A simplicial complex K is d-Leray if all reduced homology groups
H_i(L)vanish when i is equal or greater
than d for every induced subcomplex L or K.

I will discuss d-Leray complexes and how they appear in combinatorial
geometry problems, and present the
following recent result, which in a
commutative algebra formulation, settles a
conjecture by Terai.

If K is d-Leray and L is r-Leray than their union is (r+d+1)-Leray and
their intersections is (r+d)-Leray.