[Paper Review] What makes a multi-complex exact?

Satoshi Mochizuki, Seidai Yasuda|arXiv (Cornell University)|May 29, 2013

Commutative Algebra and Its Applications5 references3 citations

TL;DR

This paper establishes a sufficient condition for the total complex of an S-cube (a multi-complex indexed by the power set of a finite set) to be exact, generalizing the Buchsbaum-Eisenbud criterion for exactness of complexes over commutative rings. It introduces the notion of 'admissible cubes' via inductive monomorphism and cokernel conditions, and proves that if a cube admits a regular adjugate, its total complex satisfies the Fitting ideal grade condition, ensuring exactness.

ABSTRACT

In this paper, we give a sufficient condition which makes the total complex of a cube exact. This can be regarded as a variant of the Buchsbaum-Eisenbud theorem which gives a characterization of what makes a complex of finitely generated free modules exact in terms of the grade of the Fitting ideals of boundary maps of the complex.

Motivation & Objective

• To generalize the Buchsbaum-Eisenbud criterion for exactness of complexes to the setting of multi-complexes, specifically S-cubes.
• To define and characterize 'admissible cubes' as a categorical analog of 0-spherical complexes in the multi-complex setting.
• To establish a sufficient condition for the total complex of an S-cube to be exact using the Fitting ideal grade condition.
• To show that the existence of a regular adjugate for an S-cube implies that its total complex satisfies the Buchsbaum-Eisenbud condition.
• To apply the result to Koszul-type cubes and prove that if the product family of elements forms a regular sequence, then the original family is also a regular sequence.

Proposed method

• Define an S-cube as a contravariant functor from the power set of a finite set S to an abelian category.
• Introduce the notion of admissible cubes inductively: monomorphisms at each boundary map and cokernels forming admissible cubes in lower dimensions.
• Define a regular adjugate for an S-cube as a family of elements that satisfy specific compatibility and monomorphism conditions.
• Use the Fitting ideal construction for boundary maps of the total complex and relate their grades to the admissibility of the cube.
• Prove that if an S-cube admits a regular adjugate, then the total complex satisfies the Buchsbaum-Eisenbud grade condition.
• Apply the main theorem to the typical cube construction associated with a family of elements in a commutative ring, linking it to Koszul complexes and regular sequences.

Experimental results

Research questions

• RQ1 What condition ensures that the total complex of an S-cube is exact, generalizing the Buchsbaum-Eisenbud criterion?
• RQ2 How can the notion of a 0-spherical complex be extended to multi-complexes via the concept of admissible cubes?
• RQ3 What is the role of a regular adjugate in ensuring the exactness of the total complex of an S-cube?
• RQ4 How does the Fitting ideal grade condition relate to the admissibility of a cube in the multi-complex setting?
• RQ5 Under what conditions does the product of elements forming a regular sequence imply that the original elements also form a regular sequence?

Key findings

• An S-cube admits a regular adjugate if and only if its total complex satisfies the Buchsbaum-Eisenbud grade condition.
• If an S-cube is admissible, then its total complex is exact.
• The total complex of the typical cube associated with a family of elements in a commutative ring is isomorphic to the Koszul complex.
• If the product family {f_s g_s} forms an A-sequence and each f_s is non-invertible, then the original family {f_s} is also an A-sequence.
• The Fitting ideal grade condition for the total complex of an S-cube of free modules is equivalent to the cube being admissible when a regular adjugate exists.