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In this article we construct a categorical resolution of singularities of an excellent reduced curve $X$, introducing a certain sheaf of orders on $X$. This categorical resolution is shown to be a recollement of the derived category of coherent sheaves on the normalization of $X$ and the derived category of finite length modules over a certain artinian quasi-hereditary ring $Q$ depending purely on the local singularity types of $X$. Using this technique, we prove several statements on the Rouquier dimension of the derived category of coherent sheaves on $X$. Moreover, in the case $X$ is rational and projective we construct a finite dimensional quasi-hereditary algebra $Lambda$ such that the triangulated category of perfect complexes on $X$ embeds into $D^b(Lambda-mathsf{mod})$ as a full subcategory.
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In this paper, we develop a geometric approach to study derived tame finite dimensional associative algebras, based on the theory of non-commutative nodal curves.
We discuss tilting modules of affine quasi-hereditary algebras. We present an existence theorem of indecomposable tilting modules when the algebra has a large center and use it to deduce a criterion for an exact functor between two affine highest weight categories to give an equivalence. As an application, we prove that the Arakawa-Suzuki functor [Arakawa-Suzuki, J. of Alg. 209 (1998)] gives a fully faithful embedding of a block of the deformed BGG category of $mathfrak{gl}_{m}$ into the module category of a suitable completion of degenerate affine Hecke algebra of $GL_{n}$.
We answer a question of Takesaki by showing that the following can be derived from the thesis of N-T Shen: If A and B are sigma-unital hereditary C*-subalgebras of C such that ||p - q|| < 1, where p and q are the corresponding open projections, then A and B are isomorphic. We give some further elaborations and counterexamples with regard to the sigma-unitality hypothesis. We produce a natural one-to-one correspondence between complete order isomorphisms of C*-algebras and invertible left multipliers of imprimitivity bimodules. A corollary of the above two results is that any complete order isomorphism between sigma-unital C*-algebras is the composite of an isomorphism with an inner complete order isomorphism. We give a separable counterexample to a question of Akemann and Pedersen; namely, the space of quasi-multipliers is not linearly generated by left and right multipliers. But we show that the space of quasi-multipliers is multiplicatively generated by left and right multipliers in the sigma-unital case. In particular every positive quasi-multiplier is of the form T*T for T a left multiplier. We show that a Lie theory consequence of the negative result just stated is that the map sending T to T*T need not be open, even for very nice C*-algebras. We show that surjective maps between sigma-unital C*-algebras induce surjective maps on left, right, and quasi-multipliers. (The more significant similar result for multipliers is Pedersens Non-commutative Tietze extension theorem.) We elaborate the relations of the above results with continuous fields of Hilbert spaces and in so doing answer a question of Dixmier and Douady (yes for separable fields, no in general). We discuss the relationship of our results to the theory of perturbations of C*-algebras.
We provide a transformation formula of non-commutative Donaldson-Thomas invariants under a composition of mutations. Consequently, we get a description of a composition of cluster transformations in terms of quiver Grassmannians. As an application, we give an alternative proof of Fomin-Zelevinskys conjectures on $F$-polynomials and $g$-vectors.
There are no known failures of Bounded Negativity in characteristic 0. In the light of recent work showing the Bounded Negativity Conjecture fails in positive characteristics for rational surfaces, we propose new characteristic free conjectures as a replacement. We also develop bounds on numerical characteristics of curves constraining their negativity. For example, we show that the $H$-constant of a rational curve $C$ with at most $9$ singular points satisfies $H(C)>-2$ regardless of the characteristic.
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