Is there a direct way to compute the higher derived image sheaves of a family of $\mathbb{P}^n$s? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T23:27:22Z http://mathoverflow.net/feeds/question/27194 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27194/is-there-a-direct-way-to-compute-the-higher-derived-image-sheaves-of-a-family-of Is there a direct way to compute the higher derived image sheaves of a family of $\mathbb{P}^n$s? Greg Muller 2010-06-05T21:07:18Z 2010-06-05T21:12:42Z <p>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.</p> <p>The Serre twists in each fiber fit together into a global Serre twist, which enables graded versions of functors. Define the <strong>graded direct image</strong> of a sheaf $\mathcal{M}$ on $X$ to be the graded sheaf on $Y$</p> <p>$$\underline{f_*}(\mathcal{M}):= \bigoplus_{i\in \mathbb{Z}} f_*(\mathcal{M}(i))$$</p> <p>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</p> <p>$$\mathbb{R}\underline{f_*}(\mathcal{O}_X)$$</p> <p>I know what it should be.</p> <ul> <li>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 $V$.</li> <li>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)</li> <li>All other $\mathbb{R}^i\underline{f_*}(\mathcal{O}_X)$ should be zero.</li> </ul> <h2>The Absolute Case</h2> <p>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.</p> <p>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 </p> <p>$$\bigoplus_{(j_1,...,j_i)\subset (0,..n)} \mathbb{C}[x_0,...x_n,x_{j_1}^{-1},...x_{j_i}^{-1}]$$</p> <p>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 $$\mathbb{C}[x_0^{\pm1},...x_n^{\pm1}]\rightarrow (\mathbb{C}[x_0,...x_n])^\vee(-n-1)$$</p> <p>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.</p> <h2>More Generally</h2> <p>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. </p> <p>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.</p> <p>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.</p> <p>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.</p>