binder

Tilting equivalence

Let $k$ be field of characteristic $0$ and $\mathrm{D}^\mathrm{b}(\mathfrak{Coh}\, \mathbb{P}^2_k)$ denote the bounded derived category of coherent sheaves on $\mathbb{P}^2_k$.

Consider the full strong exceptional collection $\{\mathcal{O}(0), \mathcal{O}(1), \mathcal{O}(2)\} \subset \mathfrak{Coh}\, \mathbb{P}^2_k$ of the three twisted line bundles with their tilting object $$ T_\mathcal{O} := \mathcal{O}(0) \oplus \mathcal{O}(1) \oplus \mathcal{O}(2) \mbox{.} $$

Consider the second full strong exceptional collection $\{ \Omega^0(0), \Omega^1(1), \Omega^2(2) = \mathcal{O}(0)\} \subset \mathfrak{Coh}\, \mathbb{P}^2_k$ of twisted contangent bundles with their tilting object $$ T_\Omega := \Omega^0(0) \oplus \Omega^1(1) \oplus \Omega^2(2) \mbox{.} $$

Then $$ \mathrm{D}^\mathrm{b}(\mathfrak{Coh}\, \mathbb{P}^2_k) \simeq \mathrm{D}^\mathrm{b}(\mathrm{End}\, T_\Omega) \simeq \mathrm{D}^\mathrm{b}(\mathrm{End}\, T_\mathcal{O}) \mbox{,} $$ where for a finite dimensional algebra $A$ we denote by $\mathrm{D}^\mathrm{b}(A) := \mathrm{D}^\mathrm{b}(A\mathrm{-mod})$ the bounded derived category of the Abelian category $A\mathrm{-mod}$ of finite dimensional $A$-modules.

In this notebook we will consider $T_\Omega$ as a tilting object in $\mathrm{D}^\mathrm{b}(\mathrm{End}\, T_\mathcal{O})$ and construct tilting equivalences $$ F: \mathrm{D}^\mathrm{b}(\operatorname{End}\, T_\Omega) \rightleftarrows \mathrm{D}^\mathrm{b}(\operatorname{End}\, T_\mathcal{O}) : G \mbox{.} $$

For a finite dimensional algebra $A$ over a field $k$ we define its decomposition algebroid $A^\mathrm{dec}$ as the $k$-linear full subcategory of $A\mathrm{-mod}$ consisting of the summands of a direct sum decomposition of $A$ in indecomposables, which are necessarily projective $A$-modules. The additive closure $A^\mathrm{dec}_\oplus$ of the algebroid $A^\mathrm{dec}$ is then the closure of $A$ (considered as a linear category on one object) under direct sums and direct summands (in $A\mathrm{-mod}$). One recovers $A$ as the direct sum of the $\mathrm{Hom}$-groups of $A^\mathrm{dec}$. If $A^\mathrm{sdec}$ is a skeleton of $A^\mathrm{dec}$, then the direct sum of $\mathrm{Hom}$-groups of $A^\mathrm{sdec}$ is the basic algebra which is Morita-equivalent to $A$.

The category $A^\mathrm{dec}_\oplus \simeq A^\mathrm{sdec}_\oplus$ is a model for the additive full subcategory of projective objects in $A\mathrm{-mod}$, where the second model is skeletal.

We model $A\mathrm{-mod}$ as the functor category $[A^\mathrm{op}, k\mathrm{-vec}]$, where $A^\mathrm{op}$ is the opposite algebra of $A$ (viewed as an algebroid with one object), and $k\mathrm{-vec}$ is the (skeletal) Abelian category of finite dimensional $k$-vector spaces. This is equivalent to the category of representations of the quiver underlying the algebroid $(A^\mathrm{op})^\mathrm{dec} \simeq (A^\mathrm{dec})^\mathrm{op}$. The identification of $A^\mathrm{dec}$ with the $k$-linear full subcategory of indecomposable projective objects in the Abelian category $A\mathrm{-mod} := [A^\mathrm{op}, k\mathrm{-vec}] \simeq [(A^\mathrm{op})^\mathrm{dec}, k\mathrm{-vec}] \simeq [(A^\mathrm{dec})^\mathrm{op}, k\mathrm{-vec}]$ is then nothing but Yoneda's embedding. This embedding factors over the embedding $A^\mathrm{dec}_\oplus \hookrightarrow [(A^\mathrm{dec})^\mathrm{op}, k\mathrm{-vec}] \simeq A\mathrm{-mod}$.

We model the bounded derived category $\mathrm{D}^\mathrm{b}(A)$ by the bounded homotopy category $\mathrm{Ho}^\mathrm{b}\!\left(A^\mathrm{dec}_\oplus\right)$.

In [1]:
using CapAndHomalg
CapAndHomalg v1.0.0
Imported OSCAR's components GAP and Singular_jll
Type: ?CapAndHomalg for more information

We start by loading the GAP package DerivedCategories and some of the presetting which will give us the colorful output below:

In [2]:
LoadPackage( "DerivedCategories" )
In [3]:
ReadPackage( g"DerivedCategories", g"examples/pre_settings.g" ); GAP.Globals.ENABLE_COLORS = true
Out[3]:
true

Define the field of rationals over which our categories will be linear:

In [4]:
ℚ = HomalgFieldOfRationals()
Out[4]:
GAP: Q

Next we define the quiver:

In [5]:
q = RightQuiver( "q_𝓞",
          [ "𝓞(0)", "𝓞(1)", "𝓞(2)" ],
          [ "x0", "x1", "x2", "y0", "y1", "y2" ],
          [ 1, 1, 1, 2, 2, 2 ],
          [ 2, 2, 2, 3, 3, 3 ] )
SetLabelsAsLaTeXStrings( q, [ "x_1", "x_2", "x_3", "y_0", "y_1", "y_2" ] )
In [6]:
Qq = PathAlgebra( ℚ, q )
Out[6]:
GAP: Q * q_𝓞
In [7]:
EndT_𝓞 = Qq / [ Qq.x0*Qq.y1-Qq.x1*Qq.y0, Qq.x0*Qq.y2-Qq.x2*Qq.y0, Qq.x1*Qq.y2-Qq.x2*Qq.y1 ];
In [8]:
SetName( EndT_𝓞, g"End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) )" ); EndT_𝓞
Out[8]:
GAP: End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) )
In [9]:
Dimension( EndT_𝓞 )
Out[9]:
15

Define the opposite algebra, mainly to set its name:

In [10]:
EndT_𝓞op = OppositeAlgebra( EndT_𝓞 ); SetName( EndT_𝓞op, g"End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) )^op" ); EndT_𝓞op
Out[10]:
GAP: End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) )^op

Define the additive closure $(\mathrm{End}\, T_\mathcal{O})^\mathrm{dec}_\oplus$ of the endomorphism algebroid $(\mathrm{End}\, T_\mathcal{O})^\mathrm{dec}$:

In [11]:
QRows = QuiverRows( EndT_𝓞 )
Out[11]:
GAP: Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) )

Define the collection $\{\Omega^0(0), \Omega^1(1), \Omega^2(2)\}$ as a full strong exceptional collection in the bounded homotopy category $\mathrm{Ho}^\mathrm{b}\!\left((\mathrm{End}\, T_\mathcal{O})^\mathrm{dec}_\oplus\right) \simeq \mathrm{D}^\mathrm{b}(\operatorname{End}\, T_\mathcal{O})$:

In [12]:
a_0 = QuiverRowsObject( [ [ q."𝓞(0)", 3 ] ], QRows )
Out[12]:
GAP: <An object in Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) defined by 3 quiver vertices>
In [13]:
Show( a_0 )
$${𝓞(0)}^{\oplus3}$$
In [14]:
a_m1 = QuiverRowsObject( [ [ q."𝓞(1)", 3 ] ], QRows )
Out[14]:
GAP: <An object in Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) defined by 3 quiver vertices>
In [15]:
Show( a_m1 )
$${𝓞(1)}^{\oplus3}$$
In [16]:
a_m2 = QuiverRowsObject( [ [ q."𝓞(2)", 1 ] ], QRows )
Out[16]:
GAP: <An object in Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) defined by 1 quiver vertices>
In [17]:
Show( a_m2 )
$${𝓞(2)}^{}$$
In [18]:
d_0 = QuiverRowsMorphism(
          a_0,
          [ [ EndT_𝓞.x1, -EndT_𝓞.x0, Zero(EndT_𝓞) ],
            [ EndT_𝓞.x2, Zero(EndT_𝓞), -EndT_𝓞.x0 ],
            [ Zero(EndT_𝓞), EndT_𝓞.x2, -EndT_𝓞.x1 ] ],
          a_m1 )
Out[18]:
GAP: <A morphism in Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) defined by a 3 x 3 matrix of quiver algebra elements>
In [19]:
IsWellDefined( d_0 )
Out[19]:
true
In [20]:
Show( d_0 )
$${𝓞(0)}^{\oplus3}\xrightarrow{\begin{pmatrix}{x_2}&-{x_1}&0\\{x_3}&0&-{x_1}\\0&{x_3}&-{x_2}\end{pmatrix}}{𝓞(1)}^{\oplus3}$$
In [21]:
d_m1 = QuiverRowsMorphism(
          a_m1,
          [ [ EndT_𝓞.y0 ],
            [ EndT_𝓞.y1 ],
            [ EndT_𝓞.y2 ] ],
          a_m2 )
Out[21]:
GAP: <A morphism in Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) defined by a 3 x 1 matrix of quiver algebra elements>
In [22]:
IsWellDefined( d_m1 )
Out[22]:
true
In [23]:
Show( d_m1 )
$${𝓞(1)}^{\oplus3}\xrightarrow{\begin{pmatrix}{y_0}\\{y_1}\\{y_2}\end{pmatrix}}{𝓞(2)}^{}$$
In [24]:
Ω00 = HomotopyCategoryObject( [ d_m1, d_0 ], -1 )
Out[24]:
GAP: <An object in Homotopy category( Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) ) with active lower bound -2 and active upper bound 0>
In [25]:
IsWellDefined( Ω00 )
Out[25]:
true
In [26]:
Show( Ω00 )
$$\begin{array}{c} \\ {𝓞(0)}^{\oplus3} \\ { \color{black}\vert^{0}} \\ \begin{pmatrix}{x_2}&-{x_1}&0\\{x_3}&0&-{x_1}\\0&{x_3}&-{x_2}\end{pmatrix} \\ { \color{black} \downarrow_{\phantom{0}}} \\ {𝓞(1)}^{\oplus3} \\ { \color{black}\vert^{-1}} \\ \begin{pmatrix}{y_0}\\{y_1}\\{y_2}\end{pmatrix} \\ { \color{black} \downarrow_{\phantom{-1}}} \\ {𝓞(2)}^{}\end{array}$$
In [27]:
a_0 = QuiverRowsObject( [ [ q."𝓞(0)", 3 ] ], QRows )
Out[27]:
GAP: <An object in Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) defined by 3 quiver vertices>
In [28]:
Show( a_0 )
$${𝓞(0)}^{\oplus3}$$
In [29]:
a_m1 = QuiverRowsObject( [ [ q."𝓞(1)", 1 ] ], QRows )
Out[29]:
GAP: <An object in Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) defined by 1 quiver vertices>
In [30]:
Show( a_m1 )
$${𝓞(1)}^{}$$
In [31]:
d_0 = QuiverRowsMorphism(
          a_0,
          [ [ EndT_𝓞.x0 ],
            [ EndT_𝓞.x1 ],
            [ EndT_𝓞.x2 ] ],
          a_m1 )
Out[31]:
GAP: <A morphism in Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) defined by a 3 x 1 matrix of quiver algebra elements>
In [32]:
Show( d_0 )
$${𝓞(0)}^{\oplus3}\xrightarrow{\begin{pmatrix}{x_1}\\{x_2}\\{x_3}\end{pmatrix}}{𝓞(1)}^{}$$
In [33]:
Ω11 = HomotopyCategoryObject( [ d_0 ], 0 )
Out[33]:
GAP: <An object in Homotopy category( Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) ) with active lower bound -1 and active upper bound 0>
In [34]:
Show( Ω11 )
$$\begin{array}{c} \\ {𝓞(0)}^{\oplus3} \\ { \color{black}\vert^{0}} \\ \begin{pmatrix}{x_1}\\{x_2}\\{x_3}\end{pmatrix} \\ { \color{black} \downarrow_{\phantom{0}}} \\ {𝓞(1)}^{}\end{array}$$
In [35]:
a_0 = QuiverRowsObject( [ [ q."𝓞(0)", 1 ] ], QRows )
Out[35]:
GAP: <An object in Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) defined by 1 quiver vertices>
In [36]:
Show( a_0 )
$${𝓞(0)}^{}$$
In [37]:
d_0 = UniversalMorphismIntoZeroObject( a_0 )
Out[37]:
GAP: <A morphism in Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) defined by a 1 x 0 matrix of quiver algebra elements>
In [38]:
Show( d_0 )
$${𝓞(0)}^{}\xrightarrow{0}0$$
In [39]:
Ω22 = HomotopyCategoryObject( [ d_0 ], 0 )
Out[39]:
GAP: <An object in Homotopy category( Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) ) with active lower bound -1 and active upper bound 0>
In [40]:
Show( Ω22 )
$$\begin{array}{c} \\ {𝓞(0)}^{} \\ { \color{black}\vert^{0}} \\ 0 \\ { \color{black} \downarrow_{\phantom{0}}} \\ 0\end{array}$$
In [41]:
Ω = CreateExceptionalCollection( [ Ω00, Ω11, Ω22 ], [ "Ω^0(0)", "Ω^1(1)", "Ω^2(2)" ] )
Out[41]:
GAP: <An exceptional collection defined by the objects of the Full subcategory generated by 3 objects in Homotopy category( Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) )>
In [42]:
EndT_Ω = EndomorphismAlgebra( Ω )
Out[42]:
GAP: End( Ω^0(0) ⊕ Ω^1(1) ⊕ Ω^2(2) )
In [43]:
Dimension( EndT_Ω )
Out[43]:
12

Now construct the tilting equivalences $$ F: \mathrm{D}^\mathrm{b}(\operatorname{End}\, T_\Omega) \rightleftarrows \mathrm{D}^\mathrm{b}(\operatorname{End}\, T_\mathcal{O}) :G \mbox{.} $$ as an equivalence between the homotopy models: $$ F: \mathrm{Ho}^\mathrm{b}\!\left((\mathrm{End}\, T_\Omega)^\mathrm{dec}_\oplus\right) \rightleftarrows \mathrm{Ho}^\mathrm{b}\!\left((\mathrm{End}\, T_\mathcal{O})^\mathrm{dec}_\oplus\right) :G $$

In [44]:
F = ConvolutionFunctorFromHomotopyCategoryOfQuiverRows( Ω )
Out[44]:
GAP: Convolution functor
In [45]:
Display( F )
Convolution functor:

Homotopy category( Quiver rows( End( Ω^0(0) ⊕ Ω^1(1) ⊕ Ω^2(2) ) ) )
  |
  V
Homotopy category( Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) )
In [46]:
HoEndT_Ω = SourceOfFunctor( F )
Out[46]:
GAP: Homotopy category( Quiver rows( End( Ω^0(0) ⊕ Ω^1(1) ⊕ Ω^2(2) ) ) )
In [47]:
HoEndT_𝓞 = RangeOfFunctor( F )
Out[47]:
GAP: Homotopy category( Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) )
In [48]:
G = ReplacementFunctorIntoHomotopyCategoryOfQuiverRows( Ω )
Out[48]:
GAP: Replacement functor
In [49]:
Display( G )
Replacement functor:

Homotopy category( Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) )
  |
  V
Homotopy category( Quiver rows( End( Ω^0(0) ⊕ Ω^1(1) ⊕ Ω^2(2) ) ) )

Consider the images of the three twisted line bundles $\{\mathcal{O}(0), \mathcal{O}(1), \mathcal{O}(2)\} \subset \mathrm{Ho}^\mathrm{b}\!\left((\mathrm{End}\, T_𝓞)^\mathrm{dec}_\oplus\right)$ under the tilting equivalence $$ G: \mathrm{Ho}^\mathrm{b}\!\left((\operatorname{End}\, T_\mathcal{O})^\mathrm{dec}_\oplus \right) \rightarrow \mathrm{Ho}^\mathrm{b}\!\left((\operatorname{End}\, T_\Omega)^\mathrm{dec}_\oplus \right) \mbox{.} $$

In [50]:
𝓞0 = HoEndT_𝓞."𝓞(0)"
Out[50]:
GAP: <An object in Homotopy category( Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) ) with active lower bound 0 and active upper bound 0>
In [51]:
Show( 𝓞0 )
$$\begin{array}{c} \\ {𝓞(0)}^{}\end{array}$$
In [52]:
𝓞1 = HoEndT_𝓞."𝓞(1)"
Out[52]:
GAP: <An object in Homotopy category( Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) ) with active lower bound 0 and active upper bound 0>
In [53]:
Show( 𝓞1 )
$$\begin{array}{c} \\ {𝓞(1)}^{}\end{array}$$
In [54]:
𝓞2 = HoEndT_𝓞."𝓞(2)"
Out[54]:
GAP: <An object in Homotopy category( Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) ) with active lower bound 0 and active upper bound 0>
In [55]:
Show( 𝓞2 )
$$\begin{array}{c} \\ {𝓞(2)}^{}\end{array}$$
In [56]:
G𝓞0 = G( 𝓞0 )
Out[56]:
GAP: <An object in Homotopy category( Quiver rows( End( Ω^0(0) ⊕ Ω^1(1) ⊕ Ω^2(2) ) ) ) with active lower bound 0 and active upper bound 0>
In [57]:
Show( G𝓞0 )
$$\begin{array}{c} \\ {Ω^2(2)}^{}\end{array}$$
In [58]:
G𝓞1 = G( 𝓞1 )
Out[58]:
GAP: <An object in Homotopy category( Quiver rows( End( Ω^0(0) ⊕ Ω^1(1) ⊕ Ω^2(2) ) ) ) with active lower bound 0 and active upper bound 1>
In [59]:
Show( G𝓞1 )
$$\begin{array}{c} \\ {Ω^1(1)}^{} \\ { \color{black}\vert^{1}} \\ \begin{pmatrix}-{m_{2,3}^{1}}&-{m_{2,3}^{2}}&-{m_{2,3}^{3}}\end{pmatrix} \\ { \color{black} \downarrow_{\phantom{1}}} \\ {Ω^2(2)}^{\oplus3}\end{array}$$
In [60]:
G𝓞2 = G( 𝓞2 )
Out[60]:
GAP: <An object in Homotopy category( Quiver rows( End( Ω^0(0) ⊕ Ω^1(1) ⊕ Ω^2(2) ) ) ) with active lower bound 0 and active upper bound 2>
In [61]:
Show( G𝓞2 )
$$\begin{array}{c} \\ {Ω^0(0)}^{} \\ { \color{black}\vert^{2}} \\ \begin{pmatrix}-{m_{1,2}^{1}}&-{m_{1,2}^{2}}&-{m_{1,2}^{3}}\end{pmatrix} \\ { \color{black} \downarrow_{\phantom{2}}} \\ {Ω^1(1)}^{\oplus3} \\ { \color{black}\vert^{1}} \\ \begin{pmatrix}-{m_{2,3}^{1}}&-{m_{2,3}^{2}}&-{m_{2,3}^{3}}&0&0&0\\0&-{m_{2,3}^{1}}&0&-{m_{2,3}^{2}}&-{m_{2,3}^{3}}&0\\0&0&-{m_{2,3}^{1}}&0&-{m_{2,3}^{2}}&-{m_{2,3}^{3}}\end{pmatrix} \\ { \color{black} \downarrow_{\phantom{1}}} \\ {Ω^2(2)}^{\oplus6}\end{array}$$

Consider the images of the three twisted line bundles $\{\Omega^0(0), \Omega^1(1), \Omega^2(2)\} \subset \mathrm{Ho}^\mathrm{b}\!\left((\mathrm{End}\, T_\Omega)^\mathrm{dec}_\oplus\right)$ under the tilting equivalence $$ F: \mathrm{Ho}^\mathrm{b}\!\left((\operatorname{End}\, T_\Omega)^\mathrm{dec}_\oplus \right) \rightarrow \mathrm{Ho}^\mathrm{b}\!\left((\operatorname{End}\, T_\mathcal{O})^\mathrm{dec}_\oplus \right) \mbox{.} $$

In [62]:
FΩ00 = F( HoEndT_Ω."Ω^0(0)" )
Out[62]:
GAP: <An object in Homotopy category( Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) ) with active lower bound -2 and active upper bound 0>
In [63]:
Show( FΩ00 )
$$\begin{array}{c} \\ {𝓞(0)}^{\oplus3} \\ { \color{black}\vert^{0}} \\ \begin{pmatrix}{x_2}&-{x_1}&0\\{x_3}&0&-{x_1}\\0&{x_3}&-{x_2}\end{pmatrix} \\ { \color{black} \downarrow_{\phantom{0}}} \\ {𝓞(1)}^{\oplus3} \\ { \color{black}\vert^{-1}} \\ \begin{pmatrix}{y_0}\\{y_1}\\{y_2}\end{pmatrix} \\ { \color{black} \downarrow_{\phantom{-1}}} \\ {𝓞(2)}^{}\end{array}$$
In [64]:
Show( Ω00 )
$$\begin{array}{c} \\ {𝓞(0)}^{\oplus3} \\ { \color{black}\vert^{0}} \\ \begin{pmatrix}{x_2}&-{x_1}&0\\{x_3}&0&-{x_1}\\0&{x_3}&-{x_2}\end{pmatrix} \\ { \color{black} \downarrow_{\phantom{0}}} \\ {𝓞(1)}^{\oplus3} \\ { \color{black}\vert^{-1}} \\ \begin{pmatrix}{y_0}\\{y_1}\\{y_2}\end{pmatrix} \\ { \color{black} \downarrow_{\phantom{-1}}} \\ {𝓞(2)}^{}\end{array}$$
In [65]:
FΩ11 = F( HoEndT_Ω."Ω^1(1)" )
Out[65]:
GAP: <An object in Homotopy category( Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) ) with active lower bound -1 and active upper bound 0>
In [66]:
Show( FΩ11 )
$$\begin{array}{c} \\ {𝓞(0)}^{\oplus3} \\ { \color{black}\vert^{0}} \\ \begin{pmatrix}{x_1}\\{x_2}\\{x_3}\end{pmatrix} \\ { \color{black} \downarrow_{\phantom{0}}} \\ {𝓞(1)}^{}\end{array}$$
In [67]:
FΩ22 = F( HoEndT_Ω."Ω^2(2)" )
Out[67]:
GAP: <An object in Homotopy category( Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) ) with active lower bound 0 and active upper bound 0>
In [68]:
Show( FΩ22 )
$$\begin{array}{c} \\ {𝓞(0)}^{}\end{array}$$

Define the equivalences $$ I: \mathrm{Ho}^\mathrm{b}\!\left((\operatorname{End}\, T_\mathcal{O})^\mathrm{dec}_\oplus \right) \to \mathrm{D}^\mathrm{b}(\mathrm{End}\, T_\mathcal{O}) \mbox{,} $$ and $$ J: \mathrm{Ho}^\mathrm{b}\!\left((\operatorname{End}\, T_\mathcal{\Omega})^\mathrm{dec}_\oplus \right) \to \mathrm{D}^\mathrm{b}(\mathrm{End}\, T_\mathcal{\Omega}) \mbox{.} $$

In [69]:
I = EmbeddingFunctorIntoDerivedCategory( HoEndT_𝓞 )
Out[69]:
GAP: Equivalence functor from homotopy category onto derived category
In [70]:
Display( I )
Equivalence functor from homotopy category onto derived category:

Homotopy category( Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) )
  |
  V
Derived category( Quiver representations( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) )^op ) )
In [71]:
J = EmbeddingFunctorIntoDerivedCategory( HoEndT_Ω )
Out[71]:
GAP: Equivalence functor from homotopy category onto derived category
In [72]:
Display( J )
Equivalence functor from homotopy category onto derived category:

Homotopy category( Quiver rows( End( Ω^0(0) ⊕ Ω^1(1) ⊕ Ω^2(2) ) ) )
  |
  V
Derived category( Quiver representations( End( Ω^0(0) ⊕ Ω^1(1) ⊕ Ω^2(2) )^op ) )

Compute the images of $F(\Omega^i(i)) \in \mathrm{Ho}^\mathrm{b}\!\left((\operatorname{End}\, T_\mathcal{O})^\mathrm{dec}_\oplus \right)$ in the derived catgeory $\mathrm{D}^\mathrm{b}(\mathrm{End}\, T_\mathcal{O})$. Two of them are not in the Abelian heart of the latter:

In [73]:
IFΩ00 = I( FΩ00 )
Out[73]:
GAP: <An object in Derived category( Quiver representations( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) )^op ) )>
In [74]:
HomologySupport( IFΩ00 )
Out[74]:
GAP: [ -2 ]
In [75]:
IFΩ11 = I( FΩ11 )
Out[75]:
GAP: <An object in Derived category( Quiver representations( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) )^op ) )>
In [76]:
HomologySupport( IFΩ11 )
Out[76]:
GAP: [ -1 ]
In [77]:
IFΩ22 = I( FΩ22 )
Out[77]:
GAP: <An object in Derived category( Quiver representations( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) )^op ) )>
In [78]:
HomologySupport( IFΩ22 )
Out[78]:
GAP: [ 0 ]

Compute the images of $G(\mathcal{O}(i)) \in \mathrm{Ho}^\mathrm{b}\!\left((\operatorname{End}\, T_\Omega)^\mathrm{dec}_\oplus \right)$ in the derived catgeory $\mathrm{D}^\mathrm{b}(\mathrm{End}\, T_\Omega)$. All of them turn out to lie in the Abelian heart of the latter:

In [79]:
JG𝓞0 = J( G𝓞0 )
Out[79]:
GAP: <An object in Derived category( Quiver representations( End( Ω^0(0) ⊕ Ω^1(1) ⊕ Ω^2(2) )^op ) )>
In [80]:
HomologySupport( JG𝓞0 )
Out[80]:
GAP: [ 0 ]

This homology is concentrated in degree $0$ and hence isomorphic to the image $J( G ( \mathcal{O}(0) ) )$ in $\mathrm{D}^\mathrm{b}(\mathrm{End}\, T_\Omega)$. It is, as an object in $(\mathrm{End}\, T_\Omega)\mathrm{-mod}$ represented by a functor $((\operatorname{End}\, T_\Omega)^\mathrm{dec})^\mathrm{op} \rightarrow \mathbb{Q}\mathrm{-vec}$. The following command displays the dimensions of the images under this functor of the three objects $\{\Omega^0(0),\Omega^1(1),\Omega^2(2)\} = \operatorname{Obj}\left(\left((\operatorname{End}\, T_\Omega)^\mathrm{dec}\right)^\mathrm{op}\right)$:

In [81]:
DimensionVector( HomologyAt( JG𝓞0, 0 ) )
Out[81]:
GAP: [ 3, 3, 1 ]
In [82]:
JG𝓞1 = J( G𝓞1 )
Out[82]:
GAP: <An object in Derived category( Quiver representations( End( Ω^0(0) ⊕ Ω^1(1) ⊕ Ω^2(2) )^op ) )>
In [83]:
HomologySupport( JG𝓞1 )
Out[83]:
GAP: [ 0 ]
In [84]:
DimensionVector( HomologyAt( JG𝓞1, 0 ) )
Out[84]:
GAP: [ 6, 8, 3 ]
In [85]:
JG𝓞2 = J( G𝓞2 )
Out[85]:
GAP: <An object in Derived category( Quiver representations( End( Ω^0(0) ⊕ Ω^1(1) ⊕ Ω^2(2) )^op ) )>
In [86]:
HomologySupport( JG𝓞2 )
Out[86]:
GAP: [ 0 ]
In [87]:
HomologyAt( JG𝓞2, 0 )
Out[87]:
GAP: <10,15,6>

We take a random object in $\mathrm{Ho}^\mathrm{b}\!\left((\operatorname{End}\, T_\mathcal{O})^\mathrm{dec}_\oplus \right) \simeq \mathrm{D}^\mathrm{b}(\mathrm{End}\, T_\mathcal{O})$ and look at its image under the monad $F\circ G: \mathrm{Ho}^\mathrm{b}\!\left((\operatorname{End}\, T_\mathcal{O})^\mathrm{dec}_\oplus \right) \to \mathrm{Ho}^\mathrm{b}\!\left((\operatorname{End}\, T_\mathcal{O})^\mathrm{dec}_\oplus \right)$. We then compare the homologies of both objects:

In [88]:
a = RandomObject( HoEndT_𝓞, 2 )
Out[88]:
GAP: <An object in Homotopy category( Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) ) with active lower bound -2 and active upper bound 2>
In [89]:
Show( a )
$$\begin{array}{c} \\ {𝓞(2)}^{} \oplus {𝓞(0)}^{} \\ { \color{black}\vert^{2}} \\ \begin{pmatrix}0&0&3{𝓞(2)}\\0&0&2{x_1y_1}+{x_1y_0}\end{pmatrix} \\ { \color{black} \downarrow_{\phantom{2}}} \\ {𝓞(1)}^{} \oplus {𝓞(0)}^{} \oplus {𝓞(2)}^{} \\ { \color{black}\vert^{1}} \\ \begin{pmatrix}-8{𝓞(1)}&4{y_2}+4{y_0}&0\\-4{x_3}-8{x_2}&{x_3y_2}+3{x_2y_2}-{x_2y_1}+{x_1y_2}+4{x_1y_1}&4{𝓞(0)}\\0&0&0\end{pmatrix} \\ { \color{black} \downarrow_{\phantom{1}}} \\ {𝓞(1)}^{} \oplus {𝓞(2)}^{} \oplus {𝓞(0)}^{} \\ { \color{black}\vert^{0}} \\ \begin{pmatrix}0&2{y_2}+2{y_0}&0\\0&4{𝓞(2)}&0\\0&{x_3y_2}+{x_2y_2}+{x_2y_1}+{x_1y_2}&0\end{pmatrix} \\ { \color{black} \downarrow_{\phantom{0}}} \\ {𝓞(0)}^{} \oplus {𝓞(2)}^{} \oplus {𝓞(1)}^{} \\ { \color{black}\vert^{-1}} \\ \begin{pmatrix}4{𝓞(0)}&{x_3}+3{x_2}\\0&0\\0&4{𝓞(1)}\end{pmatrix} \\ { \color{black} \downarrow_{\phantom{-1}}} \\ {𝓞(0)}^{} \oplus {𝓞(1)}^{}\end{array}$$
In [90]:
FGa = F( G( a ) )
Out[90]:
GAP: <An object in Homotopy category( Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) ) with active lower bound 2 and active upper bound 2>
In [91]:
Show( FGa )
$$\begin{array}{c} \\ {𝓞(0)}^{}\end{array}$$

Finally we define the counit from $F \circ G \to \mathrm{Id}_{\mathrm{Ho}^\mathrm{b}\!\left((\operatorname{End}\, T_\mathcal{O})^\mathrm{dec}_\oplus \right)}$:

In [92]:
eta = CounitOfConvolutionReplacementAdjunction( Ω )
Out[92]:
GAP: Conv( Rep( - ) ) --> Id( - )
In [93]:
eta_a = eta( a )
Out[93]:
GAP: <A morphism in Homotopy category( Quiver rows( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) ) ) ) with active lower bound 2 and active upper bound 2>
In [94]:
Show( eta_a )
$$\begin{array}{ccc} \\ {𝓞(0)}^{}&-\phantom{-}{\begin{pmatrix}-\frac{2}{3}{x_1y_1}-\frac{1}{3}{x_1y_0}&{𝓞(0)}\end{pmatrix}}\phantom{-}\rightarrow&{𝓞(2)}^{} \oplus {𝓞(0)}^{} \\ \vert^{2} &&\vert^{2} \\ 0&&\begin{pmatrix}0&0&3{𝓞(2)}\\0&0&2{x_1y_1}+{x_1y_0}\end{pmatrix} \\ \downarrow_{\phantom{2}}&& \downarrow_{\phantom{2}}\\ 0&-\phantom{-}{0}\phantom{-}\rightarrow&{𝓞(1)}^{} \oplus {𝓞(0)}^{} \oplus {𝓞(2)}^{} \\ \vert^{1} &&\vert^{1} \\ 0&&\begin{pmatrix}-8{𝓞(1)}&4{y_2}+4{y_0}&0\\-4{x_3}-8{x_2}&{x_3y_2}+3{x_2y_2}-{x_2y_1}+{x_1y_2}+4{x_1y_1}&4{𝓞(0)}\\0&0&0\end{pmatrix} \\ \downarrow_{\phantom{1}}&& \downarrow_{\phantom{1}}\\ 0&-\phantom{-}{0}\phantom{-}\rightarrow&{𝓞(1)}^{} \oplus {𝓞(2)}^{} \oplus {𝓞(0)}^{} \\ \vert^{0} &&\vert^{0} \\ 0&&\begin{pmatrix}0&2{y_2}+2{y_0}&0\\0&4{𝓞(2)}&0\\0&{x_3y_2}+{x_2y_2}+{x_2y_1}+{x_1y_2}&0\end{pmatrix} \\ \downarrow_{\phantom{0}}&& \downarrow_{\phantom{0}}\\ 0&-\phantom{-}{0}\phantom{-}\rightarrow&{𝓞(0)}^{} \oplus {𝓞(2)}^{} \oplus {𝓞(1)}^{} \\ \vert^{-1} &&\vert^{-1} \\ 0&&\begin{pmatrix}4{𝓞(0)}&{x_3}+3{x_2}\\0&0\\0&4{𝓞(1)}\end{pmatrix} \\ \downarrow_{\phantom{-1}}&& \downarrow_{\phantom{-1}}\\ 0&-\phantom{-}{0}\phantom{-}\rightarrow&{𝓞(0)}^{} \oplus {𝓞(1)}^{} \\ \end{array}$$
In [95]:
IsIsomorphism( eta_a )
Out[95]:
true

Now we use the embedding the derived category:

In [96]:
Ia = I( a )
Out[96]:
GAP: <An object in Derived category( Quiver representations( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) )^op ) )>
In [97]:
suppIa = HomologySupport( Ia )
Out[97]:
GAP: [ 2 ]
In [98]:
List( suppIa, i -> HomologyAt( Ia, i ) )
Out[98]:
GAP: [ <1,0,0> ]
In [99]:
IFGa = I( FGa )
Out[99]:
GAP: <An object in Derived category( Quiver representations( End( 𝓞(0) ⊕ 𝓞(1) ⊕ 𝓞(2) )^op ) )>
In [100]:
suppIFGa = HomologySupport( IFGa )
Out[100]:
GAP: [ 2 ]
In [101]:
List( suppIa, i -> HomologyAt( IFGa, i ) )
Out[101]:
GAP: [ <1,0,0> ]
In [102]:
Length( BasisOfExternalHom( a, FGa ) )
Out[102]:
1
In [103]:
Length( BasisOfExternalHom( Ia, IFGa ) )
Out[103]:
1
In [ ]: