# What does $f^!$ do for line bundles on a curve?

Sorry for the rather basic question, I asked this on mathstackexchange but didn't get an answer.

I'm trying to understand how does the inverse exceptional image $$f^!$$ of a coherent sheaf look like.

Let's say for the sake of this question that $$f:X\rightarrow Y$$ is a finite flat morphism between schemes. In this case, my understanding is that $$f^!$$ is a functor from coherent sheaves on $$Y$$ to coherent sheaves on $$X$$, and that if $$\mathcal{F}$$ is a coherent sheaf on $$Y$$ then $$f^!\mathcal{F}$$ can be described as $$\underline{Hom}_{\mathcal{O}_Y}(f_*(\mathcal{O}_X),\mathcal{F})$$.

If $$X$$ and $$Y$$ are affine then it's not hard to work out what this does for a coherent sheaf on Y.

As another basic case, I'm trying to understand what happens if $$X$$ and $$Y$$ are smooth projective curves, and $$\mathcal{L}$$ is a line bundle on $$Y$$, in terms of divisors. Say $$D=\sum{P_i}$$ is a divisor on $$Y$$, and $$\mathcal{L}(D)$$ is the corresponding line bundle. My understanding is that if $$f$$ is etale then $$f^! = f^*$$, so that in that case $$f^!(\mathcal{L}(D))=\mathcal{L}(D')$$ where $$D'=\sum_i\sum_{f(Q)=P_i}{Q}$$.

Can we understand $$f^!(\mathcal{L}(D))$$ in general (by which I mean, when $$f$$ is assumed finite flat but not necessarily etale)? (can we say something more specific than $$f^!\mathcal{L}(D)=\underline{Hom}_{\mathcal{O}_Y}(f_*(\mathcal{O}_X),\mathcal{L}(D))$$, in terms of the divisor $$D$$)? And how should I think of the functor $$f^!$$ for general coherent sheaves, in this setting?

Thanks!

• Your question is imprecise, I will assume that $X$ and $Y$ are smooth curves. Then $f^{!}\mathscr{F}=K_X\otimes f^*(\mathscr{F}\otimes K_{Y}^{-1})$ for any coherent sheaf $\mathscr{F}$ on $Y$. See any book on Grothendieck duality.
– abx
Aug 19, 2020 at 7:40
• Whoops, meant to assume $X$ and $Y$ are smooth curves in the second part of the question. I added the edit now. Thanks! Aug 19, 2020 at 8:24
• Any recommended books on Grothendieck duality? Oct 22, 2020 at 18:45