Forcing algebras/Induced torsors/Section

As ${}T_{U}$ is a ${}V_{U}$-torsor, and as every ${}V$-torsor is represented by a unique cohomology class, there should be a natural cohomology class coming from the forcing data. To see this, let ${}R$ be a noetherian ring and ${}I={\left(f_{1},\ldots ,f_{n}\right)}$ be an ideal. Then on ${}U=D(I)$ we have the short exact sequence

$0\longrightarrow \operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}\longrightarrow {\mathcal {O}}_{U}^{n}\longrightarrow {\mathcal {O}}_{U}\longrightarrow 0.$

An element ${}f\in R$ defines an element ${}f\in \Gamma (U,{\mathcal {O}}_{U})$ and hence a cohomology class ${}\delta (f)\in H^{1}(U,\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)})$. Hence ${}f$ defines in fact a ${}\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}$-torsor over ${}U$. We will see that this torsor is induced by the forcing algebra given by ${}f_{1},\ldots ,f_{n}$ and ${}f$.

Theorem

Let ${}R$ denote a noetherian ring, let ${}I={\left(f_{1},\ldots ,f_{n}\right)}$ denote an ideal and let ${}f\in R$ be another element. Let ${}c=\delta (f)\in H^{1}(D(I),\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)})$ be the corresponding cohomology class and let

${}B=R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}-f\right)}\,$

denote the forcing algebra for these data. Then the scheme ${}\operatorname {Spec} {\left(B\right)}{|}_{D(I)}$ together with the natural action of the syzygy bundle on it is isomorphic to the torsor given by ${}c$.

Proof

We compute the cohomology class ${}\delta (f)\in H^{1}(U,\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)})$ and the cohomology class given by the forcing algebra. For the first computation we look at the short exact sequence

$0\longrightarrow \operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}\longrightarrow {\mathcal {O}}_{U}^{n}{\stackrel {f_{1},\ldots ,f_{n}}{\longrightarrow }}{\mathcal {O}}_{U}\longrightarrow 0.$

On ${}D(f_{i})$, the element ${}f$ is the image of ${}\left(0,\ldots ,0,\,{\frac {f}{f_{i}}},\,0,\ldots ,0)\right)$ (the non-zero entry is at the ${}i$th place). The cohomology class is therefore represented by the family of differences

$\left(0,\ldots ,0,\,{\frac {f}{f_{i}}},\,0,\ldots ,0,\,-{\frac {f}{f_{j}}},\,0,\ldots ,0\right)\in \Gamma (D(f_{i})\cap D(f_{j}),\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}).$

On the other hand, there are isomorphisms

$V{|}_{D(f_{i})}\longrightarrow T{|}_{D(f_{i})},\left(s_{1},\ldots ,s_{n}\right)\longmapsto \left(s_{1},\ldots ,s_{i-1},\,s_{i}+{\frac {f}{f_{i}}},\,s_{i+1},\ldots ,s_{n}\right).$

The composition of two such isomorphisms on ${}D(f_{i}f_{j})$ is the identity plus the same section as before.

$\Box$

Example

Let ${}(R,{\mathfrak {m}})$ denote a two-dimensional normal local noetherian domain and let ${}f$ and ${}g$ be two parameters in ${}R$. On ${}U=D({\mathfrak {m}})$ we have the short exact sequence

$0\longrightarrow {\mathcal {O}}_{U}\cong \operatorname {Syz} {\left(f,g\right)}\longrightarrow {\mathcal {O}}_{U}^{2}{\stackrel {f,g}{\longrightarrow }}{\mathcal {O}}_{U}\longrightarrow 0$

and its corresponding long exact sequence of cohomology,

$0\longrightarrow R\longrightarrow R^{2}{\stackrel {f,g}{\longrightarrow }}R{\stackrel {\delta }{\longrightarrow }}H^{1}(U,{\mathcal {O}}_{X})\longrightarrow \ldots .$

The connecting homomorphism ${}\delta$ sends an element ${}h\in R$ to ${}{\frac {h}{fg}}$. The torsor given by such a cohomology class ${}c={\frac {h}{fg}}\in H^{1}{\left(U,{\mathcal {O}}_{X}\right)}$ can be realized by the forcing algebra

$R[T_{1},T_{2}]/{\left(fT_{1}+gT_{2}-h\right)}.$

Note that different forcing algebras may give the same torsor, because the torsor depends only on the spectrum of the forcing algebra restricted to the punctured spectrum of ${}R$. For example, the cohomology class ${}{\frac {1}{fg}}={\frac {fg}{f^{2}g^{2}}}$ defines one torsor, but the two fractions yield the two forcing algebras ${}R[T_{1},T_{2}]/{\left(fT_{1}+gT_{2}-1\right)}$ and ${}R[T_{1},T_{2}]/{\left(f^{2}T_{1}+g^{2}T_{2}-fg\right)}$, which are quite different. The fiber over the maximal ideal of the first one is empty, whereas the fiber over the maximal ideal of the second one is a plane.

If ${}R$ is regular, say ${}R=K[X,Y]$ (or the localization of this at $(X,Y)$ or the corresponding power series ring) then the first cohomology classes are ${}K$-linear combinations of ${}{\frac {1}{x^{i}y^{j}}}$, ${}i,j\geq 1$. They are realized by the forcing algebras

$K[X,Y,T_{1},T_{2}]/{\left(X^{i}T_{1}+Y^{j}T_{2}-1\right)}.$

Since the fiber over the maximal ideal is empty, the spectrum of the forcing algebra equals the torsor. Or, the other way round, the torsor is itself an affine scheme.