arXiv subjects
Publications and source records attributed to Michael K. Brown.
We prove that equivariant matrix factorization categories associated to henselian local hypersurface rings are idempotent complete, generalizing a result of Dyckerhoff in the non-equivariant case.
We develop an analogue of Eisenbud-Floystad-Schreyer's Tate resolutions for toric varieties. Our construction, which is given by a noncommutative analogue of a Fourier- Mukai transform, works quite generally and provides a new perspective on the relationship between Tate resolutions and Beilinson's resolution of the diagonal. We also develop a Beilinson-type resolution of the diagonal for toric varieties.
We propose a notion of minimal free resolutions for differential modules, and we prove existence and uniqueness results for such resolutions. We also take the first steps toward studying the structure of minimal free resolutions of differential modules. Our main result in this direction explains a sense in which the minimal free resolution of a differential module is a deformation of the minimal free resolution of its homology; this leads to structural results that mirror classical theorems about minimal free resolutions of modules.
We construct a comparison map from the topological K-theory of the dg-category of twisted perfect complexes on certain global quotient stacks to twisted equivariant K-theory, generalizing constructions of Halpern-Leistner-Pomerleano and Moulinos. We prove that this map is an equivalence if a version of the projective bundle theorem holds for twisted equivariant K-theory. Along the way, we give a new proof of a theorem of Moulinos that the comparison map is an equivalence in the non-equivariant case.
We prove a conjecture of Shklyarov concerning the relationship between K. Saito's higher residue pairing and a certain pairing on the periodic cyclic homology of matrix factorization categories. Along the way, we give new proofs of a result of Shklyarov and Polishchuk-Vaintrob's Hirzebruch-Riemann-Roch formula for matrix factorizations.
We prove the non-commutative analogue of Grothendieck's Standard Conjecture D for the dg-category of matrix factorizations of an isolated hypersurface singularity in characteristic 0. Along the way, we show the Euler pairing for such dg-categories of matrix factorizations is positive semi-definite.
We describe a Macaulay2 package for computing Schur complexes. This package expands on the ChainComplexOperations package by David Eisenbud.
We study the topological K-theory spectrum of the dg singularity category associated to a weighted projective complete intersection. We calculate the topological K-theory of the dg singularity category of a weighted projective hypersurface in terms of its Milnor fiber and monodromy operator, and, as an application, we obtain a lift of the Atiyah-Bott-Shapiro construction to the level of spectra.
Let $k$ be a field of characteristic 0 and $\mathcal{A}$ a curved $k$-algebra. We obtain a Chern-Weil-type formula for the Chern character of a perfect $\mathcal{A}$-module taking values in $HN_0^{II}(\mathcal{A})$, the negative cyclic homology of the second kind associated to $\mathcal{A}$, when $\mathcal{A}$ satisfies a certain smoothness condition.
We prove a special case of a conjecture of Dao-Kurano concerning the vanishing of Hochster's theta pairing. The proof uses Adams operations on both topological $K$-theory and perfect complexes with support.
We define Adams operations on matrix factorizations, and we show these operations enjoy analogues of several key properties of the Adams operations on perfect complexes with support developed by Gillet-Soulé in their paper "Intersection Theory Using Adams Operations". As an application, we give a proof of a conjecture of Dao-Kurano concerning the vanishing of Hochster's theta invariant.
The goal of this article is to explain a precise sense in which Knoerrer periodicity in commutative algebra and Bott periodicity in topological K-theory are compatible phenomena. Along the way, we prove an 8-periodic version of Knoerrer periodicity for real isolated hypersurface singularities, and we construct a homomorphism from the Grothendieck group of the homotopy category of matrix factorizations of a complex (real) polynomial $f$ into the topological K-theory of its Milnor fiber (positive or negative Milnor fiber).
Let $Q$ be a commutative, Noetherian ring and $Z \subseteq \operatorname{Spec}(Q)$ a closed subset. Define $K_0^Z(Q)$ to be the Grothendieck group of those bounded complexes of finitely generated projective $Q$-modules that have homology supported on $Z$. We develop "cyclic" Adams operations on $K_0^Z(Q)$ and we prove these operations satisfy the four axioms used by Gillet and Soulé in their paper "Intersection Theory Using Adams Operations". From this we recover a shorter proof of Serre's Vanishing Conjecture. We also show our cyclic Adams operations agree with the Adams operations defined by Gillet and Soulé in certain cases.
In this paper, we argue for the need to distinguish between task and dialogue initiatives, and present a model for tracking shifts in both types of initiatives in dialogue interactions. Our model predicts the initiative holders in the next dialogue turn based on the current initiative holders and the effect that observed cues have on changing them. Our evaluation across various corpora shows that the use of cues consistently improves the accuracy in the system's prediction of task and dialogue initiative holders by 2-4 and 8-13 percentage points, respectively, thus illustrating the generality of our model.