IDEMPOTENT COMPLETION OF TRIANGULATED CATEGORIES

By Paul Balmer, M. Schlichting. 2001

318 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dfd805fba75cafe0ca5d451861c0a9c15f712356

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Citations in the corpus

On purity and applications to coderived and singularity categories (2014, background): “rem 2.1], the induced functor i¯∗: Dsing(fpG) → S(G)c is fully faithful and each X ∈ S(G)c is a retract of an object in Imi¯∗. In other words, i¯∗ is the idempotent completion functor in the sense of [BS01]. It is also well known that S(G) is compactly generated in this situation, see [Kra10, Theore”
Hermitian K-theory, derived equivalences and Karoubi's fundamental theorem (2012, background): “are the Karoubi-Grothendieck-Witt groups of A: GWi(A) = πiGW(A) = [S˜n,GW(A)]BiSp. As usual, we write GW[n](A) and GW[n] i (A) for GW(A[n]) and GWi(A[n]). Let Tbe a triangulated category. Recall from [BS01] that its idempotent completion T˜ is canonically a triangulated category. If Tis a triangulat”
From triangulated categories to Lie algebras: A theorem of Peng and Xiao (2005, unknown): “For fixed indecomposable objects X,Y and Z we will calculate the numbers FLXY , F M LZ and N LM XY Z := ∣∣∣ W L XY ×WM LZ Aut(X,Y,Z,L) ∣∣∣.”
Homological mirror symmetry for the genus two curve (2008, background): “Finally, the categories D sing(W (0)) are not split-closed in general, but we can take the split-closure D sing(W (0)), which is again naturally triangulated [8].”
Separability and triangulated categories (2011, background): “We proved in [4] that for C (pre-)triangulated, C inherits a unique structure of (pre-)triangulated category such that ι is exact.”
Formal completions and idempotent completions of triangulated categories of singularities (2009, background): “It has a natural structure of a triangulated category and the canonical functor T → T is an exact full embedding [4].”
Braiding via geometric Lie algebra actions (2010, background): “e derived category of coherent sheaves on any variety is idempotent complete. This is because the derived category of any abelian category is idempotent complete (see, for instance, Corollary 2.10 of [BS]). So all the categories we work with are idempotent complete. Suppose that (each graded piece o”
Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky versus Hanamura (2006, background): “Mgm(X − Z) → Mgm(X) → Mgm(Z)(c)[2c] → Mgm(X − Z)[1].”
Local and stable homological algebra in Grothendieck abelian categories (2007, background): “From a general result of Balmer and Schlichting [2] it follows that Dc(A) is canonically endowed with a structure of triangulated category such that the functor K (A) −→ Dc(A) is triangulated.”
Matrix Factorizations and Representations of Quivers I (2005, background): “It is known that D…(A) is again a triangulated category (see [ BS ]).”
Non-connective K-theory via universal invariants (2009, background): “1) to distinguished triangles A I −→ B P −→ C 7→ E(A) E(I) −→ E(B) E(P ) −→ E(C) −→ E(A)[1] In [35], the second named author have constructed the universal additive invariant U dg : HO(dgcat) −→ Mot dg”
Homological mirror symmetry for the quartic surface (2003, methodology): “where Xis an object of the original category and p∈Hom(X,X) an idempotent endomorphism. One thinks of this as the “direct summand im(p)”, and so Hom((X0,p0),(X1,p1)) = p1Hom(X0,X1)p0. It is proved in [6] that the idempotent completion of a triangulated category is again triangulated, in a unique way”
Symmetric products of mixed Hodge modules (2010, methodology): “can apply Theorem (1.9), and get the canonical action of Sn on π∗⊠ nM ∈ DbMHM(SnX) which is compatible with the one on the underlying Q-complexes. By the splitting of an idempotent in DbMHM(SnX) (see [BS] and also [LC] for a simpler argument) which is applied to e1 in (2.1.1), we get a direct factor”
Matrix factorizations for nonaffine LG–models (2011, methodology): “Taking the kernel of p we obtain a pair G that is isomorphic to E[1] in MF0(X,W ).”
Derived categories, resolutions, and Brown representability (2005, background): “The proof is straightforward, except for the verification of the octahedral axiom which requires some work; see [3].”
Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general) (2007, background): “It is well known that K(A) is idempotent complete; see, for example, [3].”