# What is the “see-saw exact sequence”?

Let a connected linear algebraic group $$G$$ acts on an algebraic variety $$X$$, proper over a filed $$k$$.

In the proof of Proposition 1.5. of Mumford's GIT book, he says "....consider the see-saw exact sequence: $$0 \rightarrow H^1(\mathcal{O}_{G}^{\times}) \rightarrow H^1(\mathcal{O}_{G\times X}^{\times}) \rightarrow H^0(G, R^1 (p_1)_{\ast} (\mathcal{O}_{G\times X}^{\times}))$$". I don't know "the see-saw exact sequence" and cannot find it on the web. What is it? Where are references of it?

• I find a lot of references when googling the title, e.g., here, page $2$. – Dietrich Burde May 31 at 8:11
• @DietrichBurde It is "see-saw property". Are "see-saw property" and "see-saw exact sequence" same notion? – LOCOAS May 31 at 9:08