# Derived functor giving algebraic map between moduli

Let $$X$$ be a smooth variety and $$M$$ a fine moduli space of certain kind of sheaves on $$X$$. Let $$\mathcal{E}$$ be the universal family on $$X\times M$$. Suppose there is a derived functor $$F$$ from $$D^b(X)$$ to some derived category, say $$D^b(Y)$$, or $$D^b(A-mod)$$ for some finite algebra $$A$$, so that $$F(\mathcal{E}_t)$$ is a semistable object of some kind (say semistable sheaves on $$Y$$ or semistable rep of $$A$$). This provides a map $$f:M \to N$$ where $$N$$ is the moduli of semistable objects on the target category.

My question: is this map always algebraic or is there any criterion for this map to be algebraic?

Thank you very much for your help.

If $$F$$ is a Fourier-Mukai functor, it can be applied not just fiberwise, but to the whole family. Namely, if $$K \in D^b(X\times Y)$$ is the Fourier-Mukai kernel, then by pullback it gives an object on $$X \times Y \times M = (X \times M) \times_M (Y \times M) \subset (X \times M) \times (Y \times M).$$ The corresponding Fourier-Mukai functor $$D^b(X \times M) \to D^b(Y \times M)$$ applied to $$\mathcal{E}$$, gives an object $$\mathcal{F} \in D^b(Y\times M)$$ with fibers $$F(\mathcal{E}_t)$$, which are by assumption stable sheaves. THerefore, the object $$\mathcal{F}$$ gives an algebraic map $$M \to N$$.
• Thank you for your answer. One follow up question: if the functor is a fully faithfull embedding of D^b(A-mod) into D^b(X) for a finite algebra $A$ and smooth $X$, is there a notion of Fourier- Mukai functor for this situation? Oct 23 '18 at 13:12
• Yes, in this case you replace $D^b(X \times Y)$ by $D^b(X,A \otimes O_X)$, the derived category of sheaves of $A \otimes O_X$-modules on $X$. Oct 23 '18 at 14:13
• And I suppose as in the coherent case, any fully faithful embedding of $D^b(A)$ into $D^b(X)$ would be FM in the above sense right? Thanks again! Oct 23 '18 at 14:36