2 added 2 characters in body; added 4 characters in body; deleted 5 characters in body

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.

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 $\mathcal{O}X$ the structure sheaf at $(i+1)$-many basis elements in $V^\vee$. Each of the terms in the Cech complex are $\underline{f*}$-acyclic, direct-image-acyclic, so $\underline{f_*}(\mathcal{C}^)$ \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. 1 # Is there a direct way to compute the higher derived image sheaves of a family of$\mathbb{P}^n$s? 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$ \mathcal{O}X $at$(i+1)$-many basis elements in$V^\vee$. Each of the terms in the Cech complex are$\underline{f*}$-acyclic, so$\underline{f_*}(\mathcal{C}^)$computes the derived direct image. If$(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.