Stability in the homology of Deligne–Mumford compactifications

Using the theory of ${\mathbf {FS}} {^\mathrm {op}}$ modules, we study the asymptotic behavior of the homology of ${\overline {\mathcal {M}}_{g,n}}$ , the Deligne–Mumford compactification of the moduli space of curves, for $n\gg 0$ . An ${\mathbf {FS}} {^\mathrm {op}}$ module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via copies that glue on marked projective lines, we give the homology of ${\overline {\mathcal {M}}_{g,n}}$ the structure of an ${\mathbf {FS}} {^\mathrm {op}}$ module and bound its degree of generation. As a consequence, we prove that the generating function $\sum _{n} \dim (H_i({\overline {\mathcal {M}}_{g,n}})) t^n$ is rational, and its denominator has roots in the set $\{1, 1/2, \ldots, 1/p(g,i)\},$ where $p(g,i)$ is a polynomial of order $O(g^2 i^2)$ . We also obtain restrictions on the decomposition of the homology of ${\overline {\mathcal {M}}_{g,n}}$ into irreducible $\mathbf {S}_n$ representations.

Recollements, comma categories and morphic enhancements

For each recollement of triangulated categories, there is an epivalence between the middle category and the comma category associated with a triangle functor from the category on the right to the category on the left. For a morphic enhancement of a triangulated category $\mathcal {T}$ , there are three explicit ideals of the enhancing category, whose corresponding factor categories are all equivalent to the module category over $\mathcal {T}$ . Examples related to inflation categories and weighted projective lines are discussed.

Computational Classification of Tubular Algebras

The effective method (based on Theorem 5.3) of classifying tubular algebras by the Cartan matrices of tilting sheaves over weighted projective lines with all indecomposable direct summands in some finite “fundamental domain” , by the reduction to the two elementary problems of discrete mathematics having algorithmic solutions is presented in details (see Problem A and B). The software package CART_TUB being an implementation of this method yields the precise classification of all up to isomorphism tubular algebras of a fixed tubular type p, by creating the complete lists of their Cartan matrices, and furnish their tilting realizations. In particular, the number of isomorphism classes of tubular algebras of the type p is determined (Theorem 2.3).