An explicit sign formula for the determinant of cohomology
Titel:
An explicit sign formula for the determinant of cohomology
Auteur:
Aitken, Wayne
Verschenen in:
Communications in algebra
Paginering:
Jaargang 27 (1999) nr. 2 pagina's 703-723
Jaar:
1999
Inhoud:
The natural or “cohomological bases” construction of the determinant of cohomology functor has the problem that a certain diagram that we might want to commute only commutes up to sign, and rinding the value of this sign is a non-trivial problem. The more sophisticated determinant of cohomology functor of Knudson and Mumford has the property that the corresponding diagram commutes, not just up to sign. However, their construction leaves open the question of the sign behavior for the cohomological bases construction The cohomological bases construction has the advantage of being natural, concrete, and well-adapted to studying sections of determinant of cohomology sheaves that arise from choices of bases of relative cohomology sheaves (on Zariski open sets where the sheaves in question are free). Knudson and Mumford's construction does not consider such sections, and, paradoxically, when one tries to define and study such sections with their constructions, one is again confronted with sign troubles. Being able to work with such sections is important to such applications as defining intersection sheaves in terms of determinants of cohomology This paper extends the author's work on the cohomological bases construction. The main result is an explicit formula for an adjustment to the cohomological bases construction that causes the key diagram to commute, not just up to sign. This adjustment-oriented approach is more amenable to calculations than the author's earlier unadjusted cohomological bases approach.
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