Derived equivalent varieties with differing integral Mukai-Hodge structures?

For a smooth projective complex variety X of dimension n, let Hi(X) denote its integral Hodge structure of weight i. Define ~H0(X)=⨁H2i(X)⊗Z(i) and ~H1(X)=⨁H2i+1⊗Z(i), respectively. It is known that any derived equivalence Φ:Db(X)→Db(Y) induces isomorphisms of rational Hodge structures ~H0Q(X)≅~H0Q(Y) and ~H1Q(X)≅~H1Q(Y), however these isomorphisms are defined by characteristic classes whose coefficients aren’t necessarily integral. … Read more

Short proof of the classification of representation-finite symmetric algebras up to stable equivalence

Assume K is an algebraically closed field and A a finite dimensional K-algebra. Assume additionally that A is symmetric and representation-finite. Then one has the following classification of such algebras up to stable equivalence( source without proof section 3.14 in: Andrzej Skowroński – Selfinjective algebras: finite and tame type): A is stable equivalent to a … Read more

Does derived equivalence imply dg Morita equivalence between DG algebras over field with char=0=0?

Let A, B be two DG algebras and D(A), D(B) be derived categories of DG-modules of A, B, respectively. We call A and B are dg Morita equivalent if there is an A–B bimodule T with the following properties Hi(A)≅HomD(B)(T,T[i]) for any i∈Z; T defines a compact object in D(B); For an object N∈D(B),HomD(B)(T,N[i])=0 for … Read more

Describing derived category of coherent sheaves concretely

The following question came up when I was asked to do prove equivalence of some derived categories ‘by hand'(I find it hard to define this..). How does one describe Db(X)? It seems one good idea is to use semi-orthogonal decomposition, but this seems hard even in some easy cases. For example, if X=C2, Coh(X)=Mod(k[x,y]), can … Read more

Skyscraper sheaf on a stack associated to a singular surface

Suppose $X$ is a normal projective surface with a du Val singularity. In this case, we know a crepant resolution $Y$ exists, and results of Kawamata (, Corollary 3.5) show that there is an equivalence $$ \phi: D^b(\text{Coh}(\mathcal X)) \simeq D^b(\text{Coh}(Y)) $$ where $\mathcal X$ is the smooth DM stack associated to $X$. Let $k(p)$ … Read more

Determining whether a morphism is the induced morphism?

Let F:A→B be a left exact functor between Grothendieck abelian categories. Given a morphism f:A→B in A and a morphism g:RF(A)→RF(B) in the derived category of B, is there a “standard” technique to determine whether g=RF(f)? Take “standard” to mean what you will. Answer AttributionSource : Link , Question Author : Avi Steiner , Answer … Read more

Derived categories of coherent sheaves and degenerations of abelian varieties

By the work of Burban-Drozd (, we know what happens to the derived category of coherent sheaves when an elliptic curve degenerates into a nodal curve or a cycle of projective lines, namely let $C$ be such a curve with $n$ double points, then $D^b\mathit{Coh}(C)$ is generated by $\mathcal{O}_{p_1},\dots,\mathcal{O}_{p_n},\mathcal{O}_C$, where $p_1,\dots,p_n\in C$ are smooth points. … Read more

Generators of unbounded derived categories of (quasi-)coherent sheaves

An object T in a triangulated category D is called a generator if T⊥=0, which means that for any nonzero X in D, there are i∈Z and a nonzero morphism T[−i]→X. For example, it is well known that T=O⊕O(1)⊕⋯⊕O(n) is a generator of Db(Pn), the bounded derived category of coherent sheaves on Pn. However, it … Read more

Automorphisms of graded rings and the induced action on the projective scheme

I am trying to understand the proof of proposition 4.17 in “Fourier-Mukai transforms in algebraic geometry” by D. Huybrechts about the structure of the group of autoequivalences of $D^b(X)$ in the case of ample (anti-)canonical bundle. So here is a brief outline of the set-up for my question: Suppose our autoequivalence $F$ maps $\mathcal{O}_X$ to … Read more

Representing j∗OUj_*\mathcal{O}_U as filtered colimit of perfect complexes

Let X be a quasi-compact and quasi-separated scheme, and U⊆X be a quasi-compact open subscheme. Then we can consider Rj∗OU the (derived) pushforward of the structure sheaf of U. This is a coconnective sheaf of rings (in fact it is locally of Tor amplitude in (−∞,0], since locally it can be represented as a finite … Read more