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The derived category of the heart of a t-structure - MathOverflow Suppose D D is a triangulated category and that we are given a t t -structure (D≤0,D≥0) (D ≤ 0, D ≥ 0) on D D The heart of the t t -structure, A = D≤0 ∩D≥0 A = D ≤ 0 ∩ D ≥ 0, is an abelian category Is it true in general that D = D(A) D = D (A) is the derived category of the heart of the given t t -structure on D D?
Triangulated Categories and t-Structures - LSU Theorem 8 Let C be a triangulated category with a t-structure (C≤0,C≥0), and let T be its heart T is an abelian category Moreover, the functor tH0: C → T with the following properties: (1) For any object A ∈ T , tH0(A) ’ A (2) tH0 takes distinguished triangles in C to long exact sequences in T
A brief introduction to Triangulated Categories - Liverpool —A triangulated category is a triple (T,T,D) where T is an additive category, T: T !T is an additive auto-equivalence and D is a class of candidate triangles, called distinguished triangles, satisfying the following
Derived Categories and T-Structures - The University of Warwick triangulated category, which means that the mapping cone is determined by f2Hom(A:;B:) = Ext1(A:[1];B:) In particular, the mapping cone splits if and only if fis zero Thus by above, we have so called exact triangle, which is central to the de nition of triangulated category: A !B !C !A[1] This is used to
DERIVED -CATEGORIES - Columbia University Let us review how the cone is constructed in an important example of triangulated cate-gories: the derived category D(A) of an abelian category A Given f : X →Y a map of elements of D(A), we set C(f) = X[1]⊕Y and define the differential on the complexC(f) by d = d X[1] 0 f[1] d Y The triangle maps Y →C(f) →X[1] are naturally induced
t-structure in nLab Given a t-structure (Def 1 1), its heart is the intersection A t-structure on a stable (∞,1)-category 𝒞 is a t-structure in the above sense (Def 1 1) on its underlying homotopy category (which is triangulated, see there) Therefore, a t-structure on a stable ∞ -category 𝒞 is a system of full sub- (∞,1)-categories 𝒞 ≥ n, 𝒞 ≤ n, n ∈ ℤ
Introduction - University of Notre Dame Abstract A full, extension-closed additive subcategory D of a triangulated category C in which Hom−1 C (M,N) = 0 for all M, N in D has a natural structure of exact category, with Ext1 D (M,N) ∼= Hom1 C (M,N) naturally Introduction The heart of a t-structure on a triangulated category C is a full abelian subcate-
MIT Open Access Articles On exact ∞-categories and the Theorem of the Heart of triangulated categories is Neeman’s Theorem of the Heart [18, 19, 22], which expresses an equivalence between the Neeman K-theory of a triangulated category T equipped with a bounded t-structure and the Quillen K-theory of its heart T ♥
K-THEORY FOR TRIANGULATED CATEGORIES III(A): THE THEOREM OF THE HEART* abelian category K-theory for triangulated categories II contains a proof of the special case of Theorem 1 7 1, where T is Db(A) 1 the bounded derived category of an abelian category A, and the ^-structure is the standard one In a very precise sense, the current article is better
When does a triangulated category have a heart? - MathOverflow The result doesn't specify when t-structures exist, but does give some criteria to check to see if C is the derived category of its heart In particular, you need a left-complete t-structure on C with the property that every object is right-bounded, enough projectives in the heart, and you need certain Ext groups of projective objects to vanish