# Forcing algebras/Induced torsors/Section

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

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

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

## Theorem

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

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

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

### Proof

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

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

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

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

On the other hand, there are isomorphisms

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

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

${\displaystyle \Box }$

## Example

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

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

and its corresponding long exact sequence of cohomology,

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

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

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

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 ${\displaystyle {}R}$. For example, the cohomology class ${\displaystyle {}{\frac {1}{fg}}={\frac {fg}{f^{2}g^{2}}}}$ defines one torsor, but the two fractions yield the two forcing algebras ${\displaystyle {}R[T_{1},T_{2}]/{\left(fT_{1}+gT_{2}-1\right)}}$ and ${\displaystyle {}R[T_{1},T_{2}]/{\left(f^{2}T_{1}+g^{2}T_{2}-fg\right)}}$, 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 ${\displaystyle {}R}$ is regular, say ${\displaystyle {}R=K[X,Y]}$ (or the localization of this at ${\displaystyle (X,Y)}$ or the corresponding power series ring) then the first cohomology classes are ${\displaystyle {}K}$-linear combinations of ${\displaystyle {}{\frac {1}{x^{i}y^{j}}}}$, ${\displaystyle {}i,j\geq 1}$. They are realized by the forcing algebras

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

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.