Let $X$ be a scheme with finitely many irreducible components $V\_1,\dots,V\_r\subset X$. Its normalization $X^\nu\to X$ is the morphism obtained as follows: $$X^\nu=\coprod\_iV\_i^\nu\to X\_{red}\to X.$$ Here $V\_i^\nu$ is the normalization of $V\_i$. They are the irreducible components of $X^\nu$.


However, there is an intermediate scheme, the *semi-normalization* of $X$, which fits in $$X^\nu\to X^{sn}\to X.$$ There are of course factorizations $$V\_i^\nu\to V\_i^{sn}\to V\_i.$$



> 
> *Question*. Is it true that $V\_i^{sn}$ are the irreducible components of
>  $X^{sn}$?
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> 
> 


Thanks!



Irreducible components of the seminormalization
I think your question is equivalent to this one: 



> 
> Is it true that the irreducible components of a seminormal scheme are themselves seminormal?
> 
> 
> 


A curve is seminormal if it is locally analytically isomorphic to the coordinate axis in an affine space. Any subset of that has the same property (for a smaller dimensional space) so a partial answer is



> 
> Yes, if $\dim X=1$.
> 
> 
> 


However, this fails in higher dimensions, because there exist reducible seminormal schemes whose irreducible components are not seminormal. The examples are not entirely simple. See (7.9.3) and (10.12) of [the reference below](https://books.google.com/books/about/Singularities_of_the_Minimal_Model_Progr.html?id=QwgQZHvESFQC&source=kp_cover&hl=en). So the answer is



> 
> No, in general.
> 
> 
> 


On the other hand, there is an almost obvious Chinese remainder theorem type exact sequence connecting the seminormalization of the scheme, the seminormalization of its irreducible components and the semi-normalization of the pairwise intersections of the irreducible components. (I suppose one can probably continue that sequence with more intersections, so it's perhaps more like a Cech resolution type exact sequence).


A good resource for the statements I mentioned above is section 10.2 of [this book](https://books.google.com/books/about/Singularities_of_the_Minimal_Model_Progr.html?id=QwgQZHvESFQC&source=kp_cover&hl=en) by Kollár.


