Depth formula via complete intersection flat dimension

TL;DR

This paper proves the depth formula for complexes with finite complete intersection flat dimension, extending classical results to a broader homological context and providing new conditions for its validity.

Contribution

It establishes the depth formula for complexes with finite complete intersection flat dimension under specific Tor vanishing conditions, generalizing previous module-based results.

Findings

Depth formula holds for complexes with finite complete intersection flat dimension.

The result applies to modules with finite complete intersection flat dimension under Tor vanishing.

Provides new homological criteria for the depth formula in derived categories.

Abstract

We prove the depth formula, for homologically bounded complexes $X, Y$ provided that the complete intersection flat dimension of $X$ is finite and $\sup(X\utp_RY)<\infty$. In particular, let $M$ and $N$ are two $R$-modules and the complete intersection flat dimension of $M$ is finite. Then $M$ and $N$ satisfies the depth formula, provided $\Tor^R_i(M,N)=0$ for all $i\ge 1$.