Irreducible components of affine Variety

[tqgnaruto]

I am having problems finding the irreducible components of the variety
W=V(x^2-y^2, y^2-z^2)
the 1st part gives x+y, x-y, the second y+z, y-z, but I am pretty sure they are connected!

 

[disregardthat]

So you have $$V(x^2-y^2,y^2-z^2) = V(x^2-y^2) \cap V(y^2-z^2) = \left(V(x-y) \cup V(x+y) \right) \cap \left(V(y-z) \cup V(y+z) \right)$$

Do some elementary set theory, and you should arrive at the components. They will be lines.

 

[mathwonk]

or just look at the equations as x^2 = y^2 = z^2. the solutions look pretty simple.