Modules of finite projective dimension with negative intersection multiplicities
For finitely generated modules M, N over an unramified regular local ring R a theorem of Serre asserts that \((i)\quad \chi (M,N)=0\) if \(\dim M+\dim N<\dim R,\) and \((ii)\quad \chi (M,N)>0\) if \(\dim M+\dim N=\dim R,\) provided, of course, that M, N has finite length. (\(\chi\) (M,N) denotes the intersection multiplicity of M and N.) It has been an open problem whether Serre's theorem is still true if one only assumes that R is an arbitrary local ring and one of the modules has finite projective dimension. Partial positive results have been achieved by various authors. The main objective of the article under review is to disprove the generalizations of (i) and (ii) just stated. Let K be a field, \(R=K[X_ 1,...,X_ 4]/(X_ 1X_ 4-X_ 2X_ 3)\) localized at the irrelevant maximal ideal, and \(P=(x_ 1,x_ 2)\subset R.\) The authors study a class of modules of finite length and finite projective dimension over R, and in particular produce such modules M with \(\chi (M,R/P)=\pm 1\) and, in case \(\chi (M,R/P)=1,\) \(\chi (M,P^ t)<0\) for \(t>>0\). We cannot comment the details of this elaborate construction here. The last section of the article discusses the consequences for the Grothendieck group of modules of finite length and finite projective dimension and related questions, such as the positivity of partial Euler characteristics. Meanwhile it has been proved by \textit{P. Roberts} [Bull. Am. Math. Soc., New Ser. 13, 127-130 (1985; Zbl 0585.13004)] and, independently, by \textit{H. Gillet} and \textit{C. Soulé} [C. R. Acad. Sci., Paris, Sér. I 300, 71-74 (1985; Zbl 0587.13007)] that (i) holds over complete intersections (including ramified regular local rings) if both M and N have finite projective dimension.
scientific article; zbMATH DE number 3943996
scientific article