# Which areas of arithmetic algebraic geometry can be learned as “black boxes” and are there any references where they are treated as such?

In Matthew Emerton's comment on Terry Tao's blog, he speaks about learning etale cohomology or the theory of Neron models as "black boxes". By this he means that you can learn what the theory is about and how to use it, without going into the detailed proofs of why they can be used.

Which theories (e.g. etale cohomology) can be learned as black boxes?

And where would one go (e.g. find lecture notes) to learn something like that?

Notes on something like this would ideally give you an idea of what is going on, give examples, and most importantly illustrate how they would be used to solve problems. I am mainly interested in arithmetic algebraic geometry and algebraic number theory, so I would especially like to know about "black boxes" in this direction, though "black boxes" in other areas might also be worth knowing about.

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Isn't that answered in the last paragraph of the question? – José Figueroa-O'Farrill Feb 15 '11 at 20:11
When I was student, Hironaka's theorem on the existence of resolutions of singularities was considered a black box by pretty much everyone. I suspect it is still not far too from the truth today, in spite of the many simplifications. – Donu Arapura Feb 15 '11 at 20:54
The formalism of Grothendieck's 6 operations (aka Voevsodsky's cross functors) and nearby/vanishing cycles constitutes a huge black box for Betti, étale, de Rham and p-adic and motivic (co)homology. Ayoub's thesis is a very complete SGA(if note EGA)-like reference. Deligne Cisinski's paper on triangulated categories of motives has more concise formulation. This includes the étale cohomology black box the Emerton was talking about: for example proprer base change theorem is the Exchange isomorphism $f'_!g'^* = g^*f_!$, while being Zariski local follows from $g^!f_* = f'_*g^!$ – AFK Feb 16 '11 at 1:42
One can "learn" anything to some extent as a black box. Learn definitions, statements of theorems, some applications, then try some new applications. But it is not as much fun as knowing how it works. I used Riemann Roch for years as a black box, but after reading Riemann, I understood it - what had been mysterious became clear. But life is short. Black box learning is easier surrounded by experts, absorbing useful knowledge by osmosis. 5 minutes with Hironaka helped more than staring at the paper for much longer. But to make real progress one must study too. – roy smith Feb 16 '11 at 3:45
I also think this site makes a huge contribution to black box learning. I often read the answers to other peoples questions here for the wonderful succinct expert answers so generously provided by many. – roy smith Feb 16 '11 at 3:47

I find Hodge theory pretty scary stuff with its compact inclusions of Sobolev spaces, pseudodifferential operators and parametrixes for elliptic differential operators. However it is very easy to use the results of Hodge theory as emanating from a black box. I remember how exhilarated I was by the argument that a Hopf surface, homeomorphic to $S^1 \times S^3$, could not be Kähler, and much less projective, just because its first Betti number is $b_1=1$. Whereas by Hodge theory a compact Kähler manifold $X$ has betti numbers $b_q(X)$ which are even whenever $q$ is odd.
it might be simpler to notice that $b_2=0$, so it cannot be symplectic :) – Pavol S. Feb 16 '11 at 16:32