🧬Homological Algebra Unit 7 – The Tor and Ext Functors

Key Concepts and Definitions

• Homological algebra studies algebraic structures using homology, a powerful tool for analyzing complexes and their maps
• Abelian categories provide a general framework for studying homological algebra, encompassing modules over rings and sheaves over topological spaces
• Chain complexes consist of a sequence of objects connected by morphisms, enabling the study of homology and cohomology
• Derived functors extend functors between abelian categories to complexes, preserving homological properties
  • Examples include the Tor and Ext functors, which generalize tensor products and Hom functors respectively
• Projective and injective resolutions are crucial tools for computing derived functors, replacing objects with more tractable complexes
• Long exact sequences connect homology groups of related complexes, facilitating computations and revealing structural relationships
• Spectral sequences provide a systematic way to compute homology groups of complexes, often arising from double complexes or filtrations

Historical Context and Development

• Homological algebra emerged in the 1940s, driven by the need to understand topology using algebraic tools
  • Pioneering work by Eilenberg, Mac Lane, Cartan, and Serre laid the foundations of the field
• The development of category theory in the 1940s and 1950s provided a unifying language for homological algebra
• Grothendieck's work on algebraic geometry in the 1960s showcased the power of homological methods, leading to a surge of interest in the field
• The introduction of derived categories by Verdier in the 1960s revolutionized homological algebra, providing a more flexible framework for studying complexes
• Quillen's work on model categories in the 1960s and 1970s further expanded the scope of homological algebra, connecting it to homotopy theory
• The study of perverse sheaves and intersection cohomology by Goresky, MacPherson, Beilinson, Bernstein, and Deligne in the 1970s and 1980s demonstrated the utility of homological methods in geometry and representation theory
• Recent decades have seen the application of homological algebra to diverse areas, including algebraic topology, algebraic geometry, representation theory, and mathematical physics

Tor Functor: Construction and Properties

• The Tor functor is a derived functor that generalizes the tensor product of modules
  • For modules $A$ and $B$ over a ring $R$, $\operatorname{Tor}_n^R(A,B)$ measures the failure of the tensor product to be exact
• To compute $\operatorname{Tor}_n^R(A,B)$, choose a projective resolution $P_\bullet \to A$ and tensor it with $B$ to obtain a complex $P_\bullet \otimes_R B$
  • The homology of this complex at the $n$-th position is $\operatorname{Tor}_n^R(A,B)$
• $\operatorname{Tor}_0^R(A,B)$ is isomorphic to the tensor product $A \otimes_R B$, while higher Tor groups measure the deviation from exactness
• The Tor functor is symmetric, i.e., $\operatorname{Tor}_n^R(A,B) \cong \operatorname{Tor}_n^R(B,A)$
• Tor is additive in each argument, meaning it preserves direct sums
• If either $A$ or $B$ is a flat $R$-module, then $\operatorname{Tor}_n^R(A,B) = 0$ for all $n > 0$
  • In particular, Tor vanishes when tensoring with a free module
• The Tor functor fits into various long exact sequences, such as the long exact sequence associated with a short exact sequence of modules

Ext Functor: Construction and Properties

• The Ext functor is a derived functor that generalizes the Hom functor between modules
  • For modules $A$ and $B$ over a ring $R$, $\operatorname{Ext}_R^n(A,B)$ measures the failure of the Hom functor to be exact
• To compute $\operatorname{Ext}_R^n(A,B)$, choose an injective resolution $B \to I^\bullet$ and apply the Hom functor $\operatorname{Hom}_R(A,-)$ to obtain a complex $\operatorname{Hom}_R(A,I^\bullet)$
  • The cohomology of this complex at the $n$-th position is $\operatorname{Ext}_R^n(A,B)$
• $\operatorname{Ext}_R^0(A,B)$ is isomorphic to the Hom group $\operatorname{Hom}_R(A,B)$, while higher Ext groups measure the deviation from exactness
• The Ext functor is contravariant in the first argument and covariant in the second argument
• Ext is additive in each argument, preserving direct sums
• If either $A$ is projective or $B$ is injective, then $\operatorname{Ext}_R^n(A,B) = 0$ for all $n > 0$
• The Ext functor appears in various long exact sequences, such as the long exact sequence associated with a short exact sequence of modules
• Yoneda's lemma establishes a connection between Ext groups and extensions of modules, providing a geometric interpretation of the Ext functor

Relationships Between Tor and Ext

• The Tor and Ext functors are dual in the sense that they can be derived from each other using the notion of opposite categories and the Hom-tensor adjunction
• For a short exact sequence of modules $0 \to A \to B \to C \to 0$ and a module $M$, there are long exact sequences connecting the Tor and Ext groups:
  • $\cdots \to \operatorname{Tor}_n^R(M,A) \to \operatorname{Tor}_n^R(M,B) \to \operatorname{Tor}_n^R(M,C) \to \operatorname{Tor}_{n-1}^R(M,A) \to \cdots$
  • $0 \to \operatorname{Hom}_R(M,A) \to \operatorname{Hom}_R(M,B) \to \operatorname{Hom}_R(M,C) \to \operatorname{Ext}_R^1(M,A) \to \cdots$
• The universal coefficient theorem relates homology and cohomology groups via a short exact sequence involving Tor and Ext
  • For a chain complex $C$ and an $R$-module $M$, there is a short exact sequence: $0 \to H_n(C) \otimes_R M \to H_n(C \otimes_R M) \to \operatorname{Tor}_1^R(H_{n-1}(C), M) \to 0$
• The Künneth formula expresses the homology of a tensor product of complexes in terms of the homology of the individual complexes and their Tor groups
• In the derived category, the Tor and Ext functors can be viewed as the homology of the derived tensor product and derived Hom functors, respectively

Applications in Homological Algebra

• Homological algebra provides powerful tools for studying the structure of rings and modules
  • Projective, injective, and flat dimensions of modules can be characterized using vanishing of Tor and Ext groups
• In algebraic geometry, sheaf cohomology is a fundamental invariant that can be computed using the Ext functor and Čech cohomology
  • The Serre duality theorem expresses a relationship between sheaf cohomology groups using the Ext functor
• Homological methods are essential in the study of group cohomology, which has applications to representation theory and algebraic topology
  • The group cohomology ring is defined using the Ext functor applied to the trivial module
• In algebraic topology, the Tor and Ext functors appear in the Künneth and universal coefficient theorems, relating homology and cohomology of topological spaces
• Homological algebra plays a crucial role in the representation theory of algebras and groups
  • Derived categories and derived equivalences provide a powerful framework for studying representations
• In commutative algebra, local cohomology is a homological tool for investigating local properties of rings and modules
  • Local cohomology groups can be computed using the Ext functor and Koszul complexes

Computational Techniques and Examples

• Projective and injective resolutions are the primary tools for computing Tor and Ext groups
  • For a module $M$ over a ring $R$, a projective resolution is an exact sequence $\cdots \to P_1 \to P_0 \to M \to 0$ with each $P_i$ projective
  • An injective resolution is an exact sequence $0 \to M \to I^0 \to I^1 \to \cdots$ with each $I^i$ injective
• Over a principal ideal domain (PID), projective and injective modules coincide with free modules, simplifying resolutions
  • Example: For $\mathbb{Z}$-modules (abelian groups), projective resolutions can be constructed using free abelian groups
• Over a polynomial ring $k[x_1, \ldots, x_n]$, the Koszul complex provides a canonical projective resolution of the residue field $k$
  • The Koszul complex is used to compute Tor and Ext groups over polynomial rings
• Spectral sequences are powerful tools for computing homology and cohomology groups in various settings
  • The Grothendieck spectral sequence relates the composition of derived functors, such as Tor and Ext
  • The Serre spectral sequence is used to compute homology and cohomology of fiber bundles in algebraic topology
• Computer algebra systems, such as Macaulay2 and Singular, provide tools for computing resolutions, Tor, Ext, and other homological invariants
  • These systems are particularly useful for computations over polynomial rings and in algebraic geometry

Advanced Topics and Current Research

• Derived algebraic geometry is a rapidly developing field that combines homological algebra with algebraic geometry
  • Derived schemes and stacks provide a framework for studying geometric objects with non-trivial homological properties
• Topological Hochschild homology (THH) and topological cyclic homology (TC) are important invariants in algebraic K-theory and trace methods
  • THH and TC are defined using homological constructions and have connections to the Tor and Ext functors
• Homotopical algebra extends homological algebra to more general settings, such as model categories and ∞-categories
  • Homotopical methods have applications in algebraic topology, algebraic geometry, and representation theory
• Khovanov homology is a powerful knot invariant that categorifies the Jones polynomial
  • The construction of Khovanov homology relies on homological algebra and has led to significant advances in low-dimensional topology
• Persistent homology is a tool from applied algebraic topology used to study the shape and structure of data sets
  • Persistent homology computations involve homological algebra and have applications in data analysis and machine learning
• Categorification is a process of lifting algebraic structures to a categorical level, often using homological methods
  • Categorified quantum groups and knot invariants have been the subject of intense research in recent years
• Homological mirror symmetry is a conjectural relationship between the derived categories of coherent sheaves on a Calabi-Yau manifold and the Fukaya category of its mirror manifold
  • Homological mirror symmetry has deep connections to string theory and symplectic geometry