Let
In this article a condition is given to detect the containment among thick subcategories of the bounded derived category of a commutative noetherian ring. More precisely, for a commutative noetherian ring
 Award ID(s):
 2002173
 NSFPAR ID:
 10520377
 Publisher / Repository:
 Proceedings of the American Mathematical Society
 Date Published:
 Journal Name:
 Proceedings of the American Mathematical Society
 Volume:
 151
 ISSN:
 00029939
 Page Range / eLocation ID:
 14591469
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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$(R,\mathfrak {m})$ be a Noetherian local ring of dimension$d\geq 2$ . We prove that if$e(\widehat {R}_{red})>1$ , then the classical Lech’s inequality can be improved uniformly for all$\mathfrak {m}$ primary ideals, that is, there exists$\varepsilon >0$ such that$e(I)\leq d!(e(R)\varepsilon )\ell (R/I)$ for all$\mathfrak {m}$ primary ideals$I\subseteq R$ . This answers a question raised by Huneke, Ma, Quy, and Smirnov [Adv. Math. 372 (2020), pp. 107296, 33]. We also obtain partial results towards improvements of Lech’s inequality when we fix the number of generators of$I$ . 
This is the first of our papers on quasisplit affine quantum symmetric pairs
$\big (\widetilde {\mathbf U}(\widehat {\mathfrak g}), \widetilde {{\mathbf U}}^\imath \big )$ , focusing on the real rank one case, i.e.,$\mathfrak g = \mathfrak {sl}_3$ equipped with a diagram involution. We construct explicitly a relative braid group action of type$A_2^{(2)}$ on the affine$\imath$ quantum group$\widetilde {{\mathbf U}}^\imath$ . Real and imaginary root vectors for$\widetilde {{\mathbf U}}^\imath$ are constructed, and a Drinfeld type presentation of$\widetilde {{\mathbf U}}^\imath$ is then established. This provides a new basic ingredient for the Drinfeld type presentation of higher rank quasisplit affine$\imath$ quantum groups in the sequels. 
For
$(t,x) \in (0,\infty )\times \mathbb {T}^{\mathfrak {D}}$ , the generalized Kasner solutions (which we refer to as Kasner solutions for short) are a family of explicit solutions to various Einsteinmatter systems that, exceptional cases aside, start out smooth but then develop a Big Bang singularity as$t \downarrow 0$ , i.e., a singularity along an entire spacelike hypersurface, where various curvature scalars blow up monotonically. The family is parameterized by the Kasner exponents$\widetilde {q}_1,\cdots ,\widetilde {q}_{\mathfrak {D}} \in \mathbb {R}$ , which satisfy two algebraic constraints. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be dynamically stable, that is, stable under perturbations of the Kasner initial data, given say at$\lbrace t = 1 \rbrace$ , as long as the exponents are “subcritical” in the following sense:$\underset {\substack {I,J,B=1,\cdots , \mathfrak {D}\\ I > J}}{\max } \{\widetilde {q}_I+\widetilde {q}_J\widetilde {q}_B\}>1$ . Previous works have rigorously shown the dynamic stability of the Kasner Big Bang singularity under stronger assumptions: (1) the Einsteinscalar field system with$\mathfrak {D}= 3$ and$\widetilde {q}_1 \approx \widetilde {q}_2 \approx \widetilde {q}_3 \approx 1/3$ , which corresponds to the stability of the Friedmann–Lemaître–Robertson–Walker solution’s Big Bang or (2) the Einsteinvacuum equations for$\mathfrak {D}\geq 38$ with$\underset {I=1,\cdots ,\mathfrak {D}}{\max } \widetilde {q}_I > 1/6$ . In this paper, we prove that the Kasner singularity is dynamically stable forall subcritical Kasner exponents, thereby justifying the heuristics in the literature in the full regime where stable monotonictype curvatureblowup is expected. We treat in detail the$1+\mathfrak {D}$ dimensional Einsteinscalar field system for all$\mathfrak {D}\geq 3$ and the$1+\mathfrak {D}$ dimensional Einsteinvacuum equations for$\mathfrak {D}\geq 10$ ; both of these systems feature nonempty sets of subcritical Kasner solutions. Moreover, for the Einsteinvacuum equations in$1+3$ dimensions, where instabilities are in general expected, we prove that all singular Kasner solutions have dynamically stable Big Bangs under polarized$U(1)$ symmetric perturbations of their initial data. Our results hold for open sets of initial data in Sobolev spaces without symmetry, apart from our work on polarized$U(1)$ symmetric solutions.Our proof relies on a new formulation of Einstein’s equations: we use a constantmeancurvature foliation, and the unknowns are the scalar field, the lapse, the components of the spatial connection and second fundamental form relative to a Fermi–Walker transported spatial orthonormal frame, and the components of the orthonormal frame vectors with respect to a transported spatial coordinate system. In this formulation, the PDE evolution system for the structure coefficients of the orthonormal frame approximately diagonalizes in a way that sharply reveals the significance of the Kasner exponent subcriticality condition for the dynamic stability of the flow: the condition leads to the timeintegrability of many terms in the equations, at least at the low derivative levels. At the high derivative levels, the solutions that we study can be much more singular with respect to
$t$ , and to handle this difficulty, we use$t$ weighted high order energies, and we control nonlinear error terms by exploiting monotonicity induced by the$t$ weights and interpolating between the singularitystrength of the solution’s low order and high order derivatives. Finally, we note that our formulation of Einstein’s equations highlights the quantities that might generate instabilities outside of the subcritical regime. 
Let
$R$ be a standard graded algebra over a field. We investigate how the singularities of$\operatorname {Spec} R$ or$\operatorname {Proj} R$ affect the$h$ vector of$R$ , which is the coefficient of the numerator of its Hilbert series. The most concrete consequence of our work asserts that if$R$ satisfies Serre’s condition$(S_r)$ and has reasonable singularities (Du Bois on the punctured spectrum or$F$ pure), then$h_0$ , …,$h_r\geq 0$ . Furthermore the multiplicity of$R$ is at least$h_0+h_1+\dots +h_{r1}$ . We also prove that equality in many cases forces$R$ to be CohenMacaulay. The main technical tools are sharp bounds on regularity of certain$\operatorname {Ext}$ modules, which can be viewed as Kodairatype vanishing statements for Du Bois and$F$ pure singularities. Many corollaries are deduced, for instance that nice singularities of small codimension must be CohenMacaulay. Our results build on and extend previous work by de FernexEin, EisenbudGoto, HunekeSmith, MuraiTerai and others. 
Over a local ring
$R$ , the theory of cohomological support varieties attaches to any bounded complex$M$ of finitely generated$R$ modules an algebraic variety${\mathrm {V}}_R(M)$ that encodes homological properties of$M$ . We give lower bounds for the dimension of${\mathrm {V}}_R(M)$ in terms of classical invariants of$R$ . In particular, when$R$ is Cohen–Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When$M$ has finite projective dimension, we also give an upper bound for$\dim {\mathrm {V}}_R(M)$ in terms of the dimension of the radical of the homotopy Lie algebra of$R$ . This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes, and it recovers a result of Avramov and Halperin on the homotopy Lie algebra of$R$ . Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring.