Let $A\to B$ be a morphism of (unitary commutative) rings such that $B$ is module-finite over $A$ and there exists $f\in A$ which is a nonzerodivisor in $A$ and in $B$, with $A[1/f]\to B[1/f]$ an isomorphism.


My question is: is the diagram $A\to B \rightrightarrows B\otimes\_A B$ exact ?


In geometric terms, this is more or less equivalent to the question whether descent of functions along a finite, scheme-theoretically birational morphism is possible.


Here are some comments:


1. Of course, I'm mainly interested in the non-flat case (otherwise it's just faithfully flat descent).
2. I computed several examples, including the normalization of the scheme formed by $n$ lines meeting in the origin in $\mathbf{A}^2$ (i.e. the scheme $V(x^n-y^n)\subset\mathbf{A}^2$); and the resolution $k[x^n,y^n,xy]\to k[x,y]$ of a plane quotient singularity. Any time, the answer is yes.
3. I suspect that the result might be known and maybe present somewhere in the literature. The question is related also to descent of modules for universally injective morphisms, as exposed in the Stacks Project [here](http://stacks.math.columbia.edu/tag/08WE "here") but I could not find there something really close to the statement I'm after.


Thanks for any help!


EDIT (Jan 1st, 2016): the claim that the normalization of the scheme formed by $n$ lines meeting in the origin in $\mathbf{A}^2$ is an example is not true if $n>2$. Explanations are given in [this MO answer](https://mathoverflow.net/questions/225160/pushing-forward-morphisms/225214#225214).



Descent of functions along finite birational morphisms
This is not always true. For example, it fails for the inclusion $k[x^3,x^5]\to k[x]$, where $k$ is a field. This is because $x^7$ is in the equalizer: $x^7\otimes 1 = x^2\otimes x^5=x^5\otimes x^2 = 1\otimes x^7$. 


Such examples are indeed in the literature. I saw this one in SGA3, exp V.


