3 edited tags
2 corrected to remove an inaccurate link.

Given $f:X\to Y$ a morphism of schemes (or stacks if it's not harder), I am interested in a geometric reformulation of the condition that the functor $f^*:D^b(Coh(Y))\to D^b(Coh(X))$ is full. I can only find full and faithful appearing together in the literature, and I need to extricate the two conditions. Does anyone know a simple formulation, or a good reference?

Things I know which might help:

1. For affine schemes, it seems to be well-known that $f^*$ is full and faithful if and only if $f$ is an open immersion. The explanation presented here translates full faithfulness of $f^*$ to saying that the diagonal map $X\to X\times_Y X$ is an isomorphism, which in turn gives you that $f$ is mono. Other arguments about flatness give you that it's an immersion. I'm not sure how to modify this for $f^*$ only full, as the full and faithful assumptions seem to get applied in tandem. I'm also not sure how particular this line of inquiry is to affine schemes.

Intuitively, asking for $f^*$ to be full seems alot like asking that anytime you have a sheaf $F$ on $Y$, and a section of it defined only on $X$ (i.e. a section of $f^*F$), it can be extended to a section on all of $Y$. And so that would seem to indicate that the image of $f$ should have codimension-two complement. However, that intuition only really applies to underived $f^*$, and maybe deriving $f^*$ eliminates the codimension 2 requirement? Also, this intution is assuming that f is mono, so that $f^*$ is just restriction, which I don't think is true a priori.

As a side-note: I'd be interested in the same question (geometric characterization of fullness) for $f_*$ and $f^!$.

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# Fullness of pullback functor in algebraic geometry

Given $f:X\to Y$ a morphism of schemes (or stacks if it's not harder), I am interested in a geometric reformulation of the condition that the functor $f^*:D^b(Coh(Y))\to D^b(Coh(X))$ is full. I can only find full and faithful appearing together in the literature, and I need to extricate the two conditions. Does anyone know a simple formulation, or a good reference?

Things I know which might help:

1. For affine schemes, it seems to be well-known that $f^*$ is full and faithful if and only if $f$ is an open immersion. The explanation presented here translates full faithfulness of $f^*$ to saying that the diagonal map $X\to X\times_Y X$ is an isomorphism, which in turn gives you that $f$ is mono. Other arguments about flatness give you that it's an immersion. I'm not sure how to modify this for $f^*$ only full, as the full and faithful assumptions seem to get applied in tandem. I'm also not sure how particular this line of inquiry is to affine schemes.

2. Intuitively, asking for $f^*$ to be full seems alot like asking that anytime you have a sheaf $F$ on $Y$, and a section of it defined only on $X$ (i.e. a section of $f^*F$), it can be extended to a section on all of $Y$. And so that would seem to indicate that the image of $f$ should have codimension-two complement. However, that intuition only really applies to underived $f^*$, and maybe deriving $f^*$ eliminates the codimension 2 requirement? Also, this intution is assuming that f is mono, so that $f^*$ is just restriction, which I don't think is true a priori.

As a side-note: I'd be interested in the same question (geometric characterization of fullness) for $f_*$ and $f^!$.