Proj Projective Schemes Twisting Sheaves and Ampleness — Examples

Chart coordinates. Fix $i$. A degree-zero element of $S[x_i^{-1}]$ has the form $a\mathrm{/}{x}_i^d$ with $a$ homogeneous of degree $d$, and every monomial $x_0^{a_0}\cdots x_n^{a_n}$ of degree $d$ gives $x_0^{a_0}\cdots x_n^{a_n}\mathrm{/}{x}_i^d={\prod }_{\mathrm{\ell }\mathrm{\ne }i}(x_{\mathrm{\ell }}\mathrm{/}{x}_i{)}^{a_{\mathrm{\ell }}}$; hence $S_{(x_i)}=k[x_{\mathrm{\ell }}\mathrm{/}{x}_i:\mathrm{\ell }\mathrm{\ne }i],$ the polynomial ring in the $n$ variables $x_{\mathrm{\ell }}^{(i)}:=x_{\mathrm{\ell }}\mathrm{/}{x}_i$.

The charts. Under the assumed AC [F4], by [F2] the chart $D_{+}(x_i)$ is $\mathrm{Spec}{S}_{(x_i)}=\mathrm{Spec}k[x_{\mathrm{\ell }}\mathrm{/}{x}_i:\mathrm{\ell }\mathrm{\ne }i]$, which is exactly the $i$-th standard chart $U_i=\mathrm{Spec}k[x_{\mathrm{\ell }}^{(i)}:\mathrm{\ell }\mathrm{\ne }i]$ of ${\mathbb{P}}_k^n$ under the identification $x_{\mathrm{\ell }}^{(i)}=x_{\mathrm{\ell }}\mathrm{/}{x}_i$ of [F1].

The overlap. On $D_{+}(x_i{x}_j)$ both $x_i$ and $x_j$ are invertible, so the relation $\frac{x_a}{x_j}=\frac{x_a\mathrm{/}{x}_i}{x_j\mathrm{/}{x}_i}$ is an identity of regular functions in the localised rings; expressed in the coordinates of step 1.1 it reads $x_a^{(j)}=x_a^{(i)}\mathrm{/}{x}_j^{(i)}$, which is exactly the transition formula of the published charts in [F1], together with $x_i^{(j)}=1\mathrm{/}{x}_j^{(i)}$ for $a=i$. Hence the overlapping charts are glued by the same isomorphisms.

The case $n=0$. For $n=0$ we have $S=k[x_0]$ with $S_{(x_0)}=k$ by step 1.1 with $n=0$ variables, so $\mathrm{Proj}k[x_0]=D_{+}(x_0)=\mathrm{Spec}k$, which is the one-point scheme of [F3]; equivalently ${\mathbb{P}}_k^0=\mathrm{Spec}k$ in the published charts.

Conclusion. Steps 2.1 and 2.2 identify the charts and gluing of $\mathrm{Proj}k[x_0,\dots ,x_n]$ with those of ${\mathbb{P}}_k^n$, in agreement with the canonical isomorphism of [F1], and step 2.3 settles $n=0$; the displayed coordinate change is the transition formula of the published charts. [F1, step 2.1, step 2.2, step 2.3] \qed

The irrelevant ideal is nilpotent. Here $S_{+}=k\epsilon$ consists of the multiples of $\epsilon$, and $S_{+}^2=k{\epsilon }^2=0$; hence every element of $S_{+}$ is nilpotent and $S_{+}=(\epsilon )$ is finitely generated.

The standard chart is empty. For the homogeneous element $\epsilon$ of degree $1$ we have $S_{(\epsilon )}=(S[{\epsilon }^{-1}]{)}_0$, and $S[{\epsilon }^{-1}]=0$ because $\epsilon$ is nilpotent; hence $D_{+}(\epsilon )=\mathrm{Spec}{S}_{(\epsilon )}=\mathrm{Spec}0=\mathrm{\varnothing }$, and since $\epsilon$ generates $S_{+}$ this is the only standard open.

Proj is empty. By step 1.1 every homogeneous element of $S_{+}$ is nilpotent, so [F2] gives $\mathrm{Proj}S=\mathrm{\varnothing }$; equivalently, any homogeneous prime $\mathfrak{p}\subseteq S$ contains the nilpotent $\epsilon$, hence contains $S_{+}=(\epsilon )$ and is excluded from $\mathrm{Proj}S$.

The spectrum is nonempty. In $k[\epsilon ]\mathrm{/}({\epsilon }^2)$ the element $\epsilon$ is nilpotent but nonzero, so the ideal $(\epsilon )$ is proper; every prime contains the nilpotent $\epsilon$, so $(\epsilon )$ is the unique prime ideal and $\mathrm{Spec}S=\{(\epsilon )\}\mathrm{\ne }\mathrm{\varnothing }$, in contrast with step 2.1.

Conclusion. Steps 1.1 and 2.1 show that the nilpotent irrelevant ideal produces empty Proj, while step 3.1 shows that the underlying ring still has a point; the two conclusions are consistent because Proj discards exactly the primes containing all of $S_{+}$. [F1, F2, step 2.1, step 3.1] \qed

The chart modules. The AC premise [F4] licenses the associated-sheaf and Proj charts [F1]–[F3]. For $n\in \mathbb{Z}$ the module $S(n{)}_{(x_0)}=(S(n)[x_0^{-1}]{)}_0$ consists of the classes $a\mathrm{/}{x}_0^k$ with $a\in S(n{)}_k=S_{n+k}$ homogeneous of degree $n+k$. Since $x_0$ is a unit in the localisation, every such class equals $(a\mathrm{/}{x}_0^{n+k})\cdot x_0^n$ with $a\mathrm{/}{x}_0^{n+k}\in S_{(x_0)}=k[t]$; hence $e_0:=x_0^n$ generates $S(n{)}_{(x_0)}$ over $k[t]$, and symmetrically $e_1:=x_1^n$ generates $S(n{)}_{(x_1)}$ over $k[t^{-1}]$.

The overlap. On the overlap $D_{+}(x_0{x}_1)$ the ring is $k[t,t^{-1}]$ with $t=x_1\mathrm{/}{x}_0$, so $x_1=t{x}_0$ and therefore $x_1^n=t^n{x}_0^n$ holds in the localisation of $S$ at $x_0{x}_1$ for every integer $n$, positive or negative; under the identifications of step 1.1 this is precisely the frame relation $e_1=t^n{e}_0$ on $U_0\cap U_1$.

Conclusion. The frame section $e_i$ is nowhere vanishing on $U_i$, and the transition relation $e_1=t^n{e}_0$ is exactly the change of frame of the invertible sheaf $\mathcal{O}(n)$ from the $0$-chart to the $1$-chart: for $n=0$ both frames are the constant function $1$ and the relation is $e_1=e_0$; for $n=1$ it is $e_1=t{e}_0$; for $n=-1$ it is $e_1=t^{-1}{e}_0$ with $t^{-1}=x_0\mathrm{/}{x}_1$ the coordinate on $U_1$. [F2, F3, step 1.1, step 2.1, cases: n=0 and negative n] \qed

The contraction is the identity. Take $L={\mathcal{O}}_X$ and $s=1$. The dual is $L^{-1}={\mathcal{O}}_X$ and the pairing ${\mathcal{O}}_X\otimes {\mathcal{O}}_X\to {\mathcal{O}}_X$ is multiplication, so $c_1:{\mathcal{O}}_X\to {\mathcal{O}}_X$ sends a local function $g$ to $g\cdot 1=g$; that is, $c_1={\mathrm{i}\mathrm{d}}_{{\mathcal{O}}_X}$. Consequently ${\mathcal{I}}_1=\mathrm{Im}(c_1)={\mathcal{O}}_X$, the unit ideal sheaf.

The zero scheme is empty. Let $U=\mathrm{Spec}A\subseteq X$ be any affine open; over $U$ the structure sheaf is trivialised by the identity and the local equation of the unit section is $f=1\in A$. By the local chart formula of [F1], $Z(1)\cap U=\mathrm{Spec}(A\mathrm{/}(1))=\mathrm{Spec}0=\mathrm{\varnothing }$. Since the affine opens cover $X$, the closed subscheme $Z(1)$ has no points; it is the empty closed subscheme.

The empty divisor. In the trivialisation of step 2.1 the local equation $f=1$ is a unit of $A$, hence in particular a nonzerodivisor, and ${\mathcal{I}}_1={\mathcal{O}}_X$ is an invertible sheaf of ideals; by the criterion of [F2] the zero scheme $Z(1)=\mathrm{\varnothing }$ is an effective Cartier divisor, namely the empty divisor, whose local equation is the unit $1$.

Conclusion and empty scheme. Steps 1.1, 2.1 and 3.1 give ${\mathcal{I}}_1={\mathcal{O}}_X$ and $Z(1)=\mathrm{\varnothing }$ with unit local equation, so the empty closed subscheme of $X$ is an effective Cartier divisor. If $X=\mathrm{\varnothing }$ then ${\mathcal{O}}_X$ is the zero sheaf, $\mathrm{\Gamma }(X,{\mathcal{O}}_X)=0$ and the unit section is $1=0$, the unit of the zero ring; the same computation gives ${\mathcal{I}}_1=0={\mathcal{O}}_X$ and $Z(1)=\mathrm{\varnothing }=X$, and since ${\mathcal{O}}_X$ is invertible (the zero sheaf is locally free of rank one on the empty scheme, where there is no point to test) the conclusion holds vacuously for the empty base as well. No hypothesis on $X$ beyond the trivialisations of ${\mathcal{O}}_X$ enters; in particular nilpotents or zero divisors in ${\mathcal{O}}_X$ do not affect the computation, which uses only multiplication by $1$. The Axiom of Choice [A1] is inherited from the affine quotient and gluing suppliers of [F1]; no choice is made here. [A1, F1, F2, step 1.1, step 2.1, step 3.1, cases: empty X and units as local equations] \qed

The quotient and the closed subscheme. Since $F$ is homogeneous, $I=(F)$ is a homogeneous ideal, so by [F1] the subscheme $V_{+}(I)\hookrightarrow {\mathbb{P}}_k^n$ exists and equals $\mathrm{Proj}(B\mathrm{/}I)=\mathrm{Proj}(k[x_0,\dots ,x_n]\mathrm{/}(F))$; by [F2] the ambient space $\mathrm{Proj}B$ is ${\mathbb{P}}_k^n$ with charts $D_{+}(x_i)$.

The chart ring. Fix $i$. By [F2] the chart ring of ${\mathbb{P}}_k^n$ on $D_{+}(x_i)$ is $B_{(x_i)}=k[x_a^{(i)}:a\mathrm{\ne }i]$ with $x_a^{(i)}=x_a\mathrm{/}{x}_i$. In the localisation $B[x_i^{-1}]$ the element $x_i$ is a unit and $F=x_i^e\cdot (F\mathrm{/}{x}_i^e)$, so the ideal generated by $F$ is generated by the degree-zero element $F\mathrm{/}{x}_i^e$; hence $I_{(x_i)}=(I[x_i^{-1}]{)}_0=(F\mathrm{/}{x}_i^e)$, and the chart ring of $V_{+}(I)$ on $D_{+}(x_i)$ is $B_{(x_i)}\mathrm{/}{I}_{(x_i)}=k[x_a^{(i)}:a\mathrm{\ne }i]\mathrm{/}(F\mathrm{/}{x}_i^e)$, that is, $k[x_a\mathrm{/}{x}_i:a\mathrm{\ne }i]\mathrm{/}(F\mathrm{/}{x}_i^{\mathrm{deg}F})$.

Conclusion and degenerate cases. Steps 1.1 and 2.1 identify $\mathrm{Proj}(k[x_0,\dots ,x_n]\mathrm{/}(F))$ with $V_{+}(F)\subseteq {\mathbb{P}}_k^n$ and compute its chart rings. If $F=c{x}_i^e$ is a nonzero multiple of a single coordinate power, then on the $i$-th chart the equation is the unit $c$ and the chart ring is $k[x_a^{(i)}]\mathrm{/}(c)=0$, so that chart meets $V_{+}(F)$ in the empty scheme; in particular for $n=0$ one has $F=c{x}_0^e$ and $V_{+}(F)=\mathrm{Spec}0=\mathrm{\varnothing }$, the empty hypersurface. If $F$ is not such a monomial then for every $i$ the element $F\mathrm{/}{x}_i^e$ is the dehomogenisation of $F$, a nonzero element of the polynomial ring $k[x_a^{(i)}:a\mathrm{\ne }i]$ that is not a unit because some monomial of $F$ has a positive exponent at a variable other than $x_i$; each chart is then the genuine affine hypersurface $k[x_a^{(i)}]\mathrm{/}(F\mathrm{/}{x}_i^e)$, which is a nonzero ring. The zero polynomial is excluded by hypothesis. The Axiom of Choice [A1] is inherited from the affine quotient and gluing suppliers of [F1]; no choice is made here. [A1, F1, F2, step 1.1, step 2.1, cases: empty chart and n=0] \qed

A global section has two chart expressions. Let $s\in \mathrm{\Gamma }({\mathbb{P}}_k^1,\mathcal{O}(-1))$. Under the AC premise [F6], by [F2] its restriction to $U_0$ has the form $s_0=a(t)x_0^{-1}$ with $a\in k[t]$, and its restriction to $U_1$ has the form $s_1=b(t^{-1})x_1^{-1}$ with $b\in k[t^{-1}]$; these are finite polynomials $a(t)={\sum }_{m\ge 0}{\alpha }_m{t}^m$ and $b(u)={\sum }_{m\ge 0}{\beta }_m{u}^m$ with $u=t^{-1}$.

Agreement on the overlap. By [F4] the two expressions agree on $U_0\cap U_1$, where $x_1=t{x}_0$ is invertible. Substituting $x_1^{-1}=t^{-1}{x}_0^{-1}$ turns the agreement into the identity $b(t^{-1})=ta(t)$ in $k[t,t^{-1}]$. The right-hand side is a finite sum of monomials $t^{m+1}$ with $m\ge 0$, so it involves only strictly positive powers of $t$; the left-hand side ${\sum }_{m\ge 0}{\beta }_m{t}^{-m}$ involves only nonpositive powers of $t$. Comparing coefficients in the basis $\{t^j:j\in \mathbb{Z}\}$ of $k[t,t^{-1}]$ gives ${\alpha }_m=0$ for all $m$ and ${\beta }_m=0$ for all $m$.

Vanishing of all global sections. By step 1.1 every global section is given by its two chart expressions, and by step 1.2 those expressions have $a=0$ and $b=0$; hence $s_0=0$ and $s_1=0$, so $s=0$ by [F4]. Therefore $\mathrm{\Gamma }({\mathbb{P}}_k^1,\mathcal{O}(-1))=0$.

Failure of global generation. With $\mathrm{\Gamma }({\mathbb{P}}_k^1,\mathcal{O}(-1))=0$ the evaluation morphism of [F5] is the zero morphism; since $\mathcal{O}(-1)$ is invertible and ${\mathbb{P}}_k^1\mathrm{\ne }\mathrm{\varnothing }$, it has a nonzero stalk at every point and the zero morphism is not surjective. Hence $\mathcal{O}(-1)$ is not globally generated, although it is invertible by [F3]. This is the standard contrast with the positive twists: $\mathcal{O}(1)$ is generated by its two coordinate sections $x_0,x_1$. [F3, F5, step 2.1] \qed

The rings and their degree-one parts. By [F2] the ring $S=k[x,y]$ with $\mathrm{deg}x=\mathrm{deg}y=1$ is nonnegatively graded with $S_1=kx\oplus ky$, of dimension $2$, and the Veronese $T=S^{(2)}={⨁}_n{S}_{2n}$ is the graded $k$-subalgebra $k[x^2,xy,y^2]\subseteq S$ generated by the three degree-two monomials: every $S_{2n}$ is spanned by the monomials $x^a{y}^{2n-a}=(x^2{)}^{a^{\mathrm{\prime }}}(xy{)}^b(y^2{)}^c$ with $2a^{\mathrm{\prime }}+b=a$ and $b+2c=2n-a$. Hence $T_1=S_2=k{x}^2\oplus kxy\oplus k{y}^2$, of dimension $3$, so ${\dim }_k{S}_1=2$ and ${\dim }_k{T}_1=3$.

The twists are different. By [F4], a global section of ${\mathcal{O}}_S(n)$ for $n=1$ or $2$ is a pair $a(t)x^n$ on $D_{+}(x)$ and $b(t^{-1})y^n$ on $D_{+}(y)$, where $a\in k[t]$, $b\in k[t^{-1}]$, and the overlap condition is $a(t)=t^{n}b(t^{-1})$. This holds exactly when $a(t)$ has degree at most $n$, with $b(t^{-1})=t^{-n}a(t)$. Hence ${\dim }_k\mathrm{\Gamma }({\mathbb{P}}_k^1,{\mathcal{O}}_S(1))=2$ and ${\dim }_k\mathrm{\Gamma }({\mathbb{P}}_k^1,{\mathcal{O}}_S(2))=3$. An isomorphism of these sheaves would give an isomorphism of their global-section $k$-vector spaces, which is impossible. Thus the two twists are not isomorphic.

Same Proj. Under the AC premise [F5], since $T=S^{(2)}$, [F1] with $R=S$ and $d=2$ gives a canonical isomorphism $\mathrm{Proj}S\cong \mathrm{Proj}T$ which maps the chart $D_{+}(f)$ of $\mathrm{Proj}S$ to the chart $D_{+}(f^2)$ of $\mathrm{Proj}T$, and under which ${\mathcal{O}}_T(1)={\mathcal{O}}_{\mathrm{Proj}T}(1)$ corresponds to ${\mathcal{O}}_{\mathrm{Proj}S}(2)$; by [F3] the scheme $\mathrm{Proj}S$ is the projective line ${\mathbb{P}}_k^1$.

No graded isomorphism. Suppose $\phi :S\to T$ is an isomorphism of graded $k$-algebras, that is, a $k$-algebra isomorphism with $\phi (S_n)=T_n$ for all $n$; by [F2] it restricts to a $k$-linear isomorphism $S_1\to T_1$, so ${\dim }_k{S}_1={\dim }_k{T}_1$. By step 1.1 this would require $2=3$, which is impossible; hence $S$ and $T$ are not isomorphic as graded $k$-algebras.

Conclusion. Steps 2.1 and 2.2 exhibit the two graded $k$-algebras $S=k[x,y]$ and $T=k[x^2,xy,y^2]$ with canonically isomorphic Proj but no graded isomorphism between them; steps 2.1 and 1.2 show that the isomorphism matches ${\mathcal{O}}_T(1)$ with ${\mathcal{O}}_S(2)$ and not with ${\mathcal{O}}_S(1)$, so not even the degree-one twists correspond. Since $k[x,y]$ and its Veronese are nonnegatively graded with nonzero degree-one parts, neither Proj is empty and the invariant ${\dim }_k(-{)}_1$ is defined; the case $d=1$ of [F1] is excluded here because the two rings are then equal, while $d=2\ge 1$ is the smallest regrading for which $S^{(d)}\mathrm{\ne }S$ in this example. No choice principle is used beyond the inherited Proj construction. [F1, F2, step 1.1, step 2.1, step 2.2, step 1.2, cases: d=1 excluded and d=2 smallest] \qed