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  • DERIVED CATEGORIES Contents - Columbia University
    1 Introduction in triangulated categories Next, we prove that the homotopy category of complexes in an additive category is a triangulated category Once this is done we define the derived category of an abelian category as the localization of the homotopy category with re pect to quasi-isomorphisms A good reference is Ve dier’s thesis [Ver96]
  • Derived category - Wikipedia
    Despite the level of abstraction, derived categories became accepted over the following decades, especially as a convenient setting for sheaf cohomology Perhaps the biggest advance was the formulation of the Riemann–Hilbert correspondence in dimensions greater than 1 in derived terms, around 1980
  • The Derived Category - MIT Mathematics
    Therefore we could have defined the derived category to be the localization Q~lCh(A) However, in order to prove that Q~lCh(A) exists we must first prove that Q~lK(A) exists, by giving an explicit description of the mor-phisms
  • Some remarks on L-equivalence for cubic fourfolds and hyper-Kähler . . .
    Indeed, cubic fourfolds are the archetypal example of Fano varieties of K3 type (see [17] for the formal definition and a survey on the subject) From the points of view of Hodge theory and derived categories, these manifolds share many properties with K3 surfaces and more in general with hyper-Kähler manifolds
  • Derived Categories - Merrick Cai
    In fact, they are precisely the adjoints of the inclusions of categories: τ≤N is the right adjoint to the inclusion D≤N ,→ D, and τ≥N is the left adjoint to the inclusion D≥N ,→ D
  • Derived categories and algebraic geometry
    Inspired by work on the stable homotopy category (description of the chromatic tower), Hopkins and Neeman [Ho, Neel] have given a classification of thick subcategories of the category of perfect complexes over an affine variety
  • GEOMETRIC LANGLANDS [afterGaitsgory,Raskin,] PeterScholze Contents
    of a “commutative algebra in categories acting on a category” Note that a “commutative algebra i categories” is the same thing as a symmetric monoidal category As a basic example to keep in mind, for any scheme or stack X, its (derived (∞, 1)-) category of quasicoherent sheaves Dqc(X) is a “commutative algebra in categories”
  • Derived and Triangulated Categories - Uni Bielefeld
    Note that the category R–Proj isn’t abelian, it is only an exact category D(R–proj) has for objects all cochain complexes of finitely generated projective left R-modules Db(R–proj) has for objects all bounded cochain complexes of finitely generated, projective left R-modules
  • derived category in nLab - ncatlab. org
    Often in the literature, the term derived category refers to the homotopy category, viewed only as a triangulated category The loss of information can often be problematic, but for many purposes is not important
  • Derived Categories of Coherent Sheaves
    First, we investigate the behaviour of derived categories of coherent sheaves when the underlying variety is subject to birational transformations such as flips, flops and contractions, which are the common operations applied in the minimal model program





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