The dimension of tensor products of k-algebras arising from pullbacks

All rings and algebras considered in this paper are commutative with identity elements and, unless otherwise specified, are to be assumed to be non-trivial. All ringhomomorphisms are unital. Let k be a field. We denote the class of commutative k−algebras with finite transcendence degree over k by C. Also, we shall use t.d.(A) to denote the transcendence degree of a k−algebra A over k, A[n] to denote the polynomial ring A[X1, ..., Xn], and p[n] to denote the prime ideal p[X1, ..., Xn] of A[n], where p is a prime ideal of A. Recall that an integral domain R of finite (Krull) dimension n is a Jaffard domain if its valuative dimension, dimv(R), is also n. Prüfer domains and noetherian domains are Jaffard domains. We assume familiarity with this concept, as in [1], [6] and [10]. Suitable background on pullbacks is [4], [11], [12] and [16]. Any unreferenced material is standard, as in [12] and [17].