Part II: Recollement, or gluing
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2 The Topological Setting The setting: closed in X, := X. i X j D i DX j D A t-structure on D X determines ones on D and D : D 0 = j (D 0 X ) D 0 = (i ) 1 (D 0 X ) Theorem Conversely, any t-structures on D and D uniquely determine one on D X.
3 The Topological Setting The setting: closed in X, := X. i X j D i DX j D A t-structure on D X determines ones on D and D : D 0 = j (D 0 X ) D 0 = (i ) 1 (D 0 X ) Theorem Conversely, any t-structures on D and D uniquely determine one on D X.
4 The Topological Setting The setting: closed in X, := X. i X j D i DX j D A t-structure on D X determines ones on D and D : D 0 = j (D 0 X ) D 0 = (i ) 1 (D 0 X ) Theorem Conversely, any t-structures on D and D uniquely determine one on D X.
5 Basic Example Topological Setting Example X = C = {0} = C The standard t-structures on D and D yield the standard t-structure on D X. Example Glue the standard t-structure (D 0 t-structure (D 1, D 1 )., D 0 ) to the shifted
6 The General Setting: D i DX j D 1 i and j are exact and have left and right adjoints: 2 j i = 0 but (i, i = i!, i! ) (j!, j! = j, j ) i i = id D = i! i! j j = id D = j! j! 3 For each F D X there exist (unique) dist. triangles j! j! F F i i F d j! j! F[1] i! i! F F j j F d i! i! F[1]
7 The Construction Definition Given t-structures on D, D and gluing data, = {K : j K D 0, i K D 0 }. D 0 X D 0 X = {K : j! K D 0, i! K D 0 }. We say that the new t-structure on D X is obtained by gluing or recollement. It will be nondegenerate if those on D and D are.
8 Example (X = C, = {0}) F: constant sheaf F = C X on X; H(i F) = C H(i! F) = C [ 2] G: pushforward G = j C from. H(i G) = C C [ 1] H(i! G) = 0. Standard t-structures on,, X: Recollement of (D 0 F is in the heart A X ; G is not., D 0 ) with (D 1, D 1 ): F is still in the heart; now G is too.
9 Example (X = C, = {0}) F: constant sheaf F = C X on X; H(i F) = C H(i! F) = C [ 2] G: pushforward G = j C from. H(i G) = C C [ 1] H(i! G) = 0. Standard t-structures on,, X: Recollement of (D 0 F is in the heart A X ; G is not., D 0 ) with (D 1, D 1 ): F is still in the heart; now G is too.
10 Functors Between Extensions Gluing data (i, j ) determine functors between hearts, e.g. p i = τ 0 τ 0 i : A A X. Adjunctions ( p i, p i = p i!, p i! ) ( p j!, p j! = p j, p j ). Compositions p j p i = 0 but p i p i = id A = p i!p i! p j p j = id A = p j!p j! Exact sequences p j p! j! F F p i p i F 0 0 p i p! i! F F p j p j F.
11 Functors Between Extensions p i identifies A with a full subcat of A X. For F A X : p i F is the largest quotient of F in A. p i! F is the largest subobject of F in A. Example (X = C, = {0}) Standard t-structures: Recollement of (D 0 p i C X = C p i! C X = 0., D 0 ) with (D 1, D 1 ): p i C X = 0 p i! C X = 0.
12 Minimal Extensions Topological Setting Extensions Definition The minimal extension j! F of F A is the unique G A X such that p j G = F and p i G = p i! G = 0. Example (X = C, = {0}) Standard t-structures: j! C = j! C. Recollement of (D 0, D 0 ) with (D 1 j! C = C X., D 1 ):
13 Extensions Minimal Extensions Construction In fact j! F = image[ p j! F p j F] Here: For any G A X, adjunctions give p j p! j G G p j p j G. If G = p j! F this yields p j! F p j F.
14 Extensions The Theorem on Simple Objects Theorem Any simple object of A X is of the form p i F or j! G (with simple F A or G A ). Example (X = C, = {0}) Standard t-structures: sky-scraper sheaves C x. Recollement of (D 0, D 0 ) with (D 1, D 1 ): sky-scrapers C x, x 0, and C 0 [ 1].
15 Localisation Topological Setting Extensions Given gluing data, one can make sense of the statements D = D X /D. A = A X /A. [Gluing data D D is a thick subcategory and the inclusion i has left and right adjoints.]
16 Topological Setting A t-structure on a triangulated category C recovers: an abelian category A (the heart) a cohomological functor C A. Gluing data D D X D determine a t-structure on D X from those on D, D functors between the corresponding hearts, including a notion of minimal extension.
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