CATENARY, LOCALLY EQUIDIMENSIONAL, AND TENSOR PRODUCTS OF ALGEBRAS

ABSTRACT

In this paper, we show that the property of being catenary and locally equidimensional descends by flat homomorphism. More precisely, if ϕcolonR → S is a flat homomorphism of Noetherian rings, then S is catenary and equidimensional if R is locally equidimensional, and the rings R/𝔭 ⊗  _R S, 𝔭 ∈ Min R, are catenary and locally equidimensional. Let k be a field, A a k-algebra, and K an extension field of k. Then we show that the K ⊗  _k A is universally catenary if one of the following holds:

	A is universally catenary and K a finitely generated extension field of k;
	A is Noetherian universally catenary and t.d.(K:k) < ∞;
	A is universally catenary and K ⊗  k A is Noetherian.

Key Words:

• Catenary ring
• Equidimensional ring
• Tensor product of algebras

Mathematics Subject Classification 2000:

• 13C15

ACKNOWLEDGMENT

The authors were visiting the Abdus Salam International Centre for Theoretical Physics (ICTP) during the preparation of this paper. They would like to thank the ICTP for its hospitality during their stay there. The authors would also like to thank H. Haghighi from K. N. Toosi Technical University for offering some serious comments. The first author is supported by a grant from IPM (NO. 82130213). The second author is supported by a grant from IPM (NO. 82130212).

Notes

^#Communicated by J. Kuzmanovich.