To do homological algebra with unbounded chain complexes one needs to first find a way of constructing resolutions. Spal-tenstein solved this problem for chain complexes of R-modules by truncating further and further to the left, resolving the pieces, and gluing back the partial resolutions. Our aim is to give a homotopy theoretical interpretation of this procedure, which may be extended to a relative setting. We work in an arbitrary abelian category A and fix a class of "injective objects" I. We show that Spaltenstein's construction can be captured by a pair of adjoint functors between unbounded chain complexes and towers of non-positively graded ones. This pair of adjoint functors forms what we call a Quillen pair and the above process of truncations, partial resolutions, and gluing, gives a meaningful way to resolve complexes in a relative setting up to a split error term. In order to do homotopy theory, and in particular to construct a well behaved relative derived category D(A;I), we need more: the split error term must vanish. This is the case when I is the class of all injective R-modules but not in general, not even for certain classes of injectives modules over a Noetherian ring. The key property is a relative analogue of Roos's AB4*-n axiom for abelian categories. Various concrete examples such as Gorenstein homological algebra and purity are also discussed.

2.

Pirisi, Roberto

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

We compute the cohomological invariants with coefficients in Z/pZ of the stack H-3 of hyperelliptic curves of genus 3 over an algebraically closed field.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Zalamansky, Gabriel

Leiden Univ, Snellius Bldg,Niels Bohrweg 1, NL-2333 CA Leiden, Netherlands..

The Complexity of a Flat Groupoid2018In: Documenta Mathematica, ISSN 1431-0635, E-ISSN 1431-0643, Vol. 23, p. 1157-1196Article in journal (Refereed)

Abstract [en]

Grothendieck proved that any finite epimorphism of noetherian schemes factors into a finite sequence of effective epimorphisms. We define the complexity of a flat groupoid R paired right arrows X with finite stabilizer to be the length of the canonical sequence of the finite map R -> X x (X/R) X, where X/R is the Keel-Mori geometric quotient. For groupoids of complexity at most 1, we prove a theorem of descent along the quotient X -> X/R and a theorem on the existence of the quotient of a groupoid by a normal subgroupoid. We expect that the complexity could play an important role in the finer study of quotients by groupoids.