K1 of Chevalley groups are nilpotent

Abstract Let F be a reduced irreducible root system and R be a commutative ring. Further, let G(F,R) be a Chevalley group of type F over R and E(F,R) be its elementary subgroup. We prove that if the rank of F is at least 2 and the Bass-Serre dimension of R is finite, then the quotient G(F,R)/E(F,R) is nilpotent by abelian. In particular, when G(F,R) is simply connected the quotient K1(F,R)=G(F,R)/E(F,R) is nilpotent. This result was previously established by Bak for the series A1 and by Hazrat for C1 and D1. As in the above papers we use the localisation-completion method of Bak, with some technical simplifications.

Stable hydrogen isotope ratios of lignin methoxyl groups as a palaeoclimate proxy

• Stable isotope ratios of organic compounds are valuable tools for determining the geographical origin, identity, authenticity or history of samples from a vast range of sources such as sediments, plants and animals, including humans. • Hydrogen isotope ratios (d2H values) of methoxyl groups in lignin from wood of trees grown in different geographical areas were measured using compound-specificpyrolysis isotope ratio mass spectrometry analysis. • Lignin methoxyl groups were depleted in 2H relative to both meteoric water andwhole wood. A high correlation (r2=0.91) was observed between the d2 H valuesof the methoxyl groups and meteoric water, with a relatively uniform fractionation of –216±19 recorded with respect to meteoric water over a range of d2H values from –110 in northern Norway to + 20‰ in Yemen. Thus, woods from northernlatitudes can be clearly distinguished from those from tropical regions. By contrast, the d2H values of bulk wood were only relatively poorly correlated (r 2 = 0.47) with those of meteoric water. • Measurement of the d 2H values of lignin methoxyl groups is potentially a powerful tool that could be of use not only in the constraint of the geographical origin of lignified material but also in paleoclimate, food authenticity and forensic investigations.