arXiv subjects
Publications and source records attributed to Andy Jiang.
We observe that for a quasi-compact and quasi-separated scheme the structure sheaf generates the perfect complexes if and only if the lattice of thick subcategories is distributive if and only if the affinization map is 0-affine. Examples are discussed, including an example of a quasi-projective scheme which is not quasi-affine but for which these equivalent conditions hold.
In 1981, L. Gruson and C. U. Jensen gave a new proof of the fact that, over a ring which is either Noetherian of Krull dimension $n$ or of cardinality $< \aleph_n$, the projective dimension of any flat module is at most $n$. In this short paper, we observe that their arguments apply to the setting of quasicoherent sheaves over perfect stacks. As a consequence, we show that for any perfect stack $\mathfrak{X}$ with a faithfully flat cover $p : \mathrm{Spec}(R) \to \mathfrak{X}$, where $R$ is a Noetherian $\mathbb{E}_{\infty}$-ring of finite Krull dimension or satisfies the cardinality bound $2^{|\pi_*(R)|} < \aleph_{\omega}$, $p_*(\mathcal{O}_{\mathrm{Spec}(R)})$ is a descendable algebra in $\mathrm{QCoh}({\mathfrak{X}})$.
By reading a standard formula for the ring of Grothendieck differential operators in a derived way, we construct a derived (sheaf of) ring of Grothendieck differential operators for Noetherian schemes $X$ separated and finite-type over a base $S$, when the map $X \to S$ is finite tor-amplitude. Using this ring of differential operators, we (re-)develop the theory of $D$-modules from scratch and show an equivalence of categories between $D$-modules using our definition and crystals over the infinitesimal site.
Following a formula found in the paper of Avramov, Iyengar, Lipman, and Nayak (2010) and ideas of Neeman and Khusyairi, we indicate that Grothendieck duality for finite tor-amplitude maps can be developed from scratch via the formula $f^! := \delta^*\pi_1^{\times}f^*$. Our strategy centers on the subcategory $\Gamma_{\Delta}(\mathrm{QCoh}(X \times X))$ of quasicoherent sheaves on $X \times X$ supported on the diagonal. By exclusively using this subcategory instead of the full category $\mathrm{QCoh}(X \times X)$ we give systematic categorical proofs of results in Grothendieck duality and reprove many formulas found in Neeman (2018). We also relate some results in Grothendieck duality with properties of the sheaf of (derived) Grothendieck differential operators.
We present the working equations for a reduced-scaling method of evaluating the perturbative triples (T) energy in coupled-cluster theory, through the tensor hypercontraction (THC) of the triples amplitudes ($t_{ijk}^{abc}$). Through our method we can reduce the scaling of the (T) energy from the traditional O($N^{7}$) to a more modest O($N^{5}$). We also discuss implementation details to aid future research, development, and software realization of this method. Additionally, we show that this method yields sub-millihartree (mEh) differences from CCSD(T) when evaluating absolute energies, and sub-0.1 kcal/mol energy differences when evaluating relative energies. Finally, we demonstrate that this method converges to the true CCSD(T) energy through the systematic increasing of the rank or eigenvalue tolerance of the orthogonal projector, as well as exhibiting sub-linear to linear error growth with respect to system size.
We study integrality gap (IG) lower bounds on strong LP and SDP relaxations derived by the Sherali-Adams (SA), Lovasz-Schrijver-SDP (LS+), and Sherali-Adams-SDP (SA+) lift-and-project (L&P) systems for the t-Partial-Vertex-Cover (t-PVC) problem, a variation of the classic Vertex-Cover problem in which only t edges need to be covered. t-PVC admits a 2-approximation using various algorithmic techniques, all relying on a natural LP relaxation. Starting from this LP relaxation, our main results assert that for every epsilon > 0, level-Theta(n) LPs or SDPs derived by all known L&P systems that have been used for positive algorithmic results (but the Lasserre hierarchy) have IGs at least (1-epsilon)n/t, where n is the number of vertices of the input graph. Our lower bounds are nearly tight. Our results show that restricted yet powerful models of computation derived by many L&P systems fail to witness c-approximate solutions to t-PVC for any constant c, and for t = O(n). This is one of the very few known examples of an intractable combinatorial optimization problem for which LP-based algorithms induce a constant approximation ratio, still lift-and-project LP and SDP tightenings of the same LP have unbounded IGs. We also show that the SDP that has given the best algorithm known for t-PVC has integrality gap n/t on instances that can be solved by the level-1 LP relaxation derived by the LS system. This constitutes another rare phenomenon where (even in specific instances) a static LP outperforms an SDP that has been used for the best approximation guarantee for the problem at hand. Finally, one of our main contributions is that we make explicit of a new and simple methodology of constructing solutions to LP relaxations that almost trivially satisfy constraints derived by all SDP L&P systems known to be useful for algorithmic positive results (except the La system).