Proof. By general results about multiplicities of indecomposable objects in Krull–Schmidt categories, the multiplicity $m(i_*\underline {k}[m],\pi _*\underline {k}[n])$ is equal to the rank of the pairing

(3.2) $$ \begin{align} \operatorname{Hom}(i_*\underline{k}[m],\pi_*\underline{k}_{\widetilde{Y}}[n])\times \operatorname{Hom}(\pi_*\underline{k}_{\widetilde{Y}}[n],i_*\underline{k}[m])\to \operatorname{End}(i_*\underline{k}[m])\cong k. \end{align} $$

The following commutative diagram appears in the proof of [Reference Juteau, Mautner and WilliamsonJMW, Lem. 3.4] and arises through applying base change and adjunctions. The pairing $B'$ arises through composition with the canonical map from $\omega _F[-n]$ to $\underline {k}_F[n]$ (which comes from applying the canonical natural transformation $i^!\to i^*$ to $\underline {k}_{\widetilde {Y}}$ ), whereas the pairing (3.2) appears along the bottom row.

Since F is smooth, $\omega _F\cong \underline {k}_F[2d]$ and the canonical morphism from $\omega _F[-n]$ to $\underline {k}_F[n]$ is identified with the Euler class e. Therefore, the pairing (3.2) is identified with the one stated in the proposition, completing the proof.

Proof. The bundle $\mathcal {E}$ is naturally a direct sum of $2l$ vector bundles

$$\begin{align*}\mathcal{E}\cong \bigoplus_{i=1}^l \mathcal{A}_i \oplus \bigoplus_{i=1}^l \mathcal{C}_i, \end{align*}$$

where the bundles $\mathcal {A}_i$ and $\mathcal {C}_i$ correspond to the choice of $A_i$ and $C_i$ in $\widetilde {Y}$ . The bundle $\mathcal {A}_i$ is a line bundle with first Chern class $a_i+w$ .

On $\mathbb {P}^{q-1}$ , write $\mathcal {L}$ for the tautological line bundle, and $\mathcal {T}$ for the trivial vector bundle of rank q. Then, restricted to the appropriate $\mathbb {P}^{q-1}\times \mathbb {P}^{q-1}$ , we have

$$\begin{align*}\mathcal{C}_i\cong p_1^*(\mathcal{T}/\mathcal{L})^* \otimes p_2^* \mathcal{L}. \end{align*}$$

The total Chern class of $(\mathcal {T}/\mathcal {L})^*$ is given by

$$\begin{align*}c((\mathcal{T}/\mathcal{L})^*)=\sum_{j=1}^\infty a_i^j. \end{align*}$$

The computation of $e(\mathcal {C}_i)$ proceeds via (4.4), whose proof follows easily using the splitting principle:

Let V be a rank n vector bundle, and let L be a line bundle. Then

(4.4) $$ \begin{align} e(V\otimes L)=\sum_{i=0}^n c_i(V) c_1(L)^{n-i}. \end{align} $$

Using this, we obtain

$$\begin{align*}e(\mathcal{C}_i)=\sum_{j=0}^{q-1}a_i^j z^{q-1-j}. \end{align*}$$

Together with the formula for the Euler class (=first Chern class) of $\mathcal {A}_i$ , this implies the formula in the lemma.

Proof. Let $\sigma :Y\to X$ and $\pi :Z\to X$ be two proper resolutions of singularities. Consider $\operatorname {Hom}(\sigma _*\underline {k}_Y,\pi _*\underline {k}_Z)\cong H^{BM}_{2d}(Y\times _X Z)$ , where $H^{BM}_*$ stands for Borel–Moore homology and $d=\dim _{\mathbb {C}}(X)$ . Let U be a connected open dense subset of X such that $\sigma $ and $\pi $ are isomorphisms over U. Write $j:U\to X$ for the inclusion. As $\sigma $ and $\pi $ are isomorphisms over U, there is a canonical inclusion of U into $Y\times _X Z$ . The closure $\overline U$ of U in $Y\times _XZ$ is an irreducible component of $Y\times _XZ$ , and hence defines a fundamental classFootnote ² $\alpha :=[\overline {U}]\in H^{BM}_{2d}(Y\times _XZ)\cong \operatorname {Hom}(\sigma _*\underline {k}_Y,\pi _*\underline {k}_Z)$ . This class satisfies $j^*\alpha =\mbox {id}\in \operatorname {Hom}(\underline {k}_U,\underline {k}_U)$ .

Similarly, we define $\beta \in \operatorname {Hom}(\pi _*\underline {k}_Z,\sigma _*\underline {k}_Y)$ with $j^*\beta =\mbox {id}\in \operatorname {Hom}(\underline {k}_U,\underline {k}_U)$ .

The category $D^b_c(X;k)$ is Krull–Schmidt. Therefore, since $\underline {k}_U$ is indecomposable and $j^*(\sigma _*\underline {k}_Y)\cong \underline {k}_U$ , we can write $\sigma _*\underline {k}_Y\cong A\oplus C$ where A is indecomposable and $j^*C=0$ . Similarly, we have $\pi _*\underline {k}_Z\cong B\oplus D$ with B indecomposable and $j^*D=0$ . Using these direct sum decompositions, consider the components of $\alpha $ and $\beta $ . They induce morphisms which by abuse of notation we call $\alpha :A\to B$ and $\beta :B\to A$ .

Consider $\beta \alpha \in \operatorname {End}(A)$ . Under the ring homomorphism $j^*:\operatorname {End}(A)\to \operatorname {End}(\underline {k}_U)\cong \underline {k}$ , we have $j^*(\beta \alpha )=1$ . As A is indecomposable, $\operatorname {End}(A)$ is local. This implies that $\beta \alpha $ is a unit in $\operatorname {End}(A)$ . Similarly, $\alpha \beta $ is a unit in $\operatorname {End}(B)$ . We thus have two indecomposable objects A and B, with morphisms between them $\alpha $ and $\beta $ , whose compositions in each direction are isomorphisms. This is enough to conclude $A\cong B$ . This completes the proof, with $\mathcal {E}(X;k)=A[\dim X]$ . Note that $\mathcal {E}(X;k)$ always exists because a resolution of singularities always exists.

Proof. Identify the rows of M with the sequence of polynomials

$$ \begin{align*} &(1+x)^l-1 \\ x&(1+x)^l \\ x^2&(1+x)^l \\ & \vdots \\ x^{q-l-1}&(1+x)^l \\ x^{q-l}&(1+x)^l -x^q. \end{align*} $$

If there is a linear dependence, then $A+Bx^q$ is divisible by $(1+x)^l$ for some constants A and B. Over $\mathbb {Q}$ , this is impossible since $A+Bx^q$ has distinct roots over $\mathbb {C}$ . This computes the rank over $\mathbb {Q}$ .

Over $\mathbb {F}_p$ , the polynomial $(1+x)^l$ divides $x^q+1$ , which easily leads to a linear dependence among the rows of M. It is obvious that the rank is at least $q-l$ , completing the proof in this case.

Proof. The variety Z is

$$ \begin{align*} Z&=\{(A_1,\ldots,A_l,B_1,\ldots,B_l,h,h',\ell_1,\ldots,\ell_l,\ell_1'\ldots\ell^{\prime}_l,\ell,\ell')\mid \\ &\quad A_i,B_i\in \operatorname{Hom}(\mathbb{C}^q,\mathbb{C}^q);\ h,h'\in Gr(q-1,q);\ \ell_1,\ldots,\ell_l,\ell_1'\ldots\ell^{\prime}_l,\ell,\ell'\in Gr(1,q); \\ &\qquad\qquad\qquad\qquad\qquad\qquad h,h'\subset \ker(A_i);\ \operatorname{im}(A_i)\subset \ell_i,\ell_i'\subset \ker B_i;\ \operatorname{im}(B_i)\subset \ell,\ell' \}. \end{align*} $$

We now construct a stratification of Z. For each $I\subset \{1,2,\ldots ,l\}\sqcup \{s,t\}$ , we define a stratum $Z_I$ consisting of tuples $(A_1,\ldots ,A_l,B_1,\ldots ,B_l,h,h',\ell _1,\ldots ,\ell _l,\ell _1'\ldots \ell ^{\prime }_l,\ell ,\ell ')\in Z$ subject to the conditions $\ell _i=\ell _i'$ if $i\in I$ , $h=h'$ if $s\in I$ , $\ell =\ell '$ if $t\in I$ , $\ell _i\neq \ell _i'$ if $i\notin I$ , $h\neq h'$ if $s\notin I$ , and $\ell \neq \ell '$ if $t\notin I$ .

Each stratum $Z_I$ is a vector bundle over a product of spaces that are either $\mathbb {P}^{q-1}$ or $(\mathbb {P}^{q-1}\times \mathbb {P}^{q-1})\setminus \Delta $ , where $\Delta $ is the diagonal. Therefore, $H_i^{BM}(Z_I;\mathbb {Z})$ is free over $\mathbb {Z}$ and vanishes when i is odd. Since the $Z_I$ stratify Z, the same is true for the Borel–Moore homology of Z.

Proof. Let I and J be two subsets of $\{1,2,\ldots ,l\}$ . Let

(6.1) $$ \begin{align} Y_{I,J}=\{(A_1,\ldots,A_l,B_1,\ldots,B_l)\in Y\mid A_i=0 \iff i\in I,\ B_j=0\iff j\in J\}. \end{align} $$

This is the stratification of Y into G-orbits. We show that this gives the desired stratification of Y. In the moduli interpretation of Y, inside each G-orbit, there is a unique representation $M_{IJ}$ of Q, up to isomorphism.

The module $M_{IJ}$ decomposes as a direct sum $M_{IJ}=X_I\oplus Y_J\oplus Z$ , where $X_I$ is an indecomposable with dimension 1 at the leftmost vertex, $Y_J$ is an indecomposable with dimension 1 at the rightmost vertex, and Z is a direct sum of simple modules, except when $I=\{1,2,\ldots ,l\}$ or $J=\{1,2,\ldots ,l\}$ , when $X_I$ or $Y_J$ , respectively, do not appear in the decomposition.

Put an order on these indecomposables where the simple at the leftmost vertex comes earliest in the order, then $X_I$ , then the simples at the middle vertices, then $Y_J$ , then the simple at the rightmost vertex. With this ordering, we can decompose $M_{IJ}$ into indecomposables, $M_{IJ}=\oplus _{i=1}^m M_i^{\oplus n_i}$ where $\operatorname {Hom}_Q(M_i,M_j)=0$ if $i<j$ .

Therefore, $\operatorname {End}_Q(M_{IJ})^\times $ surjects onto $\prod _i \operatorname {Mat}_{n_i}(\operatorname {End}_Q(M_i))^\times $ with unipotent kernel. Each $\operatorname {End}_Q(M_i)$ is isomorphic to $\mathbb {C}$ .

The quotient stack $[Y_{IJ}/G]$ is isomorphic to $[\mathrm {pt}/\operatorname {End}_Q(M_{IJ})^\times ]$ . This shows that $H^*_G(Y_{IJ};\mathcal {L})$ is a free k-module and vanishes in odd degrees since these properties hold for the stack $[\mathrm {pt}/GL_n(\mathbb {C})]$ .

It is clear that $\pi $ is a proper G-equivariant resolution of singularities, and thus is stratified for the stratification into G-orbits. It is even because every fiber of $\pi $ is a product of projective spaces.

Proof. Let $n=(q-1)(l+2)+ql$ be the common dimension of Y and $\widetilde {Y}$ . Decompose $\pi _*\underline {\mathbb {Z}}_p[n]$ into indecomposables

$$\begin{align*}\pi_*\underline{\mathbb{Z}}_p[n]\cong \bigoplus_t \mathcal{E}_t^{n_t}. \end{align*}$$

The endomorphism ring of $\pi _*\underline {\mathbb {Z}}_p[n]$ is $\operatorname {End}(\pi _*\underline {\mathbb {Z}}_p[n])\cong H_{2n}^{BM}(Z;\mathbb {Z}_p)$ , which governs the decomposition into irreducibles.

The spectral sequence with $E_2^{p,q}=H^p(BG;H_{-q}^{BM}(Z))$ converges to the G-equivariant Borel–Moore homology $H_*^G(Z)$ . This spectral sequence is concentrated in even degrees by Lemma 6.2 and hence degenerates at the $E_2$ page. Therefore, $H_{2n}^G(Z)$ surjects onto $H_{2n}^{BM}(Z)$ . As a consequence, each $\mathcal {E}_t$ is the deequivariantization of an indecomposable G-equivariant sheaf, which is an equivariant parity sheaf by Lemma 6.3.

Again by Lemma 6.2, the convolution algebra $H_{2n}^{BM}(Z;\mathbb {Z}_p)$ surjects onto $H_{2n}^{BM}(Z;\mathbb {F}_p)$ . Therefore, each $\mathcal {E}_t\otimes \mathbb {F}_p$ is also indecomposable.

Let i be the inclusion of $\{0\}$ in Y. By Proposition 3.2 and Lemma 6.1, the multiplicity of $i_*\underline {k}[l-2]$ in $\pi _*\underline {k}[n]$ is $q-l+1$ when $k=\mathbb {Q}_p$ and $q-l$ when $k=\mathbb {F}_p$ . Therefore, there exists a unique t such that

(6.2) $$ \begin{align} m(i_*\underline{\mathbb{Q}}_p[l-2],\mathcal{E}_t\otimes \mathbb{Q}_p)=1 \end{align} $$

and $\mathcal {E}_t$ is not a skyscraper sheaf at 0.

This setup is symmetric under the action of the symmetric group $S_l$ which permutes the indices $\{1,2,\ldots ,l\}$ of elements of Y. So, by uniqueness of t, $\mathcal {E}_t$ is $S_l$ -invariant. From the classification of equivariant parity sheaves in [Reference Juteau, Mautner and WilliamsonJMW], $\mathcal {E}_t$ is up to homological shift the parity extension of a constant sheaf on an $S_l$ -invariant G-orbit on Y.

There are only two $S_l$ -invariant intermediate G-orbits in Y. They are $Y_{\{1,2,\ldots ,l\},\emptyset }$ and $Y_{\emptyset ,\{1,2,\ldots ,l\}}$ in the notation of (6.1).

Let W be the closure of $Y_{\{1,2,\ldots ,l\},\emptyset }$ . Let us assume that $\mathcal {E}_t$ has support W. Define

$$\begin{align*}\widetilde{W}=\{(\ell,B_1,B_2,\ldots ,B_l)\mid \ell\in Gr(1,q), B_i\in \operatorname{Hom}(\mathbb{C}^q,\ell)\}. \end{align*}$$

Write $\sigma $ for the map from $\widetilde {W}$ to ${W}$ . This is a G-equivariant even resolution of singularities. Therefore, in the G-equivariant derived category, $\mathcal {E}_t$ must appear as a direct summand of $\sigma _*\underline {\mathbb {Z}}_p$ (up to homological shift) as it is the unique indecomposable G-equivariant parity sheaf extending the constant sheaf and thus the same statement holds when we forget the equivariant structure.

The space $\widetilde {W}$ is the total space of a vector bundle $\mathcal {F}\cong \mathcal {O}(1)^{\oplus lq}$ over $\mathbb {P}^{q-1}$ , where the zero section is the fiber over $0\in Y$ . The Euler class $e(\mathcal {F})$ is a power of the Chern class $c_1(\mathcal {O}(1))$ . In computing the intersection form (3.1) for the resolution $\sigma $ , once irrelevant rows are deleted, one is left with the antidiagonal matrix J, which has the same rank over $\mathbb {Q}$ and $\mathbb {F}_p$ .

Since the rank does not change, we deduce that $m(i_*\underline {\mathbb {Q}} _p[l-2],\mathcal {E}_t\otimes \mathbb {Q}_p)=0$ , contradicting (6.2). Therefore, $\mathcal {E}_t$ cannot have support equal to W.

Similarly, $\mathcal {E}_t$ cannot have support equal to the closure of the other $S_l$ -equivariant intermediate stratum $Y_{\emptyset ,\{1,2,\ldots ,l\}}$ .

Therefore, $\mathcal {E}_t$ is an extension of the shifted constant sheaf on the open stratum in Y. As the stalk of $\mathcal {E}_t$ at 0 is free over $\mathbb {Z}_p$ and satisfies a parity vanishing property, it is nonzero in degree $l-2$ using (6.2). Therefore, $\mathcal {E}_t\otimes \mathbb {F}_p$ has nonzero stalk cohomology at 0 in degree $l-2$ . Thus,

$$\begin{align*}\mathcal{E}_t\otimes \mathbb{F}_p \not \cong {{}^p\tau}_{\leq l-3}(\mathcal{E}_t\otimes \mathbb{F}_p). \end{align*}$$

Since $\mathcal {E}_t\otimes \mathbb {F}_p$ is an indecomposable extension of the shifted constant sheaf on the open stratum on Y and occurs as a summand of $\pi _*\underline {\mathbb {F}_p}[n]$ , by Theorem 5.1, $\mathcal {E}_t\otimes \mathbb {F}_p\cong \mathcal {E}(Y;\mathbb {F}_p)$ . By the reduction to the slice argument made in the previous section, this completes the proof of Theorem 2.1.