easier to solve
  • Results obtained algebraically can then be interpreted back in the topological setting, providing insights into the original problem

Derived Functors in Homological Algebra

Derived Functors and Their Importance

• Derived functors are a way to extend functors between abelian categories (such as modules over a ring) to take into account additional homological information
• Derived functors measure the failure of a functor to be exact, i.e., to preserve exact sequences
  • For a left exact functor $F$, its right derived functors $R^iF$ measure the deviation from being right exact
  • For a right exact functor $G$, its left derived functors $L_iG$ measure the deviation from being left exact
• Derived functors are important tools in homological algebra and have applications in various areas of mathematics, including algebraic geometry and representation theory

Ext and Tor Functors

• Ext and Tor are two fundamental examples of derived functors in homological algebra
• For $R$-modules $A$ and $B$, $Ext_R^i(A,B)$ is the $i$-th right derived functor of the $Hom_R(-,B)$ functor applied to $A$
  • $Ext_R^i(A,B)$ measures the failure of the functor $Hom_R(-,B)$ to be exact when applied to $A$
  • $Ext_R^1(A,B)$ classifies extensions of $B$ by $A$, i.e., short exact sequences of the form $0 \to B \to E \to A \to 0$
• For $R$-modules $A$ and $B$, $Tor_i^R(A,B)$ is the $i$-th left derived functor of the tensor product functor $- \otimes_R B$ applied to $A$
  • $Tor_i^R(A,B)$ measures the failure of the functor $- \otimes_R B$ to be exact when applied to $A$
  • $Tor_0^R(A,B)$ is isomorphic to the tensor product $A \otimes_R B$