Hilbert's Syzygy Theorem in the bigraded case I've been recently wondering how to prove the existence of a Hilbert polynomial for finitely generated bigraded modules $M$ over a polynomial ring $R=k[X_0,...,X_n,Y_0,...,Y_m]$ with the usual bigraded structure; concretely, is there a polynomial $P\in \mathbb{Q}[T,S]$ such that $P(a,b)=\text{dim}_k M_{a,b}$ for all sufficiently large $a,b\in \mathbb{Z}$?
I know that the usual proof of this fact for the graded case uses the existence of a finite free resolution of the module. So I've been trying to apply the same technique for the bigraded case: I take a resolution of $M$ of the form $$ 0\rightarrow K \rightarrow L_{n+m+1}\rightarrow ... \rightarrow L_{0}\rightarrow M\rightarrow 0$$ where $L_i$ are finitely generated free modules (now, free in the sense of having a basis of bihomogeneous elements), $K$ is finitely generated and all the maps are compatible with the bigrading and of bidegree (0,0). We can view this resolution as a graded one (take, for instance, $M_r=\sum_{a+b=r} M_{a,b}$ as the $r$-degree piece of $M$), and then Hilbert's Syzygy Theorem asserts that $K$ is also free $\textbf{as graded module}$. I've trying to prove that in fact we can take a basis of $K$ consisting of bihomogeneous elements, but I didn't succeed (in fact I don't even know if what I'm trying to prove is true). So here's the question:
Can we assure that a finite free resolution of this sort exists for bigraded modules (that is, in which every term is freely generated by bihomogeneous elements)? Is there an analogous Hilbert Syzygy Theorem for bigraded modules? And finally (and importantly!), can this be extended to the multigraded case?
Thank you all a lot in advance.
 A: There are several questions above.  The answer below gives one multigraded finite free resolution of multigraded finitely generated modules over the multigraded polynomial ring that are saturated, i.e., equal to the multigraded module of a coherent sheaf on the associated product of projective schemes.  Since the Hilbert polynomial of a finitely generated multigraded module equals the Hilbert polynomial of its saturation, this suffices for the application to multigraded Hilbert polynomials.
One useful result is Beilinson's resolution of the diagonal.  Since this is completely functorial, it helps with these types of questions.  Let $V$ be a $k$-vector space of finite dimension $n$.  Let $r$ be an integer with $0<r<n$. Denote by $$(G\to \text{Spec}\ k,\ q^\dagger:V^\vee\otimes_k \mathcal{O}_G \twoheadrightarrow S^\vee)$$ a universal pair of a $k$-scheme together with a surjective homomorphism of locally free sheaves whose target has rank $r$.  The adjoint homomorphism fits into a short exact sequence of locally free sheaves, $$0 \to S \xrightarrow{q} V\otimes_k \mathcal{O}_G \xrightarrow{p} Q \to 0.$$ On the self-product in the category of $k$-schemes, $G\times G$, there is an associated homomorphism of locally free sheaves, $$\text{pr}_1^*p\circ \text{pr}_2^*q:\text{pr}_2^*S \to V\otimes_k \mathcal{O}_{G\times G} \to \text{pr}_1^* Q.$$ By adjointness of Hom and tensor product, this is equivalent to a homomorphism of locally free sheaves, $$\alpha:\mathcal{E}
 \to \mathcal{O}_{G\times G}, \ \ \mathcal{E}:= \text{pr}_1^*Q^\vee \otimes_{\mathcal{O}_{G\times G}} \text{pr}_2^*S.$$ Beilinson's observation is that the image of $\alpha$ is precisely the ideal sheaf of the diagonal in $G\times G$, which is itself everywhere locally cut out by a regular sequence of length $m=\text{dim}\ G = \text{rank}\ \mathcal{E}=r(n-r)$.  Thus, the Koszul complex of $\alpha$ gives a finite, locally free resolution of the structure sheaf of the diagonal, $$\left( K_\ell(\alpha) := \bigwedge^\ell_{\mathcal{O}_{G\times G}}\mathcal{E} \right)_{0\leq \ell \leq m}, \ \ (d_\ell:K_\ell(\alpha) \to K_{\ell-1}(\alpha))_{1\leq \ell \leq m}. $$
Note that the exterior power $\bigwedge^\ell \ \mathcal{E}$ has a direct sum decomposition as a tensor power of Schur functors $\text{pr}_1^*\mathbb{S}_{\lambda}(Q^\vee)\otimes_{\mathcal{O}_{G\times G}}\text{pr}_2^*\mathbb{S}_{\lambda'}(S),$ where $\lambda$ runs over all partitions of $\ell$ into integers no greater than $r$ and with no more than $n-r$ parts.
The same holds for any product of projective spaces, i.e., for an integer $\rho \geq 1$ and an ordered $\rho$-tuple of ordered pairs $(V_i,r_i)$ as above, for the product scheme $X=G_1\times \dots \times G_\rho$ with its coordinate projections $\pi_i:X\to G_i$, on the product $k$-scheme $X\times X$, one locally free resolution of the structure sheaf of the diagonal is the Koszul complex of the homomorphism of locally free sheaves, $$\beta: \mathcal{F} \to \mathcal{O}_{X\times X}, \ \ \mathcal{F}:= \bigoplus_{i=1}^\rho (\pi_i\circ \text{pr}_1,\pi_i\circ \text{pr}_2)^*\mathcal{E}_i ,$$
whose components are the maps $(\pi_i\circ \text{pr}_1,\pi_i\circ \text{pr}_2)^*\alpha_i$.  Note that each locally free sheaf $\bigwedge^\ell \mathcal{F}$ has a canonical direct sum decomposition into tensor products of pullbacks of locally free sheaves $\bigwedge^{\ell_i} \mathcal{E}_i$, indexed by all ordered $\rho$-tuples of nonnegative integers $(\ell_1,\dots,\ell_\rho)$, such that $\ell_1+\dots+\ell_\rho$ equals $\ell$ and each $\ell_i\leq m_i$.
One application of this is the following.
Theorem [Beilinson resolution].  Let $\mathcal{G}$ be a coherent sheaf on $X$ such that for every ordered $\rho$-tuple $(\ell_1,\dots,\ell_\rho)$ as above, all higher cohomology vanishes for the tensor product of $\mathcal{G}$ and the pullbacks $\pi_i^*\mathbb{S}_{\lambda_i}(Q^\vee_i)$, where each $\lambda_i$ is a partition of $\ell_i$ into integers no greater than $r_i$ with at most $n_i-r_i$ parts.  Then the complex $\text{pr}_2^*(K_\bullet(\beta)\otimes_{\mathcal{O}_{X\times X}}\text{pr}_1^*\mathcal{G})$ is a locally free resolution of $\mathcal{G}$ whose terms are direct sums of tensor products of locally free sheaves $\mathbb{S}_{\lambda'_i}(S_i)$.  This resolution is functorial in $\mathcal{G}$.
A particularly important case is when each $G_i$ is just a projective space, i.e., every $r_i$ equals $1$.  In this case, the terms of the locally free resolution above are direct sums of tensor products of invertible sheaves $\pi_i^*\mathcal{O}_{\mathbb{P} V_i}(-d_i)$ for $0\leq d_i < n_i$.  Since the tensor product of $\mathcal{G}$ with the invertible sheaves $\pi_i^*\mathcal{O}_{\mathbb{P} V_i}(e_i)$ satisfies the hypothesis above for all integers $e_i \gg 0$, this gives the following corollary.
Corollary.  On a product of projective spaces, $X=\mathbb{P} V_1 \times \dots \mathbb{P} V_\rho$, for every coherent sheaf $\mathcal{G}$, for all ordered $\rho$-tuple of integers $(e_1,\dots,e_\rho)$ with $e_i\gg 0$, there is a functorial locally free resolution of the tensor product of $\mathcal{G}$ and the invertible sheaves $\pi_i^*\mathcal{O}_{\mathbb{P} V_i}(e_i)$ as above.  After tensoring this locally free resolution with the tensor product of the invertible sheaves $\pi_i^*\mathcal{O}_{\mathbb{P} V_i}(-e_i)$, this gives a functorial locally free resolution of $\mathcal{G}$ whose terms are direct sums of tensor products of invertible sheaves $\pi_i^*\mathcal{O}_{\mathbb{P} V_i}(-d_i-e_i)$ with $0\leq d_i < n_i$.
Translating this back into the language of multigraded modules over the multigraded homogeneous coordinate ring of $X$, this gives a functorial finite free resolution by direct sums of shifts of the multigraded homogeneous coordinate ring.
As mentioned in my comments above, one needs much less if one only needs to prove the existence of a multigraded analogue of the Hilbert polynomial.  However, the functoriality in the Beilinson resolution is frequently useful.
