Let \(k\) be an algebraically closed field of characteristic zero. Orlov’s representation theorem states that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties over \(k\) is a Fourier-Mukai functor. It is unknown if the hypothesis on fully faithfulness is necessary for this to be true. V. A. Lunts and D. O. Orlov [J. Am. Math. Soc. 23, No. 3, 853–908 (2010; Zbl 1197.14014)] extended the theorem to quasi-coherent sheaves: Let \(X/k\) be a projectve scheme s.t. \(\mathcal O_X\) has no zero-dimensional torsion, and let \(Y\) be a quasi-compact separated scheme. Then every fully faithful exact functor \(\Psi:\text{Perf}(X)\rightarrow D(\text{Qcoh}(Y))\) is isomorphic to the restriction of a Fourier-Mukai functor associated to an object in \(D(\text{Qcoh}(X\times Y))\).
A main result of this article is that the extended result above is false if the condition that \(\Psi\) is fully faithful is dropped. This is proved using scalar extensions of derived categories which are treated in the main body of the article.
If \(\mathfrak a\) is a \(k\)-linear category and \(B\) is a \(k\)-algebra, \(\mathfrak a_B\) denotes the category of \(B\)-objects in \(\mathfrak a\), i.e. pairs \((M,\rho)\) with \(M\in\text{Ob}(\mathfrak a)\), \(\rho:B\rightarrow\mathfrak a(M,M)\) a \(k\)-algebra homomorphism.
The first author studies the forgetful functor \(F:D^b(\mathcal C_B)\rightarrow D^b(\mathcal C)_B\) for a field extension \(B/k=L/k\). She proved a surjectivity result for \(\text{trdeg}L/k<2\).
The first result in the article is the following which is proved using \(A_\infty\)-techniques:
{Proposition B} Assume that \(\mathcal C\) is a Grothendieck category. If \(B/k\) has Hochschild dimension \(\leq 2\), then \(F\) is essentially surjective. If \(B/k\) has Hochschild dimension \(\leq 1\), \(F\) is (in addition) full. If \(B/k\) has Hochschild dimension \(0\), \(F\) is an equivalence of categories.
The next result proves that one cannot hope to substantially improve proposition B:
{Theorem C} Let \(X/k\) be a smooth connected projective variety which is not a point, a projective line or an elliptic curve. Then there exists a finitely generated field extension \(L/k\) of transcendence degree 3 together with an object \(Z\in D^b(\text{Qcoh(X))}_L\) which is not in the essential image of \(F\).
To prove this result, the authors prove a similar result for representations of wild quivers. Theorem C excludes the case where \(X\) is a curve of genus \(<1\). This is solved by introducing the essential dimension, the minimal number of parameters required to define any family of indecomposable objects. It follows from this that if \(X\) is a curve of genus \(<1\) and \(\mathbb C=\text{Qcoh}(X)\), if \(L/k\) is a field extension, then the essential image of \(F\) contains all objects in \(D^b(\text{Qcoh}(X))_L\) whose objects is in \(\text{coh}(X_L)\subset\text{Qcoh}(X_L)\simeq\text{Qcoh}(X)_L\).
A counterexample to proposition A (dropping the faithfulness) is obtained by using the following:
{Theorem D}. Let \(X,Y\) be connected smooth projective schemes. Let \(i_\eta:\eta\rightarrow X\) be the generic point of \(X\), and let \(L=k(\eta)\) be the function field of \(X\). Assume that \(D^b(\text{Qcoh}(Y)_L)\) contains an object \(Z\) which is not in the essential image of \(D^b(\text{Qcoh}(Y)_L)\). Define \(\Psi\) as the composition \(\text{Perf}(X)\overset{i^\ast_\eta}\rightarrow D(L)\overset{L\mapsto Z}\rightarrow D(\text{Qcoh}(Y))\). Then \(\Psi\) is not the restriction of a Fourier-Mukai functor.
The main issue of the article is to prove the results above, and to give some guiding examples. The article starts by giving the necessary tools, including moduli spaces of representations of algebras, where the properties of the Formanek center and the concepts of Azumaya algebras as matrix algebras for the étale topology are involved, i.e. they are used to define a representable functor, thereby defining a moduli. This introduction to the theory is explicit and contains explicit results from algebra.
The algebraic theory is then lifted to vector bundles on curves, and the necessary results on stability is considered. For example, the following essential result of G. Faltings [J. Algebr. Geom. 2, No. 3, 507–568 (1993; Zbl 0790.14019)] is used:
Let \(X\) be a smooth projective curve. A bundle \(\mathcal E\) on \(X\) is semi-stable if there exists a non-zero bundle \(\mathcal F\) such that \(\mathcal F\perp\mathcal E\).
Then the homological identities is given, using Hochshild cohomology. This gives a theory of lifting field actions in the hereditary case, and counterexamples to this. Then counterexamples to lifting in the geometric case can be given, as indicated in the introduction.
After that, Non-Fourier-Mukai functors are defined, and some general results about them are given.
Finally, using all the introduced theory, the liftings can be studied using \(A_\infty\)-actions. This includes enhancing categories, studying \(A_\infty\)-schemes and \(A_\infty\)-modules by their \(A_\infty\)-morphisms. This not only proves the stated results, but gives very nice examples of the necessity of the \(A_\infty\)-theory.
A very nice and complete article, working in the recent edge of derived algebraic geometry.