Papers

We develop a general first-order theory of Schur apolarity for the study of secant varieties of flag varieties in arbitrary homogeneous embeddings. Extending the classical apolarity–fat-point correspondence for Veronese varieties, we show that in the Schur setting the algebraic square of the apolar ideal need not coincide with the geometric double-point conditions. We introduce a geometric Schur square whose relevant component is the conormal space, yielding a Schur Dual Terracini Lemma. Our construction recovers classical apolarity in the symmetric case. A slot-by-slot Consistency Theorem realizes these intrinsic conditions as multigraded double points. As an application, we determine the dimensions of all secant varieties of $\operatorname{Fl}(1,2;V_n)$ embedded by $\mathcal{O}(1,1)$: the only defective cases are $\sigma_2(\operatorname{Fl}(1,2;V_3))$ and $\sigma_3(\operatorname{Fl}(1,2;V_4))$, both of defect one.

We define symplectic Dirac operators on homogeneous spaces and study their representation-theoretic role. For an invariant polarization, the symplectic Dirac operator decomposes into two symplectic Dolbeault operators. We compute their commutator as the natural symplectic analogue of the square of the classical Dirac operator. Our first main result gives a necessary and sufficient condition for this commutator to satisfy a Parthasarathy-type formula. We further prove that, whenever this condition fails, no cubic perturbation of the symplectic Dolbeault operators can yield such a formula, in contrast with Kostant's cubic Dirac operator in the orthogonal setting. As applications, we establish an ${\mathfrak s}{\mathfrak l}_2$-structure generated by the symplectic Dolbeault operators and derive Dirac-type inequalities for unitary representations of Hermitian symmetric spaces labelled by the levels of the symmetric powers of the antiholomorphic tangent space at the identity. The level-zero inequality recovers the standard Parthasarathy-Dirac inequality, while the higher levels inequalities yield new constraints. For $SU(1,n)$, we show that, for representations with a specific Kraljević corner, the level-one inequality strengthens all basic Parthasarathy inequalities of the first kind for particular $K$-types, precisely those satisfying an explicit highest-weight condition.

We give several characterizations of an inhomogeneous basis of the ring of symmetric functions whose evaluations are the character values of the irreducible representations of the rook monoid (symmetric inverse semigroup). Using Schur–Weyl duality, we show that the transition coefficients of this basis with the power symmetric basis are the characters of the propagating partition algebra (dual symmetric inverse monoid algebra). In addition, the structure coefficients of this basis are equal to the coefficients in the smash (or Heisenberg) product of Schur functions.

The $F$-invariant in cluster algebras is a combinatorial invariant that unifies the $E$-invariant from additive categorification and the $\mathfrak{d}$-invariant from monoidal categorification. In this paper, we study its refinement, the partial $F$-invariant, and establish its mutation formula under changes of the initial seed. As an application, we prove a conjecture of Reading, which asserts that the non-compatible cluster variables can be separated by sign-coherence of $g$-vectors upon varying the initial seed. We further show that, for cluster monomials, the partial $F$-invariants coincide with both the partial $E$-invariants for reachable decorated representations of quivers with potentials and the pole orders of normalized $R$-matrices (or partial $\mathfrak{d}$-invariants) for finite-dimensional reachable simple modules over quantum affine algebras. As consequences, we obtain a combinatorial formula for the pole orders for reachable simple modules in terms of $q$-characters; we verify the conjectural explicit formula for the pole orders between Kirillov–Reshetikhin modules.

We give a new equivalent formulation of Verdier's octahedral axiom. An initial square in a pre-triangulated category determines an obstruction in a quotient of ordinary Hom groups. Its vanishing is equivalent to the existence of a good completion, and $\mathrm{TR4}$ is equivalent to this vanishing for the squares associated with composable morphisms. As a consequence, if two pre-triangulations have the same underlying additive category and one is triangulated, then the other satisfies $\mathrm{TR4}$ precisely when their relative inverse Heller comparison admits an exact lift along every short exact sequence in the Freyd category. As applications, we prove that the exotic pre-triangulated category of Díaz Cabrera–Muro is in fact a triangulated category over every algebraically closed field. We construct scalar families for type-$A_{3a-1}$ preprojective algebras over arbitrary fields and for a range of Dynkin preprojective algebras in characteristic two; in each family the zero parameter is the unique triangulated member. The local equations also yield separable descent, including canonical triangulations for non-modular equivariantizations and for the Markman–Mehrotra K3 deformation categories.

Let $A$ be a split finite-dimensional algebra over a field whose radical $J$ satisfies $J^3=0$, and let $s$ be the number of isomorphism classes of simple $A$-modules. We prove that a non-projective module $M$ with ${\rm Ext}_A^i(M,A)=0$ for all $i>0$ has a non-zero self-extension in some degree between $1$ and $3s+1$. In particular, $A$ satisfies the Auslander–Reiten conjecture, which asserts that every self-orthogonal generator is projective. As a consequence, every finite-dimensional algebra over an algebraically closed field with radical cube zero satisfies the Auslander-Reiten conjecture.

We study the continuous homology and cohomology of semisimple Lie groups with coefficients in arbitrary unitary representations. The case of irreducible representations was determined by Vogan and Zuckerman; we focus on reducible representations. The (co)homology splits into its Hausdorff and torsion parts. The Hausdorff part is governed by containment of irreducible cohomological representations. We show that the torsion part is governed by weak containment of irreducible cohomological representations. Precisely, we show that a unitary representation admits non-zero torsion if and only if there is a cohomological point which is not isolated in its support. We discuss in length examples of rank-$1$ groups, completely determining unitary cohomology for the group $\mathrm{SO}^\circ(n,1)$. For a simple Lie group, the first and fourth named authors showed that the first degree in which it obtains non-trivial cohomology for some unitary representation with no invariant vectors is related to the rank of the group. We discuss the analogous question regarding torsion cohomology, and show that the corresponding first degree could be much higher: in the presence of property (T), it is bounded below by the square root of the dimension of the symmetric space. A technical device that we use is the restriction to well chosen dense subgroups which satisfy finiteness properties, which we call cohomological witnesses. We combine it with results on the unitary cohomology of groups which satisfy finiteness properties. These results are of an independent interest.

Let $T$ be a finite group and let $kB_T$ denote the shifted Burnside biset functor over a commutative ring $k$. We study the essential algebra $\widehat{kB_T}(G)$ of $kB_T$ at an arbitrary finite group $G$. We first establish criteria determining when a covering subgroup of $G\times G\times T$ factors, with respect to the star product, through a group of order strictly smaller than $|G|$; the first criterion places no hypothesis on $T$. As an application, we describe $\widehat{kB_T}(G)$ completely whenever no nontrivial quotient of $G$ is isomorphic to a section of $T$, extending the coprime case treated by Romero. When $T$ is abelian and $|T|$ is invertible in $k$, we prove that $\widehat{kB_T}(G)$ decomposes as a direct sum of matrix algebras over the group algebras of the groups $Out_T(A)$, where $A$ runs over the linkage classes of reduced subgroups of $G\times T$, and we parametrize the simple modules of $\widehat{kB_T}(G)$.

Let $v$ be an indeterminate. We give a presentation of the mirabolic quantum Schur algebra $\mathcal{MS}_v(2,d)$ in terms of generators and defining relations. More precisely, we determine the kernel of the natural epimorphism from Rosso's mirabolic quantum $\mathfrak{sl}_2$, $\mathbf U^{\mathrm{mir}}_v(2)=\langle e,f,k^{\pm 1},\ell\rangle$, onto $\mathcal{MS}_v(2,d)$. Setting $P_d(k)=\prod_{r=0}^{d}(k-v^{2r-d})$ and $Q_d(k)=\prod_{r=1}^{d}(k-v^{2r-d})$, we prove $\mathcal{MS}_v(2,d)\cong \mathbf U^{\mathrm{mir}}_v(2)/ \left\langle P_d(k),(\ell-1)Q_d(k)\right\rangle$. We further determine the split Wedderburn decomposition of this quotient and provide an equivalent presentation in terms of weight idempotents.

We study cohomological Hall algebras of curves and their actions on the homology of Quot schemes. We introduce the virtual homology of Quot schemes and show that it is preserved by both creation and annihilation actions. We prove that the torsion CoHA is isomorphic to a shuffle algebra and, equivalently, to a braided symmetric algebra associated with a Yang-Baxter operator. We use this description to determine the ideal of tautological relations for punctual Quot schemes and obtain a new basis for their cohomology rings. Finally, we introduce a universal way to double the torsion CoHA and show that it acts naturally on the virtual homology of Quot schemes of arbitrary type.

We study iYangians, namely twisted Yangians in the Drinfeld current presentation, associated with quasi-split symmetric pairs of type $\mathsf{ADE}$ with nontrivial diagram involution. We establish minimalistic presentations in terms of degree-zero and degree-one generators. Type $\mathsf A_{2n}$ is treated separately: the isolated rank-two case requires two additional relations, whereas in higher rank these relations are forced by the neighboring noncentral orbit. As a by-product, we strengthen Theorem 3.1 in arxiv:2511.07136 by proving that the extra relation in the minimalistic presentation of split iYangian is redundant whenever $\mathfrak g$ has rank at least two. We also construct explicit injective homomorphisms from quasi-split iYangians into Yangians, identify their images as right coideal subalgebras, and obtain isomorphisms between the Drinfeld and $J$ presentations. Finally, we derive triangular estimates for the Drinfeld currents and their coproducts and apply them to the ${}^\imath\ell$-weights arising by restriction from finite-dimensional Yangian modules.

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and, for a dominant integral weight $\lambda$, let $\Theta_\lambda$ be the multiplicity-free sum of the weights of the irreducible module $V(\lambda)$. The $\Theta_\mu$ form a $\mathbb{Z}$-basis of the $W$-invariants, so there are unique integers $a(\mu,\lambda)$, the atomic numbers, with $\mathrm{ch}\,V(\lambda) = \sum_\mu a(\mu,\lambda)\Theta_\mu$ over dominant $\mu$. We ask when they are nonnegative. They are the Moebius transform of the weight-multiplicity function on the dominant-weight poset, which the crosscut theorem turns into an alternating sum of at most $2^{\mathrm{rank}\,\Phi}$ multiplicities indexed by subsets of the covers of $\mu$ in $[\mu,\lambda]$. We prove that $a(\mu,\lambda)\ge 0$ whenever every connected component of the Dynkin diagram is a path, and identify the coefficient with the dimension of an explicit weight space: it is cut out of $V(\lambda)$ by alternately taking kernels of raising operators and cokernels of lowering operators, one per cover, in order along the path. Type $D_4$ shows the hypothesis is necessary. Put $\beta = \alpha_1 + 2\alpha_2 + 2\alpha_3 + 2\alpha_4$; for every dominant $\mu$ with $\langle\mu,\alpha_2^\vee\rangle = 1$ and $\mu+\beta$ dominant, $a(\mu,\mu+\beta)$ is $-2$ if $\langle\mu,\alpha_1^\vee\rangle = 0$ and $-1$ otherwise. Restriction to the support of $\lambda-\mu$ carries this family into every irreducible type with a trivalent node. Thus an irreducible finite root system has all atomic numbers nonnegative exactly when its Dynkin diagram is a path, namely in types $A_n$, $B_n$, $C_n$, $F_4$, $G_2$. Deep in the dominant chamber, in an explicit range, $a(\mu,\lambda)$ is the Kostant partition number of $\lambda-\mu$ for the nonsimple positive roots; negative atomic numbers are therefore confined to boundary slabs.

Let $d \geqslant 1$ be an integer, $\mathscr{M}$ a suitable $d$-abelian category. There is a notion of higher torsion classes in $\mathscr{M}$, also known as $d$-torsion classes. We generalise a classic theorem of Hoshino by providing criteria for a $d$-torsion class $\mathscr{U}$ to be splitting, notably that $\mathscr{U}$ is stable under $\tau_d^-$, the inverse $d$-Auslander–Reiten translation. We apply the criteria to show that under additional assumptions, which are satisfied in Dynkin type $A$, the elements of the spine of the lattice of $d$-torsion classes are precisely the splitting $d$-torsion classes. The spine consists of the elements which belong to a chain of maximal length.

Line operators of 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$ of type $A_{N-1}$ in a half Omega background provide a physical model for $SL_N$ skein algebras. In this paper we extend this correspondence to $GL_N$ skein algebras and prove that the $GL_2$ skein algebra of the punctured torus is isomorphic to the $GL_2$ spherical double affine Hecke algebra ${{\mathbf{S}\ddot{\mathbf{H}}}}^{q,t}_2$. The construction is based on $\mathfrak{q}$-nonabelianization for the $SU(2)$ $\mathcal{N}=2^*$ gauge theory, which realizes the skein algebra as a quantum torus algebra associated with the Seiberg-Witten curve. Different regions of the Coulomb branch yield distinct presentations of ${{\mathbf{S}\ddot{\mathbf{H}}}}^{q,t}_2$. These range from the Macdonald $q$-difference module in Fenchel-Nielsen (weak coupling) charts to cluster-type realizations in Fock-Goncharov (strong coupling) charts. Transitions between these descriptions are governed by the vanilla BPS spectrum of the $\mathcal{N}=2^*$ theory via framed wall-crossing, which provides a unified physical framework for several representations of spherical DAHA.

Gheibi, Jorgensen and Takahashi recently introduced the quasi-projective dimension, a homological invariant that extends the classical projective dimension. In this paper, we investigate this invariant from the perspective of Gorenstein homological algebra. First, we show that the quasi-projective dimension coincides with the Gorenstein projective dimension under certain conditions. Second, we introduce and study the quasi-Gorenstein projective dimension as a Gorenstein analogue of the quasi-projective dimension.

Let $G$ be a connected reductive complex algebraic group and let $q\in\mathbb{C}^{\times}$ be not a root of unity. We prove that every $(G,q)$-connection on a formal punctured disc admits a $(G,q)$-oper structure.

In this paper, we attempt to develop a general notion of representation theory for multiplicative Lie algebras. Consequently, we study equivalent, irreducible, and completely reducible representations. We also introduce the notion of the action of a multiplicative Lie algebra on a vector space and show that there is a one-to-one correspondence between the action and the representation. We also studied the extended structure of a multiplicative Lie algebra through a group.

For a symmetric weight $w(x)$ on a finite discrete lattice and its Christoffel transforms $x^{2}w(x),x^{2}(x^{2}w(x)),\dots$, we prove that, at each Christoffel step, the conjugated projection associated with the transformed Christoffel–Darboux kernel differs from the original orthogonal projection by a rank-one operator on the subspace of functions vanishing at the origin. This provides a general mechanism for transferring local asymptotic results from an orthogonal polynomial ensemble to its Christoffel-transformed counterpart as the lattice size tends to infinity. As a main application, we study local fluctuations of random Young diagrams arising from skew $(\mathrm{Sp}_{2n},\mathrm{Sp}_{2k})$ Howe duality. The corresponding particle ensemble is obtained from the Krawtchouk orthogonal polynomial ensemble on a quadratic lattice by a Christoffel transform. We identify four asymptotic regimes of local fluctuations in the limit $n,k\to\infty$ with $n/k\to c\in(0,\infty)$. Besides the universal bulk fluctuations governed by the discrete sine kernel and universal Airy fluctuations at the right edge of the limit shape, we obtain the discrete Hermite kernel in the critical regime $(k-n)/\sqrt{n+k}\longrightarrow r\in\mathbb{R}$, and the discrete hard-wall sine kernel at the left corner.

Cyclotomic Brauer or cyclotomic Nazarov–Wenzl algebras arise in higher Schur–Weyl dualities involving parabolic categories $\mathcal O$ for Lie algebras of types $B,C$ and $D$. Their connection with Kac–Moody-type categorification is substantially less developed than the corresponding type $A$ theory for cyclotomic Hecke algebras. We construct a categorical bridge between affine Brauer-type representation theory and the half-integral quasi-split type $\operatorname{AIII}$ $\imath$-Kac–Moody $2$-category of Bao–Shan–Wang–Webster. More precisely, an action of the affine Brauer category on a locally Schurian category, with dot spectrum exactly $\frac{1}{2}+\mathbb Z$, determines a generalized nilpotent $2$-representation of the even component $\mathfrak U^{\imath}_{+}$. Conversely, every nilpotent $2$-subrepresentation of an ambient locally Schurian $2$-representation of $\mathfrak U^{\imath}_{+}$ carries a compatible affine Brauer action whose dot spectrum is contained in $\frac{1}{2}+\mathbb Z$. Applying these constructions to cyclotomic quotients, we prove that the locally unital algebra attached to a $\mathbf u$-admissible cyclotomic Brauer category is isomorphic to the locally unital algebra attached to the corresponding cyclotomic quotient of the principal $2$-representation of $\mathfrak U^{\imath}_{+}$. Consequently, the associated cyclotomic Brauer (or cyclotomic Nazarov–Wenzl) algebras acquire natural $\mathbb Z$-gradings. To our knowledge, this is the first such categorical realization of $\mathbf u$-admissible half-integral cyclotomic Brauer algebras by means of an $\imath$-Kac–Moody $2$-category. It provides the categorical and graded framework toward a Brauer-type extension of the Brundan–Kleshchev–Ariki theory in which coideal algebras and $\imath$-canonical bases are expected to replace ordinary quantum groups and canonical bases.

Let $V$ be a compact connected manifold and $G$ a simply connected solvable Lie group. We study $G$-Lie foliations on $V$ from the point of view of $\mathcal{D}$-module theory, following the approach initiated by Dathe. To each singular foliation $\mathcal{I}$, we associate the $\mathcal{D}_X$-module $\mathcal{M}_{\mathcal{I}} = \mathcal{D}_X / \mathcal{D}_X \cdot \mathcal{I}$ and the derived ring $\mathcal{D}_{\mathcal{I}} := R\mathcal{H}om_{\mathcal{D}_X}(\mathcal{M}_{\mathcal{I}}, \mathcal{M}_{\mathcal{I}})$. We compute the D-irregularity $\mathrm{D\text{-}irr}(\mathcal{I})$ for three classes of solvable Lie foliations: regular homogeneous foliations, the Meigniez foliation with non-polycyclic holonomy group, and an explicit foliation on a compact 5-dimensional manifold. We show that the non-polycyclicity of the holonomy group $\Gamma$ is reflected in the non-vanishing of higher cohomology groups of $\mathcal{D}_{\mathcal{I}}$, establishing a new connection between the geometry of the holonomy group and $\mathcal{D}$-module invariants.

This short article resolves a question about Arthur parameters highlighted in previous work of Balodis, Cunningham, Fiori and Zhang, in the case of $\mathrm{GL}_n/F$ where $F/\mathbb{Q}_p$ is a $p$-adic field. Namely, we prove that every Arthur parameter $\psi$ of $\mathrm{GL}_n/F$, can be factored into Arthur parameters $\psi\cong \psi_1\oplus\cdots \oplus\psi_r$ such that the orbit $C_\psi=C_{\psi_1}\times\cdots\times C_{\psi_r}\subseteq V_{\lambda_{\psi_1}}\times \cdots \times V_{\lambda_{\psi_r}}$ in the associated Vogan variety has smooth closure if and only if each $\bar{C}_{\psi_i}$ is smooth for $i=1, \ldots, r$, and that $C_{\psi_i}$ has smooth closure if and only if it is open or closed.

Let \(\Lambda\) be a finite-dimensional Koszul algebra. Using a description of the linear part of minimal projective resolutions in terms of Koszul duality, we prove that \(\Lambda\) is absolutely Koszul if and only if its Koszul dual \(\Lambda^{!}\) is graded left co-coherent. We further show that an absolutely Koszul algebra of finite global linearity defect has finite graded finitistic dimension. Finally, we refine Koszul duality for finite-dimensional absolutely Koszul algebras. As a further application, we answer a question of Green et al.~\cite{12} by means of the Koszul dual algebra.

We study bounded derived categories of semisimple tensor categories and their semisimple module categories. These categories may have infinitely many simple isomorphism classes. We classify bounded monoidal $t$-structures by integer-valued characters of the universal grading group and an integer deviation. We also classify compatible module $t$-structures using stabilizer subgroups of the module components. The corresponding hearts are tensor and module equivalent to the original categories. It follows that a monoidal derived equivalence and a coherent derived module equivalence recover both the tensor category and its module category.

We study homogeneous forms with invariant Jacobian under group actions. For finite-dimensional irreducible complex representations, we show that the canonical decomposition defined by the center of the form is either trivial or gives a system of imprimitivity, which determines the module structure of Jacobian $J(f)$. We apply the theory to symmetric groups and classify homogeneous forms with invariant Jacobian on the natural permutation module.

Based on the conjectured (and checked for tree-type quivers) quiver Yangian/Coulomb branch algebra correspondence, we give a quiver Yangian interpretation of the monopole formulas for 3D $\mathcal N=4$ good $A$-type quiver gauge theories, also known as the $T_\rho(SU(N))$ theories. Using the algebra action on the $\frac12$-BPS vortices, we reorganize the generators of the truncated shifted quiver Yangian into boundary-adapted generators, and obtain the classical relations in terms of $U(N)$ Casimirs on the Slodowy slice $\mathcal S^{\mathfrak{gl}_N}_\rho$, whose coordinates are given by these boundary-adapted generators. We also show that the boundary-adapted generators and the Casimir relations have the correct fugacities as predicted by the Hall-Littlewood expression of the monopole formula for $T_\rho(SU(N))$, hence reconstructing it as the Hilbert series of the truncated shifted quiver Yangian.

Let $\mathbf{U}$ be a quantum group associated with a symmetric Cartan datum, let $\dot{\mathbf{U}}$ be its modified form, and let $\dot{\mathbf{B}}$ be the canonical basis of $\dot{\mathbf{U}}$. Lusztig conjectured that the structure constants of the multiplication, comultiplication, and bilinear form in $\dot{\mathbf{U}}$ with respect to $\dot{\mathbf{B}}$ belong to $\mathbb{N}[v,v^{-1}]$. We introduce the thickening realization, which relates $\dot{\mathbf{B}}$ to the canonical basis of the negative part of a larger quantum group $\tilde{\mathbf{U}}^-$. More precisely, it identifies the relevant structure constants in $\dot{\mathbf{U}}$ with the structure constants in $\tilde{\mathbf{U}}^-$. As consequences, we prove, for arbitrary symmetric Cartan datum, $\dot{\mathbf{B}}$ has the positivity properties for the comultiplication and bilinear form, as well as the positivity for the multiplication whenever one factor is spherical parabolic. In particular, Lusztig's conjecture holds for simply-laced finite type. We also prove the canonical bases of a broad class of tensor products of integrable modules have the positivity properties for the transition matrices and actions by $\dot{\mathbf{B}}$.

For irreducible components $C_\sigma$ of Lusztig's nilpotent varieties of type $A$ with graded dimension $(1,2,\dots,n,n-1,\dots,1)$ arising from permutations $\sigma\in S_n$, we characterize the property that $C_\sigma\oplus C_\sigma$ is an irreducible component combinatorially in terms of $\sigma$. The permutations that occur, which we call pseudosmooth, are described by a recursion on direct sums and deleting suitable corners, whose terminal cases are the permutations obtained from \[ 3412,\quad 4231,\quad 35142,\quad 42513,\quad 45312,\quad 426153,\quad 463152,\quad 526413 \] by inflating the entries into consecutive decreasing blocks. The main new input is a hereditary property valid for decompositions of multisegments with disjoint extreme points.

Let $G$ be a finite group acting properly by lattice automorphisms on a complete simplicial fan $\Sigma$. An open question due to Stanley asked whether the (ungraded) representation carried by the cohomology $H^*(X_{\Sigma})$ of the associated toric variety $X_{\Sigma}$ is isomorphic to a permutation representation of $G$. We prove that Stanley's question has an affirmative answer for all smooth projective toric varieties without the properness assumption on the action. The proof is inspired by toric mirror symmetry.

In this paper, we study when the restriction of an irreducible representation of a reductive Lie group $G$ to a reductive subgroup $H$ is discretely decomposable. T.\ Kobayashi gave a necessary condition for discrete decomposability in terms of associated varieties. In a previous paper, we proved that the condition is also sufficient if $(G, H)$ is a symmetric pair. The purpose of this paper is to prove the sufficiency for any reductive subgroup $H$ under some hypothesis on $G$ (e.g., $G$ is linear). As an application, we show that discrete decomposability is equivalent to $\mathfrak{h}$-admissibility when the compact factor of $\mathfrak{h}$ is maximal in some sense.

Let $R$ be a commutative noetherian local ring with residue field $k$. Denote by $\operatorname{\mathsf{D^b}}(R)$ the bounded derived category of finitely generated $R$-modules. In this paper, we introduce the notion of ranks of triangulated categories and study Verdier quotients of $\operatorname{\mathsf{D^b}}(R)$ from the viewpoint of this invariant. Our main result determines the rank of the Verdier quotient $\operatorname{\mathsf{D^b}}(R)/\operatorname{\mathsf{thick}}(R\oplus k)$ in terms of the ranks of the categories of maximal Cohen-Macaulay modules on the punctured spectrum.