First some context. In most algebraic number theory textbooks, the notion of discriminant and different of an extension of number fields $L/K$, or rather, of the corresponding extension $B/A$ of their rings of algebraic integers is defined. The discriminant, an ideal of $A$, is the ideal generated by the discriminant of the quadratic form $\text{tr}(xy)$ on $B$. The different, an ideal of $B$, is the inverse of the fractional ideal $c$ of $L$ defined by $c=\{x \in L, \text{tr}(xy) \in A \ \forall y \in B\}$. The norm of the different is the discriminant.

Now the discriminant makes sense in a much more general context, say for any extension of (commutative) rings $B/A$ that is finite projective, since the trace map $\tr$ makes sense in this context. My question is: is there a standard definition of the different in this context? if so, where can I find it in the literature, if possible with the basic results about it?

I am pretty sure the answer to the first question is yes, but I have not been able to find a reference. The problem when I try to use google or MathSciNet seems to be that "different" is not a very discriminant name: almost every paper in mathematics contains it.

Let me propose an answer to my own question: we could define the different of $B/A$ by the Fitting ideal of the universal $B$-modules of differentials $\Omega_{B/A}$. The fact that it gives the correct definition in the number field cases is [Serre, Local Fields, chapter III, Prop. 14], and moreover it behaves well under base change. This definition may very possibly be a remembrance of something I had heard in an earlier life. But even if it is the correct definition, I'd like to know a reference where it is stated.

First some context. In most algebraic number theory textbooks, the notion of discriminant and different of an extension of number fields $L/K$, or rather, of the corresponding extension $B/A$ of their rings of algebraic integers is defined. The discriminant, an ideal of $A$, is the ideal generated by the discriminant of the quadratic form $\text{tr}(xy)$ on $B$. The different, an ideal of $B$, is the inverse of the fractional ideal $c$ of $L$ defined by $c=\{x \in L, \text{tr}(xy) \in A \ \forall y \in B\}$. The norm of the different is the discriminant.

Now the discriminant makes sense in a much more general context, say for any extension of (commutative) rings $B/A$ that is finite projective, since the trace map $\operatorname{tr}$ makes sense in this context. My question is: is there a standard definition of the different in this context? if so, where can I find it in the literature, if possible with the basic results about it?

I am pretty sure the answer to the first question is yes, but I have not been able to find a reference. The problem when I try to use google or MathSciNet seems to be that "different" is not a very discriminant name: almost every paper in mathematics contains it.

Let me propose an answer to my own question: we could define the different of $B/A$ by the Fitting ideal of the universal $B$-modules of differentials $\Omega_{B/A}$. The fact that it gives the correct definition in the number field cases is [Serre, Local Fields, chapter III, Prop. 14], and moreover it behaves well under base change. This definition may very possibly be a remembrance of something I had heard in an earlier life. But even if it is the correct definition, I'd like to know a reference where it is stated.