Abstract We study the relationship between higher Du Bois singularities and K -regularity, a notion that measures the double struck upper A Superscript 1 A 1 $\mathbb A^1$ -invariance of the algebraic K -groups. Building on this relationship, we establish a strengthened form of Vorst’s conjecture for local complete intersections in characteristic zero. Our work also provides tools to construct new examples that illustrate various phenomena in the study of K -regularity. The main inputs for our results are vanishing theorems for the Du Bois complexes.
Abstract We study the basic locus of Shimura varieties associated to the group of spinor similitudes of a quadratic space over double struck upper Q Q ${{\mathbb{Q}}}$ with level structure given by the stabilizer of a vertex lattice. We give a description of the underlying reduced scheme of the associated Rapoport–Zink space, generalizing results of Howard–Pappas and Oki, in the case of self-dual, and almost self-dual, level structure.
Abstract We study the classification of double struck upper Z Z $\mathbb{Z}$ -DGAs (differential graded algebras) with polynomial homology double struck upper F Subscript p Baseline left bracket x right bracket F p [ x ] $\mathbb{F}_p[x]$ with StartAbsoluteValue x EndAbsoluteValue greater than 0 | x | > 0 $\lvert x \rvert \gt0$ , motivated by computations in algebraic K -theory. This classification problem was left open in work of Dwyer, Greenlees, and Iyengar. We prove that there are infinitely many such DGAs for even StartAbsoluteValue x EndAbsoluteValue | x | $\lvert x\rvert$ and that for StartAbsoluteValue x EndAbsoluteValue greater than or equals 2 p minus 2 | x | ≥ 2 p − 2 $\lvert x \rvert \geq 2p-2$ any such DGA is formal as a ring spectrum. Through this, we obtain examples of triangulated categories with infinitely many DG enhancements and a classification of prime DG division rings. Combining our results with earlier work of the second author and Tamme, we obtain new (relative) algebraic K -theory computations for rings such as the mixed characteristic coordinate axes double struck upper Z left bracket x right bracket divided by p x Z [ x ] / p x $\mathbb{Z}[x]/px$ and the group ring double struck upper Z left bracket upper C Subscript p Sub Superscript n Subscript Baseline right bracket Z [ C p n ] $\mathbb{Z}[C_{p^n}]$ .
Abstract We prove a conjecture of Manin and Orlov: the moduli space script upper M overbar Subscript 0 comma n M ¯ 0 , n $\overline{\mathcal{M}}_{0,n}$ of stable rational curves with n marked points admits a full exceptional collection that is equivariant under the action of the symmetric group upper S Subscript n S n $S_n$ permuting the marked points. In particular, its upper K 0 K 0 $K_0$ -group with integral coefficients is a permutation upper S Subscript n S n $S_n$ -lattice.
Abstract We define relative Donaldson–Thomas invariants of local log Calabi–Yau 4-folds, under the assumption that the derived Hilbert scheme on the anti-canonical divisor is a derived critical locus. We prove a degeneration formula and apply it to compute zero-dimensional invariants on double struck upper C Superscript 4 C 4 $\mathbb{C}^4$ and on any local curve. This verifies a conjecture proposed by the first author and Kool.
