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We classify central extensions of a reductive group $G$ by $\mathcal{K}_3$ and $B\mathcal{K}_3$, the sheaf of third Quillen $K$-theory groups and its classifying stack. These turn out to be parametrized by the group of Weyl-invariant quadratic forms on the cocharacter lattice valued in $k^\times$ and the group of integral Weyl-invariant cubic forms on the cocharacter lattice respectively., 25 pages

We continue the program started in Manon (2010) [M1] to understand the combinatorial commutative algebra of the projective coordinate rings of the moduli stack M C , p → ( SL 2 ( C ) ) of quasi-parabolic SL 2 ( C ) principal bundles on a generic marked projective curve. We find general bounds on the degrees of polynomials needed to present these algebras by studying their toric degenerations. In particular, we show that the square of any effective line bundle on this moduli stack yields a Koszul projective coordinate ring. This leads us to formalize the properties of the polytopes used in proving our results by constructing a category of polytopes with term orders. We show that many of results on the projective coordinate rings of M C , p → ( SL 2 ( C ) ) follow from closure properties of this category with respect to fiber products.

In this paper we study ideals generated by quite general sets of 2-minors of an $m \times n$-matrix of indeterminates. The sets of 2-minors are defined by collections of cells and include 2-sided ladders. For convex collections of cells it is shown that the attached ideal of 2-minors is a Cohen--Macaulay prime ideal. Primality is also shown for collections of cells whose connected components are row or column convex. Finally the class group of the ring attached to a stack polyomino and its canonical class is computed, and a classification of the Gorenstein stack polyominoes is given., 29 pages, 32 figures

We give a proof of a conjecture on the connected components of moduli spaces of Kisin module, which is valid also in the case p = 2 .

Let A denote the ring of differential operators on the affine line with its two usual generators t and d/dt given degrees +1 and -1 respectively. Let X be the stack having coarse moduli space the affine line Spec k[z] and isotropy groups Z/2 at each integer point. Then the category of graded A-modules is equivalent to the category of quasi-coherent sheaves on X. Version 2: corrected typos and deleted appendix at referee's suggestion., Comment: Keywords:Weyl algebra, graded module category, category equivalence, graded principal ideal domain, quotient stack

We show boundedness for PT-semistable objects of any Chern classes on a smooth projective three-fold $X$. Then we show that the stack of objects in the heart $\langle \Coh_{\leq 1}(X), \Coh_{\geq 2}(X)[1] \rangle$ satisfies a version of the valuative criterion for completeness. In the remainder of the paper, we give a series of results on how to compute cohomology with respect to this heart., 23 pages

In this article we study the modular properties of a family of cyclic coverings of P^1 of degree N, in all odd characteristics. We compute the moduli space of the corresponding algebraic stack over Z[1/2], as well as the Picard groups over algebraically closed fields. We put special emphasis on the study of the fibre of the stack at a prime of wild ramification; in particular we show that the moduli space has good reduction at such a prime., Comment: 23 pages; 2 figures; 2nd revised version

Continuing M. Domokos and H. Lenzing [ J. Algebra 228 (2000), 738–762], we investigate the relative invariants and moduli spaces introduced in A. D. King [ Quart. J. Math. Oxford Ser. (2) 45 (1994), 515–530] which are related to a separating family of stable tubes in the module category of a concealed-canonical algebra. The class of concealed-canonical algebras consists of representation-infinite algebras of tame or wild representation type and contains in particular the representation theory of extended Dynkin quivers. We present a uniform characteristic-free approach avoiding case-by-case analysis. We introduce the notion of admissible weight for such algebras; these are the weights (i.e., elements of the dual of the Grothendieck group of the module category) attached to the infinite moduli spaces for families of modules from the separating subcategory. By using an explicit description of the semigroup of admissible weights the corresponding notion of semistability is analyzed and is related to the formation of perpendicular categories. We construct relative invariants belonging to admissible weights and prove that the constructed system is complete in the case of canonical algebras. It turns out that for any concealed-canonical algebra and for any admissible weight all the corresponding moduli spaces are isomorphic to some projective space. Moreover, any such moduli space can be naturally identified with the moduli space belonging to a special weight, called the rank. As a consequence we show that in the case of a tame concealed algebra, any infinite moduli space for families of modules is a projective space, and all fields of rational invariants on irreducible components of representation spaces are purely transcendental.