A small question on commutative algebra

Question

Recently I read the book Intersection Theory by Fulton. I think one property in his book relies on this commutative algebra conclusion. I'm not sure whether it is right. Assume all rings are of finite type over $K$. Given $A\to B\to B/I$, where $B$ and $B/I$ are both flat over $A$. $I$ is generated by a regular sequence of lenth $d$ . And $A\to C$ any other homomorphism. Question: is the ideal $C\otimes _AI$ of $C\otimes_AB$ generated a regular sequence of lenth $d$ ? Thank you for your consideration.

Answer 1

I think it is right. Write down the Koszul complex for the regular sequence, which generates $I$. THe regularity of the sequence, is equivalent to vanishing of all the homology groups of the complex, except for $H_0$, which is isomorphic to $B/I$. Tensoring with $C$ over $A$, you obtain the Kozsul complex for the same sequence but in $C\otimes_A B$. Since $B$ is $A$-flat, then again by looking at the homology, you deduce that the sequence remains regular in $C\otimes_A B$. The flatness of $B/I$ over $A$ implies that the ideal generated by the sequence in $C\otimes_A B$ is equal to $C\otimes_A I$.

Edit: unknown is right, i have been spending too much time with local rings. However, the argument together with matsumura Thm 16.8 on p.131 shows that the length of a maximal regular sequence in $C\otimes_A I$ is equal to the length of a regular sequence in $I$.

Edit2: After a while, I had actually decided to look at what Fulton writes. In his appendix A.5 he talks not of regular sequences, but of regular sections, which are defined in terms of vanishing of Kozsul complex, and basically my proof is his Lemma A.5.3.

Answer 2

but it is notably that tHe regularity of the sequence, is NOT equivalent to vanishing of all the homology groups of the complex, except for H0! For examples see Mastumura or the appendix of the book mensioned above! There are sequence that are not regular but vanishing of all the homology groups of the complex, except for H0 holds! And there are examples that a permutation of a regular sequence is not a regular sequence!