Let $V\rightarrow Y$ be a vector bundle of rank $n+1$ over $Y$, with $Y$ reasonably nice (I care about the case of smooth, irreducible affine). Let $X=\mathbb{P}(V)$ be the projectivization of $V$, so that $f:X\rightarrow Y$ is a flat family of $\mathbb{P}^n$s.
The Serre twists in each fiber fit together into a global Serre twist, which enables graded versions of functors. Define the graded direct image of a sheaf $\mathcal{M}$ on $X$ to be the graded sheaf on $Y$
$$ \underline{f_*}(\mathcal{M}):= \bigoplus_{i\in \mathbb{Z}} f_*(\mathcal{M}(i))$$
This is still a left exact functor, so I can right derive it. I want to compute the derived direct image of the structure sheaf
$$ \mathbb{R}\underline{f_*}(\mathcal{O}_X) $$
I know what it should be.
- The group $\underline{f_*}(\mathcal{O}_X)$ should be $Sym_Y(V^\vee)$, the sheaf of graded algebras on $Y$ which is the symmetric algebra of the dual vector bundle $V^*$ over $V$.
- The group $\mathbb{R}^n\underline{f_*}(\mathcal{O}_X)$ should be $(Sym_Y(V^\vee))^\vee(-n-1)$, the graded dual to the graded sheaf of modules $Sym_Y(V^\vee)$. This means that in graded degree $-n-1$, we have $(Sym_Y(V^\vee))_0^\vee$, in graded degree $-n-2$, we have $(Sym_Y(V^\vee))_1^\vee$, etc. In particular, it vanishes in graded degree $-n$ and above. (This group might need to be tensored with a line bundle on $Y$, I'm not entirely sure)
- All other $\mathbb{R}^i\underline{f_*}(\mathcal{O}_X)$ should be zero.
The Absolute Case
I know how to compute this in the absolute case, when $Y$ is a point. There, I can choose a basis for $V^\vee$, which produces a Cech complex $\mathcal{C}$ which in each degree i is the direct sum over all localizations of the structure sheaf at $(i+1)$-many basis elements in $V^\vee$. Each of the terms in the Cech complex are direct-image-acyclic, so $\underline{f_*}(\mathcal{C}^\vee)$ computes the derived direct image.
If $(x_0,...x_n)$ is a basis for $V^\vee$, then $Sym_Y(V^\vee)=\mathbb{C}[x_0,...x_n]$, and the ith term of $\underline{f_*}(\mathcal{C})$ is isomorphic to
$$ \bigoplus_{(j_1,...,j_i)\subset (0,..n)} \mathbb{C}[x_0,...x_n,x_{j_1}^{-1},...x_{j_i}^{-1}] $$
There is then a natural map $$ \mathbb{C}[x_0,...x_n]\rightarrow \bigoplus_{j\in (0,..n)} \mathbb{C}[x_0,...x_n,x_j^{-1}] $$ given by the alternating sum over the natural inclusions. There is also a natural pairing $$ \mathbb{C}[x_0^{\pm1},...x_n^{\pm1}]\times \mathbb{C}[x_0,...x_n]\rightarrow \mathbb{C}(-n-1) $$ which takes $f\in \mathbb{C}[x_0^{\pm1},...x_n^{\pm1}]$ and $g\in \mathbb{C}[x_0,...x_n]$ to $res(fg)$, where $res$ is the coefficient of the monomial $x_0^{-1}x_1^{-1}...x_n^{-1}$ (which is a degree $-n-1$ element). This natural pairing gives an adjoint map $$ \mathbb{C}[x_0^{\pm1},...x_n^{\pm1}]\rightarrow (\mathbb{C}[x_0,...x_n])^\vee(-n-1) $$
Then, some computation yields that the composition of these two maps is a distinguished triangle in the derived category. $$ \mathbb{C}[x_0,...x_n]\rightarrow \underline{f_*}(\mathcal{C})\rightarrow (\mathbb{C}[x_0,...x_n])^\vee(-n-1)[-n] $$ This establishes the above structure of $\mathbb{R}\underline{f_*}(\mathcal{O}_X)$ in this case.
More Generally
For $Y$ not a point, it is still possible to perform an identical computation whenever the vector bundle $V$ is trivial. However, I am running into difficulty when $V$ is not trivial. The structure of the Cech complex depends very much on the choice of a basis, so I can't seem to patch together local results.
When $V$ is generated by global sections, I can come up with an analogous Cech complex where I localize by subsets of some basis of global sections. However, outside the trivial case, this Cech complex will be longer than I want, and there doesn't seem to be a nice residue map like above.
Note! I do not want to use Serre duality/Grothendieck duality, because the above computation of $\mathbb{R}\underline{f_*}(\mathcal{O}_X)$ seems to be the starting point for most proofs of these dualities.
My interest is in a nearby non-commutative version of this question, and I am trying to use an analog of the above computation to prove the corresponding duality theorem.
