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On Koszul complex of a supermodule

Darío Sánchez Gómez, Fernando Sancho de Salas

Abstract

This paper is devoted to an exposition of the Koszul complex of a supermodule and its Berezinian from an intrinsic and as general as possible point of view. As an application, an analogue to Bott's formula in the supercommutative setting computing the cohomology of twisted differential p-forms sheaves on the projective superspaces is given.

On Koszul complex of a supermodule

Abstract

This paper is devoted to an exposition of the Koszul complex of a supermodule and its Berezinian from an intrinsic and as general as possible point of view. As an application, an analogue to Bott's formula in the supercommutative setting computing the cohomology of twisted differential p-forms sheaves on the projective superspaces is given.
Paper Structure (13 sections, 28 theorems, 173 equations)

This paper contains 13 sections, 28 theorems, 173 equations.

Table of Contents

  1. Generalities on supercomutative algebra
  2. Supermodules
  3. Superalgebras
  4. Kahler differentials and De Rham complex
  5. Superschemes and the relative dualizing complex on superschemes
  6. De Rham and Koszul complexes of a supermodule
  7. Acyclicity of super Koszul complex and super De Rham complex
  8. Universality of Koszul complex
  9. The Berezinians
  10. The Berezin function
  11. The super Bott formula
  12. Cohomology of projective superspaces
  13. Cohomology of Bott's super formula

Key Result

Proposition 1.8

Theorems & Definitions (74)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Example 1.4
  • Example 1.5
  • Definition 1.6
  • Definition 1.7
  • Proposition 1.8
  • Definition 1.9
  • Example 1.10
  • ...and 64 more