Matrix Points on Varieties

Technical Summary: Matrix Points on Varieties

Problem Statement
The paper investigates the cohomology of $C_n(X)$, the moduli space of commuting $n$-by-$n$ matrices satisfying the equations defining a quasi-projective scheme $X$ over a field $k$. This space represents the functor $S \mapsto \text{Hom}_{k\text{-alg}}(R, M_n(S))$ for an affine scheme $X = \text{Spec } R$. While $C_n(X)$ generalizes classical commuting varieties (e.g., commuting tuples in $\mathfrak{gl}_n$ or $GL_n$), its rational cohomology had not been previously computed, even for simple cases like matrices avoiding specific eigenvalues. The authors frame this problem through the lens of "non-commutative Weil restriction," viewing the inclusion $k \hookrightarrow M_n(k)$ as a non-commutative analogue of a degree $n$ separable field extension.

Methodology
The authors employ a combination of algebraic geometry, representation theory, and combinatorics:

1. Geometric Comparison via Semi-Simple Counterpart:
   The core strategy involves constructing a "semi-simple counterpart" $S_n(X)$, defined as the quotient $(X^n \times GL_n/T_n) / S_n$, where $S_n$ acts diagonally. Here, $GL_n/T_n$ represents the space of ordered direct sum decompositions of $k^n$. A natural map $\sigma: S_n(X) \to C_n(X)$ is defined, which sends a semi-simple framed sheaf to its corresponding matrix point.
   The authors analyze the fibers of the natural projections $p_1: S_n(X) \to \text{Sym}^n X$ and $p_2: C_n(X) \to \text{Sym}^n X$. They prove that under specific conditions (characteristic zero or $X$ being locally homogeneous), the map $\sigma$ induces an isomorphism on the derived direct images of the constant sheaf ($R p_{2*} \mathbb{Q}_\ell \cong R p_{1*} \mathbb{Q}_\ell$). This relies on showing that the fibers of $p_2$ (involving framed sheaves with nilpotent support) have trivial cohomology under the given assumptions, effectively reducing the cohomology of $C_n(X)$ to that of the more tractable $S_n(X)$.

2. Combinatorial Computation:
   Once the cohomology is reduced to $S_n(X)$, the authors utilize the theory of graded characters of the symmetric group $S_n$ and symmetric functions. They define the Frobenius character map and utilize the principal specialization of symmetric polynomials. By analyzing the $S_n$-action on the cohomology of $X^n$ and the flag variety $GL_n/T_n$, they derive explicit formulas for the Poincaré polynomials.

3. Generating Series and Zeta Functions:
   The paper establishes a connection between the generating series of the Poincaré polynomials of $C_n(X)$ (and the associated stack $\mathcal{C}_n(X)$) and the Betti zeta function of $X$, $\zeta^B_X(t)$. This parallels Macdonald's classical results on the cohomology of symmetric products.

4. Hermitian Variant:
   For schemes defined over $\mathbb{R}$, the authors adapt the framework to the space of commuting Hermitian matrices $HC_n(X)$, using the unitary group $U_n$ and its maximal torus $U_{T_n}$ to construct a semi-simple counterpart $HS_n(X)$.

Key Contributions and Results

• Cohomological Isomorphism (Theorem 1.2): Under the assumption that $k$ has characteristic zero or $X$ is locally homogeneous, the map $\sigma: S_n(X) \to C_n(X)$ induces an isomorphism on $\ell$-adic cohomology:
  $$H^i_{\text{ét}}(C_n(X), \mathbb{Q}_\ell) \cong H^i_{\text{ét}}(S_n(X), \mathbb{Q}_\ell)$$
  This validates the heuristic that $C_n(X)$ behaves cohomologically like the Weil restriction of $X^n$ along the "non-commutative extension" $k \to M_n(k)$.

• Explicit Poincaré Polynomials (Theorem 1.3): The authors provide a combinatorial formula for the Poincaré polynomial of $C_n(X)$:
  $$P_u(C_n(X)) = \phi_n(u^2) \text{sp}_{u^2}(\text{ch}_u(X^n))$$
  where $\phi_n(q) = \prod_{i=1}^n (1-q^i)$, $\text{sp}$ denotes principal specialization, and $\text{ch}_u$ is the graded $S_n$-character of the cohomology of $X^n$.

• Generating Series Identity (Theorem 1.4): They prove an infinite product identity relating the generating series of the Poincaré polynomials of the stack $\mathcal{C}_n(X)$ to the Betti zeta function of $X$:
  $$1 + \sum_{n=1}^\infty P_u(\mathcal{C}_n(X)) t^n = \prod_{i=0}^\infty \zeta^B_X(u^{2i}t)$$
  This generalizes Macdonald's formula for symmetric products.

• Equivariant Formality (Corollary 2.12): The paper establishes that the $GL_n$-action on $C_n(X)$ is equivariantly formal. Consequently, the equivariant cohomology of the stack $\mathcal{C}_n(X)$ is the product of the cohomology of $BGL_n$ and $C_n(X)$.

• Hermitian Result (Theorem 1.5): For $X$ over $\mathbb{R}$, the map $\sigma_H: HS_n(X) \to HC_n(X)$ induces an isomorphism on rational cohomology, providing Poincaré polynomials for real varieties of commuting Hermitian matrices.

Significance and Claims
The paper claims to resolve the previously open problem of computing the rational cohomology of $C_n(X)$. It asserts that the space $C_n(X)$, despite being a complex non-commutative moduli space, is cohomologically equivalent to a simpler semi-simple quotient $S_n(X)$ under mild conditions. This equivalence allows for the application of symmetric function theory to derive explicit Betti numbers.

The authors highlight that their combinatorial formula offers a "low complexity" method to compute Betti numbers for spaces that were previously difficult to solve using classical methods (e.g., matrices avoiding specific eigenvalues). Furthermore, the work connects the geometry of matrix points to the theory of Weil restrictions and provides a new perspective on the cohomology of commuting varieties, confirming a heuristic derived from non-commutative geometry. The results also extend to the equivariant setting, identifying the cohomology of the stack of coherent sheaves with that of a symmetric product stack.