Let \(f: X \to Y\) be a morphism of schemes.
Show by example that if \(\mcf\) is coherent on \(X\), then \(f_* \mcf\) need not be coherent on \(Y\), even if \(X\) and \(Y\) are varieties over a field \(k\).
Show that a closed immersion is a finite morphism.
If \(f\) is a finite morphism of noetherian schemes, and if \(\mcf\) is coherent on \(X\), then \(f_* \mcf\) is coherent on \(Y\).
Solution
1. For part (a), an example is\[ \boxed{f:\AA_k^2\longrightarrow\AA_k^1,\quad (x,y)\longmapsto x,\quad \mcf=\OO_{\AA_k^2}}. \]
Take \(k\) algebraically closed, \(A=k[x]\), and \(B=k[x,y]\). The morphism \(f\) is induced by the inclusion \(A\hookrightarrow B\). Both affine spaces are varieties, and \(\OO_{\AA_k^2}\) is coherent because \(B\) is noetherian and free of rank one over itself.
For every \(g\in A\), the inverse image of \(D_Y(g)\) is \(D_X(g)\), so \[ (f_*\OO_X)(D_Y(g))=B_g. \] The restriction maps are the localization maps. Thus \(f_*\OO_X\cong\widetilde B\), with \(B\) regarded as an \(A\)-module. This module is not finitely generated: the \(y\)-degrees of any finite list of polynomials have an upper bound \(d\), and their \(k[x]\)-linear combinations still have \(y\)-degree at most \(d\), whereas \(y^{d+1}\) does not. By the affine characterization of coherent sheaves over a noetherian ring, \(f_*\OO_X\) is not coherent [@Har10a, Proposition II.5.4].
2. Every closed immersion is finite, proving part (b).
Let \(f:X\to Y\) be a closed immersion, and take an affine open \(U=\Spec A\subseteq Y\). The affine closed-subscheme theorem gives an ideal \(I\subseteq A\) and an isomorphism \[ f^{-1}(U)\cong\Spec(A/I) \] over \(U\); under this isomorphism the restricted morphism is induced by \(A\to A/I\) [@Har10a, Corollary II.5.10]. The \(A\)-module \(A/I\) is generated by \(1+I\). Thus every member of an affine cover of \(Y\) has affine inverse image with coordinate ring finite as an \(A\)-module, which is the definition of a finite morphism.
3. A finite morphism of noetherian schemes carries coherent sheaves to coherent sheaves under direct image, proving part (c).
Choose an affine cover of \(Y\) as in the definition of finiteness. On a member \(U=\Spec A\), write \(V=f^{-1}(U)=\Spec B\), where the structural map is \(\varphi:A\to B\) and \(B\) is a finite \(A\)-module. Since \(\mcf\) is coherent, its restriction to \(V\) is \(\widetilde M\) for a finite \(B\)-module \(M\) [@Har10a, Proposition II.5.4]. Let \(N\) be \(M\) regarded as an \(A\)-module through \(\varphi\).
For \(a\in A\), direct image and the sections of an associated sheaf give \[ \begin{aligned} (f_*\mcf)(D_U(a)) &=\mcf(D_V(\varphi(a)))\\ &\cong M_{\varphi(a)}\cong N_a. \end{aligned} \] The last isomorphism identifies the fraction \(m/\varphi(a)^n\) with \(m/a^n\). These identifications commute with restriction, so \((f_*\mcf)|_U\cong\widetilde N\).
Choose \(A\)-module generators \(b_1,\ldots,b_r\) of \(B\) and \(B\)-module generators \(m_1,\ldots,m_s\) of \(M\). Writing each coefficient in a sum \(\sum_j c_jm_j\) as an \(A\)-linear combination of the \(b_i\) shows that the finitely many products \(b_im_j\) generate \(N\) over \(A\). Since \(A\) is noetherian, \(\widetilde N\) is coherent. This holds on the chosen affine cover, so \(f_*\mcf\) is coherent on \(Y\).
4. Q.E.D.
Steps 1, 2, and 3 prove parts (a), (b), and (c), respectively.