Dimension of Tensor Product for Flat Extensions

Question

Suppose that $A,B,$ and $C$ are commutative unital rings, $A\to B$ is flat, and $A\to C$ is any map. I am trying to determine whether $$ \dim B\otimes_AC=\dim B+\dim C-\dim A $$ Any counterexamples or references? I am taking the Krull dimension of the zero ring to be -1 (in case a tensor product is 0).

Thanks for any feedback.

Edit: By $\dim$ I mean Krull dimension.

can one add assumptions like "finite"? – user 1 Mar 23 '15 at 07:15 — user 1, Mar 23 '15 at 07:15

score 4 · Accepted Answer · answered Mar 23 '15 at 08:28

Take $A=k,$ a field and $B=k(x), C=k(y)$ where $x,y$ are transcendental over $k.$ Then the ring $k(x)\otimes_kk(y)$ is an one dimensional ring (Qing Liu, Algebraic Geometry and Arithmetic Curves, Ch. 3, Ex. 1.9) and this clearly contradicts the stated equality.

score 4 · Answer 2 · answered Mar 23 '15 at 09:02

4

Take $A=\mathbb Z,$ and $B=C=\mathbb Q$. Then $B\otimes_AC=\mathbb Q$ and your formula gives: $0=0+0-1$, absurd.

answered Mar 23 '15 at 09:02

user26857

52,094

This is a far better example than the one I gave. Very nice and elementary. – Krish Mar 23 '15 at 09:48

Dimension of Tensor Product for Flat Extensions

2 Answers2