Let \(X\) be a scheme over a locally noetherian scheme \(Y\), and let \(\mcl, \mcm\) be two very ample invertible sheaves on \(X\) relative to \(Y\). Show that \(\mcl \tensor \mcm\) is also very ample.
Let \(f: X \to Y\) and \(g: Y \to Z\) be two morphisms of schemes, with \(Z\) locally noetherian. Let \(\mcl\) be a very ample invertible sheaf on \(X\) relative to \(Y\), and let \(\mcm\) be a very ample invertible sheaf on \(Y\) relative to \(Z\). Show that \(\mcl \tensor f^* \mcm\) is a very ample invertible sheaf on \(X\) relative to \(Z\).
(Erratum to the unrestricted source convention)
Exercise II.5.12 does not state the locally noetherian hypotheses added here. The definition preceding Remark II.5.16.1 uses open-in-closed immersions, not closed-in-open immersions [@Har10a, Chapter II, §5 and Exercise II.5.12]. With that definition, unrestricted part (b) is false. The locally noetherian hypotheses are sufficient conditions for both assertions, not a claim of necessity in part (a).
Let \(k\) be a field, put \(A=k[x_1,x_2,\ldots]\), and let \[ V=\Spec A,\qquad U=\bigcup_{n\ge1}D(x_n). \] On \(D(x_n)\) take the ideal \[ I_n=(x_1^n,\ldots,x_{n-1}^n,x_n-1,x_{n+1},x_{n+2},\ldots)A_{x_n}. \] For \(m\ne n\), localizing \(I_n\) at \(x_m\) gives the unit ideal, so these ideals glue to a quasi-coherent ideal on \(U\). Its closed subscheme is \[ W=\coprod_{n\ge1}W_n,\qquad W_n=\Spec R_n,\qquad R_n=k[x_1,\ldots,x_{n-1}]/(x_1^n,\ldots,x_{n-1}^n). \] In the structural map \(A\to R_n\), the variable \(x_n\) maps to \(1\) and every \(x_i\) with \(i>n\) maps to \(0\). Each \(W_n\) has one point, denoted \(w_n\). This is the nonfactorizable immersion example. The following argument strengthens its conclusion: \(W\) has no open immersion over \(A\) into any finite-type \(A\)-scheme.
Suppose such an open immersion existed. A finite affine cover of the finite-type target would contain an affine member whose intersection with \(W\) contains \(W_n\) for an infinite set \(E\subseteq\NN_{>0}\). Since \(W_n\) has only one point, its nonempty intersection with that member is all of \(W_n\). We would obtain an open immersion \[ W_E=\coprod_{n\in E}W_n\longrightarrow\Spec B \] with \(B\) a finite-type \(A\)-algebra. Quotient \(B\) by the kernel of \(B\to\Gamma(W_E,\OO_{W_E})=\prod_{n\in E}R_n\). Every element of that kernel restricts to zero on \(W_E\), so this quotient preserves the open subscheme \(W_E\). Thus we may assume \(B\subseteq\prod_{n\in E}R_n\).
The map \(A\to B\) is injective. Indeed, a nonzero polynomial in finitely many \(x_i\) survives in \(R_n\) whenever \(n\in E\) exceeds every variable index and every exponent appearing in it. Choose a presentation \(B=A[t_1,\ldots,t_q]/J\) and put \(K=\operatorname{Frac}A\). The ideal \(J_K\subset K[t_1,\ldots,t_q]\) is proper and has finitely many generators. Choose \(m\) so that the coefficients of these generators all belong to \(K_m=k(x_1,\ldots,x_m)\), and put \(A_m=k[x_1,\ldots,x_m]\). Let \(J^0\subset K_m[t_1,\ldots,t_q]\) be their generated ideal and set \[ J_m=J^0\cap A_m[t_1,\ldots,t_q]. \] Then \[ J_K\cap A[t_1,\ldots,t_q]=J_m A[t_1,\ldots,t_q]. \] To verify this equality, first pass from \(A_m\) to \(K_m\) and expand polynomials coefficientwise in the remaining \(x_i\). The module \[ \bigl(K_m[t_1,\ldots,t_q]/J^0\bigr)[x_{m+1},x_{m+2},\ldots] \] is free over \(K_m[x_{m+1},x_{m+2},\ldots]\), by choosing a \(K_m\)-basis of the coefficient algebra. Consequently localization of that polynomial base to \(K\) is injective, and contraction is computed coefficientwise as claimed.
The noetherian ring \(A_m[t_1,\ldots,t_q]\) makes \(J_m\) finitely generated, say by \(F_1,\ldots,F_a\). For each \(F_j\in J_K\), choose a nonzero \(d_j\in A\) such that \(d_jF_j\in J\). Choose \(n\in E\) larger than \(m\), than every variable index in the finitely many \(d_j\), and than their total degrees. Each \(d_j\) has nonzero image in \(R_n\). Since \(d_jF_j=0\) in \(R_n\), the image of \(F_j\) cannot be a unit in this local ring. It therefore has zero residue at \(w_n\).
It follows that \(w_n\) lies in the closed subscheme \[ H=\Spec\bigl(A[t_1,\ldots,t_q]/J_m A[t_1,\ldots,t_q]\bigr) \subseteq\Spec B. \] Its coordinate ring is \[ \bigl(A_m[t_1,\ldots,t_q]/J_m\bigr)[x_{m+1},x_{m+2},\ldots]. \] Because \(n>m\), multiplication by the monic polynomial \(x_n-1\) is injective on this ring and remains injective in its nonzero local ring at \(w_n\). Thus \(x_n-1\) is nonzero in \(\OO_{H,w_n}\). But \(W_E\) is open in \(\Spec B\), so \(\OO_{\Spec B,w_n}=R_n\), in which \(x_n-1=0\). Its image in the quotient local ring \(\OO_{H,w_n}\) must also be zero, a contradiction. This proves the nonembedding assertion.
Now take \(f:W\hookrightarrow U\) to be the closed immersion and \(g:U\hookrightarrow V\) the open immersion, with \(\mcl=\OO_W\) and \(\mcm=\OO_U\). Both sheaves are very ample over their respective bases using \(\PP^0\) and the source immersion convention. Their proposed tensor product is \(\OO_W\). If it were very ample over \(V\), there would be an immersion of \(W\) as an open subscheme of a closed subscheme of some \(\PP_V^N\). That closed subscheme is of finite type over \(A\), contradicting the nonembedding assertion. Hence the unrestricted conclusion of part (b) fails; failure of a particular Segre map is not the only obstruction.
Hint
Use a Segre embedding.
Solution
Use very ampleness and the immersion convention in which the source is an open subscheme of a closed subscheme of the target. The locally noetherian hypotheses give a corrected formulation of the source exercise; the erratum treats its unrestricted version.
1. For a scheme \(B\) and nonnegative integers \(r,s\), the Segre closed immersion\[ \sigma:\PP_B^r\times_B\PP_B^s\longrightarrow\PP_B^{(r+1)(s+1)-1} \]satisfies\[ \sigma^*\OO(1)\cong q_1^*\OO_{\PP_B^r}(1)\otimes q_2^*\OO_{\PP_B^s}(1), \]where \(q_1,q_2\) are the projections.
The products \(X_iY_j\) of the homogeneous coordinate sections generate the displayed tensor product of invertible sheaves and define \(\sigma\). On the target chart with \(Z_{ij}\ne0\), its inverse image is \(D_+(X_i)\times_B D_+(Y_j)\), and the coordinate map sends \[ Z_{ab}/Z_{ij}\longmapsto (X_a/X_i)(Y_b/Y_j). \] The coordinates with \(b=j\) and \(a=i\) recover all affine coordinates of the two factors, so the map is surjective on coordinate rings. Thus \(\sigma\) is a closed immersion. The construction by generating sections identifies its pullback of \(\OO(1)\) with their invertible sheaf; equivalently, the chart frames are \(X_iY_j\), with transition functions \((X_a/X_i)(Y_b/Y_j)\). This construction commutes with base change, so the calculation over affine opens proves the assertion over every \(B\); see the Segre embedding and P-AGH2511CARTPROD.
2. A locally closed immersion into a locally noetherian scheme is an immersion in the source convention.
Let \(h:V\to W\) be closed in an open subscheme \(W'\subseteq W\). For every affine open \(T\subseteq W\), the open subset \(T\cap W'\) of the noetherian space \(T\) is quasi-compact. Its closed subscheme \(h^{-1}(T)\) is therefore quasi-compact. Thus \(h\) is quasi-compact. The factorization for quasi-compact immersions writes \(h\) as an open immersion followed by a closed immersion. Concretely, its closed factor is defined by the quasi-coherent ideal \(\ker(\OO_W\to h_*\OO_V)\); its restriction to \(W'\) is the ideal defining \(V\), so \(V\) is open in that closed factor.
3. The assertion of part (a) holds.
Choose \(Y\)-immersions \[ i:X\longrightarrow\PP_Y^r,\qquad j:X\longrightarrow\PP_Y^s \] with \(i^*\OO(1)\cong\mcl\) and \(j^*\OO(1)\cong\mcm\). The graph \(\Gamma_j:X\to X\times_Y\PP_Y^s\) is a closed immersion because \(\PP_Y^s\to Y\) is separated. The map \(i\times\operatorname{id}:X\times_Y\PP_Y^s\to\PP_Y^r\times_Y\PP_Y^s\) is a base change of \(i\). Their composite is \((i,j)\), which is a locally closed immersion since closed and open immersions are stable under base change and locally closed immersions are stable under composition. Consequently \[ h=\sigma\circ(i,j):X\longrightarrow\PP_Y^{(r+1)(s+1)-1} \] is a locally closed immersion. The target is locally noetherian, so step 2 makes \(h\) an immersion in the source convention. Step 1 gives \[ h^*\OO(1)\cong i^*\OO(1)\otimes j^*\OO(1)\cong\mcl\otimes\mcm. \] This realizes the required very ample sheaf.
4. The assertion of part (b) holds.
Choose immersions over the indicated bases \[ i:X\longrightarrow\PP_Y^r,\qquad j:Y\longrightarrow\PP_Z^s \] with \(i^*\OO_{\PP_Y^r}(1)\cong\mcl\) and \(j^*\OO_{\PP_Z^s}(1)\cong\mcm\). Write \(\pi:\PP_Y^r\to Y\) for the projection and \(a:\PP_Y^r\to\PP_Z^r\) for the base-change map, so \(\pi\circ i=f\). The identification \(\PP_Y^r\cong\PP_Z^r\times_Z Y\) gives the base change \[ k=(a,j\circ\pi):\PP_Y^r\longrightarrow\PP_Z^r\times_Z\PP_Z^s \] of \(j\). The composite \[ h=\sigma\circ k\circ i:X\longrightarrow\PP_Z^{(r+1)(s+1)-1} \] is a locally closed immersion. Its target is locally noetherian, so step 2 gives the required open-in-closed factorization. The twisting sheaf on relative projective space commutes with base change, giving \(a^*\OO_{\PP_Z^r}(1)\cong\OO_{\PP_Y^r}(1)\) [@Har10a, Chapter II, §5]. Therefore step 1 gives \[ \begin{aligned} h^*\OO(1) &\cong i^*a^*\OO_{\PP_Z^r}(1)\otimes (j\circ f)^*\OO_{\PP_Z^s}(1)\\ &\cong\mcl\otimes f^*\mcm. \end{aligned} \] Thus this sheaf is very ample relative to \(Z\).
5. With the closed-in-open immersion convention, the same two assertions hold over arbitrary base schemes.
The constructions in steps 3 and 4 already give locally closed immersions and the asserted pullbacks over arbitrary bases. Only the conversion in step 2 used local noetherianity. Omitting that conversion proves the arbitrary-base formulation when very ampleness is witnessed by a closed immersion into an open subscheme of a finite-dimensional relative projective space.
6. Q.E.D.
Steps 3 and 4 prove the corrected assertions in the source convention, and step 5 proves their arbitrary-base closed-in-open formulation.