Cone Construction Research Articles

Iterated Mapping Cones on the Koszul Complex and Their Application to Complete Intersection Rings

Iterated Mapping Cones on the Koszul Complex and Their Application to Complete Intersection Rings

Filtered formal groups, Cartier duality, and derived algebraic geometry

We develop a notion of formal groups in the filtered setting and describe a duality relating these to a specified class of filtered Hopf algebras. We then study a deformation to the normal cone construction in the setting of derived algebraic geometry. Applied to the unit section of a formal group $\widehat{\mathbb{G}}$, this provides a $\mathbb{G}_m$-equivariant degeneration of $\widehat{\mathbb{G}}$ to its tangent Lie algebra. We prove a unicity result on complete filtrations, which, in particular, identifies the resulting filtration on the coordinate algebra of this deformation with the adic filtration on the coordinate algebra of $\widehat{\mathbb{G}}$. We use this in a special case, together with the aforementioned notion of Cartier duality, to recover the filtration on the filtered circle of [MRT19]. Finally, we investigate some properties of $\widehat{\mathbb{G}}$-Hochschild homology set out in loc. cit., and describe "lifts" of these invariants to the setting of spectral algebraic geometry.

Mapping cones of monomial ideals over exterior algebras

Let K be a field, V a finite dimensional K-vector space and E the exterior algebra of V. We analyze iterated mapping cone over E. If I is a monomial ideal of E with linear quotients, we show that the mapping cone construction yields a minimal graded free resolution F of I via the Cartan complex. Moreover, we provide an explicit description of the differentials in F when the ideal I has a regular decomposition function. Finally, we get a formula for the graded Betti numbers of a new class of monomial ideals including the class of strongly stable ideals.

Nerves and cones of free loop-free ω-categories

We show that the complicial nerve construction is homotopically compatible with two flavors of cone constructions when starting with an $\omega$-category that is suitably free and loop-free. An instance of the result recovers the fact that the standard $m$-simplex is equivalent to the complicial nerve of the $m$-oriental.

Very well–covered graphs by Betti splittings

A very well–covered graph is an unmixed graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. We study these graphs by means of Betti splittings and mapping cone constructions. We show that the cover ideals of Cohen–Macaulay very well–covered graphs are splittable. As a consequence, we compute explicitly the minimal graded free resolution of the cover ideals of such a class of graphs and prove that these graphs have homological linear quotients. Finally, we conjecture the same is true for each power of the cover ideal of a Cohen–Macaulay very well–covered graph, and settle it in the bipartite case.

Transition from volcano-sagging to volcano-spreading

Many volcanoes exhibit evidence of long-term deformation driven by gravity. This deformation influences the development of magmatic plumbing systems and volcanic vent locations, and it manifests as earthquakes and ground surface motions. Two end-member deformation styles are volcano sagging (flexure) and volcano spreading. Previous modeling studies indicated that the mechanical and geometric properties of basement rocks below the volcano can control whether a volcano spreads or sags, but these works differed on exactly how. Furthermore, evidence in nature that a volcano may progress from sagging to spreading as it grows has not been tested experimentally. Using Digital Image Correlation (DIC) and a revised scaling approach, we here revisit the classic experiments in which a sand–plaster cone is emplaced upon a basement comprising an upper brittle layer and a lower ductile layer. In a first experiment series, we emplaced the cone instantaneously. By normalizing both brittle and ductile layer thicknesses to cone height, we provide new insight into the dimensionless geometric controls on spreading and sagging. In a novel second experiment series, we emplaced the cone incrementally. We show that as a growing cone migrates through the dimensionless geometric parameter space, its deformation style changes successively in time from sagging to spreading. DIC shows that the horizontal velocity of the cone flank increases linearly or non-linearly during cone construction, but it decreases exponentially after construction ceases. Despite their geometric simplification and the omission of several sedimentological, magmatic, and regional-tectonic processes, our models' results are compatible with a range of observations of volcanoes on Earth and Mars. In particular, they help to explain: (i) why lithosphere-scale loading results in sagging rather than spreading, and (ii) why ocean island volcanoes of >2-3 km in height develop stellate forms and rift zones.

A scanning algorithm for odd Khovanov homology

We adapt Bar-Natan's scanning algorithm for fast computations in (even) Khovanov homology to odd Khovanov homology. We use a mapping cone construction instead of a tensor product, which allows us to deal efficiently with the more complicated sign assignments in the odd theory. The algorithm has been implemented in a computer program. We also use the algorithm to determine the odd Khovanov homology of 3-strand torus links.

Koszul homology of quotients by edge ideals

We show that the Koszul homology algebra of a quotient of a polynomial ring by the edge ideal of a forest is generated by the lowest linear strand. This provides a large class of Koszul algebras whose Koszul homology algebras satisfy this property. We obtain this result by constructing the minimal graded free resolution of a quotient by such an edge ideal via the so called iterated mapping cone construction and using the explicit bases of Koszul homology given by Herzog and Maleki. Using these methods we also recover a result of Roth and Van Tuyl on the graded Betti numbers of quotients by edge ideals of trees.

Geochronology and Petrogenesis of Early Pleistocene Dikes in the Changbai Mountain Volcanic Field (NE China) Based on Geochemistry and Sr-Nd-Pb-Hf Isotopic Compositions

Changbai Mountains intraplate volcanism (NE China) developed above the 500 km deep stagnant portion of the Pacific slab in the last 20 Ma. The more recent activity includes a shield-forming stage (2.8–0.3 Ma), the Tianchi cone construction stage (1.5–0.01 Ma), and a caldera-forming stage (0.2 Ma-present). Detailed studies on the petrogenesis of the volcanic products between the first two stages and the possible role of geodynamics and local tectonics in controlling the volcanism, however, are lacking. Here, we present structural and whole-rock geochemical and zircon Hf isotopic data on Pleistocene dikes of the Changbai Mountains at the transition from the shield-forming to the Tianchi stage with the aim to constrain their age and the source(s) of their parental magma. The dikes represent the shallower feeding system of monogenetic cones and have a NW-SE strike, which is also the preferred strike of the major fault affecting the area and along which the Changbai Mountains monogenetic scoria cones align. The dikes have a potassic affinity and a trachybasaltic composition. Their zircon U–Pb age is 1.19–1.20 Ma (Calabrian). The trachybasalts are enriched in Rb, Ba, Th, U, Nb, Ta, K, Pb, and LREE and slightly depleted in Sr, Zr, Hf, Ti, and HREE with a weak negative Eu/Eu* (δEu = 0.96–0.97). Trace elements and isotopic compositions are compatible with an OIB-type source with an EMI signature. The calculated (87Sr/86Sr)i(=0.705165–0.705324), (143Nd/144Nd)i(= 0.512552–0.512607, εNd(t) = −0.58 to −1.65), and Hf model ages (TDM2) of 1768–1562 Ma suggest that the trachybasaltic dikes were contaminated by a Mesoproterozoic, relatively basic lower crust. The source of the Calabrian trachybasalts consists of asthenospheric melts modified by a subcontinental lithospheric mantle. These melts upwell from depth and stop at the crust-mantle interface where underplating processes favor the assimilation of ancient lower crust material. During the ascent to the surface along deep-seated crustal discontinuities, these magmas weakly differentiate.

Betti Numbers of Weighted Oriented Graphs

Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we investigate the Betti numbers of $I(\mathcal{D})$ via upper-Koszul simplicial complexes, Betti splittings and the mapping cone construction. In particular, we provide recursive formulas for the Betti numbers of edge ideals of several classes of weighted oriented graphs. We also identify classes of weighted oriented graphs whose edge ideals have a unique extremal Betti number which allows us to compute the regularity and projective dimension for the identified classes. Furthermore, we characterize the structure of a weighted oriented graph $\mathcal{D}$ on $n$ vertices such that $\textrm{pdim } (R/I(\mathcal{D}))=n$ where $R=k[x_1,\ldots, x_n]$.

Para-Sasaki-like Riemannian manifolds and new Einstein metrics

We determine a new class of paracontact paracomplex Riemannian manifolds derived from certain cone construction, called para-Sasaki-like Riemannian manifolds, and give explicit examples. We define a hyperbolic extension of a paraholomorphic paracomplex Riemannian manifold, which is a local product of two Riemannian spaces of equal dimension, and show that it is a para-Sasaki-like Riemannian manifold. If the original paraholomorphic paracomplex Riemannian manifold is a complete Einstein space of negative scalar curvature, then its hyperbolic extension is a complete Einstein para-Sasaki-like Riemannian manifold of negative scalar curvature. Thus, we present new examples of complete Einstein Riemannian manifolds of negative scalar curvature.

Large Mg[sbnd]Fe isotope fractionation in volcanic rocks from northeast China: The role of chemical weathering and magma compositional effect

This study presents Mg and Fe isotopic data for a suite of volcanic rocks including basalt, trachyte and comendite from Changbaishan, northeast China. Our results show that the Millennium Eruption (ME) trachytes and comendites have δ26Mg ranging from −0.37 to 0.14‰. Combined with literature data, the positive correlation between δ26Mg and 87Sr/86Sr ratios and the almost constant value for Nd isotopes suggest that the heavy Mg isotopic composition most likely derived from involvement of weathered products of basalt instead of the old supracrustal/basement rocks. Iron isotopic composition is significantly varied throughout our suite of samples, with δ56Fe values of 0.17 ± 0.04‰ and 0.16 ± 0.03‰ for basalts, 0.09 to 0.23‰ for cone-construction trachytes, and 0.23 to 0.37‰ for the ME samples. The slightly higher δ56Fe than MORB observed in basalts reflect the role of subducted materials that could elevate source Fe3+/ΣFe ratios via redox reaction. The trachytes from the cone construction, thought to be significantly affected by chemical weathered product, have a restricted Fe isotopic variation, while those ME samples with less weathered product input have overall higher δ56Fe values. This contrasting Fe isotopic signature suggests that the paleo-weathering event likely occurred under oxidized environment, leaving the residue with little Fe isotope fractionation. By contrast, the good correlations between Fe isotopes and indicators of magmatic differentiation (e.g., tFe2O3, SiO2, Mg#) in all rocks reflect that fractional crystallization under an open system most likely accounts for the Fe isotopic variation, which was enhanced by the compositional effect as suggested by the broadly positive relationship between δ56Fe and (Na + K)/(Ca + Mg). This study therefore highlights the potential applications of MgFe isotopes as great tracers of redox condition and ancient climate.

Volcanic processes and magmatic evolution of Tianchi volcano, Changbaishan

Abstract The Changbaishan volcanic field located on the Gaima (Gaema, Gaiman) Plateau has witnessed plateau-forming eruptions along with the uplift of the Gaima Plateau. The Tianchi basaltic lava shield volcano was formed at the main peak of Changbaishan, with cone construction eruptions that formed a huge and steep trachytic composite cone on the gentle lava shield. At the peak of the Millennium Eruption (ME), the height of the eruption column reached 25 km and the bulk volume of tephra was about 120 km 3 . The ME eventually formed Tianchi caldera, after which several eruptions occurred, albeit of a much smaller scale. The magmas involved in the shield-forming eruptions are characterized by both alkalic series trachybasalt and basaltic trachyandesite and subalkalic tholeiite and basaltic andesite. In the cone-construction and ignimbrite-forming eruption stages, the magma is completely composed of alkalic series trachyte and comendite. The largest negative Eu anomalies observed in ME magmas indicate that plagioclase was strongly crystallized and differentiated.

On the deformation groupoid of the inhomogeneous pseudo‐differential Calculus

In 1974, Folland and Stein constructed an inhomogeneous pseudo-differential calculus based on analysis on the Heisenberg group. This Heisenberg calculus was generalized by several authors, to any subbundle of the tangent bundle. van Erp and Yuncken, following Debord and Skandalis showed that this calculus can be recovered using a deformation groupoid alla tangent groupoid of Connes. Using functoriality of the deformation to the normal cone construction, we give an elementary construction of this groupoid. We then extend it to the general case of a filtration of the tangent bundle by an iterated deformation.

A special cone construction and its connections to structured tensors and their spectra

In this work we construct a cone comprised of a group of tensors hypermatrices satisfying a special condition, and we study its relations to structured tensors such as M-tensors and H-tensors. We also investigate its applications to spectra of certain Z-tensors. We obtain an inequality for the spectral radius of certain tensors when the order m is odd.

Preprojective analogue of the cone construction

We formulate a relative, representation theoretic, notion of the algebraic cone construction. This motivates a generalization of the cone corresponding to a preprojective algebra.

Eruptive chronology and tectonic context of the late Pleistocene Tres Vírgenes volcanic complex, Baja California Sur (México)

Based on a detailed geological map and stratigraphy aided by new 230Th/238U and 206P/238U age determinations we established the eruptive chronology of the Tres Vírgenes Volcanic Complex (TVVC). The complex consists of three northeast-southwest aligned stratovolcanoes, which from older to younger are El Viejo, El Azufre, and La Virgen and associated cinder cones and domes. The alignment was fed from fissures that mark the southern trace of the left-lateral Cimarron Fault. The TVVC was built upon the basement rocks of the Peninsular Ranges Batholith (~99 Ma), the Santa Lucia Formation (21.6 Ma), the Esperanza basalt (7.6 Ma), and the 1.1 Ma ignimbrite of El Aguajito caldera. The TVVC began its formation at ca. 300 ka mainly with dacitic lava flows and domes that formed El Viejo volcano (4.2 km3). At around 173 ka, activity migrated to the southwest to build El Azufre volcano (3.8 km3) with the emission of dacitic lavas and several phases of dome construction and destruction with the generation of pyroclastic density currents. El Azufre's activity ended ~128 ka with the extrusion of the summit dome. At about 112 ka, volcanism migrated 4.2 km to the southwest to begin the construction of La Virgen Volcano through the emission of andesitic and dacitic lavas that built the largest cone of the complex until ~22 ka (31.2 km3). This volcano was built upon several stages of cone construction with the emission of lava flows and peripheral dacitic domes and cinder cones. Between 128 and 112 ka six cinder cones vented along the Cimarron Fault between the La Virgen and El Azufre volcanoes. By assuming a constant emission of magma the total volume of the TVVC (39 km3) was erupted at an average rate of 0.13 km3/kyr. Of this volume, effusive eruptions dominated the evolution of the complex (89%) of which 62% are dacites and 38% andesites. The TVVC rocks have compositions of 50–67 wt% in silica, and low-medium K (0.5–2.5 wt% K2O). Trace element signatures of the TVVC rocks (Sr/Y <40, La/Yb <10, Nb-Ti negative anomalies and relative enrichments of LILE and LREE) suggest a post-subduction mixture between adakitic and calc-alkaline magmas. Xenoliths of granodiorites hosted in lavas and La Virgen tephra suggest that magma chambers feeding the complex have stagnated at depths >2 km within the Peninsular Ranges Batholith as confirmed by geobarometry, aeromagnetic data, and the drill-holes of the geothermal field. Today, the Tres Vírgenes geothermal field generates 10 MW of electricity although future exploration of several nearby areas may increase such capacity.

K-Theory of Locally Compact Modules over Rings of Integers

Abstract We generalize a recent result of Clausen; for a number field with integers $\mathcal{O}$, we compute the K-theory of locally compact $\mathcal{O}$-modules. For the rational integers this recovers Clausen’s result as a special case. Our method of proof is quite different; instead of a homotopy coherent cone construction in $\infty$-categories, we rely on calculus of fraction type results in the style of Schlichting. This produces concrete exact category models for certain quotients, a fact that might be of independent interest. As in Clausen’s work, our computation works for all localizing invariants, not just K-theory.

Smooth projective varieties with a torus action of complexity 1 and Picard number 2

We give an explicit description of all smooth varieties with a torus action of complexity one having Picard number at most two. As a consequence, we classify in every dimension the smooth (almost) Fano varieties with a torus action of complexity one having Picard number two. It turns out that all the Fano examples are obtained via an iterated generalized cone construction from a series of smooth varieties of dimension at most seven.

Challenges due to problematic soils: a case study at the crossroads of geotechnology and sustainable pavement solutions

Geotechnical characteristics of subsoils should be adequately incorporated in the rehabilitation strategies of existing pavements which have performed poorly due to problematic subsurface conditions. However, there appears to be a disconnect between the advances in our understanding of the mechanics of soft or problematic soils and the rehabilitation design of the overlying pavement structure, leading to repeated cycles of premature distresses, underperformance, and failures. A case study is presented for the rehabilitation of a flexible pavement built over soft organic soils in Southeastern Florida, USA. The study incorporates forensic investigation of the deteriorated pavement structure, subsurface investigations with cone penetration testing, design and construction of reinforced overlays in field test sections, and long-term performance monitoring with non-destructive dynamic tests. Efforts are made to correlate site characteristics with pavement performance. Based on the secondary compression behavior of the organic soils, cement deep mixing criteria are proposed for a more durable and sustainable solution.