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Sarah Scherotzke

Publications and source records attributed to Sarah Scherotzke.

30 records · Page 2Linked to original sources
arXiv2015

Derived loop stacks and categorification of orbifold products

The existence of interesting multiplicative cohomology theories for orbifolds was first suggested by string theorists. Orbifold products have been intensely studied by mathematicians for the last fifteen years. In this paper we focus on the virtual orbifold product that was first introduced in Lupercio et al. (2007). We construct a categorification of the virtual orbifold product that leverages the geometry of derived loop stacks. By work of Ben-Zvi, Francis and Nadler, this reveals connections between virtual orbifold products and Drinfeld centers of monoidal categories, thus answering a question of Hinich.

math.AG↗
arXiv2014

The nonequivariant coherent-constructible correspondence and tilting

The coherent-constructible correspondence is a relationship between coherent sheaves on a toric variety X, and constructible sheaves on a real torus T. This was discovered by Bondal, and explored in the equivariant setting by Fang, Liu, Treumann and Zaslow. In this paper we collect partial results towards a proof of the non equivariant coherent-constructible correspondence. Also, we give applications to the construction of tilting complexes in the derived category of toric DM stacks.

math.AG↗
arXiv2013

Desingularizations of quiver Grassmannians via graded quiver varieties

Inspired by recent work of Cerulli-Feigin-Reineke on desingularizations of quiver Grassmannians of representations of Dynkin quivers, we obtain desingularizations in considerably more general situations and in particular for Grassmannians of modules over iterated tilted algebras of Dynkin type. Our desingularization map is constructed from Nakajima's desingularization map for graded quiver varieties.

math.AG↗
arXiv2013

Graded quiver varieties and derived categories

Inspired by recent work of Hernandez-Leclerc and Leclerc-Plamondon we investigate the link between Nakajima's graded affine quiver varieties associated with an acyclic connected quiver Q and the derived category of Q. As Leclerc-Plamondon have shown, the points of these varieties can be interpreted as representations of a category, which we call the (singular) Nakajima category S. We determine the quiver of S and the number of minimal relations between any two given vertices. We construct a delta-functor Phi taking each finite-dimensional representation of S to an object of the derived category of Q. We show that the functor Phi establishes a bijection between the strata of the graded affine quiver varieties and the isomorphism classes of objects in the image of Phi. If the underlying graph of Q is an ADE Dynkin diagram, the image is the whole derived category; otherwise, it is the category of "line bundles over the non commutative curve given by Q". We show that the degeneration order between strata corresponds to Jensen-Su-Zimmermann's degeneration order on objects of the derived category. Moreover, if Q is an ADE Dynkin quiver, the singular category S is weakly Gorenstein of dimension 1 and its derived category of singularities is equivalent to the derived category of Q.

math.RT↗
arXiv2011

The Integral Cluster Category

Integral cluster categories of acyclic quivers have recently been used in the representation-theoretic approach to quantum cluster algebras. We show that over a principal ideal domain, such categories behave much better than one would expect: They can be described as orbit categories, their indecomposable rigid objects do not depend on the ground ring and the mutation operation is transitive.

math.RT↗
arXiv2010

Finite and bounded Auslander-Reiten Components in the Derived Category

We analyze Auslander-Reiten components for the bounded derived category of a finite-dimensional algebra. We classify derived categories whose Auslander-Reiten quiver has either a finite stable component or a stable component with finite Dynkin tree class or a bounded stable component. Their Auslander-Reiten quiver is determined. We also determine components that contain shift periodic complexes.

math.RT↗
arXiv2009

Examples of Auslander-Reiten components in the bounded derived Category

We deduce a necessary condition for Auslander-Reiten components of the bounded derived category of a finite dimensional algebra to have Euclidean tree class by classifying certain types of irreducible maps in the category of complexes. This result shows that there are only finitely many Auslander-Reiten components with Euclidean tree class up to shift. Also the Auslander-Reiten quiver of certain classes of Nakayama are computed directly and it is shown that they are piecewise hereditary. Finally we state a condition for $\Z[A_{\infty}]$-components to appear in the Auslander-Reiten quiver generalizing a result in \cite{W}.

math.RT↗
arXiv2009

Rank Varieties for Hopf Algebras

We construct rank varieties for the Drinfel'd double of the Taft algebra and for U_q(sl2). For the Drinfel'd double when n=2 this uses a result which identifies a family of subalgebras that control projectivity of A-modules whenever A is a Hopf algebra satisfying a certain homological condition. In this case we show that our rank variety is homeomorphic to the cohomological support variety. We also show that Ext^*(M,M) is finitely generated over the cohomology ring of the Drinfel'd double for any finitely-generated module M.

math.RT↗
arXiv2008

Euclidean components for a class of self-injective algebras

We determine the length of composition series of projective modules of G-transitive algebras with an Auslander-Reiten component of Euclidean tree class. Furthermore we show that modules with certain length of composition series are periodic. We apply these results to G-transitive blocks of the Universal enveloping of restricted p-Lie algebras and prove that G-transitive principal blocks only allow components with Euclidean tree class if p=2. Finally we deduce conditions for a smash product of a local basic algebra with a commutative semi-simple group algebra to have components with Euclidean tree class, depending on the components of the Auslander-Reiten quiver of the basic algebra. We also develop some properties of the bilinear form dim Hom_A(-,-) for the representation ring G(A) of any finite-dimensional algebra A.

math.RT↗
arXiv2008

Formulas for primitive Idempotents in Frobenius Algebras and an Application to Decomposition maps

In the first part of this paper we present explicit formulas for primitive idempotents in arbitrary Frobenius algebras using the entries of representing matrices coming from projective indecomposable modules with respect to a certain choice of basis. The proofs use a generalisation of the well known Frobenius-Schur relations for semisimple algebras. The second part of this paper considers $\Oh$-free $\Oh$-algebras of finite $\Oh$-rank over a discrete valuation ring $\Oh$ and their decomposition maps under modular reduction modulo the maximal ideal of $\Oh$, thereby studying the modular representation theory of such algebras. Using the formulas from the first part we derive general criteria for such a decomposition map to be an isomorphism that preserves the classes of simple modules involving explicitly known matrix representations on projective indecomposable modules. Finally we show how this approach could eventually be used to attack a conjecture by Gordon James in the formulation of Meinolf Geck for Iwahori-Hecke-Algebras, provided the necessary matrix representations on projective indecomposable modules could be constructed explicitly.

math.RT↗
arXiv2008

Classification of pointed rank one Hopf algebras

In this paper we classify the finite-dimensional pointed rank one Hopf algebras which are generated as algebras by the first element of the coradical filtration over a field of prime characteristic.

math.RA↗