2.1

Let $\mathcal {G}$ be an algebraic group over ${\mathbb F}$ and $\mathcal {H}$ a normal subgroup of $\mathcal {G}$ . We consider $\mathbf {Rep}(\mathcal {G}/\mathcal {H})$ as a full subcategory of $\mathbf {Rep}(\mathcal {G})$ . Let ${\mathfrak {g}}$ and ${\mathfrak {h}}$ be, respectively, the Lie algebras of $\mathcal {G}$ and $\mathcal {H}$ , both considered as representations of $\mathcal {G}$ via the adjoint action. The abelianization ${\mathfrak {h}}^{\mathrm {ab}}$ belongs to the subcategory $\mathbf {Rep}(\mathcal {G}/\mathcal {H})$ , as $\mathcal {H}$ acts on it trivially.

Let X be an object of $\mathbf {Rep}(\mathcal {G}/\mathcal {H})$ . Then there exists a functorial (in X) map

(2.1)

defined as follows:

Step one: The canonical isomorphism between (Yoneda) and the group cohomology $H^1(\mathcal {G}, X)$ (for algebraic groups) composed with the restriction map for the subgroup $\mathcal {H}\mathrel {\unlhd } \mathcal {G}$ from group cohomology gives a map

where X is considered as an additive algebraic group, and the action of $\mathcal {G}$ on $\mathrm {Hom}_{\mathbf {AlgGr}}(\mathcal {H}, X)$ is via its natural action on X and its conjugation action on $\mathcal {H}$ . In the article, we will need to work with this map explicitly, so here we shall give its description more concretely (we may take this concrete description as the definition). Given represented by an extension

choose a linear section s of to identify $Z\cong X\oplus {\mathbb F}$ as vector spaces. Express the action of $\mathcal {H}$ on Z in terms of this decomposition. Since $\mathcal {H}$ acts trivially on X, every element h of $\mathcal {H}$ will act on Z through an element of $\mathrm {GL}(X\oplus {\mathbb F})$ of the form

$$\begin{align*}\begin{pmatrix} 1 & h_{12}\\ 0& 1\end{pmatrix}. \end{align*}$$

Sending $h\mapsto h_{12}$ , we obtain a morphism of algebraic groups

This map is easily seen to be independent of both the choice of the section s and the choice of the representative of $\mathcal {Z}$ . Thus, so far, we have defined a map from to $\mathrm {Hom}_{\mathbf {AlgGr}}(\mathcal {H}, X)$ .

Identifying Z as $X\oplus {\mathbb F}$ via a choice of a section and writing elements of $\mathrm {GL}(Z)$ as $2\times 2$ matrices again, given $g\in \mathcal {G}$ and $h\in \mathcal {H}$ , the action of the element $ghg^{-1}$ of $\mathcal {G}$ on Z is through the element

$$\begin{align*}\begin{pmatrix} g_{11} & g_{12}\\ 0& 1\end{pmatrix} \begin{pmatrix} 1 & h_{12}\\ 0& 1\end{pmatrix} \begin{pmatrix} g_{11}^{-1} & -g_{11}^{-1}g_{12} \\ 0& 1\end{pmatrix} = \begin{pmatrix} 1 & g_{11}h_{12}\\ 0& 1\end{pmatrix}, \end{align*}$$

where $g_{11}=g_X$ is the action of g on X. It follows that , so that is $\mathcal {G}$ -equivariant. We have thus indeed defined a map

(2.2)

Its functoriality in X can be checked by a direct computation from the construction.

Step two: The map (2.1) is obtained by composing (2.2) with

$$\begin{align*}\mathrm{Hom}_{\mathbf{AlgGr}}(\mathcal{H}, X)^{\mathcal{G}} \xrightarrow{\mathrm{Lie}} \mathrm{Hom}_{\mathbf{Lie}}({\mathfrak{h}}, X)^{\mathcal{G}} \stackrel{(\ast)}{\cong} \mathrm{Hom}_{{\mathbb F}}({\mathfrak{h}}^{\mathrm{ab}}, X)^{\mathcal{G}} = \mathrm{Hom}_{\mathcal{G}}({\mathfrak{h}}^{\mathrm{ab}}, X), \end{align*}$$

where the first map passes on to the Lie algebras ( $\mathbf {Lie}$ = the category of Lie algebras over ${\mathbb F}$ ) and $(\ast )$ is because the Lie algebra X is abelian.Footnote ²

2.2

From this point on in the article, we fix a Tannakian category ${\mathbf {T}}$ over ${\mathbb F}$ . By a fiber functor, we shall mean a fiber functor over ${\mathbb F}$ . Let $\omega $ be a fiber functor for ${\mathbf {T}}$ . For any object X of ${\mathbf {T}}$ , we denote the Tannakian group of X with respect to $\omega $ (i.e., the group $\underline {\mathrm {Aut}}^{\otimes }(\omega |_{\langle X\rangle ^{\otimes }})$ of tensor automorphisms of the restriction of $\omega $ to $\langle X\rangle ^{\otimes }$ , see [Reference Deligne and Milne6]) by $\mathcal {G}(X,\omega )$ . By the main theorem of Tannakian formalism [Reference Deligne and Milne6, Theorem 2.11], the functor $\omega $ gives an equivalence of tensor categories

$$\begin{align*}\langle X\rangle^{\otimes} \rightarrow \mathbf{Rep}(\mathcal{G}(X,\omega)). \end{align*}$$

We identify $\mathcal {G}(X,\omega )$ as a subgroup of $\mathrm {GL}(\omega X)$ via its action on X. Via the equivalence above, the internal Hom ${\underline {\mathrm {End}}}(X)={\underline {\mathrm{Hom}}}(X,X)$ then becomes $\mathrm {End}_{\mathbb F}(\omega X)$ equipped with the restriction of the adjoint action of $\mathrm {GL}(\omega X)$ to the subgroup $\mathcal {G}(X,\omega )$ .

From here until the end of Section 2, we fix an object M of ${\mathbf {T}}$ and an object N of $\langle M\rangle ^{\otimes }$ . There is a surjective restriction map

$$\begin{align*}\mathcal{G}(M,\omega) \rightarrow \mathcal{G}(N,\omega), \end{align*}$$

the kernel of which we shall denote by $\mathcal {H}(M,N,\omega )$ . An object X of $\langle M\rangle ^{\otimes }$ belongs to the subcategory $\langle N\rangle ^{\otimes }$ if and only if the action of $\mathcal {H}(M,N,\omega )$ on $\omega X$ is trivial.

We shall denote the Lie algebra of $\mathcal {H}(M,N,\omega )$ by ${\mathfrak {h}}(M,N,\omega )$ . It is a Lie subalgebra of $\mathrm {End}_{\mathbb F}(\omega X)$ . The adjoint action of $\mathcal {G}(M,\omega )$ restricts to an action on ${\mathfrak {h}}(M,N,\omega )$ , which via the main theorem of Tannakian formalism gives rise to a Lie subobject

$$\begin{align*}{\mathfrak{h}}(M,N) \subset {\underline{\mathrm{End}}}(M) \end{align*}$$

whose image under $\omega $ is the Lie subalgebra ${\mathfrak {h}}(M,N,\omega )$ of $\mathrm {End}_{\mathbb F}(\omega X)$ . Moreover, the subobject ${\mathfrak {h}}(M,N)$ of ${\underline {\mathrm {End}}}(M)$ is independent of the choice of the fiber functor $\omega $ . See, for instance, [Reference Eskandari and Murty10, Section 2].

2.3

Let X be an object of the subcategory $\langle N\rangle ^{\otimes }$ of $\langle M\rangle ^{\otimes }$ . Choose a fiber functor $\omega $ for ${\mathbf {T}}$ . In view of the equivalence of categories $\langle M\rangle ^{\otimes } \xrightarrow { \, \omega \, } \mathbf {Rep}(\mathcal {G}(M,\omega ))$ , the map (2.1) for $\mathcal {G}=\mathcal {G}(M,\omega )$ and $\mathcal {H}=\mathcal {H}(M,N,\omega )$ gives us a map

that is functorial in the object X of $\langle N\rangle ^{\otimes }$ . Note that the object ${\mathfrak {h}}(M,N)^{\mathrm {ab}}$ belongs to $\langle N\rangle ^{\otimes }$ because $\mathcal {H}(M,N,\omega )$ acts on $\omega {\mathfrak {h}}(M,N)^{\mathrm {ab}}$ trivially.

Proof This is a consequence of the fact, due to Deligne [Reference Deligne5, Sections 1.12 and 1.13], that any two fiber functors are isomorphic over an algebraic closure $\overline {{\mathbb F}}$ of ${\mathbb F}$ . The proof was given in details in [Reference Eskandari8] (see Lemma 3.2.1 therein) when ${\mathfrak {h}}(M,N)$ is the Lie algebra of the unipotent radical of the fundamental group of M. The argument goes through essentially identically to establish the more general statement given here. Given two fiber functors $\omega $ and $\omega '$ , let $\alpha $ be an isomorphism of $\otimes $ -functors $\overline {\omega }\rightarrow \overline {\omega '}$ , where throughout, we use the bar symbol to refer to the extension of scalars from ${\mathbb F}$ to $\overline {{\mathbb F}}$ . For brevity, let us tentatively write ${\mathfrak {h}}$ for ${\mathfrak {h}}(M,N)$ , and $\mathcal {G}$ , $\mathcal {H}$ (resp. $\mathcal {G}'$ , $\mathcal {H}'$ ) for $\mathcal {G}(M,\omega )$ , $\mathcal {H}(M,N,\omega )$ (resp. the counterparts for $\omega '$ ). Then $\alpha $ gives an isomorphism $\overline {\mathcal {G}}\rightarrow \overline {\mathcal {G}'}$ restricting to an isomorphism $\overline {\mathcal {H}}\rightarrow \overline {\mathcal {H}'}$ . Then there is a commutative diagram

Here, the horizontal maps are all of the form of equation (2.1) of Section 2.1. The vertical maps in the top and bottom squares are by extension of scalars; that the ones on the left (from $\mathrm {Ext}^1_{\mathcal {G}}$ to $\mathrm {Ext}^1_{\overline {\mathcal {G}}}$ and the counterpart for $\mathcal {G}'$ ) are injective will not matter for the proof, but it is by the fact that invariants behave well with respect to extension of scalars [Reference Milne13, Proposition 4.31]. The isomorphisms marked by $\omega $ and $\omega '$ are given by Tannakian formalism. The isomorphism $\alpha $ on the right is given by the isomorphism of functors $\alpha $ . The isomorphism marked by $\alpha $ on the left is obtained at the level of extensions by replacing the $\overline {\mathcal {G}}$ -representation $\overline {\omega X}$ by the $\overline {\mathcal {G}'}$ -representation $\overline {\omega ' X}$ via $\alpha $ , keeping the middle vector space unchanged, and transferring the $\overline {\mathcal {G}}$ -action on it to a $\overline {\mathcal {G}'}$ -action via $\alpha $ . That the triangles on the left and right commute is because $\alpha $ is an isomorphism of functors. That the top and bottom squares commute is easily seen from the definition of the map (2.1). That the middle square is commutative is seen via again the definition of the horizontal maps, on noting that the map $\overline {\omega {\mathfrak {h}}}\rightarrow \overline {\omega ' {\mathfrak {h}}}$ given by the morphism of functors $\alpha $ coincides with the map obtained by first considering the isomorphism $\overline {\mathcal {H}}\rightarrow \overline {\mathcal {H}'}$ given by $\alpha $ , and then passing to the Lie algebras.

The independence of $\Psi ^{M,N}_X$ from the choice of $\omega $ now follows from the commutativity of the diagram and injectivity of the top and bottom vertical arrows on the right.

2.4

For convenience and referencing purposes, let us summarize the discussion so far. Fix an object M of ${\mathbf {T}}$ ( = a Tannakian category over ${\mathbb F}$ ) and an object N of $\langle M\rangle ^{\otimes }$ . Then for every object X of $\langle N\rangle ^{\otimes }$ , we have constructed a map

(2.3)

which is functorial with respect to morphisms $X\rightarrow X'$ between two objects $X,X'$ of $\langle N\rangle ^{\otimes }$ . Given an extension class $\mathcal {Z}$ of by X in $\langle M\rangle ^{\otimes }$ , the image of $\mathcal {Z}$ under $\Psi ^{M,N}_X$ is computed as follows: Choose a representative extension for $\mathcal {Z}$ with its middle denoted by Z. Choose a fiber functor $\omega $ and a linear section of the surjection to obtain a decomposition $\omega Z\cong \omega X\oplus {\mathbb F}$ as vector spaces. The action of $\mathcal {H}(M,N,\omega )$ on $\omega Z$ expressed in terms of this decomposition gives rise to a $\mathcal {G}(M,\omega )$ -equivariant morphism of algebraic groups $\mathcal {H}(M,N,\omega )\rightarrow \omega X$ . Passing to the Lie algebras and the abelianization, we obtain a morphism of $\mathcal {G}(M,\omega )$ -representations $\omega {\mathfrak {h}}(M,N)^{\mathrm {ab}} = {\mathfrak {h}}(M,N,\omega )^{\mathrm {ab}}\rightarrow \omega X$ . This map is $\omega $ of $\Psi ^{M,N}_X(\mathcal {Z})$ . This recipe for defining $\Psi ^{M,N}_X(\mathcal {Z})$ is indeed well-defined and in particular, is independent of the choice of a fiber functor $\omega $ .

2.5

Under mild conditions (which will be satisfied for the purposes of this article), the kernel of the map $\Psi ^{M,N}_X$ can be intrinsically described, as follows.

Proof We calculate $\Psi ^{M,N}_X$ using the fiber functor $\omega $ . By construction (see Sections 2.1 and 2.4), $\Psi ^{M,N}_X$ is the composition

where we have abbreviated $\mathcal {G}(M,\omega )$ , $\mathcal {H}(M,N,\omega )$ , and ${\mathfrak {h}}(M,N),$ respectively, as $\mathcal {G}$ , $\mathcal {H}$ , and ${\mathfrak {h}}$ . By the hypothesis of connectedness of $\mathcal {H}(M,N,\omega )$ , the map denoted by $\mathrm {Lie}$ (which passes to the Lie algebras) is injective. Thus, an extension is in the kernel of $\Psi ^{M,N}_X$ if and only if $\omega \mathcal {Z}$ is in the kernel of the map $(\ast )$ of the diagram. The result now follows from the first three terms of the inflation-restriction exact sequence for cohomology of algebraic groups together with the interpretation of $H^1(\mathcal {G}(M,\omega )/\mathcal {H}(M,N,\omega ), \omega X)$ as . In the interest of completeness and since we already have an explicit definition of $(\ast )$ at hand, we give a more explicit version of the argument: Denoting the middle object of a representative of $\mathcal {Z}$ by Z, upon recalling the construction of the map (2.2) from Section 2.1, we see that $\Psi ^{M,N}_X(\mathcal {Z})=0$ if and only if $\omega \mathcal {Z}$ admits an $\mathcal {H}(M,N,\omega )$ -equivariant section, which in turn (since $X\in \langle N\rangle ^{\otimes }$ ) is equivalent to the action of $\mathcal {H}(M,N,\omega )$ on $\omega Z$ being trivial. The latter is equivalent to Z being in $\langle N\rangle ^{\otimes }$ , or in other words, $\mathcal {Z}$ being in .

2.6

We now consider an important special case. Suppose that $\langle N\rangle ^{\otimes }$ is the full subcategory of all the semisimple objects of $\langle M\rangle ^{\otimes }$ , or that equivalently, $\mathcal {H}(M,N,\omega )$ for one or all choices of $\omega $ is the unipotent radical of $\mathcal {G}(M,\omega )$ . Then for every semisimple object X of $\langle M\rangle ^{\otimes }$ , the map $\Psi ^{M,N}_X$ of equation (2.3) is an isomorphism. Indeed, the injectivity of $\Psi ^{M,N}_X$ follows from Lemma 2.5.1, as $\mathcal {H}(M,N,\omega )$ is connected and $\langle N\rangle ^{\otimes }$ is semisimple. As for surjectivity, since $\mathcal {H}(M,N,\omega )$ is unipotent, the map $\mathrm {Lie}$ of the diagram of the proof of Lemma 2.5.1 is an isomorphism, so the surjectivity amounts to the surjectivity of the map marked as $(\ast )$ in the same diagram. The surjectivity of this map can be seen via the inflation-restriction sequence of group cohomology (since $\mathcal {G}(M,\omega )/\mathcal {H}(M,N,\omega )\cong \mathcal {G}(N,\omega )$ is reductive and hence $H^2(\mathcal {G}(N,\omega ),-)$ vanishes). One can also easily check surjectivity of $(\ast )$ in this case just from our explicit definition from Section 2.1. See [Reference Eskandari8, Section 3.1] for details.

3.1

From here until the end of Section 4, we fix the following data:

• – an object M of ${\mathbf {T}}$ , equipped with a finite increasing filtration $(F_r M)_{r\in {\mathbb Z}}$ with

  $$\begin{align*}0=F_0M \subsetneq F_1M \subsetneq F_2M\subsetneq \cdots \subsetneq F_kM=M \hspace{.4in}(k\geq 1). \end{align*}$$

As usual, we denote

$$\begin{align*}\mathrm{Gr}^FM = \bigoplus\limits_r \mathrm{Gr}^F_r M = \bigoplus\limits_r \frac{F_rM}{F_{r-1}M}. \end{align*}$$

We will adopt the notation of Section 2.2 for Tannakian fundamental groups, their subgroups, Lie algebras, etc. Throughout, we shall set

$$\begin{align*}{\mathfrak{u}}(M) := {\mathfrak{h}}(M,\mathrm{Gr}^FM) \end{align*}$$

and for every fiber functor $\omega $ ,

$$\begin{align*}\mathcal{U}(M,\omega) := \mathcal{H}(M,\mathrm{Gr}^FM,\omega) \end{align*}$$

(see Section 2.2). The fundamental group $\mathcal {G}(M,\omega )$ of M respects the filtration $F_\bullet \omega M:=\omega F_\bullet M$ , and $\mathcal {U}(M,\omega )$ is the subgroup of $\mathcal {G}(M,\omega )$ consisting of the elements that act trivially on $\mathrm {Gr}^F\omega M$ . In particular, $\mathcal {U}(M,\omega )$ is unipotent. If $\mathrm {Gr}^FM$ is semisimple, then $\mathcal {U}(M,\omega )$ will be the unipotent radical of $\mathcal {G}(G,\omega )$ . In what follows, unless otherwise indicated, we will not assume that $\mathrm {Gr}^FM$ is semisimple.

3.2

The filtration $F_\bullet $ on M induces a finite increasing filtration on the internal Hom ${\underline {\mathrm {End}}}(M)$ . For any fiber functor $\omega $ and any integer n, the image of $F_n{\underline {\mathrm {End}}}(M)$ under $\omega $ is the subspace of $\omega {\underline {\mathrm {End}}}(M) = \mathrm {End}_{\mathbb F}(\omega M)$ consisting of all the linear maps $f: \omega M\rightarrow \omega M$ such that $f(F_r \omega M) \subset F_{r+n}\omega M$ for all r. That the subobject $F_n{\underline {\mathrm {End}}}(M)$ defined this way is independent of the choice of $\omega $ can be seen using the fact, due to Deligne [Reference Deligne5], that every two fiber functors are isomorphic after base change to an algebraic closure of ${\mathbb F}$ . From the description of $F_\bullet {\underline {\mathrm {End}}}(M),$ it is easily seen that

$$\begin{align*}[F_n{\underline{\mathrm{End}}}(M), F_m{\underline{\mathrm{End}}}(M)] = F_{n+m}{\underline{\mathrm{End}}}(M). \end{align*}$$

In particular, $F_{-1}{\underline {\mathrm {End}}}(M)$ is a Lie subobject of ${\underline {\mathrm {End}}}(M)$ with derived algebra $F_{-2}{\underline {\mathrm {End}}}(M)$ .

3.3

Let $\omega $ be a fiber functor. By a splitting of $\omega M,$ we shall mean the data of a (linear) section $s_r$ of the quotient map $F_r\omega M \rightarrow \mathrm {Gr}^F_r\omega M$ for each integer r. Given a choice of a splitting $(s_r)$ of $\omega M$ , for each $r,$ we use the section $s_r$ to obtain an isomorphism $F_r \omega M \cong F_{r-1}\omega M \oplus \mathrm {Gr}^F_r\omega M$ as vector spaces. Putting these isomorphisms together, we obtain isomorphisms

$$\begin{align*}\omega M = F_k \omega M \cong F_{k-1} \omega M\oplus \mathrm{Gr}^F_k\omega M \cong F_{k-2} \omega M\oplus \mathrm{Gr}^F_{k-1}\omega M\oplus \mathrm{Gr}^F_k\omega M \cong \cdots \end{align*}$$

and thus an isomorphism

(3.1) $$ \begin{align} \omega M \cong \bigoplus\limits_{r=1}^k \mathrm{Gr}^F_r\omega M \end{align} $$

of vector spaces. This isomorphism induces isomorphisms

(3.2) $$ \begin{align} \frac{F_n\omega M}{F_m\omega M} \cong \bigoplus\limits_{r=m+1}^n \mathrm{Gr}^F_r\omega M. \end{align} $$

for all $m<n$ , all compatible with the natural inclusion and projection maps.

Once a splitting of $\omega M$ is chosen, we make identifications given by (3.1) and (3.2). We will then write elements of $\mathrm {End}_{\mathbb F}(\omega M)$ and $\mathrm {GL}(\omega M)$ as $k\times k$ matrices. Similarly, elements of $\mathrm {End}_{\mathbb F}(\frac {F_n\omega M}{F_m\omega M})$ and more generally, $\mathrm {Hom}_{\mathbb F}(\frac {F_n\omega M}{F_m\omega M}, \frac {F_{n'}\omega M}{F_{m'}\omega M})$ will become matrices. The entry $f_{ij}$ of an element $f=(f_{ij})\in \mathrm {End}_{\mathbb F} (\omega M)$ is the component of f in $\mathrm {Hom}_{\mathbb F}(\mathrm {Gr}^F_j\omega M, \mathrm {Gr}^F_i\omega M)$ via the decomposition

(3.3) $$ \begin{align} \mathrm{End}_{\mathbb F} \omega M \cong \mathrm{End}_{\mathbb F} (\mathrm{Gr}^F\omega M) = \bigoplus\limits_{1\leq i,j\leq k} \mathrm{Hom}_{\mathbb F}(\mathrm{Gr}^F_j\omega M, \mathrm{Gr}^F_i\omega M). \end{align} $$

The group $\mathcal {G}(M,\omega )$ is contained in the subgroup of $\mathrm {GL}(\omega M)$ consisting of upper triangular invertible matrices. If $\sigma =(\sigma _{ij})$ is an element of $\mathcal {G}(M,\omega )$ , the automorphism $\sigma _{\frac {F_nM}{F_mM}}$ (i.e., the action of $\sigma $ on $\frac {F_nM}{F_mM}$ ) of $F_n\omega M/F_m\omega M$ is simply given by truncating the matrix $(\sigma _{ij})$ to the part with $m< i,j\leq n$ .

The unipotent group $\mathcal {U}(M,\omega )$ is the intersection of $\mathcal {G}(M,\omega )$ and the subgroup of $\mathrm {GL}(\omega M)$ consisting of upper triangular elements with the identity maps on the diagonal. The Lie algebra $\omega F_{-1}{\underline {\mathrm {End}}}(M)$ consists of strictly upper triangular elements of $\mathrm {End}_{\mathbb F}(\omega M)$ . The subalgebra $\omega F_{-2}{\underline {\mathrm {End}}}(M)$ consists of those matrices $(f_{ij})$ for which $f_{ij}=0$ if $j-i\leq 1$ , that is, it consists of strictly upper triangular matrices with zeros on the superdiagonal.

3.4

The goal of this paragraph is to define a canonical surjective morphism

$$\begin{align*}\pi: F_{-1}{\underline{\mathrm{End}}}(M) \rightarrow \bigoplus\limits_{r} {\underline{\mathrm{Hom}}}(\mathrm{Gr}^F_{r+1}M, \mathrm{Gr}^F_{r}M) \end{align*}$$

with

$$\begin{align*}\ker(\pi) = F_{-2}{\underline{\mathrm{End}}}(M). \end{align*}$$

This map will play an important role in the article. Let $\omega $ be a fiber functor for ${\mathbf {T}}$ . We will first explicitly define the linear map $\omega \pi $ , and then verify that $\omega \pi $ indeed comes from a morphism $\pi $ that is independent of the choice of $\omega $ .

Choose a splitting of $\omega M$ in the sense of Section 3.3. Use the isomorphism (3.3) given by this splitting to write the elements of $\mathrm {End}_{\mathbb F}(\omega M)$ as $k\times k$ matrices. Let $\omega \pi $ be the linear map

$$ \begin{align*} \omega F_{-1}{\underline{\mathrm{End}}}(M) & = \bigoplus\limits_{\stackrel{i,j}{j-i\geq 1}} \mathrm{Hom}_{\mathbb F}(\mathrm{Gr}^F_{j}\omega M, \mathrm{Gr}^F_{i}\omega M) \xrightarrow{ \ \omega\pi \ } \\ & \bigoplus\limits_{r} \ \mathrm{Hom}_{\mathbb F}(\mathrm{Gr}^F_{r+1}\omega M, \mathrm{Gr}^F_{r}\omega M) = \bigoplus\limits_{r} \ \omega {\underline{\mathrm{Hom}}}(\mathrm{Gr}^F_{r+1}M, \mathrm{Gr}^F_{r}M) \end{align*} $$

that sends

$$\begin{align*}f=(f_{i,j}) \mapsto (f_{r,r+1}) \end{align*}$$

(the indices compatible with those used in the decompositions). That is, $\omega \pi $ is the projection onto the superdiagonal entries. The notation $\omega \pi $ for this map will be justified momentarily in the next lemma. Clearly, $\omega \pi $ is surjective and

$$\begin{align*}\ker(\omega\pi) = \bigoplus\limits_{\stackrel{i,j}{j-i\geq 2}} \mathrm{Hom}_{\mathbb F}(\mathrm{Gr}^F_{j}\omega M, \mathrm{Gr}^F_{i}\omega M) = \omega F_{-2}{\underline{\mathrm{End}}}(M). \end{align*}$$

Lemma 3.4.1 The map

$$\begin{align*}\omega \pi: \omega F_{-1}{\underline{\mathrm{End}}}(M) \rightarrow \bigoplus\limits_{r} \ \omega {\underline{\mathrm{Hom}}}(\mathrm{Gr}^F_{r+1}M, \mathrm{Gr}^F_{r}M) \end{align*}$$

is independent of the choice of splitting of $\omega M$ and is the image under $\omega $ of a morphism

$$\begin{align*}F_{-1}{\underline{\mathrm{End}}}(M) \rightarrow \bigoplus\limits_{r} {\underline{\mathrm{Hom}}}(\mathrm{Gr}^F_{r+1}M, \mathrm{Gr}^F_{r}M) \end{align*}$$

denoted by $\pi $ . Moreover, the morphism $\pi $ is independent of the choice of fiber functor $\omega $ .

Proof We will give an intrinsic description of $\pi $ . All the claims made in the construction below can be checked easily (upon taking a fiber functor $\omega $ and a splitting of $\omega M$ , if needed).

For each r, there is a canonical surjective map

(3.4) $$ \begin{align} {\underline{\mathrm{Hom}}}(\frac{M}{F_rM}, F_rM) \rightarrow {\underline{\mathrm{Hom}}}(\mathrm{Gr}^F_{r+1}M, \mathrm{Gr}^F_{r}M) \end{align} $$

given by functoriality properties of ${\underline {\mathrm{Hom}}}$ (say, first apply ${\underline {\mathrm{Hom}}}(-,F_rM)$ to $\mathrm {Gr}^F_{r+1}M\hookrightarrow M/F_rM$ , then ${\underline {\mathrm{Hom}}}(\mathrm {Gr}^F_{r+1}M,-)$ to $F_rM\twoheadrightarrow \mathrm {Gr}^F_{r}M$ ). The kernel of (3.4) is the subobject of ${\underline {\mathrm{Hom}}}(\frac {M}{F_rM}, F_rM)$ , denoted by $F_{-2}{\underline {\mathrm{Hom}}}(\frac {M}{F_rM}, F_rM)$ , that after applying a fiber functor $\omega $ consists of all linear maps $\omega M/F_r\omega M \xrightarrow {f} F_r\omega M$ satisfying $f(\mathrm {Gr}^F_{r+1}\omega M)\subset F_{r-1}\omega M$ . Taking the direct sum of the maps (3.4) over $r,$ we obtain a surjective map

(3.5) $$ \begin{align} \bigoplus\limits_r {\underline{\mathrm{Hom}}}(\frac{M}{F_rM}, F_rM) \rightarrow \bigoplus\limits_r {\underline{\mathrm{Hom}}}(\mathrm{Gr}^F_{r+1}M, \mathrm{Gr}^F_{r}M). \end{align} $$

On the other hand, for each $r,$ there is an obvious canonical inclusion

$$\begin{align*}{\underline{\mathrm{Hom}}}(\frac{M}{F_rM}, F_rM) \hookrightarrow F_{-1}{\underline{\mathrm{End}}}(M), \end{align*}$$

again given by functoriality of ${\underline {\mathrm{Hom}}}$ . The summation map

$$\begin{align*}\sum: \ \bigoplus\limits_r {\underline{\mathrm{Hom}}}(\frac{M}{F_rM}, F_rM) \rightarrow F_{-1}{\underline{\mathrm{End}}}(M) \end{align*}$$

is surjective. Moreover, the map (3.5) vanishes on the kernel of the summation map above. The map $\pi $ is then the map fitting in the following commutative diagram:

The image of $\pi $ under any fiber functor $\omega $ is the map denoted by $\omega \pi $ earlier.

Let

$$\begin{align*}\pi_{r,r+1} : F_{-1}{\underline{\mathrm{End}}}(M) \rightarrow {\underline{\mathrm{Hom}}}(\mathrm{Gr}^F_{r+1}M, \mathrm{Gr}^F_{r}M) \end{align*}$$

be the composition of $\pi $ with the projection to ${\underline {\mathrm{Hom}}}(\mathrm {Gr}^F_{r+1}M, \mathrm {Gr}^F_{r}M)$ , so that $\pi =(\pi _{r,r+1})$ . After choosing a fiber functor $\omega $ and a splitting of $\omega M$ , the map $\omega \pi _{r,r+1}$ sends an element $f=(f_{ij})\in \omega F_{-1}{\underline {\mathrm {End}}}(M)$ to the superdiagonal entry $f_{r,r+1}$ .

3.5

The elements of $\mathcal {U}(M,\omega )$ stabilize the filtration $F_\bullet \omega M$ and induce identity on $\mathrm {Gr}^F\omega M$ . Thus,

$$\begin{align*}{\mathfrak{u}}(M) \subset F_{-1}{\underline{\mathrm{End}}}(M). \end{align*}$$

Define a filtration $F_\bullet {\mathfrak {u}}(M)$ on ${\mathfrak {u}}(M)$ by setting

$$\begin{align*}F_n{\mathfrak{u}}(M) := {\mathfrak{u}}(M)\cap F_n{\underline{\mathrm{End}}}(M). \end{align*}$$

Then $\mathrm {Gr}^F {\mathfrak {u}}(M)$ is concentrated in negative degrees. Our focus in this article is on the graded component $\mathrm {Gr}^F_{-1}{\mathfrak {u}}(M)$ . In view of Section 3.4, we have a commutative diagram

(3.6)

with exact rows. We identify $\mathrm {Gr}^F_{-1} {\mathfrak {u}}(M)$ as a subobject of $\bigoplus \limits _r {\underline {\mathrm{Hom}}}(\mathrm {Gr}^F_{r+1}M, \mathrm {Gr}^F_{r}M)$ via the vertical map on the right (this is the embedding in equation (1.2) of Section 1). Our goal is to determine

$$\begin{align*}\mathrm{Gr}^F_{-1} {\mathfrak{u}}(M) = \pi({\mathfrak{u}}(M)) \subset \bigoplus\limits_r {\underline{\mathrm{Hom}}}(\mathrm{Gr}^F_{r+1}M, \mathrm{Gr}^F_{r}M). \end{align*}$$

4.1 The maps $\Psi _X$

Out tool to study $\mathrm {Gr}^F_{-1}{\mathfrak {u}}(M)$ will be the maps of the form $\Psi ^{M,N}_X$ constructed in Section 2, taking $N=\mathrm {Gr}^FM$ . By Sections 2.1–2.4, for every object X of $\langle \mathrm {Gr}^FM\rangle ^{\otimes }$ , we have a canonical map

(4.1)

which is functorial in X. This map can be computed by choosing any fiber functor $\omega $ , following the procedure outlined in Section 2.4 (with $\mathcal {H}(M,N,\omega )= \mathcal {U}(M,\omega )$ ). Since $\mathcal {U}(M,\omega )$ is unipotent, by Lemma 2.5.1, we have

If $\mathrm {Gr}^FM$ is semisimple, then the map $\Psi _X$ is injective for every object X of $\langle \mathrm {Gr}^F M\rangle ^{\otimes }$ (in fact, it is an isomorphism because we are then in the situation of Section 2.6).

The quotient map ${\mathfrak {u}}(M)\rightarrow {\mathfrak {u}}(M)^{\mathrm {ab}}$ induces an injection

$$\begin{align*}\mathrm{Hom}_{\mathbf{T}}({\mathfrak{u}}(M)^{\mathrm{ab}}, X) \hookrightarrow \mathrm{Hom}_{\mathbf{T}}({\mathfrak{u}}(M), X). \end{align*}$$

For every object X of $\langle \mathrm {Gr}^FM\rangle ^{\otimes }$ , we use this injection to consider $\Psi _X$ as a map

(4.2)

(functorial in X and with the same kernel as (4.1)). The image of an extension class $\mathcal {Z}$ of by X in $\langle M\rangle ^{\otimes }$ under (4.2) is computed using the same procedure as in Section 2.4, except we skip the last step of passing to the abelianization: Denoting the middle object of a representative of $\mathcal {Z}$ by Z, the logarithm of the morphism $\mathcal {U}(M,\omega )\rightarrow \omega X$ that describes the action of $\mathcal {U}(M,\omega )$ on $\omega Z$ is equal to the image under $\omega $ of the morphism ${\mathfrak {u}}(M)\rightarrow X$ corresponding to $\mathcal {Z}$ under (4.2).

4.2 The key computation

In this section, we establish the key technical component of the proof of Theorem 1.1.1. Following Section 1, we shall set

$$\begin{align*}V_{r,r+1}:= {\underline{\mathrm{Hom}}}(\mathrm{Gr}^F_{r+1}M, \mathrm{Gr}^F_{r}M) \hspace{.3in}\text{and} \hspace{.2in} V:=\bigoplus\limits_r V_{r,r+1}. \end{align*}$$

Recall the definitions of the extensions $\mathcal {E}_r$ and $\mathcal {E}$ from Section 1: The filtration $F_\bullet M$ gives rise to an extension

(4.3)

for each r. The element

is the extension class corresponding to (4.3) under the canonical isomorphism

The extension class

is the element corresponding to the tuple under the canonical isomorphism

Recall from Section 3.4 that we have a map

$$\begin{align*}\pi: F_{-1}{\underline{\mathrm{End}}}(M) \rightarrow V,\end{align*}$$

whose component in $V_{r,r+1}$ was denoted by $\pi _{r,r+1}$ . Denote the map obtained by restricting the domain of $\pi $ (resp. $\pi _{r,r+1}$ ) to ${\mathfrak {u}}(M)$ also by $\pi $ (resp. $\pi _{r,r+1}$ ). Recall that after applying a fiber functor $\omega $ and choosing a splitting of $\omega M$ , the map $\pi _{r,r+1}$ simply sends $(f_{ij})\mapsto f_{r,r+1}$ .

Finally, recall that for every object X of $\langle \mathrm {Gr}^FM\rangle ^{\otimes }$ , we have a map $\Psi _X$ as in (4.2) of Section 4.1.

Proof

1. (a) Fix r. Let $\omega $ be a fiber functor. Recalling the explicit description of the canonical isomorphism for any objects N and L of a Tannakian category (see, e.g., [Reference Eskandari and Murty9, Section 3.2]), $\mathcal {E}_r$ is given by the extension

   (4.4)

   

where $E_{r,r+1}$ is the object ${\underline {\mathrm{Hom}}}(\mathrm {Gr}^F_{r+1}M, F_{r+1}M/F_{r-1}M)^\dagger $ in the notation of [Reference Eskandari and Murty9]: This is the subobject of ${\underline {\mathrm{Hom}}}(\mathrm {Gr}^F_{r+1}M, F_{r+1}M/F_{r-1}M)$ whose image under $\omega $ consists of all linear maps $g: \mathrm {Gr}^F_{r+1}\omega M\rightarrow F_{r+1}\omega M/F_{r-1}\omega M$ such that the composition

$$\begin{align*}\mathrm{Gr}^F_{r+1}\omega M \xrightarrow{ \ g \ } \displaystyle{\frac{F_{r+1}\omega M}{F_{r-1}\omega M}} \ \twoheadrightarrow \ \mathrm{Gr}^F_{r+1}\omega M\end{align*}$$

(the latter being the canonical surjection) is a scalar multiple $\lambda (g)\cdot \mathrm {Id}$ of the identity map. The injection $V_{r,r+1} \hookrightarrow E_{r,r+1}$ is given by functoriality of ${\underline {\mathrm{Hom}}}$ , and the surjection after applying $\omega $ is simply the map $\lambda : g\mapsto \lambda (g)$ . See [Reference Eskandari and Murty9, Section 3.2] for more details.

Choose a splitting of $\omega M$ in the sense of Section 3.3 to identify the vector space $\omega M$ with $\mathrm {Gr}^F\omega M$ . Let $s_{r+1}$ be the section of the quotient map $F_{r+1}\omega M \rightarrow \mathrm {Gr}^F_{r+1}\omega M$ used to form the splitting. Then $s_{r+1}$ gives a section $\overline {s_{r+1}} = s_{r+1} \pmod {F_{r-1}\omega M}$ of the map $F_{r+1}\omega M/F_{r-1}\omega M \rightarrow \mathrm {Gr}^F_{r+1}\omega M$ , which in turn gives rise to a section of the map $\lambda $ . We will use this section (which is given by $1\mapsto \overline {s_{r+1}}$ ) to compute $\Psi _{V_{r,r+1}}(\mathcal {E}_r)$ following the steps outlined at the end of Section 4.1.

Identify

(4.5) $$ \begin{align} \omega E_{r,r+1} \cong \mathrm{Hom}_{\mathbb F}(\mathrm{Gr}^F_{r+1}\omega M, \mathrm{Gr}^F_{r}\omega M) \oplus {\mathbb F} \end{align} $$

via the section of $\lambda $ above. The map $\overline {s_{r+1}}$ gives an isomorphism $F_{r+1}\omega M/F_{r-1}\omega M \cong \mathrm {Gr}^F_{r}\omega M \oplus \mathrm {Gr}^F_{r+1}\omega M$ , which agrees with the one in equation (3.2) of Section 3.3 for $n=r+1$ and $m=r-1$ and our choice of splitting of $\omega M$ . This gives an isomorphism

$$\begin{align*}\begin{aligned} \mathrm{Hom}_{\mathbb F}(\mathrm{Gr}^F_{r+1}\omega M, \displaystyle{\frac{F_{r+1}\omega M}{F_{r-1}\omega M}}) & \cong \mathrm{Hom}_{\mathbb F}(\mathrm{Gr}^F_{r+1}\omega M, \mathrm{Gr}^F_{r}\omega M)\\ & \oplus \mathrm{Hom}_{\mathbb F}(\mathrm{Gr}^F_{r+1}\omega M, \mathrm{Gr}^F_{r+1}\omega M), \end{aligned} \end{align*}$$

compatible with (4.5) via the natural inclusion maps (embedding ${\mathbb F}$ as the scalar endomorphisms of $\mathrm {Gr}^F_{r+1}\omega M$ ).

Use our splitting to write endomorphisms of $\omega M$ as $k\times k$ matrices. Let $f=(f_{ij})\in \omega {\mathfrak {u}}(M)$ (a strictly upper triangular matrix). Set $\sigma :=\exp (f)\in \mathcal {U}(M,\omega )$ . Then f and $\sigma $ have the same superdiagonal entries because $f^n\in \omega F_{-2}{\underline {\mathrm {End}}}(M)$ for $n\geq 2$ . The action of $\sigma $ on $\omega {\underline {\mathrm{Hom}}}(\mathrm {Gr}^F_{r+1}M, F_{r+1}M/ F_{r-1}M)$ is by sending $g:\mathrm {Gr}^F_{r+1}\omega M \rightarrow F_{r+1}\omega M/ F_{r-1}\omega M$ to $\sigma _{F_{r+1}M/ F_{r-1}M}\circ g\circ {\sigma _{\mathrm {Gr}^F_{r+1}M}}^{-1}=\sigma _{F_{r+1}M/ F_{r-1}M}\circ g,$ where as before, $\sigma _X$ means the action of $\sigma $ on $\omega X$ for an object X of $\langle M\rangle ^{\otimes }$ (note that $\sigma _{\mathrm {Gr}^F_{r+1}M}=1$ because $\sigma \in \mathcal {U}(M,\omega )$ ). Writing endomorphisms of $F_{r+1}\omega M/F_{r-1}\omega M$ as $2\times 2$ matrices via our isomorphism $F_{r+1}\omega M/F_{r-1}\omega M \cong \mathrm {Gr}^F_{r}\omega M \oplus \mathrm {Gr}^F_{r+1}\omega M$ , the matrix $\sigma _{F_{r+1}M/ F_{r-1}M}$ is obtained from $\sigma =(\sigma _{ij})$ by truncation to the part $r\leq i,j\leq r+1$ , so that

$$\begin{align*}\sigma_{F_{r+1}M/ F_{r-1}M}=\begin{pmatrix} 1 & f_{r,r+1}\\ 0 & 1\end{pmatrix}.\end{align*}$$

Given any $g=(g_{r,r+1}, \lambda )\in \omega E_{r,r+1}$ , we have

$$\begin{align*}\sigma\cdot g = \sigma_{F_{r+1}M/ F_{r-1}M}\circ g = \begin{pmatrix} 1 & f_{r,r+1}\\ 0 & 1\end{pmatrix}\begin{pmatrix} g_{r,r+1} \\ \lambda\end{pmatrix} \end{align*}$$

(the computation done in $\mathrm {Hom}_{\mathbb F}(\mathrm {Gr}^F_{r+1}\omega M, \displaystyle {\frac {F_{r+1}\omega M}{F_{r-1}\omega M}})$ ). Thus, the morphism $\mathcal {U}(M,\omega )\rightarrow \mathrm {Hom}_{\mathbb F}(\mathrm {Gr}^F_{r+1}\omega M,\mathrm {Gr}^F_{r}\omega M)$ of algebraic groups that corresponds to the action of $\mathcal {U}(M,\omega )$ on $E_{r,r+1}$ is simply given by $\sigma \mapsto f_{r,r+1}$ . Taking logarithms, the image under $\omega $ of the morphism $\Psi _{V_{r,r+1}}(\mathcal {E}_r)$ is the map

$$\begin{align*}\omega {\mathfrak{u}}(M)\rightarrow \mathrm{Hom}_{\mathbb F}(\mathrm{Gr}^F_{r+1}\omega M,\mathrm{Gr}^F_{r}\omega M) \hspace{.5in} f\mapsto f_{r,r+1}. \end{align*}$$

Upon recalling the description of $\pi _{r,r+1}$ from Section 3.4, this gives the result.

1. (b) By the functoriality of the map $\Psi _X$ in $X,$ we have a commutative diagram

   

   where the vertical identifications are via the canonical isomorphisms. Part (b) follows from part (a) since $\mathcal {E}$ (resp. $\pi $ ) corresponds to the tuple $(\mathcal {E}_{r,r+1})$ (resp. $(\pi _{r,r+1})$ ) under the vertical identifications.

4.3 Proof of Theorem 1.1.1

We now deduce Theorem 1.1.1. Recall from (3.6) in Section 3.5 that $\mathrm {Gr}^F_{-1}{\mathfrak {u}}(M)=\pi ({\mathfrak {u}}(M))$ . Let W be a subobject of V. We need to show that one has $\pi ({\mathfrak {u}}(M))\subset W$ if and only if the pushforward of $\mathcal {E}$ along the quotient map $V\rightarrow V/W$ lives in the subcategory $\langle \mathrm {Gr}^FM\rangle ^{\otimes }$ . Tentatively, denote the quotient map $V\rightarrow V/W$ by $\varphi $ . We have a commutative diagram

(4.6)

where $\varphi _\ast $ denotes pushforward of extensions along $\varphi $ . The kernel of the map $\Psi _{V/W}$ is the subgroup (see Section 4.1). Thus, $\varphi _\ast (\mathcal {E})$ is in the subcategory $\langle \mathrm {Gr}^FM\rangle ^{\otimes }$ if and only if $\Psi _{V/W}(\varphi _\ast (\mathcal {E}))=0$ if and only if $\varphi \circ (\Psi _V(\mathcal {E})) = 0$ . Thanks to Proposition 4.2.1, the latter is equivalent to $\pi ({\mathfrak {u}}(M))\subset W$ .

Remark 4.3.1 Suppose $\mathrm {Gr}^FM$ is semisimple. Then Theorem 1.1.1 asserts that $\mathrm {Gr}^F_{-1}{\mathfrak {u}}(M)$ is the smallest subobject of V such that the pushforward of $\mathcal {E}$ along the quotient map $V\rightarrow V/\mathrm {Gr}^F_{-1}{\mathfrak {u}}(M)$ splits. Since $\mathrm {Gr}^FM$ is semisimple, this can be equivalently formulated as follows: $\mathrm {Gr}^F_{-1}{\mathfrak {u}}(M)$ is the intersection of the kernels of all endomorphisms of V that annihilate $\mathcal {E}$ :

$$\begin{align*}\mathrm{Gr}^F_{-1}{\mathfrak{u}}(M) \ = \bigcap\limits_{\stackrel{\phi\in \mathrm{End}_{\mathbf{T}}(V)}{\phi_\ast(\mathcal{E})=0}} \ker(\phi). \end{align*}$$

4.4 A maximality criterion

In this section, we use Theorem 1.1.1 to give a criterion for when ${\mathfrak {u}}(M)$ is equal to $F_{-1}{\underline {\mathrm {End}}}(M)$ . Before we state the result, let us recall a definition due to Bertrand [Reference Bertrand3]: An extension $\mathcal {Z}$ of an object Y by an object X in ${\mathbf {T}}$ is called totally nonsplit (or totally unsplit) if, considering $\mathcal {Z}$ as an extension of by ${\underline {\mathrm{Hom}}}(Y,X)$ , for every proper subobject W of ${\underline {\mathrm{Hom}}}(Y,X),$ the pushforward of $\mathcal {Z}$ along the quotient map ${\underline {\mathrm{Hom}}}(Y,X)\rightarrow {\underline {\mathrm{Hom}}}(Y,X)/W$ is nonsplit. If ${\underline {\mathrm{Hom}}}(Y,X)$ is semisimple, then $\mathcal {Z}$ is totally nonsplit if and only if the annihilator of $\mathcal {Z}$ in $\mathrm {End}_{\mathbf {T}}({\underline {\mathrm{Hom}}}(Y,X))$ (for the $\mathrm {End}_{\mathbf {T}}({\underline {\mathrm{Hom}}}(Y,X))$ -module ) is trivial.

Theorem 1.1.1 has the following consequence.

The equivalence of statements (i) and (iii) in part (a) of Proposition 4.4.1 will play an important role in Section 5. The following definition (motivated by statement (iii) of Proposition 4.4.1(a)) will be convenient.

By Proposition 4.4.1(a), we have ${\mathfrak {u}}(M)=F_{-1}{\underline {\mathrm {End}}}(M)$ if and only if the extension $\mathcal {E}$ of by V is totally disjoint from the subcategory $\langle \mathrm {Gr}^FM\rangle ^{\otimes }$ .

4.5 Variants – I

In the remainder of Section 4, we will discuss two variants of Theorem 1.1.1 and some consequences. This content will not be needed in Section 5. A reader who wishes to skip to Section 5 may do so.

One has the following variant (or generalization) of Theorem 1.1.1 about the projections of $\pi ({\mathfrak {u}}(M))$ to the direct summands of V.

Consider a subquotient $F_nM/F_mM$ of M with $0\leq m<n\leq k$ . The filtration $F_\bullet $ induces a filtration on $F_nM/F_mM$ . We can apply the constructions and results of the article to this filtered object. We obtain an object ${\mathfrak {u}}(F_nM/F_mM)\subset F_{-1}{\underline {\mathrm {End}}}(F_nM/F_mM)$ whose image under any fiber functor is the Lie algebra of the subgroup of the Tannakian fundamental group of $F_nM/F_mM$ that acts trivially on $\mathrm {Gr}^F(F_nM/F_mM)\cong \bigoplus _{m<r\leq n}\mathrm {Gr}^F_r(M)$ . We also have a natural map

(4.7) $$ \begin{align} {\mathfrak{u}}(M) \rightarrow {\mathfrak{u}}(F_nM/F_mM) \end{align} $$

induced by the restriction map from the Tannakian group of M to the Tannakian group of $F_nM/F_mM$ . This map fits in a commutative diagram

where the bottom arrow is the map that after applying a fiber functor $\omega $ , sends a linear map $f\in \mathrm {End}_{\mathbb F}(\omega M)$ in $F_{-1}\omega {\underline {\mathrm {End}}}(M)$ to the induced map in $\mathrm {End}_{\mathbb F}(\omega F_nM/\omega F_mM)$ (this makes sense because f is in $F_{-1}\omega {\underline {\mathrm {End}}}(M)$ ). One may ask if the top arrow is always surjective. As an application of Proposition 4.5.1, we can see that this is not true in general, as we now explain.

For brevity, let $(m,n)$ denote the set of integers $\{r: m< r< n\}$ . Consider the map $\pi $ for $F_nM/F_mM$ (see Section 3.4). Including $F_nM/F_mM$ as a superscript in the notation to avoid confusion, this is a map

$$\begin{align*}\pi^{F_nM/F_mM} : F_{-1}{\underline{\mathrm{End}}}(F_nM/F_mM) \rightarrow V_{(m,n)} = \bigoplus\limits_{m<r<n} V_{r,r+1} \end{align*}$$

(the notation $V_{(m,n)}$ is as in Proposition 4.5.1). There is a commutative diagram

(4.8)

where $\pi ^M$ means our $\pi $ from before (for M) and $\pi ^M_{(m,n)}={(\pi _{r,r+1})}_{m<r<n}$ (again, following the notation of Proposition 4.5.1). By Proposition 4.5.1, $\pi ^M_{(m,n)}({\mathfrak {u}}(M))$ is the smallest subobject of $V_{(m,n)}$ such that the pushforward of $\mathcal {E}_{(m,n)}$ along the quotient $V_{(m,n)}\rightarrow V_{(m,n)}/\pi ^M_{(m,n)}({\mathfrak {u}}(M))$ belongs to $\langle \mathrm {Gr}^FM\rangle ^{\otimes }$ . On the other hand, applying the same result (or Theorem 1.1.1) to $F_{n}M/F_{m}M,$ we see that $\pi ^{F_{n}M/F_{m}M}({\mathfrak {u}}(F_{n}M/F_{m}M))$ is the smallest subobject of $V_{(m,n)}$ such that the pushforward of $\mathcal {E}_{(m,n)}$ along the quotient $V_{(m,n)}\rightarrow V_{(m,n)}/\pi ^{F_{n}M/F_{m}M}({\mathfrak {u}}(F_{n}M/F_{m}M))$ belongs to $\langle \mathrm {Gr}^F(F_nM/F_mM)\rangle ^{\otimes }$ . Already in the case $n-m=2$ with $n=r+1$ and $m=r-1$ (where $F_nM/F_mM$ has only two graded components $\mathrm {Gr}^F_{r+1}M$ and $\mathrm {Gr}^F_r M$ ), one can now easily use these characterizations to construct an example where the restriction map ${\mathfrak {u}}(M)\rightarrow {\mathfrak {u}}(F_{r+1}M/F_{r-1}M)$ is not surjective.

On the other hand, one has the following corollary.

4.6 Variants – II

Suppose $\mathrm {Gr}^FM$ is semisimple. In this case, we will give an equivalent formulation of Theorem 1.1.1 that will be useful to give a refinement of Proposition 4.4.1. Thanks to semisimplicity of $\mathrm {Gr}^FM$ , the object ${\mathfrak {u}}(F_{r+1}M/F_{r-1}M)$ is the smallest subobject of $V_{r,r+1}$ such that the pushforward of $\mathcal {E}_r$ along the quotient map $V_{r,r+1}\rightarrow V_{r,r+1}/{\mathfrak {u}}(F_{r+1}M/F_{r-1}M)$ splits (this is by the case $k=2$ of Theorem 1.1.1, which was also proved earlier in [Reference Eskandari and Murty9, Theorem 3.3.1]). Equivalently, ${\mathfrak {u}}(F_{r+1}M/F_{r-1}M)$ is the smallest subobject of $V_{r,r+1}$ such that $\mathcal {E}_r$ is in the image of the map

(4.10)

given by pushforward along the inclusion ${\mathfrak {u}}(F_{r+1}M/F_{r-1}M)\hookrightarrow V_{r,r+1}$ . Since $V_{r,r+1}$ is semisimple, the map (4.10) is injective. With abuse of notation, we use the same notation for $\mathcal {E}_r$ and the element of that pushes forward to it.

Under the assumption of semisimplicity of $\mathrm {Gr}^FM$ , Theorem 1.1.1 can be equivalently formulated as follows.

Proposition 4.6.1 Suppose $\mathrm {Gr}^FM$ is semisimple. Set

$$\begin{align*}\hat{V}:=\bigoplus\limits_r \, {\mathfrak{u}}(F_{r+1}M/F_{r-1}M) \subset V. \end{align*}$$

For each r, consider $\mathcal {E}_r$ as an extension of by ${\mathfrak {u}}(F_{r+1}M/F_{r-1}M)$ , and consider $\mathcal {E}=(\mathcal {E}_r)$ as an extension of by $\hat {V}$ . Then $\mathrm {Gr}^F_{-1}{\mathfrak {u}}(M)$ is the smallest subobject of $\hat {V}$ such that the pushforward of the extension $\mathcal {E}$ along the quotient map $\hat {V}\rightarrow \hat {V}/\mathrm {Gr}^F_{-1}{\mathfrak {u}}(M)$ splits.

As an immediate consequence of Proposition 4.6.1, one gets the following result, which refines Proposition 4.4.1 in the case when $\mathrm {Gr}^FM$ is semisimple and $\hat {V}\neq V$ .

Note that for semisimple $\mathrm {Gr}^FM$ , one has $\hat {V}=V$ if and only if for every r, the extension $\mathcal {E}_r\in \mathrm {Ext}^1_{\mathbf {T}}(\mathrm {Gr}^F_{r+1}M, \mathrm {Gr}^F_{r}M)$ is totally nonsplit.

5.1 Statement of the result

In the remainder of the article, we assume that our Tannakian category ${\mathbf {T}}$ is equipped with a functorial filtration $W_\bullet $ (called the weight filtration) with similar properties to the weight filtration on rational mixed Hodge structures: indexed by ${\mathbb Z}$ , increasing and finite on every object, functorial, exact, and compatible with the tensor structure. More explicitly, this means that every object M of ${\mathbf {T}}$ is equipped with a filtration $(W_n M)_{n\in {\mathbb Z}}$ with $W_{n-1}M\subset W_nM$ for all $n\in {\mathbb Z}$ , $W_n M = 0$ for $n\ll 0$ , and $W_n M = M$ for $n\gg 0$ , such that the following conditions (i)–(iii) hold: (i) For every objects M and N and morphism $f:M\rightarrow N$ in ${\mathbf {T}}$ , we have $f(W_n M)\subset W_n N$ for every n; (ii) the functors $W_n: {\mathbf {T}}\rightarrow {\mathbf {T}}$ (defined by $M\mapsto W_n M$ on objects and acting on a morphism $f: M\rightarrow N$ by restricting it to $f: W_nM\rightarrow W_nN$ ) are exact for every n; and (iii) for every objects M and N and every integer n,

$$\begin{align*}W_n(M\otimes N) = \sum\limits_{p+q=n}W_pM\otimes W_qN. \end{align*}$$

We call an object M of ${\mathbf {T}}$ pure if there exists an integer n such that $W_{n-1}M=0$ and $W_nM=M$ . A weight of an object M is an integer n such that $W_{n-1}M\neq W_nM$ . A nonzero pure object is an object with exactly one weight.

For every nonzero object M of ${\mathbf {T}}$ , we apply the constructions and results of the article by taking the increasing filtration $(F_r M)_{r\in {\mathbb Z}}$ to be given by the weight filtration: more precisely, if the weights of M are $p_1<\cdots <p_k$ , set $F_rM:=W_{p_r}M$ for $1\leq r\leq k$ and $F_0M=0$ . We thus obtain an object

$$\begin{align*}{\mathfrak{u}}(M)\subset F_{-1}{\underline{\mathrm{End}}}(M)=W_{-1}{\underline{\mathrm{End}}}(M).\end{align*}$$

As an application of Theorem 1.1.1, we will prove the following result about the structure of the set of isomorphism classes of objects M of ${\mathbf {T}}$ with a given associated graded and maximal ${\mathfrak {u}}(M)$ .

Theorem 5.1.1 Fix objects $A_1,\ldots , A_k$ of ${\mathbf {T}}$ , where $A_r$ is nonzero and pure of weight $p_r$ , and $p_1<\cdots < p_k$ . Set $A := \bigoplus _r A_r$ . Let $S^\ast (A)$ be the set of equivalence classes of objects M of ${\mathbf {T}}$ such that $\mathrm {Gr}^WM$ is isomorphic to A and ${\mathfrak {u}}(M)=W_{-1}{\underline {\mathrm {End}}}(M)$ , where two such objects M and $M'$ are considered equivalent if they are isomorphic in ${\mathbf {T}}$ (note that we do not keep any trace of a choice of isomorphism between $\mathrm {Gr}^WM$ and A). Then there exist sets $S^\ast _\ell (A)$ ( $1\leq \ell \leq k-1$ ) and maps

(5.1) $$ \begin{align} S^\ast(A)\cong S^\ast_{k-1}(A) \rightarrow S^\ast_{k-2}(A) \rightarrow\cdots \rightarrow S^\ast_1(A) \end{align} $$

such that the following statements hold:

1. (i) For $2\leq \ell \leq k-1$ , every nonempty fiber of the map $S_\ell ^\ast (A)\rightarrow S_{\ell -1}^\ast (A)$ is a torsor over

   $$\begin{align*}\bigoplus\limits_{r} \mathrm{Ext}^1_{\mathbf{T}}(A_{r+\ell},A_r) \end{align*}$$

   (where r runs from $1$ to $k-\ell $ ).

2. (ii) Let $2\leq \ell \leq k-1$ . Then the map $S_\ell ^\ast (A)\rightarrow S_{\ell -1}^\ast (A)$ is surjective if

   $$\begin{align*}\bigoplus\limits_{r} \mathrm{Ext}^2_{\mathbf{T}}(A_{r+\ell},A_r) \end{align*}$$

   vanishes.

3. (iii) (Description of $S^\ast _1(A)$ ) Let

   $$\begin{align*}\bigm(\bigoplus_r \mathrm{Ext}^1_{\mathbf{T}}(A_{r+1},A_r)\bigm)^\ast \end{align*}$$

   be the subset of $\bigoplus _r \mathrm {Ext}^1_{\mathbf {T}}(A_{r+1},A_r)$ consisting of every extension tuple $\mathcal {E}=(\mathcal {E}_r)$ such that considering $\mathcal {E}$ as an element of in the natural way, $\mathcal {E}$ is totally disjoint from the subcategory $\langle A\rangle ^{\otimes }$ (see Definition 4.4.2). Then there is a canonical bijection

   $$\begin{align*}S^\ast_1(A) \ \cong \ \bigm(\bigoplus_r \mathrm{Ext}^1_{\mathbf{T}}(A_{r+1},A_r)\bigm)^\ast \bigm/ \mathrm{Aut}(A),\end{align*}$$

   where the action of $\mathrm {Aut}(A)=\prod _r \mathrm {Aut}(A_r)$ on extensions is by pushforwards and pullbacks (i.e., $\sigma =(\sigma _r)\in \prod _r \mathrm {Aut}(A_r)$ sends $(\mathcal {E}_r)$ to $((\sigma _{r+1}^{-1})^\ast (\sigma _{r})_\ast \mathcal {E}_r)$ ).

This result was proved in [Reference Eskandari7, Section 5, Theorem 5.4.3] in the special case when A is “graded-independent” (in the sense of Definition 5.3.1 of [Reference Eskandari7]) and semisimple. Our goal in the rest of the article is to establish Theorem 5.1.1. This will be done in Section 5.3 after we review some relevant constructions of [Reference Eskandari7] in Section 5.2 below. In what follows until the end of the article, A (and k, the $A_r$ and $p_r$ ) are fixed and as described in the statement of Theorem 5.1.1.

5.2 Recollections on generalized extensions

Following [Reference Eskandari7], we denote by $S'(A)$ the set of equivalence classes of pairs $(M,\phi )$ consisting of an object M of ${\mathbf {T}}$ and an isomorphism $\phi : \mathrm {Gr}^WM\rightarrow A$ , where two such pairs $(M,\phi )$ and $(M',\phi ')$ are considered equivalent if there exists a morphism (automatically, an isomorphism) $f: M\rightarrow M'$ such that $\phi '\circ \mathrm {Gr}^Wf=\phi $ . Denote by $S(A)$ the set of isomorphism classes of objects M of ${\mathbf {T}}$ such that $\mathrm {Gr}^WM$ is isomorphic to A. Note that no trace of a choice of an isomorphism $\mathrm {Gr}^WM\rightarrow A$ is kept in $S(A)$ (in contrast to the set $S'(A)$ ). There is a natural action of $\mathrm {Aut}(A)$ on $S'(A)$ (given by twisting $\phi $ in a pair $(M,\mathrm {Gr}^WM\xrightarrow {\phi } A)$ ), and the set $S(A)$ can be identified with the quotient of $S'(A)$ by $\mathrm {Aut}(A)$ . The set $S^\ast (A)$ of Theorem 5.1.1 is contained in $S(A)$ .

We introduced a new approach to study $S'(A)$ and $S(A)$ in [Reference Eskandari7, Section 3]. This approach, which in loc. cit. we called induction on the level, appears to have better naturalness properties than the obvious approach of induction on the number of weights k (see [Reference Eskandari7, Remark 3.3.2]). The inductive approach on the level is a natural generalization of the formalism of blended extensions (extensions panachées [Reference Grothendieck11]). Here, we will briefly sketch this approach to the extent needed to prove Theorem 5.1.1. We refer the reader to Sections 3 and 4 of [Reference Eskandari7] for more details and the proofs of everything we will say below in this section.

We constructed sets $S^{\prime }_\ell (A)$ for $1\leq \ell \leq k-1$ and maps

(5.2) $$ \begin{align} S'(A)\cong S^{\prime}_{k-1}(A) \rightarrow S^{\prime}_{k-2}(A) \rightarrow\cdots \rightarrow S^{\prime}_1(A)\cong \bigoplus\limits_{r}\mathrm{Ext}^1_{\mathbf{T}}(A_{r+1},A_r) \end{align} $$

such that for every $\ell $ , every nonempty fiber of $S^{\prime }_\ell (A)\rightarrow S^{\prime }_{\ell -1}(A)$ is canonically a torsor over

$$\begin{align*}\bigoplus\limits_{r} \mathrm{Ext}^1_{\mathbf{T}}(A_{r+\ell},A_r). \end{align*}$$

Moreover, the map $S^{\prime }_\ell (A)\rightarrow S^{\prime }_{\ell -1}(A)$ is surjective if

$$\begin{align*}\bigoplus\limits_{r} \mathrm{Ext}^2_{\mathbf{T}}(A_{r+\ell},A_r) \end{align*}$$

vanishes. We also constructed quotients $S_\ell (A)$ of $S^{\prime }_\ell (A)$ for $1\leq \ell \leq k-1$ such that (5.2) descends to a sequence of maps

(5.3) $$ \begin{align} S(A)\cong S_{k-1}(A) \rightarrow S_{k-2}(A) \rightarrow\cdots \rightarrow S_1(A)\cong \ \bigm(\bigoplus\limits_{r} \mathrm{Ext}^1_{\mathbf{T}}(A_{r+1},A_r)\bigm)\bigm/\mathrm{Aut}(A), \end{align} $$

where the action of the automorphism group $\mathrm {Aut}(A)=\prod _r \mathrm {Aut}(A_r)$ on a tuple of extension classes is by pushforwards and pullbacks, as described at the end of Theorem 5.1.1(iii). (See Theorem 3.3.1(a–d) of [Reference Eskandari7] for all of this.)

We will sketch the constructions of (5.2) and (5.3) as they will be relevant. We shall use the word “depth” for what was called “level” in [Reference Eskandari7], since the former might be a better choice of wording. Given an object M and an isomorphism $\phi : \mathrm {Gr}^WM\rightarrow A$ , setting $M_{m,n}:=W_{p_n}M/W_{p_m}M$ for $0\leq m<n\leq k$ with $p_0:=p_1-1$ (so that $W_{p_0}M=0$ ), the natural inclusions and projections between the $M_{m,n}$ give rise to a diagram

(5.4)

where we have used the isomorphism $\phi $ to replace $M_{r-1,r}$ by $A_r$ for every r. A generalized extension of depth $k-1$ (or full depth) of A is an abstract version of the data of the diagram above (see Definition 3.4.1 of [Reference Eskandari7]). More generally, for $1\leq \ell \leq k-1$ , a generalized extension of depth $\ell $ of A is an abstract version of a truncated version of the diagram above, consisting of the first $\ell $ diagonals above the diagonal formed by the $A_r$ . (By the i-th diagonal here, we mean the objects $M_{m,n}$ with $n-m=i+1$ .) The precise definition is recalled below. Note that in this definition and in what follows after, for convenience, in the context of indices, we will often use the phrase “the eligible range” to mean the range in which the indices in question make sense.

We visualize a generalized extension of depth $\ell =k-1$ (resp. $\ell <k-1$ ) by a diagram of the form (5.4) (resp. a truncated version of the diagram that includes the $\ell $ diagonals above the $A_r$ ). For example, the diagram below shows a generalized extension of depth $\ell =2$ in the case $k=4$ :

Given two generalized extensions $(M_{\bullet ,\bullet })$ and $(N_{\bullet ,\bullet })$ of the same depth, a morphism between them is defined as a collection of morphisms $M_{m,n}\rightarrow N_{m,n}$ in ${\mathbf {T}}$ that commute with the structure arrows of $(M_{\bullet ,\bullet })$ and $(N_{\bullet ,\bullet })$ . For each $1\leq \ell \leq k-1$ , the set $S_\ell (A)$ of (5.3) (resp. $S^{\prime }_\ell (A)$ of (5.2)) is defined as the collection of all generalized extensions of depth $\ell $ of A modulo the equivalence relation given by declaring two generalized extensions to be equivalent if there exists an isomorphism between them (resp. an isomorphism between them that is identity on the $A_r$ ). In particular, there is a natural surjection $S^{\prime }_\ell (A)\rightarrow S_\ell (A)$ for each $\ell $ . The identification $S^{\prime }_1(A) \cong \bigoplus _r \mathrm {Ext}^1_{\mathbf {T}}(A_{r+1},A_r)$ sends the equivalence class of a generalized extension $(M_{m,n})_{n-m\leq 2}$ of depth 1 (i.e., with just one diagonal above the $A_r$ ) to the element $(\mathcal {E}_r)$ of $\bigoplus _r \mathrm {Ext}^1_{\mathbf {T}}(A_{r+1},A_r)$ , where $\mathcal {E}_r$ is given by

(5.6)

(the maps being structure arrows of our generalized extension; note that this is indeed an extension by axiom (ii) of Definition 5.2.1). The canonical bijection $S'(A)\cong S_{k-1}'(A)$ is given by sending the equivalence class of a pair $(M,\mathrm {Gr}^WM\xrightarrow {\phi } A)$ to the class of the generalized extension (5.4) of full depth with $M_{m,n}:=W_{p_n}M/W_{p_m}M$ and $\mathrm {Gr}^W_{p_r}M$ having been replaced by $A_r$ via $\phi $ . This descends to a canonical bijection $S(A)\cong S_{k-1}(A)$ . The maps $S^{\prime }_\ell (A)\rightarrow S^{\prime }_{\ell -1}(A)$ and $S_\ell (A)\rightarrow S_{\ell -1}(A)$ are given by truncation, that is, by erasing the top diagonal ( = the $\ell $ -th diagonal above the $A_r$ ) of a generalized extension of depth $\ell $ . With abuse of notation, we denote the truncation maps $S^{\prime }_\ell (A)\rightarrow S^{\prime }_{\ell -1}(A)$ and $S_\ell (A)\rightarrow S_{\ell -1}(A)$ both by $\Theta _\ell $ .

Notably, the group actions that describe how the fibers of the truncation map $S_\ell (A)\rightarrow S_{\ell -1}(A)$ descend from those of $S^{\prime }_\ell (A)\rightarrow S^{\prime }_{\ell -1}(A)$ were described in [Reference Eskandari7] (see Section 3.10 therein). In particular, by Proposition 4.2.3 of [Reference Eskandari7] (also see Remark 4.2.5 therein), given $\epsilon \in S_{\ell -1}(A)$ and any $\epsilon '\in S^{\prime }_{\ell -1}(A)$ above $\epsilon $ , if $\epsilon $ (equivalently, $\epsilon '$ ) is the class of a generalized extension $(M_{\bullet ,\bullet })$ of depth $\ell -1$ such that all the extensions (5.5) are totally nonsplit, then the fiber of $S^{\prime }_\ell (A)\rightarrow S^{\prime }_{\ell -1}(A)$ above $\epsilon '$ descends bijectively to the fiber of $S_\ell (A)\rightarrow S_{\ell -1}(A)$ above $\epsilon $ :

(5.7)

This last fact will be important for us in Section 5.3.

As mentioned earlier, the reader can consult [Reference Eskandari7, Sections 3 and 4] for the details and the proofs of all of what was discussed above. Note that the diagrams of generalized extensions above are the reflected versions of those in [Reference Eskandari7] (reflected over the diagonal of the $A_r$ ), with the role of horizontal and vertical arrows switched. The pictures as shown here are more consistent with our matrix notation for elements of Tannakian groups of objects with increasing filtrations (where the matrices are upper triangular).

5.3 Proof of Theorem 5.1.1

We can now prove Theorem 5.1.1. The notation ( $A_r$ , $p_r$ , A, $S^\ast (A)$ and $(\bigoplus _r \mathrm {Ext}^1_{\mathbf {T}}(A_{r+1},A_r))^\ast $ ) is as introduced in the statement of the theorem. We shall set

$$\begin{align*}V:=\bigoplus_{r} {\underline{\mathrm{Hom}}}(A_{r+1},A_r). \end{align*}$$

This notation is consistent with the notation of Section 4 in the following way: If M is an object of ${\mathbf {T}}$ with $\mathrm {Gr}^WM$ isomorphic to A, a choice of an isomorphism $\mathrm {Gr}^WM\rightarrow A$ allows us to identify V defined here with its namesake in Section 4 and Theorem 1.1.1 for the filtration $(F_r M)$ on M given by $F_rM=W_{p_r}M$ .

We start the proof by a lemma.

Proof Part (b) is immediate from (a) so we only need to prove the latter. Thanks to the functoriality properties of ${\underline {\mathrm{Hom}}}$ , the isomorphism $\phi =(\phi _r): A'\rightarrow A$ induces an isomorphism $V'\rightarrow V$ , which we denote also by $\phi $ ; given a fiber functor $\omega $ , the isomorphism $\omega \phi : \omega V'\rightarrow \omega V$ sends $f\in \mathrm {Hom}_{\mathbb F}(\omega A^{\prime }_{r+1},\omega A^{\prime }_r)$ to $(\omega \phi _r)\circ f\circ (\omega \phi _{r+1})^{-1}$ . We have a commutative diagram

where the horizontal arrows are the canonical isomorphisms and $\phi _\ast $ is pushforward along $\phi : V'\rightarrow V$ . Thus, to establish the lemma, it suffices to verify that is totally disjoint from $\langle A\rangle ^{\otimes }$ if and only if is such. To verify this, let $\mathcal {E}'$ be represented by an extension

Given $W\subset V$ , consider the following two extension classes: (i) the pushforward of $\mathcal {E}'$ along the quotient $V'\rightarrow V'/\phi ^{-1}(W)$ and (ii) the pushforward of $\phi _\ast (\mathcal {E}')$ along the quotient $V\rightarrow V/W$ . These two extension classes are, respectively, given by the first and second rows of the commutative diagram

In particular, the two extension classes have isomorphic middle objects and hence are in $\langle A\rangle ^{\otimes }$ at the same time. Thus, $\mathcal {E}'$ is totally disjoint from $\langle A\rangle ^{\otimes }$ if and only if $\phi _\ast (\mathcal {E}')$ is such.

With Lemma 5.3.1 in hand, we proceed to give the definition of the sets $S^\ast _\ell (A)$ of Theorem 5.1.1. We shall use the notation introduced in Section 5.2 for generalized extensions. For $1\leq \ell \leq k-1$ , the set $S^\ast _\ell (A)$ is the pre-image of

(5.9) $$ \begin{align} \bigm(\bigoplus_r \mathrm{Ext}^1_{\mathbf{T}}(A_{r+1},A_r)\bigm)^\ast \bigm/ \mathrm{Aut}(A) \ \ \subset \ \ \bigm(\bigoplus_r \mathrm{Ext}^1_{\mathbf{T}}(A_{r+1},A_r)\bigm) \bigm/ \mathrm{Aut}(A) \end{align} $$

(which makes sense by Lemma 5.3.1(b)) under the composition

$$\begin{align*}S_\ell(A)\rightarrow \cdots \rightarrow S_1(A) \stackrel{(\dagger)}{\cong} \bigm(\bigoplus_r \mathrm{Ext}^1_{\mathbf{T}}(A_{r+1},A_r)\bigm) \bigm/ \mathrm{Aut}(A) \end{align*}$$

in (5.3). The requirement given in statement (iii) of Theorem 5.1.1 is satisfied thanks to the bijection $(\dagger )$ . The map $S^\ast _\ell (A)\rightarrow S^\ast _{\ell -1}(A)$ in the statement of the theorem is just the restriction of the truncation map $S_\ell (A)\rightarrow S_{\ell -1}(A)$ in (5.3).

Our next task is to argue that the bijection $S(A)\xrightarrow {\simeq } S_{k-1}(A)$ in (5.3) restricts to a bijection between $S^\ast (A)$ and $S^\ast _{k-1}(A)$ . By construction, the composition

$$\begin{align*}S(A)\rightarrow S_{k-1}(A)\rightarrow \cdots \rightarrow S_1(A) \cong \bigm(\bigoplus\limits_{r} \mathrm{Ext}^1_{\mathbf{T}}(A_{r+1},A_r)\bigm)\bigm/\mathrm{Aut}(A)\end{align*}$$

is the map that sends the isomorphism class of an object M to the $\mathrm {Aut}(A)$ -orbit of the tuple of extension classes

$$\begin{align*}(W_{p_{r+1}}M/W_{p_{r-1}} M) \in \bigoplus_r \mathrm{Ext}^1_{\mathbf{T}}(\mathrm{Gr}^W_{p_{r+1}}M, \mathrm{Gr}^W_{p_{r}}M), \end{align*}$$

with each $\mathrm {Gr}^W_{p_r}M$ being replaced by $A_r$ via a choice of isomorphism $\phi : \mathrm {Gr}^WM\rightarrow A$ . By Proposition 4.4.1(a) and Lemma 5.3.1(a), this sends $S^\ast (A)$ into (5.9). Thus, the bijection $S(A)\rightarrow S_{k-1}(A)$ restricts to an injection $S^\ast (A)\rightarrow S^\ast _{k-1}(A)$ . On the other hand, given $\epsilon \in S^\ast _{k-1}(A)$ , let M be an object of ${\mathbf {T}}$ with $\mathrm {Gr}^WM\simeq A$ such that the class of M in $S(A)$ is sent to $\epsilon $ by the bijection $S(A)\rightarrow S_{k-1}(A)$ . Since the truncation of $\epsilon $ to depth 1 lives in (5.9), the element $(W_{p_{r+1}}M/W_{p_{r-1}} M)$ of

(5.10)

is totally disjoint from $\langle A\rangle ^{\otimes }$ (first, we see this for the tuple $(W_{p_{r+1}}M/W_{p_{r-1}} M)$ considered as an element of via a choice of isomorphism $\phi : \mathrm {Gr}^WM\rightarrow A$ , and then by Lemma 5.3.1(a) for $(W_{p_{r+1}}M/W_{p_{r-1}} M)$ as an element of (5.10)). It now follows from Proposition 4.4.1(a) that ${\mathfrak {u}}(M)=W_{-1}{\underline {\mathrm {End}}}(M)$ , so that the class of M is indeed in $S^\ast (A)$ . We have proved that the bijection $S(A)\cong S_{k-1}(A)$ restricts to a bijection $S^\ast (A)\cong S^\ast _{k-1}(A)$ .

It remains to verify requirements (i) and (ii) of the statement of the theorem. Let $\epsilon \in S^\ast _{\ell -1}(A)$ . By construction, the fiber of $S^\ast _\ell (A)\rightarrow S^\ast _{\ell -1}(A)$ above $\epsilon $ is the same as the fiber of $S_\ell (A)\rightarrow S_{\ell -1}(A)$ above $\epsilon $ . We will see that if $(M_{\bullet ,\bullet })$ is a generalized extension of depth $\ell -1$ representing $\epsilon $ , then all the extensions (5.5) are totally nonsplit. Taking this for granted for the moment, choosing $\epsilon '\in S^{\prime }_{\ell -1}(A)$ above $\epsilon $ , by Proposition 4.2.3 of [Reference Eskandari7] (also see Remark 4.2.5 therein), the fiber of $S_\ell (A)\rightarrow S_{\ell -1}(A)$ above $\epsilon $ is in a canonical bijection with the fiber of $S^{\prime }_\ell (A)\rightarrow S^{\prime }_{\ell -1}(A)$ above $\epsilon '$ (see diagram (5.7)). Statements (i) and (ii) of Theorem 5.1.1 thus follow from the structure of the fibers of the truncation map $S^{\prime }_\ell (A)\rightarrow S^{\prime }_{\ell -1}(A)$ given in [Reference Eskandari7, Theorem 3.3.1(b,c)] and recalled earlier in Section 5.2.

To complete the proof, let $(M_{\bullet ,\bullet })$ be a generalized extension of A of depth $\ell -1$ representing $\epsilon \in S^\ast _{\ell -1}(A)$ . Fixing $m,n$ in the eligible range, we need to show that the extension (5.5) is totally nonsplit. The data of the generalized extension $(M_{\bullet ,\bullet })$ allow us to identify $\mathrm {Gr}^WM_{m,n}\cong \bigoplus _{m+1\leq r\leq n}A_r$ (see [Reference Eskandari7, Lemma 3.5.1(b)]). Let

$$\begin{align*}(\mathcal{E}_r)_{1\leq r\leq k-1}\in \bigoplus_r \mathrm{Ext}^1_{\mathbf{T}}(A_{r+1},A_r)\end{align*}$$

be the truncation of $(M_{\bullet ,\bullet })$ to depth one (i.e., $\mathcal {E}_r$ is given by (5.6)). By definition of $S^\ast _{\ell -1}(A)$ , this truncation belongs to $\bigm (\bigoplus _r \mathrm {Ext}^1_{\mathbf {T}}(A_{r+1},A_r)\bigm )^\ast $ . That is, considered as an element of , the extension $(\mathcal {E}_r)_{1\leq r\leq k-1}$ is totally disjoint from $\langle A\rangle ^{\otimes }$ . On recalling the definition of total disjointness (Definition 4.4.2), we see easily that this forces the extension

to be totally disjoint from $\langle A\rangle ^{\otimes }$ . A fortiori, the latter extension is totally disjoint from $\langle \mathrm {Gr}^WM_{m,n}\rangle ^{\otimes }$ . By [Reference Eskandari7, Lemma 3.7.3(b)] (applied to the generalized extension of $\bigoplus _{m+1\leq r\leq n}A_r$ of full depth obtained by cropping $(M_{\bullet ,\bullet })$ to the part to the left and below $M_{m,n}$ , see diagram (5.4)), the extension $(\mathcal {E}_r)_{m< r<n}$ above coincides with the extension

where we have used the identification $\mathrm {Gr}^WM_{m,n}\cong \bigoplus _{m+1\leq r\leq n}A_r$ given by the data of our generalized extension to identify $\mathrm {Gr}^W_{p_{r}}M_{m,n}$ with $A_r$ . In view of Lemma 5.3.1(a), it follows that the element

is totally disjoint from $\langle \mathrm {Gr}^WM_{m,n}\rangle ^{\otimes }$ . Thus, criterion (iii) of Proposition 4.4.1(a) holds for $M_{m,n}$ , so that ${\mathfrak {u}}(M_{m,n})=W_{-1}{\underline {\mathrm {End}}}(M_{m,n})$ . By Lemma 5.2.1 of [Reference Eskandari7] (see also Remark 5.2.4 therein), this implies that every extension coming from the weight filtration of $M_{m,n}$ , and in particular the one in (5.4), is totally nonsplit.