Rational points and derived equivalence

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over $\mathbb {Q}$ and $\mathbb {F}_q(t)$ , and conclude with a pair of hyperkähler 4-folds over $\mathbb {Q}$ . The latter is independently interesting as a new example of a transcendental Brauer–Manin obstruction to the Hasse principle. The source code for the various computations is supplied as supplementary material with the online version of this article.

On the Abuaf-Ueda Flop via Non-Commutative Crepant Resolutions

The Abuaf-Ueda flop is a 7-dimensional flop related to G₂ homogeneous spaces. The derived equivalence for this flop was first proved by Ueda using mutations of semi-orthogonal decompositions. In this article, we give an alternative proof for the derived equivalence using tilting bundles. Our proof also shows the existence of a non-commutative crepant resolution of the singularity appearing in the flopping contraction. We also give some results on moduli spaces of finite-length modules over this non-commutative crepant resolution.

Derived Equivalences for the Flops of Type C2 and ${A}_{4}^{G}$ via Mutation of Semiorthogonal Decomposition

We give a new proof of the derived equivalence of a pair of varieties connected by the flop of type C2 in the list of Kanemitsu (2018), which is originally due to Segal (Bull. Lond. Math. Soc., 48 (3) 533–538, 2016). We also prove the derived equivalence of a pair of varieties connected by the flop of type ${A}_{4}^{G}$ A 4 G in the same list. The latter proof follows that of the derived equivalence of Calabi–Yau 3-folds in Grassmannians Gr(2,5) and Gr(3,5) by Kapustka and Rampazzo (Commun. Num. Theor. Phys., 13 (4) 725–761 2019) closely.

Equivalences between Calabi–Yau manifolds and roofs of projective bundles

It is conjectured that many birational transformations, called K-inequalities, have a categorical counterpart in terms of an embedding of derived categories. In the special case of simple K-equivalence (or more generally K-equivalence), a derived equivalence is expected: we propose a method to prove derived equivalence for a wide class of such cases. This method is related to the construction of roofs of projective bundles introduced by Kanemitsu. Such roofs can be related to candidate pairs of derived equivalent, L-equivalent and non isomorphic Calabi–Yau varieties, we prove such claims in some examples of this construction. In the same framework, we show that a similar approach applies to prove derived equivalence of pairs of Calabi–Yau fibrations, we provide some working examples and we relate them to gauged linear sigma model phase transitions.

On the finiteness of the derived equivalence classes of some stable endomorphism rings

We prove that the stable endomorphism rings of rigid objects in a suitable Frobenius category have only finitely many basic algebras in their derived equivalence class and that these are precisely the stable endomorphism rings of objects obtained by iterated mutation. The main application is to the Homological Minimal Model Programme. For a 3-fold flopping contraction $$f :X \rightarrow {\mathrm{Spec}\;}\,R$$ f : X → Spec R , where X has only Gorenstein terminal singularities, there is an associated finite dimensional algebra $$A_{{\text {con}}}$$ A con known as the contraction algebra. As a corollary of our main result, there are only finitely many basic algebras in the derived equivalence class of $$A_{\text {con}}$$ A con and these are precisely the contraction algebras of maps obtained by a sequence of iterated flops from f. This provides evidence towards a key conjecture in the area.

A Representation Theoretic Study of Non-Commutative Symmetric Algebras

We study Van den Bergh's non-commutative symmetric algebra 𝕊nc(M) (over division rings) via Minamoto's theory of Fano algebras. In particular, we show that 𝕊nc(M) is coherent, and its proj category ℙnc(M) is derived equivalent to the corresponding bimodule species. This generalizes the main theorem of [8], which in turn is a generalization of Beilinson's derived equivalence. As corollaries, we show that ℙnc(M) is hereditary and there is a structure theorem for sheaves on ℙnc(M) analogous to that for ℙ1.

DIFFERENTIAL GRADED ENDOMORPHISM ALGEBRAS, COHOMOLOGY RINGS AND DERIVED EQUIVALENCES

In this paper, we will consider derived equivalences for differential graded endomorphism algebras by Keller's approaches. First, we construct derived equivalences of differential graded algebras which are endomorphism algebras of the objects from a triangle in the homotopy category of differential graded algebras. We also obtain derived equivalences of differential graded endomorphism algebras from a standard derived equivalence of finite dimensional algebras. Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. Then we give an affirmative answer to a problem of Dugas (A construction of derived equivalent pairs of symmetric algebras, Proc. Amer. Math. Soc. 143 (2015), 2281–2300) in some special case.

Külshammer ideals of algebras of quaternion type

For a symmetric algebra [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text] Külshammer constructed a descending sequence of ideals of the center of [Formula: see text]. If [Formula: see text] is perfect, this sequence was shown to be an invariant under derived equivalence and for algebraically closed [Formula: see text] the dimensions of their image in the stable center were shown to be invariant under stable equivalence of Morita type. Erdmann classified algebras of tame representation type which may be blocks of group algebras, and Holm classified Erdmann’s list up to derived equivalence. In both classifications, certain parameters occur in the classification, and it was unclear if different parameters lead to different algebras. Erdmann’s algebras fall into three classes, namely of dihedral, semidihedral and of quaternion type. In previous joint work with Holm, we used Külshammer ideals to distinguish classes with respect to these parameters in case of algebras of dihedral and semidihedral type. In the present paper, we determine the Külshammer ideals for algebras of quaternion type and distinguish again algebras with respect to certain parameters.