11.2 Koszul complexes

Koszul complexes are powerful tools in commutative algebra, linking sequences of ring elements to homological properties. They're built using exterior algebras and help us understand regular sequences, which are crucial for studying ideals and modules.

These complexes shine when it comes to minimal free resolutions, especially for modules defined by regular sequences. They let us calculate important stuff like depth, projective dimension, and Betti numbers, giving us a deeper look into the structure of rings and modules.

Koszul Complexes and Regular Sequences

Construction and Properties of Koszul Complexes

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• Koszul complex is a chain complex associated with a sequence of elements $x_1, \ldots, x_n$ in a commutative ring $R$
  • Constructed using the exterior algebra $\Lambda_R(R^n)$ on a free $R$-module of rank $n$
  • Differential is defined using the sequence elements and the exterior algebra multiplication
• Regular sequence is a sequence $x_1, \ldots, x_n$ in a commutative ring $R$ such that $x_i$ is a non-zero-divisor in $R/(x_1, \ldots, x_{i-1})$ for each $i$
  • Important property for studying the homological behavior of ideals and modules
  • Examples include the variables in a polynomial ring (regular sequence) and the maximal ideal in a regular local ring (regular sequence)
• Exterior algebra $\Lambda_R(M)$ over an $R$-module $M$ is the quotient of the tensor algebra by the ideal generated by elements of the form $m \otimes m$ for $m \in M$
  • Graded algebra with a wedge product that is alternating and satisfies the graded Leibniz rule
  • Used in the construction of the Koszul complex and its differentials

Minimal Free Resolutions and Koszul Complexes

• Minimal free resolution of an $R$-module $M$ is an exact sequence of free $R$-modules that resolves $M$ with the property that each differential matrix has entries in the maximal ideal
  • Encodes homological information about the module, such as its projective dimension and Betti numbers
  • Koszul complex provides a minimal free resolution for cyclic modules defined by a regular sequence
• Koszul complex associated with a regular sequence $x_1, \ldots, x_n$ is a minimal free resolution of the quotient ring $R/(x_1, \ldots, x_n)$
  • Acyclicity of the Koszul complex is equivalent to the sequence being regular
  • Allows for the computation of invariants like depth, projective dimension, and Betti numbers for quotients by regular sequences

Koszul Homology and Cohomology

Definitions and Basic Properties

• Koszul homology $H_*(x_1, \ldots, x_n; R)$ is the homology of the Koszul complex associated with a sequence $x_1, \ldots, x_n$ in a commutative ring $R$
  • Measures the deviation of the sequence from being a regular sequence
  • Vanishes if and only if the sequence is regular
• Koszul cohomology $H^*(x_1, \ldots, x_n; R)$ is the cohomology of the Koszul complex, obtained by applying the functor $\operatorname{Hom}_R(-, R)$ to the Koszul complex
  • Dual notion to Koszul homology, with analogous properties and interpretations
  • Provides a cohomological characterization of regular sequences

Applications and Interpretations

• Betti numbers of a module $M$ over a local ring $R$ are the ranks of the free modules appearing in a minimal free resolution of $M$
  • Measure the complexity of the module and its homological behavior
  • Can be computed using the Koszul homology or cohomology when the module is defined by a regular sequence
• Koszul homology and cohomology have connections to other invariants and constructions in commutative algebra, such as:
  • Depth and projective dimension of modules
  • Local cohomology and local duality
  • Gorenstein rings and complete intersections (Koszul homology vanishes for complete intersection ideals)
• Koszul complexes and their (co)homology provide a bridge between the algebraic properties of sequences and the homological properties of modules and rings
  • Enable the study of regular sequences, which play a central role in commutative algebra
  • Allow for the computation of important invariants and the understanding of structural properties of rings and modules