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We define and study cohomological tensor functors from the category

The first significant buildup in atmospheric oxygen, the Great Oxidation Event (GOE), began in the early Paleoproterozoic in association with global glaciations and continued until the end of the Lomagundi carbon isotope excursion ca. 2,060 Ma. The exact timing of and relationships among these events are debated because of poor age constraints and contradictory stratigraphic correlations. Here, we show that the first Paleoproterozoic global glaciation and the onset of the GOE occurred between ca. 2,460 and 2,426 Ma, ∼100 My earlier than previously estimated, based on an age of 2,426 ± 3 Ma for Ongeluk Formation magmatism from the Kaapvaal Craton of southern Africa. This age helps define a key paleomagnetic pole that positions the Kaapvaal Craton at equatorial latitudes of 11° ± 6° at this time. Furthermore, the rise of atmospheric oxygen was not monotonic, but was instead characterized by oscillations, which together with climatic instabilities may have continued over the next ∼200 My until ≤2,250–2,240 Ma. Ongeluk Formation volcanism at ca. 2,426 Ma was part of a large igneous province (LIP) and represents a waning stage in the emplacement of several temporally discrete LIPs across a large low-latitude continental landmass. These LIPs played critical, albeit complex, roles in the rise of oxygen and in both initiating and terminating global glaciations. This series of events invites comparison with the Neoproterozoic oxygen increase and Sturtian Snowball Earth glaciation, which accompanied emplacement of LIPs across supercontinent Rodinia, also positioned at low latitude.

We prove a category equivalence between algebraic supergroups and Harish-Chandra pairs over a commutative ring which is 22-torsion free. The result is applied to reconstruct the Chevalley Z\\mathbb {Z}-supergroups constructed by Fioresi and Gavarini (2012) and by Gavarini (2014). For a wide class of algebraic supergroups we describe their representations by using their super-hyperalgebras.

Aluminotaipingite-(CeCa), (Ce6Ca3)Al(SiO4)3[SiO3(OH)]4F3, is a new member of the cerite-supergroup minerals, whose general chemical formula is A9XM[TO3O]7Z3, (A = REE, Ca, Sr, Na and ∎; X = ∎, Ca, Na and Fe2+; M = Mg, Fe2+, Fe3+, Al and Mn; T = Si and P; O = O and OH; Z = ∎, OH and F). It was found in cavities of a leucogranitic orthogneiss at the Casette quarry, Montoso, Bagnolo Piemonte, Cuneo Province, Piedmont, Italy. Crystals of aluminotaipingite-(CeCa) are light pink to pink, transparent or semi-transparent, with a vitreous lustre. It forms pyramidal crystals up to 0.07 mm in size and observed forms are {0 0 1}, {1 0 2̄}. The tenacity is brittle, no distinct cleavage is observed and the fracture is uneven. The mineral does not fluoresce in long- or short-wave ultraviolet light. The streak is white. Hardness (Mohs) = 5. The calculated density is 4.476 g cm-3. The mineral is trigonal, space group R3c, with a = 10.658(3), c = 37.865(9) Å, V = 3725(2) Å3 and Z = 6. The eight strongest powder X-ray diffraction lines are [dobs, Å (I, %) (h k l)]: 8.38(29)(0 1 2), 4.499(28)(2 0 2), 3.282(41)(2 1 4), 2.936(100)(0 2 10), 2.816(51)(1 2 8), 2.669(37)(2 2 0), 2.207 (29)(3 0 12) and 1.935(35)(2 3 8). The structure was refined to R =0.0306 for 2297 reflections with I >;2σ(I). The crystal structure of aluminotaipingite-(CeCa) contains two nine-fold coordinated sites (A1 and A2), which are occupied mainly by lanthanides, and a third nine-fold coordinated A3 site containing almost equal amounts of lanthanides and Ca. The X site is vacant and at the octahedral M site aluminium prevails over Fe3+. Among the three independent T sites, T2 belongs to a (SiO4)4- anion, whereas T1 and T3 belong to (SiO3OH)3- anions. Fluorine is involved in coordination with the A1 and A3 sites.

After nearly a billion years with no evidence for glaciation, ice advanced to equatorial latitudes at least twice between 717 and 635 Mya. Although the initiation mechanism of these Neoproterozoic Snowball Earth events has remained a mystery, the broad synchronicity of rifting of the supercontinent Rodinia, the emplacement of large igneous provinces at low latitude, and the onset of the Sturtian glaciation has suggested a tectonic forcing. We present unique Re-Os geochronology and high-resolution Os and Sr isotope profiles bracketing Sturtian-age glacial deposits of the Rapitan Group in northwest Canada. Coupled with existing U-Pb dates, the postglacial Re-Os date of 662.4 ± 3.9 Mya represents direct geochronological constraints for both the onset and demise of a Cryogenian glaciation from the same continental margin and suggests a 55-My duration of the Sturtian glacial epoch. The Os and Sr isotope data allow us to assess the relative weathering input of old radiogenic crust and more juvenile, mantle-derived substrate. The preglacial isotopic signals are consistent with an enhanced contribution of juvenile material to the oceans and glacial initiation through enhanced global weatherability. In contrast, postglacial strata feature radiogenic Os and Sr isotope compositions indicative of extensive glacial scouring of the continents and intense silicate weathering in a post–Snowball Earth hothouse.

Fluoralforsite, ideally Ba 5 (PO 4 ) 3 F, (space group P 6 3 / m (#176), Z = 2, a = 10.0031(2) Å, c = 7.5382(2) Å and V = 653.23(3) Å 3 ), is a new mineral species of the apatite group – a Ba-analogue of fluorapatite and a F-analogue of alforsite. It was discovered in rankinite paralava filling cracks in pyrometamorphic gehlenite hornfels near the tributary of wadi Zohar and Gurim Anticline, Hatrurim Basin, Negev Desert, Israel. Fluoralforsite occurs in small intergranular spaces between large gehlenite and garnet crystals and in enclaves inside large rankinite crystals with other Ba minerals such as walstromite, zadovite, bennesherite, gurimite, mazorite, barioferrite and baryte. It forms tiny transparent, colourless crystals up to 50 μm with a white streak and a vitreous lustre. The cleavage was not observed. It exhibits a brittle tenacity and a conchoidal fracture. The estimated Mohs hardness is 4–4½, and its calculated density is 4.57 g/cm –3 . Fluoralforsite is uniaxial (–) with refractive indices (589 nm) n ω = 1.689(3) and n ɛ = 1.687(3). The empirical crystal-chemical formula for the holotype calculated on the basis of 8 cations is: (Ba 3.81 Ca 0.97 Na 0.07 K 0.05 Sr 0.05 Fe 0.05 ) Σ5 (P 5+ 2.32 V 5+ 0.29 S 6+ 0.22 Si 0.17 ) Σ3 O 12 (F 0.85 Cl 0.13 ) Σ0.98 . The crystal structure was refined from single-crystal X-ray diffraction data with R 1 = 0.0192. The structural investigation indicated an ordered arrangement of Ba/Ca at the M 1 site within individual columns running along the c -axis, but a disordered distribution among adjacent columns throughout the structure, which enables the maintenance of the P 6 3 / m space group. Fluoralforsite was formed at the final stage of crystallisation as a result of a reaction between the primary mineral assemblages and residual melt.

Mangani-eckermannite, ideally NaNa 2 (Mg 4 Mn 3+ )Si 8 O 22 (OH) 2 , is a new member of the amphibole supergroup found at Tanohata Mine, Shimohei District, Iwate Prefecture, Japan. It occurs as prismatic crystals up to 0.3 × 0.2 mm and their aggregates up to 1 mm intergrown with braunite, vittinkiite and quartz. Mangani-eckermannite is cherry-red to very dark red and reddish-brown in thicker grains. It is translucent with a pinkish white streak and vitreous lustre. It is brittle, fracture is stepped along crystal elongation and uneven across a crystal. Cleavage is perfect on 110. Mohs hardness is 6. D meas = 3.16(2) and D calc = 3.186 g/cm 3 . The mineral is optically biaxial (–), with α = 1.645(3), β = 1.668(2), γ = 1.675(3) (589 nm); 2V meas = 60(10)°, 2V calc = 57°. The empirical formula derived from electron microprobe analysis, secondary-ion mass spectrometry and single-crystal structure refinement and calculated on the basis of 24 (O+OH) atoms per formula unit (apfu) is A (Na 0.74 K 0.24 □ 0.02 )Σ1.00 B (Na 1.52 Ca 0.24 Mn 2+ 0.24 ) Σ2.00 C (Mg 2.54 Mn 2+ 1.45 Mn 3+ 0.71 Fe 3+ 0.26 Ti 0.04 ) Σ5.00 T (Si 7.97 Al 0.03 ) Σ8.00 O 22 W [(OH) 1.52 O 0.48 ] Σ2.00 . Mangani-eckermannite is monoclinic, space group C 2/ m , a = 9.9533(4), b = 18.1440(7), c = 5.2970(2) Å, β = 103.948(4)°, V = 928.39(6) Å 3 and Z = 2. The strongest lines of the powder X-ray diffraction pattern [ d , Å ( I , %)( hkl )] are: 8.52(100)(110); 4.54(25)(040); 3.41(29)(131); 3.16(23)(310,201); 2.721(37)(151); 2.533(26)($\\bar{2}$02). The crystal structure was refined to R 1 = 0.0264 for 1001 independent reflections with I > 2σ( I ). The place of mangani-eckermannite in the nomenclature of the amphibole supergroup is discussed and the status of mangano-ferri-eckermannite as a valid mineral species and successor of ‘kôzulite’ is questioned.

The source of gold in the ca. 2.66 Ga Black Reef Formation (BRF) has been investigated and constrained through petrographic, mineralogical, geochemical, and high-resolution three-dimensional reflection seismic data combined with drill core and underground geological mapping. The BRF is a strong seismic marker and consists of carbonaceous shale, quartz arenite, and conglomerate. Gold grade in the BRF is primarily controlled by the nature of the host conglomerates. Most of the gold in the BRF conglomerate occurs in native form, and its morphology is highly heterogeneous. Gold was initially introduced through mechanical recycling of underlying Witwatersrand reefs, followed by short-range (millimeter- to centimeter-scale) postdepositional alteration/remobilization associated with the Bushveld Complex and the Vredefort meteorite impact. Although the BRF was subjected to high postdepositional fluid circulation facilitated by high fracture density, the volume of dissolved gold was probably too small to form a large gold deposit, except in areas around the Black Reef/Witwatersrand reefs subcrop positions. Findings from this study demonstrate the importance of both sedimentological controls and impact-related structures in the formation of paleoplacer gold deposits during Neoarchean times.

An attempt has been made to illustrate the evolution of pelitic granulite from south of the Balaram-Abu road, which lies in the South Delhi Terrane (SDT) of the Aravalli-Delhi Mobile Belt (ADMB), using geochemistry and geochronology. The current work offers a plausible explanation for the protolith of pelitic granulite, nature of the sediments and its provenance. The elemental geochemistry of the pelitic granulites reveals that the protolith is an arkosic to shaley type. The rare earth elements pattern shows that there is a negative Eu anomaly and a small excess of LREE over HREE. This means that the source of sediments probably has the same elements as the upper crust. However, the amounts of Sr, Nd and Pb vary a lot, which shows that the sediments supplied from two different types of sources (felsic and mafic) in different proportions from a Proterozoic terrain. The monazite geochronology indicates that the metamorphic overprint occurred between 797 Ma and 906 Ma. Additionally, the ages correlate to the debris that was formed between the 1188 Ma and 1324 Ma from magmatic/sedimentary sources for pelitic granulite. The present research provides a more in-depth understanding of the evolutionary history of the pelitic granulite that comprises the SDT in the ADMB region during the Proterozoic era.

The new columbite-supergroup mineral dmitryvarlamovite, ideally Ti 2 (Fe 3+ Nb)O 8 , was discovered in weathered alkaline metasomatic assemblages formed after late Riphaean sedimentary carbonate rocks of the Verkhne-Shchugorskoe deposit, Middle Timan Mts., Russia. The associated minerals are columbite-(Fe), pyrochlore-group minerals, monazite-(Ce), xenotime-(Y), baryte, pyrite, drugmanite and plumbogummite. Dmitryvarlamovite occurs as isolated anhedral equant grains up to 0.5 mm across. The colour of dmitryvarlamovite is black, the streak is black and the lustre is submetallic. The new mineral is brittle, with the mean VHN hardness of 753 kg mm –2 corresponding to the Mohs’ hardness of 6. No cleavage is observed. The fracture is conchoidal. The calculated density is 4.891 g⋅cm –3 . In reflected light, dmitryvarlamovite is light grey; no pleochroism is observed. The reflectance values ( R min , % / R max , % / λ, nm) are: 19.8/20.3/470, 18.3/18.9/546, 17.8/18.5/589 and 17.3/17.8/650. The chemical composition is (electron microprobe data, with iron divided into Fe 2 O 3 and FeO based on the charge balance, wt.%): MnO 0.11, FeO 1.51, V 2 O 3 0.89, Cr 2 O 3 0.28, Fe 2 O 3 19.26, TiO 2 37.72, Nb 2 O 5 40.08, total 99.85. The IR and Raman spectra indicate the absence of H-, C- and N-bearing groups. The empirical formula is (Fe 2+ 0.08 V 3+ 0.05 Cr 3+ 0.01 Fe 3+ 0.92 Ti 1.79 Nb 1.15 ) Σ4.00 O 8 . The crystal structure was determined using single-crystal X-ray diffraction data and refined to R = 0.048. Dmitryvarlamovite is orthorhombic, space group P 2 1 2 1 2, a = 4.9825(6), b = 4.6268(4), c = 5.5952(6) Å and V = 5.5952(6) Å 3 ( Z = 1). The structure is related to those of wolframite-group minerals but differs in the scheme of cation ordering. The crystal-chemical formula derived based on the structural data is (Ti 0.57 Nb 0.21 Fe 3+ 0.15 Fe 2+ 0.04 V 0.02 Cr 0.01 ) 2 (Nb 0.36 Ti 0.33 Fe 3+ 0.31 ) 2 O 8 . The strongest lines of the powder X-ray diffraction pattern [ d , Å ( I , %) ( hkl )] are: 3.58 (40) (011), 2.911 (100) (111), 2.809 (40) (002), 2.497 (38) (020), 2.447 (29) (103), 1.7363 (32) (103) and 1.7047 (29) (220). Dmitryvarlamovite is named after Dmitry Anatol'evich Varlamov (b. 1965).