Frobenius and spherical codomains and neighbourhoods (Q2190801)

Triangulated categories and exact functors between them play an important role in different areas of mathematics, such as algebraic geometry, representation theory, and topology. Often, an exact functor \(F\colon \mathcal A\to \mathcal B\) has both, a left and a right adjoint, \(L,R\colon \mathcal B\to \mathcal A\). For example, this is always the case for Fourier-Mukai functors between bounded derived categories of coherent sheaves on smooth projective varieties. If \(F\) is an equivalence, then these two adjoints coincide (up to isomorphism), as both are given by the inverse of \(F\). However, already in the case that \(F\) is a fully faithful functor with both adjoints (such a functor is called an \textit{exceptional} functor), it often happens that \(L\not\cong R\). If there is an isomorphism \(L\cong R\), the exceptional functor is called \textit{Frobenius}. In the paper under review, the authors study the \textit{Frobenius codomain} of an exceptional functor \(F\colon \mathcal A\to \mathcal B\). This is the maximal full triangulated subcategory \(\operatorname{Frb}(F)\subset \mathcal B\) such that \(F\colon \mathcal A\to \operatorname{Frb}(F)\) is a Frobenius functor; in other words, such that \(L_{\mid\operatorname{Frb}(F)}\cong R_{\mid\operatorname{Frb}(F)}\). One of the main results is the existence (via concrete construction) of this maximal subcategory; see Theorems A and 3.2.3. Another main result is the decomposition \[ \operatorname{Frb}(F)=\operatorname{im}(F)\oplus (\ker R \cap \ker L)\,. \] see Theorem 3.2.5. In addition, the Frobenius codomain (and a variant, namely the \textit{Frobenius neighbourhood} of an object in the image of \(F\)) is computed in basic examples. In the second part of the paper, a similar study as in the first part of the paper is carried out, with exceptional functors being replaced by \textit{sphere-like} functors and Frobenius functors being replaced by \textit{spherical functors}. Spherical functors are important since they induce twist autoequivalences of their target categories. In fact, every autoequivalence can be realised as the twist by an appropriate spherical functor, as proved by Segal [Int. Math. Res. Not. 2018, No. 10, 3137-3154 (2018, Zbl 1415.18005)]. The authors study the \textit{spherical codomain} of a spherelike functor \(F\colon \mathcal A\to \mathcal B\). This is the maximal triangulated subcategory \(\operatorname{Sph}(F)\subset \mathcal B\) such that \(F\colon \mathcal A\to \operatorname{Sph}(F)\) is a spherical functor; in other words, such that \(CL[1]_{\mid\operatorname{Sph}(F)}\cong R_{\mid\operatorname{Sph}(F)}\) where \(C\colon \mathcal A\to \mathcal A\) is the cotwist autoequivalence of \(F\). Again, there is an existence result by concrete construction, a decomposition result, and computations of examples. In summary, the authors carry out a thorough investigation of a natural and interesting question, namely: how far has the codomain of an exceptional (sphere-like) functor to be restricted for the functor to become Frobenius (spherical)?