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  • 1.
    Chacholski, Wojciech
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Neeman, Amnon
    Australian Natl Univ, Math Sci Inst, Ctr Math & Its Applicat, Canberra, ACT 0200, Australia..
    Pitsch, Wolfgang
    Univ Autonoma Barcelona, Dept Matemat, Bellaterra 08193, Cerdanyola Del, Spain..
    Scherer, Jerome
    Ecole Polytech Fed Lausanne, Inst Math, Stn 8, CH-1015 Lausanne, Switzerland..
    Relative Homological Algebra Via Truncations2018In: Documenta Mathematica, ISSN 1431-0635, E-ISSN 1431-0643, Vol. 23, p. 895-937Article in journal (Refereed)
    Abstract [en]

    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.).
    Cohomological Invariants Of Genus Three Hyperelliptic Curves2018In: Documenta Mathematica, ISSN 1431-0635, E-ISSN 1431-0643, Vol. 23, p. 969-996Article in journal (Refereed)
    Abstract [en]

    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.

  • 3.
    Romagny, Matthieu
    et al.
    Univ Rennes 1, IRMAR, Campus Beaulieu, F-35042 Rennes, France..
    Rydh, David
    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.

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