The first set of questions can be found here: Understanding (the wiki page on) Verdier duality

I'm fairly confident that I understand something wrong, so I'll write down here clearly what my set of beliefs is about what is right, and you feel free to shoot down any falsehood:

Let $X$ be our geometric object, be it a topological space, variety, scheme, or what have you. I will do two cases, one Poincare duality, and the other Serre duality.

### Serre duality

I will assume $X$ is nice (a variety, projective, smooth,... of dimension $n$). Here I will look at the (abelian) category of coherent $O_X$-modules. In this case the "dualizing module" is $\omega_X[n]$. I take that to mean that $[\mathcal{E},\mathcal{F}]$ is dual to $[\mathcal{F},\mathcal{E}\otimes \omega_X[n]]$.

I also believe $H^k(X,\mathcal{F})\cong [O_X,\mathcal{F}[k]]$. Going from there, the rest is easy: $H^k(X,\mathcal{F})\cong [O_X,\mathcal{F}[k]]\cong [$dual of $\mathcal{F},O_X[k]]$, which is dual to $[O_X[k],\omega_X[n]\otimes$ dual of $\mathcal{F}]\cong [O_X, \omega_X \otimes$ dual of $\mathcal{F}[n-k]] \cong H^{n-k}(X,\omega_X \otimes $ dual of $\mathcal{F})$.

Great! However...

### Poincare duality

Again assume $X$ is nice (orientable compact smooth manifold of dimension $n$). Let $K$ be field. Here I will look at the (abelian) category of $K$-vector spaces. Here the "dualizing sheaf" is $K[-n]$. I will continue somewhat similarly to the Serre duality case. I interpret the dualizing sheaf as meaning that $[\mathcal{E},\mathcal{F}]$ is dual to $[\mathcal{F},\mathcal{E}\otimes K[-n]]$ (here $\mathcal{E}$ and $\mathcal{F}$ are $K$-vector spaces).

For whatever reason, I believe that $H_k(X,\mathcal{E})\cong[\mathcal{E}[-k],D_X]$ ($D_X$ is the dualizing sheaf in general. But what does this even mean in general? For example in the Serre duality case, what would $H_k(X,\mathcal{F})$ even mean?). So:

$H_k(X,K) \cong [K[-k],K[-n]] \cong [K,K[k-n]] \cong H^{k-n}(X,K)$. Wait. What? Makes no sense!

If we use the "duality" it still makes no sense:

$[K,K[k-n]] \cong [K[k-n],K[-n]] \cong [K,K[-k]]\cong H^{-k}(X,K)$. What? Huh?

So in conclusion, I desperately want to have a handle on this, but I clearly don't. Hopefully just a nudge in the right direction would lead me to a better understanding of this yoga.

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