GAP (QPA) - Chapter 11: Projective resolutions and the bounded derived category

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11 Projective resolutions and the bounded derived category
11.1 Projective and injective complexes
11.1-1 InjectiveResolution
11.1-2 IsProjectiveComplex
11.1-3 IsInjectiveComplex
11.1-4 ProjectiveResolution
11.2 The bounded derived category
11.2-1 ProjectiveResolutionOfComplex
11.2-2 ProjectiveToInjectiveComplex
11.2-3 TauOfComplex
11.2-4 Example
11.2-5 StarOfMapBetweenProjectives

### 11 Projective resolutions and the bounded derived category

What is implemented so far for working with the bounded derived category \mathcal{D}^{b}( \modc A ). We use the isomorphism \mathcal{D}^{b}( \modc A ) \cong \mathcal{K}^{-,b}(\proj A), and will hence need a way to describe complexes where all objectives are projective (or, dually, injective).

#### 11.1 Projective and injective complexes

##### 11.1-1 InjectiveResolution

`‣ InjectiveResolution`( M ) | ( operation )

Arguments: M -- a module.

Returns: The injective resolution of M with M in degree -1.

##### 11.1-2 IsProjectiveComplex

`‣ IsProjectiveComplex`( C ) | ( property )

Arguments: C -- a complex.

Returns: true if C is either a finite complex of projectives or an infinite complex of projectives constructed as a projective resolution (`ProjectiveResolutionOfComplex` (11.2-1)), false otherwise.

A complex for which this property is true, will be printed in a different manner than ordinary complexes. Instead of writing the dimension vector of the objects in each degree, the indecomposable direct summands are listed (for instance `P1`, `P2` … , where P_i is the indecomposable projective module corresponding to vertex i of the quiver). Note that if a complex is both projective and injective, it is printed as a projective complex.

##### 11.1-3 IsInjectiveComplex

`‣ IsInjectiveComplex`( C ) | ( property )

Arguments: C -- a complex.

Returns: true if C is either a finite complex of injectives or an infinite complex of injectives constructed as D\mathrm{Hom}_{A}(-,A) of a projective complex (`ProjectiveToInjectiveComplex` (11.2-2)), false otherwise.

A complex for which this property is true, will be printed in a different manner than ordinary complexes. Instead of writing the dimension vector of the objects in each degree, the indecomposable direct summands are listed (for instance `I1`, `I2` … , where I_i is the indecomposable injective module corresponding to vertex i of the quiver). Note that if a complex is both projective and injective, it is printed as a projective complex.

##### 11.1-4 ProjectiveResolution

`‣ ProjectiveResolution`( M ) | ( operation )

Arguments: M -- a module.

Returns: The projective resolution of M with M in degree -1.

#### 11.2 The bounded derived category

Let \mathcal{D}^{b}( \modc A ) denote the bounded derived category. If C is an element of \mathcal{D}^{b}( \modc A ), that is, a bounded complex of A-modules, there exists a projective resolution P of C which is a complex of projective A-modules quasi-isomorphic to C. Moreover, there exists such a P with the following properties:

- P is minimal (in the homotopy category).

- C is bounded, so C_i = 0 for i < k for a lower bound k and C_i = 0 for i > j for an upper bound j. Then P_i = 0 for i < k, and P is exact in degree i for i > j.

The function `ProjectiveResolutionOfComplex` computes such a projective resolution of any bounded complex. If A has finite global dimension, then \mathcal{D}^{b}( \modc A ) has AR-triangles, and there exists an algorithm for computing the AR-translation of a complex C \in \mathcal{D}^{b}( \modc A ):

- Compute a projective resolution P' of C.

- Shift P' one degree to the right.

- Compute I = D\mathrm{Hom}_{A}(P',A) to get a complex of injectives.

- Compute a projective resolution P of I.

Then P is the AR-translation of C, sometimes written \tau(C). The following documents the QPA functions for working with complexes in the derived category.

##### 11.2-1 ProjectiveResolutionOfComplex

`‣ ProjectiveResolutionOfComplex`( C ) | ( operation )

Arguments: C -- a finite complex.

Returns: A projective complex P which is the projective resolution of C, as described in the introduction to this section.

If the algebra has infinite global dimension, the projective resolution of C could possibly be infinite.

##### 11.2-2 ProjectiveToInjectiveComplex

`‣ ProjectiveToInjectiveComplex`( P ) | ( operation )

`‣ ProjectiveToInjectiveFiniteComplex`( P ) | ( operation )

Arguments: P -- a bounded below projective complex.

Returns: An injective complex I = D\mathrm{Hom}_{A}(P,A).

P and I will always have the same length. Especially, if P is unbounded above, then so is I. If P is a finite complex (that is; `LengthOfComplex(P)` is an integer) then the simpler method `ProjectiveToInjectiveFiniteComplex` is used.

##### 11.2-3 TauOfComplex

`‣ TauOfComplex`( C ) | ( operation )

Arguments: C -- a finite complex over an algebra of finite global dimension.

Returns: A projective complex P which is the AR-translation of C.

This function only works when the algebra has finite global dimension. It will always assume that both the projective resolutions computed are finite.

##### 11.2-4 Example

The following example illustrates the above mentioned functions and properties. Note that both `ProjectiveResolutionOfComplex` and `ProjectiveToInjectiveComplex` return complexes with a nonzero positive part, whereas `TauOfComplex` always returns a complex for which `IsFiniteComplex` returns true. Also note that after the complex `C` in the example is found to have the `IsInjectiveComplex` property, the printing of the complex changes.

The algebra in the example is kQ/I, where Q is the quiver 1 \longrightarrow 2 \longrightarrow 3 and I is generated by the composition of the arrows. We construct C as the stalk complex with the injective I_1 in degree 0.

    gap> alg;
    <Rationals[<quiver with 3 vertices and 2 arrows>]/
    <two-sided ideal in <Rationals[<quiver with 3 vertices and 2 arrows>]>,
      (1 generators)>>
    gap> cat := CatOfRightAlgebraModules(alg);
    <cat: right modules over algebra>
    gap> C := StalkComplex(cat, IndecInjectiveModules(alg)[1], 0);
    0 -> 0:(1,0,0) -> 0
    gap> ProjC := ProjectiveResolutionOfComplex(C);
    --- -> 0: P1 -> 0
    gap> InjC := ProjectiveToInjectiveComplex(ProjC);
    --- -> 1: I2 -> 0: I1 -> 0
    gap> TauC := TauOfComplex(C);
    0 -> 1: P3 -> 0
    gap> IsProjectiveComplex(C);
    false
    gap> IsInjectiveComplex(C);
    true
    gap> C;
    0 -> 0: I1 -> 0

##### 11.2-5 StarOfMapBetweenProjectives

`‣ StarOfMapBetweenProjectives`( f, list_i, list_j ) | ( operation )

`‣ StarOfMapBetweenIndecProjectives`( f, i, list_j ) | ( operation )

`‣ StarOfMapBetweenDecompProjectives`( f, list_i, list_j ) | ( operation )

Arguments: f -- a map between to projective modules P = \bigoplus P_i and Q = \bigoplus Q_j, each of which were constructed as direct sums of indecomposable projective modules; list_i -- describes the summands of P; list_j -- describes the summands of Q. If P = P_1 \oplus P_3 \oplus P_3 (where P_i is the indecomposable projective representation in vertex i), then list_i is [1,3,3].

Returns: The map f^* = \Hom_A(f,A): \Hom_A(Q,A) \rightarrow \Hom_A(P,A) in A^{\mathrm{op}} (where A is the original algebra).

The function `StarOfMapBetweenProjectives` is supposed to be called from within the `ProjectiveToInjectiveComplex` method, and might not do as expected when called from somewhere else.

The other similarly named functions are called from within the first.

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