We study cluster categories arising from marked surfaces (with punctures and nonempty boundaries). By constructing skewed-gentle algebras, we show that there is a bijection between tagged curves and string objects. Applications include interpreting
dimensions of Ext1 as intersection numbers of tagged curves and Auslander-Reiten translation as tagged rotation. An important consequence is that the cluster(-tilting) exchange graphs of such cluster categories are connected.
We study endomorphism algebras of 2-term silting complexes in derived categories of hereditary finite dimensional algebras, or more generally of Ext-finite hereditary abelian categories. Module categories of such endomorphism algebras are known to occur as hearts of certain bounded t-structures in such derived categories. We show that the algebras occurring are exactly the algebras of small homological dimension, which are algebras characterized by the property that each indecomposable module either has injective dimension at most one, or it has projective dimension at most one.
We introduce the notions of a D-standard abelian category and a K-standard additive category. We prove that for a finite dimensional algebra A, its module category is D-standard if and only if any derived autoequivalence on Ais standard, that is, isomorphic to the derived tensor functor by a two-sided tilting complex. We prove that if the subcategory of projective A-modules is K-standard, then the module category is D-standard. We provide new examples of D-standard module categories.
We show that if X is a cocompact G-CW-complex such that each isotropy
subgroup Gσ is L(2)-good over an arbitrary commutative ring k, then X satisfies some fixed-point
formula which is an L(2)-analogue of Brown’s formula in 1982. Using this result we present a fixed
point formula for a cocompact proper G-CW-complex which relates the equivariant L(2)-Euler
characteristic of a fixed point CW-complex Xs and the Euler characteristic of X/G. As corollaries,
we prove Atiyah’s theorem in 1976, Akita’s formula in 1999 and a result of Chatterji-Mislin in
2009. We also show that if X is a free G-CW-complex such that C∗(X) is chain homotopy
equivalent to a chain complex of finitely generated projective Zπ1(X)-modules of finite length and
X satisfies some fixed-point formula over Q or C which is an L(2)-analogue of Brown’s formula, then
χ(X/G)=χ(2)(X). As an application, we prove that the weak Bass conjecture holds for any finitely
presented group G satisfying the following condition: for any finitely dominated CW-complex Y
with π1(Y )=G, Y satisfies some fixed-point formula over Q or C which is an L(2)-analogue of
We prove that the cover ideals of all unimodular hypergraphs have the nonincreasing
depth function property. Furthermore, we show that the index of depth stability of
these ideals is bounded by the number of variables.
We develop a local cohomology theory for FI^m-modules, and show that it in many ways mimics the classical theory for multi-graded modules over a polynomial ring. In particular, we define an invariant of FI^m-modules using this local cohomology theory which closely resembles an invariant of multi-graded modules over Cox rings defined by Maclagan and Smith. It is then shown that this invariant behaves almost identically to the invariant of Maclagan and Smith.
We give bounds for various homological invariants (including Castelnuovo-Mumford regularity, degrees of local cohomology, and injective dimension) of finitely generated VI-modules in the non-describing characteristic case. It turns out that the formulas of these bounds for VI-modules are the same as the formulas of corresponding bounds for FI-modules.
Using the Nakayama functor, we construct an equivalence from a Serre quotient category of a category of finitely generated modules to a category of finite-dimensional modules. We then apply this result to the categories FI_G and VI_q, and answer positively an open question of Nagpal on representation stability theory.
In this paper we study representation theory of the category FI^m introduced by Gadish which is a product of copies of the category FI, and show that quite a few interesting representational and homological properties of FI can be generalized to FI^m in a natural way. In particular, we prove the representation stability property of finitely generated FI^m -modules over fields of characteristic 0.