In the paper under review, the author describes a method to construct a semiorthogonal decomposition of the equivariant derived category of a quasi-projective \(k\)-scheme \(X\) under the action of an algebraic group \(G\), starting from a semiorthogonal decomposition on the standard derived category.
Suppose that \(G\) acts on \(X\) via an action \(a:G\times X\longrightarrow X\), and let \(p_{2}:G\times X\longrightarrow X\) be the projection. An equivariant sheaf on \(X\) is a pair \((F,\theta)\) where \(F\) is a sheaf on \(X\) and \(\theta:p_{2}^{*}F\longrightarrow F\) is a morphism verifying some cocycle condition depending on \(a\). We write \(D(X)\) (resp. \(D^{G}(X)\)) for the derived category of quasi-coherent sheaves (resp. equivariant quasi-coherent sheaves), \(D^{b}(X)\) (resp. \(D^{b}(\text{coh}^{G}(X))\)) for the bounded derived category of coherent sheaves (resp. equivariant coherent sheaves) and \(D^{\text{perf}}(X)\) (resp. \(D^{\text{perf},G}(X)\)) for the category of perfect complexes in \(D(X)\) (resp. in \(D^{G}(X)\)).
In section 2, the author provides a description of \(D^{G}(X)\) using the language of cosimplicial categories and comonads. A comonad on a category \(\mathcal{C}\) is a triple \(\mathrm{T}=(T,\epsilon,\delta)\) given by a functor \(T:\mathcal{C}\longrightarrow\mathcal{C}\) and two natural transformation \(\epsilon:T\longrightarrow id_{\mathcal{C}}\) and \(\delta:T\longrightarrow T^{2}\) satisfying some commutativity condition. A comodule over a comonad \(\mathrm{T}\) on \(\mathcal{C}\) is a pair \((F,h)\) where \(F\) is an object of \(\mathcal{C}\), and \(h:F\longrightarrow TF\) is a morphism verifying some commutativity condition. The comodules over the comonad \(\mathrm{T}\) form a category \(\mathcal{C}_{\mathrm{T}}\). If \(\mathcal{C}\) is abelian and \(\mathrm{T}\) is a comonad on it, then \(\mathcal{C}_{\mathrm{T}}\) is abelian. In general, if \(\mathcal{C}\) is triangulated, we do not have \(\mathcal{C}_{\mathrm{T}}\) tringulated.
Now, using cosimplicial categories and results of A. D. Elagin contained in [Sb. Math. 202, No. 4, 495–526 (2011; Zbl 1234.18006)], the author shows that if \(p:X\longrightarrow S\) is a morphism such that \(\mathcal{O}_{S}\longrightarrow R_{p*}\mathcal{O}_{X}\) is a split embedding, then the canonical functor \(\Phi:D(S)\longrightarrow D(X)_{\mathrm{T}_{p}}\) sending a complex \(H\) to the pair \((p^{*}H,h)\) (where \(h\) is the canonical adjunction morphism \(p^{*}H\longrightarrow p^{*}p_{*}p^{*}H\)), is an equivalence. The same is true if we replace \(D(X)\) with \(D^{b}(X)\) or \(D^{\text{perf}}(X)\). If \(G\) is linearily reductive and \(S=X/\!/G\), the author gets finally equivalences \(D^{G}(X)\simeq D(X)_{\mathrm{T}_{p}}\), \(D^{b}(\text{coh}^{G}(X))\simeq D^{b}(X)_{\mathrm{T}_{p}}\) and \(D^{\text{perf},G}(X)\simeq D^{\text{perf}}(X)_{\mathrm{T}_{p}}\), where \(\mathrm{T}_{p}\) is the comonad \((p^{*}p_{*},\epsilon,\delta)\) (and \(\epsilon\) and \(\delta\) are the canonical adjunction morphisms).
In section 3, the author shows that if \(\mathrm{T}\) is a comonad on a tringulated category \(\mathcal{C}\) such that \(\mathcal{C}_{\mathrm{T}}\) has a triangulated structure, then a semiorthogonal decomposition \(\langle A_{1},\dots,A_{n}\rangle\) of \(\mathcal{C}\) induces a semiorthogonal decomposition on \(\mathcal{C}_{\mathrm{T}}\) whenever \(\mathrm{T}\) is upper triangular with respect to \(\langle A_{1},\dots,A_{n}\rangle\), i. e. \(\mathrm{T}A_{k}\subseteq\langle A_{1},\dots,A_{k}\rangle\) for every \(k\). As an application of this, if \(p:X\longrightarrow S\) is a flat morphism and \(\mathcal{O}_{S}\longrightarrow Rp_{*}\mathcal{O}_{X}\) is a split embedding, then any semiorthogonal decomposition \(\langle A_{1},\dots,A_{n}\rangle\) of \(D(X)\) such that the functor \(\mathrm{T}_{p}=p^{*}p_{*}\) is upper triangular induces a semiorthogonal decomposition \(\langle B_{1},\dots,B_{n}\rangle\) of \(D(S)\), where \(B_{k}\) is given by all complexes \(H\) such that \(p^{*}H\in A_{k}\).
To prove similar statement for \(D^{b}(X)\) and \(D^{\text{perf}}(X)\), one needs to extend a semiorthogonal decomposition on these categories to semiorthogonal decompositions of \(D(X)\), which is possible by A. Kuznetsov, [Compos. Math. 147, No. 3, 852–876 (2011; Zbl 1218.18009)]. In section 5, the case of \(p:X\longrightarrow X/\!/S\) is studied, and the author shows that \(\mathrm{T}_{p}\) is upper semitriangular with respect to the semiorthogonal decompostion \(\langle A_{1},\dots,A_{n}\rangle\) if \(A_{i}\) is preserved by \(G\), i. e. \(p_{2}^{*}A_{i}=a^{*}A_{i}\) for every \(i\).
The conclusion is provided in Theorems 6.1, 6.2 and 6.3, saying that if \(\langle A_{i},\dots,A_{n}\rangle\) is a semiorthogonal decomposition of \(D(X)\) (or \(D^{\text{perf}}(X)\), \(D^{b}(X)\)) whose factors are preserved by \(G\), then it induces a semiorthogonal decomposition \(\langle B_{1},\dots,B_{n}\rangle\) on \(D^{G}(X)\) (or \(D^{\text{perf},G}(X)\), \(D^{b}(\text{coh}^{G}(X))\)), where the objects of \(B_{k}\) are precisely those which are in \(A_{i}\) once the equivariant structure is forgotten.
To conclude, the author applies his construction to produce explicit semiorthogonal decompositions for \(D^{\text{perf},G}(X)\) when \(X\) is the projective bundle over a quasi-projective \(k-\)scheme, and when \(X\) is the blow-up of a quasi-projective \(k-\)scheme along a smooth subscheme, in both cases starting from the semiorthogonal decompositions constructed by D. O. Orlov [Russ. Acad. Sci., Izv., Math. 41, No. 1, 133–141 (1993; Zbl 0798.14007)].