Let \(f: X \to Y\) be a finite morphism of projective schemes of the same dimension over a field \(k\), and let \(\omega_Y^\circ\) be a dualizing sheaf for \(Y\).
Show that \(f^! \omega_Y^\circ\) is a dualizing sheaf for \(X\), where \(f^!\) is defined as in (Ex. 6.10).
If \(X\) and \(Y\) are both nonsingular, and \(k\) algebraically closed, conclude that there is a natural trace map \(t: f_* \omega_X \to \omega_Y\).
Solution
Put \(n=\dim X=\dim Y\) and let \(t_Y:H^n(Y,\omega_Y^\circ)\to k\) be the trace belonging to the given dualizing sheaf. For a quasi-coherent sheaf \(F\) on \(X\), write \[ a_F:H^n(X,F)\xrightarrow{\cong}H^n(Y,f_*F) \] for the natural affine-morphism cohomology comparison of P-AGH341AFFINEMORPH. Let \(\varepsilon:f_*f^!\omega_Y^\circ\to\omega_Y^\circ\) be evaluation at \(1\), as constructed in P-AGH3610FINITEFLATDUAL.
1. The coherent sheaf \(H=f^!\omega_Y^\circ\) has a trace\[ t_H=t_Y\circ H^n(\varepsilon)\circ a_H:H^n(X,H)\longrightarrow k. \]
The sheaves \(f_*\OO_X\) and \(\omega_Y^\circ\) are coherent, so their sheaf Hom is coherent by P-AGH363EXTCOHERENT, in degree zero. Under the affine equivalence defining \(f^!\), it corresponds to a coherent sheaf on \(X\). Indeed, on \(Y=\Spec A\), the module \(\Hom_A(B,N)\) is finite over \(A\) when \(B,N\) are finite over the noetherian ring \(A\), and its finite \(A\)-generators also generate it as a \(B\)-module. Thus \(H\) is coherent. The displayed composition uses natural maps on cohomology and is a well-defined \(k\)-linear trace.
2. The pair \((H,t_H)\) is a dualizing sheaf for \(X\), proving (a).
1.Thus the isomorphism is precisely the pairing required in the definition of a dualizing sheaf, not just an abstract vector-space isomorphism.This proves (a) without flatness or smoothness assumptions on \(f\).
3. If \(X\) and \(Y\) are nonsingular and pure-dimensional of the same dimension over algebraically closed \(k\), the trace in (b) is\[ \boxed{t:f_*\omega_X\xrightarrow{f_*\theta} f_*f^!\omega_Y\xrightarrow{\varepsilon}\omega_Y,} \]where \(\theta:\omega_X\xrightarrow{\cong}f^!\omega_Y\) is the trace-compatible dualizing-sheaf isomorphism.
1.Uniqueness of a dualizing sheaf with its trace [@Har10a, Proposition III.7.2] supplies the unique trace-compatible isomorphism \(\theta\).Evaluation at \(1\) is intrinsic to the finite algebra \(f_*\OO_X\), so the displayed morphism is natural and involves no choice of local coordinates or bases.By its construction, \(t_Y\circ H^n(t)\circ a_{\omega_X}\) is the canonical trace on \(X\).
4. The same assertion holds for nonsingular schemes with components of different dimensions, with canonical sheaves understood componentwise.
A noetherian nonsingular scheme is a finite disjoint union of nonsingular integral open-and-closed components: its local rings are domains, so distinct irreducible components cannot meet. Write these components as \(X_a\) and \(Y_b\), and on a component \(Z\) of dimension \(d\) use \(\omega_Z=\Omega_{Z/k}^d\). Every \(X_a\) maps to a single \(Y_b\). Its closed image has dimension \(\dim X_a\), since the morphism onto its image is finite; affinely this is equality of dimensions for an integral ring extension. Thus \(\dim X_a\le\dim Y_b\).
For equal dimensions, step 3 applied to \(X_a\to Y_b\) gives its canonical trace. For strictly smaller dimension, the image is a proper closed subset of the integral \(Y_b\). Every morphism from the coherent sheaf \(f_*\omega_{X_a}\) to the invertible sheaf \(\omega_{Y_b}\) is zero: the source has zero generic stalk and its local sections are torsion, whereas the target is torsion-free. Use this uniquely possible zero map on that component. Summing these finitely many component maps constructs the natural trace on all of \(Y\). This also covers the statement when nonsingular projective schemes are not assumed equidimensional.
5. Q.E.D.
Steps <1>1–<1>2 construct and verify the dualizing sheaf in (a), and steps <1>3–<1>4 construct the canonical trace in (b).