A description of Mult_i(A¹,⋯,Aⁿ) by generators and relations

Thomas William Hungerford

Vol. 16 (1966), No. 1, 61–76

DOI: 10.2140/pjm.1966.16.61

Abstract

If R is a ring (with unit) and A_R,R¹A_R²,⋯,_RA_R,Rⁿ⁻¹Aⁿ are R-(bi)modules, then Mult_i^R,n(A¹,⋯,Aⁿ) is defined to be the i-th left derived functor of the multiple tensor product A¹ ⊗⋯ ⊗ Aⁿ(⊗ = ⊗_R); i.e., H_i(K¹ ⊗⋯ ⊗ Kⁿ), where each K^r is a projective resolution of A^r.

The purpose of this paper is to give a description of Mult_i^R,n(A¹,⋯,Aⁿ) in terms of generators and relations, analogous to that given by MacLane in the case n = 2 [and Mult_i = Tor_i^R(A¹,A²)].

Mathematical Subject Classification

Primary: 16.90

Secondary: 18.20

Milestones

Received: 2 July 1964

Published: 1 January 1966