Multiple nitrogen reservoirs in a protoplanetary disk at the epoch of comet and giant planet formation

P. Hily-Blant^1
V. Magalhaes de Souza^2
J. Kastner^3
T. Forveille^1

Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
IRAM, 300 rue de la Piscine, Grenoble, France
Chester F. Carlson Center for Imaging Science, School of Physics & Astronomy, and Laboratory for Multiwavelength Astrophysics,
Rochester Institute of Technology, 54 Lomb Memorial Drive, Rochester NY 14623 USA

The isotopic ratio of nitrogen measured in primitive Solar System bodies shows a broad range of values, the origin of which remains unknown. One key question is whether these isotopic reservoirs of nitrogen predate the comet formation stage or are posterior to it. Another central question is elucidating the processes that can produce the observed variations in the $^{14}$N/$^{15}$N isotopic ratio. Disks that orbit pre-main-sequence (TTauri) stars provide unique opportunities for observing the chemical content of analogs of the protosolar nebula and therefore for building a comprehensive scenario that can explain the origin of nitrogen in the Solar System and in planet-forming disks. With ALMA, it has become possible to measure isotopic ratios of nitrogen-bearing species in such environments. We present spectrally and spatially resolved observations of the hyperfine structure of the 4-3 rotational transition of HCN and its main isotopologs H$^{13}$CN and HC$^{15}$N in the disk orbiting the 8 Myr old TTauri star TW Hya. The sensitivity allows directly measuring the HCN/H$^{13}$CN and HCN/HC$^{15}$N abundance ratios with minimal assumptions. Averaged spatially over the disks, the ratios are 86$\pm$4 and 223$\pm$21, respectively. The latter value is significantly lower than the CN/C$^{15}$N ratio of 323$\pm$30 in this disk and thus provides the first evidence that two isotopic reservoirs of nitrogen are present in a disk at the stage of giant planet and comet formation. Furthermore, we find clear evidence for an increase in the ratio of HCN to HC$^{15}$N with radius. The ratio in the outer disk, at 45 au, is 339$\pm$28, in excellent agreement with direct measurements in the local interstellar medium, and with the bulk nitrogen isotopic ratio predicted from galactic evolution calculations. In the comet formation region at $r=20$ au, the ratio is a factor $\approx3$ lower, 121$\pm$11. This radial increase qualitatively agrees with the scenario in which selective photodissociation of N$_2$ is the dominant fractionation process. However, our isotopic ratios and kinetic temperature of the HCN-emitting layers quantitatively disagree with models of nitrogen chemistry in disks.

§ INTRODUCTION

The isotopic ratio of nitrogen, , in various bodies of the Solar System (meteorites, comets, planets, etc.) shows the largest variations among the most abundant constituents (carbon and oxygen), with values ranging from $\sim$50 in submicron grains that are immersed in a chondrite matrix (so-called hotspots) to 441 in the Sun and Jupiter <cit.>. When, where, and how these isotopic reservoirs of nitrogen formed during the evolution from a molecular cloud to planetary systems remains to be determined. Comets provide a special case among primitive cosmomaterials: regardless of the cometary type (short or long period) and of the carrier of nitrogen that is observed in their coma (CN, HCN, NH2, NH3, N2, or NO), the isotopic ratio is consistently found to be $\approx$140 <cit.>. This is lower by a factor three than the ratio of the bulk proto-Sun, which is 441. This well-established observational fact remains unexplained.

Integrated intensity maps of HCN, H, and HC (from left to right) in Jy/beam. The maps are centered at $\alpha$=11:01:51.81150, $\delta$=-34:42:17.2636 (J2000.0). The visibilities were projected on the same spatial grid and convolved to a synthesized beam of HPBW $0.26\arcsec\times0.21\arcsec$. The emission is centered on the systemic velocity of 2.83 and is integrated over 5 for HCN and 2 for the less abundant isotopologs.

Evolved protoplanetary disks at the stage of giant planet and comet formation provide unique opportunities for investigating how and when this -rich reservoir of nitrogen was built up. The nitrogen isotopic ratio was measured via rare isotopologs[Hereafter, the three isotopologs H$^{12}$C$^{14}$N, $\rm H^{13}C^{14}N$, and $\rm H^{12}C^{15}N$, are denoted HCN, H, and HC, respectively.] of HCN, specifically, H and HC,  for a small sample of disks with the ALMA interferometer <cit.>, yielding an HCN/HC  between 83$\pm$32 and 156$\pm$71, and a mean value of 111$\pm$19, under the assumption that HCN/H=70. On the other hand, a direct measurement, one that does not involve assuming a / ratio, of CN/C in the TW Hya system yielded a significantly higher ratio, 323$\pm$30 (). Based on this value and on directly measured values in the dense local interstellar medium (ISM), a value of =330$\pm$30 was proposed for the present-day bulk solar neighborhood. This also agrees excellently well with predictions from the galactic chemical evolution model of <cit.> for the local ISM. A new picture thus emerged in which HCN and CN trace two isotopic reservoirs of nitrogen that are present in analogs of the protosolar nebula (PSN) at the time of comet formation, with HCN representing a -enriched reservoir of nitrogen similar to that recorded in Solar System comets. However, HCN/HC was only measured indirectly, and moreover, this was not done in TW Hya, therefore clear-cut evidence for this scenario was lacking.

To elucidate how the various nitrogen reservoirs in the Solar System were formed requires identifying the processes that lead to variations of   <cit.>. The efficiency of chemical mass fractionation, which is the only process expected in cold and shielded gas <cit.>, has recently been called into question both theoretically <cit.> and observationally by measuring HCN/HC  in the L1498 pre-stellar core <cit.>. The obtained value, 338$\pm$28, agrees very well with the present-day elemental ratio of $\sim$330 proposed in  and thus supports the prediction that chemical fractionation is not an efficient process even in cold pre-stellar cores. Further credence was provided by subsequent direct measurements in HC3N and HC5N toward similar sources <cit.>. The concordance between observations and models may be only an apparent one, however, because sharp deviations are recurrently obtained for N2H+ <cit.>.

The other fractionation process, selective photodissociation of N2, which is expected in unshielded gas, has been investigated in the context of protoplanetary disks and molecular clouds <cit.>. The opacity of N is far lower than that of N2, therefore more  atoms are released upon dissociation than . This leads to an atomic isotopic ratio that is lower than the bulk. Gas-phase and/or surface chemistry then propagates this deviation into other species. A  correlation (positive or negative) between the UV flux and the  is therefore expected in some species. For HCN in disks, the HCN/HC  is predicted to increase with decreasing UV flux.

In this Letter, we aim at providing the requisite evidence for two distinct N reservoirs within the disk that orbits the nearby TTauri star TW Hya through an analysis of the emission of HCN and isotopologs, based on archival ALMA data (2016.1.00629.S, Cleeves et al). We also explore the possibility of a radial gradient in the HCN/HC  to determine whether selective photodissociation is the driving fractionation process in PSN analogs.

§ OBSERVATIONS AND DATA REDUCTION

The (4-3) rotational transitions of HCN, H, and HC at 354.5, 345.4, and 344.2 GHz, respectively, have been mapped with the ALMA interferometer with a spectral resolution of 61 KHz or 0.053 (see Appendix <ref>). All the lines were observed simultaneously, thus mitigating cross-calibration biases. The standard pipeline of CASA 5.4.0 was used to generate the complex visibilities, which were then self-calibrated in Gildas[IRAM memo available at <http://www.iram.fr/IRAMFR/ARC/documents/filler/casa-gildas.pdf>.]. The final sensitivities are 2.3, 1.9, and 2.0 mJy/beam per 61 kHz channel for HCN, H, and HC respectively. The resulting integrated intensity maps are shown in tdv.

The HCN(4-3) spectra (see wspectra) contain two weak hyperfine (hf) transitions of equal intensity, each carrying 2.1% of the total flux, and shifted by -1.672 and +1.360 with respect to our reference frequency 354505.477 MHz. They surround a strong, non-Gaussian, central feature made of three overlapping hf transitions representing 95.8% of the flux. The intrinsically weaker intensity of H and HC compared to HCN leads to one single feature only, which in the case of H  carries 95.8% of the total flux. For HC, the splitting is entirely unresolved and the single feature thus carries 100% of the rotational transition flux (see spectro).

§ RESULTS

Spectra of HCN, H, and HC (from top to bottom) from the Keplerian mask at 2.83 (systemic velocity) and averaged within annuli centered at radii $r=$5 to 55 au. The specific intensity scale is K using K/(Jy/beam) conversion factors of 178.7, 188.1, and 189.0 for HCN, H, and HC, respectively.

The conversion of line flux into abundance ratios is a simple procedure provided the lines are optically thin and the excitation temperatures are known. Even for optically thin transitions, however, the overlap in velocity space of lines that are emitted at different locations in the disk, which is a result of the Keplerian rotation of the disk, complicates the conversion from flux into abundance ratios because this mixes different regions of the disk, broadens the lines, and may also weaken the assumption that the emitting species are cospatial. The latter is especially true in the case of species with very different abundances, such as nitrogen isotopologs. Mitigating both the optical depth and Keplerian blurring effects thus offers clear advantages.

The nearly face-on configuration of the TW Hya disk naturally reduces the Keplerian blurring. In addition, the contribution from different locations along the line of sight to the emerging lines is limited to the vertical direction. Moreover, the spectrally resolved hf multiplet of HCN provides two weak lines that are most likely optically thin. Their flux is thus directly proportional to their column density.

Because the emission of all three isotopologs is well detected and spatially resolved, one goal might be measuring the HCN isotopic ratios as a function of radius, provided that line broadening can be minimized. This implies that the spatial area over which spectra are averaged needs to be limited. Another complementary approach consists of deriving a spatially averaged isotopic ratio by measuring line flux ratios over the whole disk. This method provides higher sensitivity, but Keplerian line broadening must still be controlled. We applied both approaches and detail them in the following.

§.§ Spatially averaged line fluxes

In order to limit the Keplerian line-broadening, the spectra were averaged over several groups of pixels that were selected for their closely similar projected Keplerian velocity. In practice, a 2D mask was thus determined by selecting the disk area where the 354507.455 MHz hyperfine emission of HCN is higher than 40 mJy/beam in each velocity channel. Although the overlap of different Keplerian velocities is well controlled, it is still noticeable close to the disk center, and we further masked out the spectra located at radii smaller than 0.3. The resulting mask recovered the usual dipole-like pattern, but with a central hole due to the radial filter (see kmask). As a result, the spectra averaged within each Keplerian velocity mask have a high signal-to-noise ratio (see kspectra), allowing the two weak hf transitions of HCN to be properly fitted. The line fluxes of the two weak hf lines at 354503.869 and 354507.455 MHz were measured through Gaussian fitting and their averaged value, corrected for their summed relative intensity (4.2%), was then used as an estimator of the HCN flux (see kgauss).

§.§ Radially dependent fluxes

In order to measure the flux ratios as a function of distance from the central TTauri star and still avoid Keplerian line-broadening, a radial mask was superimposed on the previously described Keplerian masks. To calculate disk radial offsets in au, we adopted a distance of 59.5$\pm$0.9 pc to TW Hya <cit.>. Spectra contained within concentric annuli, centered at radii $r=5$ to 55 au in steps of 5 au (one-third of the half-power beam width, HPBW), and of constant thickness $\delta r=$5 au,  were averaged. The outer radius was imposed by the HC detection limit (see tdv). The number of spectra in each annulus increased with $r$, which compensated for the natural decrease of the signal-to-noise ratio with radius. In the process, spectra were weighted according to their flux rms. As in the spatially averaged approach, the line fluxes of HCN, H, and HC  were measured through Gaussian fitting.

Radially averaged spectra corresponding to the Keplerian mask at 2.83 are shown in wspectra. The velocity confusion at small radii is evident, while at radii above 20 au, it is efficiently removed by our masking procedure. Average spectra in all Keplerian channels are displayed in spectra_r_vk. While the peak intensity of HCN hardly changes beyond 20 au, the peak intensities of H and HC decrease when moving outward. Also, all lines become narrower as the radius increases (see fwhm). For HCN, the FWHM is 1.5$\pm0.7$  at $r\le0.2\arcsec$, decreasing to 0.34$\pm$0.02 at 0.9.

§ ISOTOPIC RATIOS

Left: HCN/HC and HCN/H isotopic ratios in each Keplerian channel derived from the spatially averaged procedure (see spatial). The colored bands indicate the average value $\pm1\sigma$. The spatially averaged CN/C ratio from  is shown as a hatched rectangle. Right: Radial dependence (see radial) of the HCN/H and HCN/HC ratios. Each box is 5 au wide (one-third of the HPBW) and $\pm1\sigma$ in height. The full line shows the best fit of a power law to the  ratio, with a slope of 1.3. The gray shaded areas are the present-day  and / elemental ratios in the local ISM (330$\pm$30 () and 65$\pm$10 <cit.>, respectively).

Isotopic ratios derived from our two approaches, the spatially averaged and the radially dependent methods (see also wratios).

HCN/HC    HCN/H    H/HC

Average                    223$\pm$21         86$\pm$4     2.6$\pm$0.2

$r$ (au)=20     121.4$\pm$10.6     85.6$\pm$ 5.2     1.4$\pm$0.1

25     140.4$\pm$14.6     84.2$\pm$ 5.7     1.7$\pm$0.1

30     176.4$\pm$24.1     75.9$\pm$ 7.0     2.4$\pm$0.3

35     248.7$\pm$24.6     77.9$\pm$ 5.5     3.2$\pm$0.4

40     307.8$\pm$14.2     81.2$\pm$ 4.8     3.7$\pm$0.3

45     338.7$\pm$28.2     82.5$\pm$ 4.2     4.2$\pm$0.4

50     334.9$\pm$55.0     85.9$\pm$ 5.9     3.9$\pm$0.7

55     256.4$\pm$91.2     88.7$\pm$11.1     2.9$\pm$1.1

Because the three molecules have very similar spectroscopic constants, their rotational flux ratios are equal to their column density ratios to within 1%, provided the lines are optically thin and have the same excitation temperature. Under these assumptions, which we verify below, we obtain the isotopic ratios directly from the fluxes. The results are shown in wratios and are listed in Tables <ref> and <ref>.

For HCN/HC, we find ratios between 185 to 252 across the Keplerian channels (typical uncertainty of 5-10%) with an average isotopic ratio $223\pm21$. Here and elsewhere, uncertainties on the absolute flux include a conservative 10% error (statistical) from amplitude calibration. The quoted uncertainty on average values is the largest of the weighted uncertainties and the dispersion of the averaged values. The lowest values (185, 186, and 198) are obtained in the three central Keplerian channels, and the average ratio in the eight remaining channels is 233$\pm$12. HCN/H is more uniform, with values ranging from 82 to 92, which averages to 86$\pm$4. The corresponding average of H/HC  is 2.6$\pm$0.2, with values ranging from 2.22 to 2.75. The low values of HCN/HC  might arise because the opacity of HCN is substantial even in its weak hf lines. However,  the almost constant HCN/H suggests that these lines are instead optically thin. Still, assuming a constant isotopic ratio of 233 and adopting the usual expression $\tau/(1-\exp(-\tau))$ for the opacity correction for the integrated intensity, with $\tau$ the opacity <cit.>, an opacity of 0.4 for the weakest hf lines, corresponding to a rotational opacity $\tau_{12}\approx 20$, would be needed to explain the flux ratios in these three channels. Alternatively, assuming a constant H/HC ratio of 2.7 based on the outer channels (see kratios), the total H opacity would be $\tau_{13}=0.3$ in the three middle channels. It is thus likely that the opacity of the weakest hf line of HCN(4-3) and of H(4-3) is $\approx 0.3$, which is not entirely negligible, but still fulfills our optically thin assumption. We note that in these three channels, the HCN/H would change by less than 4% if these corrections were applied.

The radially dependent approach shows that the HCN-to-HC isotopic ratio varies with radius while HCN/H remains constant within 1$\sigma$. More specifically, the former ratio increases from 121 at 20 au to 339 at 45 au, and may decrease at larger distances. The nitrogen isotopic ratio in HCN close to the star is thus a factor 2.7 lower than the spatially averaged CN/C, 323$\pm$30, whereas between 45 to 50 au, the two ratios become consistent within 1$\sigma$. The ratios change by less than 4% when the three central Keplerian channels are excluded. The best fit of a power law to the HCN/HC gives 117$\pm$10$\times(r/20\,\au)^{1.3\pm0.13}$, indicating a shallower dependence with radius than a pure geometrical dilution of the UV flux from the central star ($\propto r^{-2}$).

A key assumption in our analysis is that a single excitation temperature characterizes the weak hf lines of HCN and the overlapping hf lines of H and of HC. However, line overlap may lead to unequal excitation temperatures among the hf manifold of HCN <cit.>. We have tested our assumption (see Appendix <ref>) by computing the excitation temperatures of HCN, H, and HC lines while taking line overlap in the excitation into account. Our calculations assume uniform physical conditions and cover the density and gas temperature regimes from 6 to 11 and 5 to 30 K, respectively. We found that deviations from a single excitation temperature are only obtained at densities below 37 (see tex). On the other hand, the full width at half-maximum (FWHM) of the HCN, H, and HC lines at a radius of 0.6 ($\approx$35 au) and beyond is $\approx$0.35, which places an upper limit on the kinetic temperature of $(FWHM/2.35)^2\times\mu \mh/(2k_B)\approx$35 K <cit.>. Referring to the TW Hya model of <cit.>, this upper limit indicates that the density is at these radii higher than 9, which means that it is in a regime where lines are expected to be thermalized. This validates our basic assumption. Our upper limit on the kinetic temperature is also well below the gas temperature of $\sim200$ K where the HCN abundance is the highest in the generic models of <cit.>.

§ DISCUSSION

The spatially averaged directly measured CN/C and HCN/HC  in the TW Hya circumstellar disk provide clear-cut confirmation of at least two isotopic reservoirs of nitrogen, as anticipated by . In CN, the ratio is 1.4$\pm$0.2 times higher than in HCN, and their difference is 90$\pm$36. Based on the $\sim$8 Myr age of TW Hya, these results indicate that multiple isotopic reservoirs are present at the key protoplanetary disk evolutionary stages when comets and giant planets are likely forming.

In the comet formation zone between 20 and 30 au, the HCN/HC  is 121$\pm$11, which is a factor of 2.8 compared to the outer disk at 55 au, where the isotopic ratio of HCN matches the revised elemental ratio of 330 well. The -enrichment in the region inside of 30 au is reminiscent of the threefold -enrichment of comets compared to the bulk proto-Sun, providing a strong indication that HCN traces the fractionated reservoir that is recorded by comets. <cit.> also found marginal evidence for an increase of the HCN-based  ratio with radius in the inner $\sim$50 au of the V4046 Sgr disk. We also note that the inner disk ratio confirms the mean HCN/HC ratio of 111$\pm$19 () that was obtained indirectly by <cit.> assuming HCN/H=70$\pm20$. Adopting our measured averaged of 86 for the latter would bring the former ratio to 136$\pm$23. We also very tentatively observe a decrease of the ratio beyond 55 au, which could indicate chemical processes in a gas not exposed to strong UV radiation and where depletion is limited. Measurements with a high signal-to-noise ratio beyond 55 au are needed to confirm this behavior, however.

The average HCN/H, $86\pm4$, is significantly higher than the / ratio of $65\pm10$ in the local ISM <cit.>, but agrees surprisingly well with the value of 89$\pm1$ in the PSN <cit.>. This indicates that HCN is slightly depleted in  in TW Hya, opposite to what was found in the L1498 pre-stellar core <cit.>, suggesting that the underlying process may not be the same in these two environments. The depletion of HCN in  is at variance with the models of <cit.>, however, which instead predict that HCN is enriched in  at radii below $\sim$100 au. Optically thick H emission could artificially increase the HCN/H provided that the weak HCN hf transition remains thin. However, the H opacity required to decrease the ratio to the elemental value is 0.55, which is higher than the value of 0.3 derived previously for the central Keplerian channels. This also implies that the total opacity of HCN would be 35, or 0.75 for the weakest HCN hf, which is twice higher than the value of 0.3 that is required to bring the spatially averaged HCN/HC  to a constant value (see Sect. <ref>). We note that high / ratios have already been measured in carbon monoxide from ices toward young stellar objects <cit.>.

The increasing HCN/HC with radius may indicate that selective photodissociation represents the dominant fractionation process in disks <cit.>. In these models, the HCN-to-HC column density ratio first decreases with radius before it increases back to the elemental ratio as a consequence of UV flux attenuation that is due to a combination of geometrical dilution and dust extinction. We note that the minimum ratio in the grid of models of <cit.> is $\approx$240–280 for disks around TTauri stars, which would translate into 180–210 when 330 (rather than 441) is assumed for the elemental ratio. This is still higher than the lowest ratio reported here and might indicate that the UV flux is underestimated in these models. However, dedicated models of TW Hya are required because the precise radial dependence of the isotopic ratio strongly depends on the physical parameters of the model (disk mass, grain size, and dust-to-gas mass ratio).

Overall, the increasing HCN/HC  from 121 up to the bulk value in the outer disk supports a scenario in which a single reservoir of interstellar, not fractionated, nitrogen dominates at large radii in PSN analogs. This nitrogen reservoir is enriched in  by the selective photodissociation of N2 at smaller distance from the central star.

PHB thanks D. Petry for his help in processing the ALMA data, and A. Faure for useful comments and discussions. We also thank the referee and the editor for valuable comments that greatly clarified the paper. This study makes use of the ADS/JAO.ALMA#2016.1.00629.S data. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. This work has received financial support from the Institut Universitaire de France and from Labex OSUG@2020. JHK's research on protoplanetary disks orbiting young stars near Earth is supported by NASA Exoplanets      Research Program grants NNX16AB43G and 80NSSC19K0292 to RIT. This study made use of the CDMS database <cit.>.

§ OBSERVATIONS

The observations were performed with the ALMA interferometer (2016.1.00629.S, PI:I. Cleeves) and were spread over six epochs (2016 December 30, 2017 July 5, 2017 July 10, 2017 July 15, 2017 July 21, and 2017 July 22). J1037-2934 was used as a bandpass, flux, and phase calibrator. The phase was also calibrated with J1103-3251, and J1058+0133 was also used for bandpass calibration.

H and HC were tuned in the lower side-band and HCN in the upper side-band, allowing simultaneous observations for the three species. We used the standard pipeline of CASA 5.4.0 to generate the first sets of complex visibilities, which were then exported into the Gildas format using the Gildas-Casa filer of Guilloteau, Chapillon et al. (see the IRAM memo available at <http://www.iram.fr/IRAMFR/ARC/documents/filler/casa-gildas.pdf>). Self-calibration was then applied to generate the final set of visibilities.

Spectroscopy shows that the three hyperfine (hf) transitions of HCN(4-3) are separated by less than 0.13 and are not resolved. Their total relative intensity is 95.8%. The hf transition at 354505.846 MHz is not detected given its 0.03% relative strength.  The properties of the hf lines of HCN and H are summarized in spectro, while for HC, the unresolved hyperfine splitting is not given. We also show in kmask channel maps of HCN strong central feature, H, and HC. The velocity range covers the main and overlapping hf lines of HCN.

Spectroscopic information on the observed transitions.

Rest Freq.$^a$     \(\delta v\)$^b$     $^c$     $g_u$$^d$     R. I.$^e$

354503.869      1.360     1.284 (-4)     9     2.083 (-2)

354505.367      0.094     1.886 (-3)     7     2.381 (-1)

354505.477      0.000     1.925 (-3)     9     3.125 (-1)

354505.523     -0.039     2.054 (-3)    11     4.075 (-1)

354505.846     -0.312     2.620 (-6)     7     3.307 (-4)

354507.455     -1.672     1.650 (-4)     7     2.083 (-2)

345338.160     -1.398      1.190(-4)      9      2.089(-2)

345339.660     -0.095      1.740(-3)      7      2.376(-1)

345339.770      0.000      1.780(-3)      9      3.125(-1)

345339.810      0.035      1.900(-3)     11      4.077(-1)

345340.140      0.321      2.420(-6)      7      3.305(-4)

345341.750      1.719      1.530(-4)      7      2.089(-2)

344200.109      0.000       1.8793(-3)     9     1

Numbers are written in the form $a(b) = a\tdix{b}$.

$^a$ Rest frequency.
$^b$ $\delta v$ is the velocity shift of each hf transition within a given multiplet.

$^c$ Coefficient for spontaneous decay.

$^d$ Upper level total degeneracy.

$^e$ The relative intensity of each hf line, normalized to a sum of 1.

§ KEPLERIAN MASKS

§.§ Results from the spatially averaged procedure

In kmask (top row), we show the Keplerian mask defined as explained in the main text. Each mask is overlaid in the channel maps of the three isotopologs, showing the usual dipole-like pattern <cit.>. Comparing the rows showing the main feature and the weak hf line of HCN, we note that the extended HCN emission is filtered out in the present study, as is the case of H, although to a lesser extent. The spectra averaged within each mask, shown in kspectra, were fit with Gaussian profiles, the results of which are summarized in kgauss. The obtained fluxes are then used to derive the spatially averaged isotopic ratios listed in kratios.

§.§ Radial dependence procedure

By combining two masks, one in velocity space (Keplerian mask) and one in the radial direction, we are able to study the radial dependence of the  isotopic ratio in HCN toward the TW Hya system. In wspectra, we show the spectra averaged within each annulus, but we concentrate on the central Keplerian channel at 2.83 (systemic velocity). Figure <ref> is complementary in that it shows the spectra averaged at three radii, but within all Keplerian masks.

As in the radially averaged approach, the spectral line intensity and shape at any radius  do not change appreciably in the various Keplerian velocity channels. In contrast, the dependence on radius is significant, especially for H and most importantly for HC. As can be seen in fwhm and as mentioned in the main text, all lines become narrower as the radius increases (see fwhm).

Keplerian masks (top row) overlaid on (from top to bottom) the HCN (main feature and weak hf line at 345507.455 MHz), H, and HC, channel maps. The masks (top row) were obtained for a threshold of 40 mJy/beam applied to the weak hf transition at 354507.455 MHz.

Spatially averaged spectra (Jy/beam) within each Keplerian velocity channel. HCN, H, and HC are shown from top to bottom. The Keplerian channels are those of kmask.

Spectra of HCN (black), H (red, $\times 5$), and HC (green, $\times 10$) averaged within each Keplerian velocity channel (see, e.g., kgauss) and within annuli of constant thickness 5 au, centered at 30, 40, and 50 au (from top to bottom). The specific intensity scale is K using K/(Jy/beam) conversion factors of 178.7, 188.1, and 189.0 for HCN, H, and HC, respectively. The spectra are shifted vertically for the sake of legibility.

Results of the Gaussian fits to the spatially averaged spectra of kspectra.

$v_K-v_0^\S$    2HCN$^a$     2H$^b$     2HC$^c$

$W^\S$     $v^\S$
$W^\S$     $v^\S$
$W^\S$     $v^\S$

-0.32     779.2(12.0)    2.515(1)       9.8(1)     2.494(3)     3.3(1)     2.491(5)

-0.26     803.2(13.1)    2.568(1)       9.8(2)     2.541(3)     3.6(1)     2.536( 7)

-0.21     835.1(13.9)    2.634(1)       9.6(2)     2.612(4)     3.7(2)     2.611(10)

-0.16     871.0(12.7)    2.707(1)     10.1(2)     2.698(3)     3.7(2)     2.711(11)

-0.10     884.6(12.2)    2.753(1)     10.6(2)     2.746(4)     4.1(2)     2.756(10)

-0.05     892.4(13.7)    2.799(1)     10.9(2)     2.788(5)     4.5(2)     2.784(11)

0.00     850.0(14.2)    2.856(1)     10.4(2)     2.834(4)     4.6(2)     2.806(11)

0.05     838.8(14.1)    2.899(1)     10.0(2)     2.877(4)     4.5(2)     2.840(12)

0.11     847.7(14.7)    2.956(1)       9.7(2)     2.937(4)     3.9(2)     2.921(12)

0.16     870.9(16.9)    3.010(1)       9.6(2)     2.988(4)     3.6(2)     2.990(10)

0.21     881.2(16.1)    3.067(1)       9.6(2)     3.035(3)     3.5(2)     3.048( 9)

0.27     870.2(15.5)    3.117(1)       9.6(2)     3.080(4)     3.5(2)     3.113(10)

0.32     859.7(14.2)    3.166(1)       9.8(2)     3.138(3)     3.7(2)     3.165(11)

Statistical uncertainties (at the 1$\sigma$ level) are written in parentheses in units of the last digit.

$^\S$ Keplerian minus systemic velocity (), $W$ is the integrated flux in mJy/beam , and $v$ the central velocity in .

$^a$ For HCN, the integrated intensity is computed as the total flux of the two weak lines divided by the sum of their relative intensities, 4.2%, while the center velocity is that of the main central component.

$^{b, c}$ The hf lines are not separated, and a single Gaussian was fit to the the main line.

Isotopic ratios from the spatially averaged spectra.

-0.32     779.2     12.0      9.8     0.1     3.3     0.1     79.5     1.5     236.1      8.0     2.97     0.09

-0.26     803.2     13.1      9.8     0.2     3.6     0.1     82.0     2.1     223.1      7.2     2.72     0.09

-0.21     835.1     13.9      9.6     0.2     3.7     0.2     87.0     2.3     225.7     12.8    2.59     0.15

-0.16     871.0     12.7     10.1    0.2     3.7     0.2     86.2     2.1     235.4     13.2    2.73     0.16

-0.10     884.6     12.2     10.6    0.2     4.1     0.2     83.5     2.0     215.8     10.9    2.59     0.14

-0.05     892.4     13.7     10.9    0.2     4.5     0.2     81.9     2.0     198.3      9.3     2.42     0.12

0.00     850.0     14.2     10.4    0.2     4.6     0.2     81.7     2.1     184.8      8.6     2.26     0.11

0.05     838.8     14.1     10.0    0.2     4.5     0.2     83.9     2.2     186.4      8.9     2.22     0.11

0.10     847.7     14.7      9.7     0.2     3.9     0.2     87.4     2.4     217.4     11.8    2.49     0.14

0.16     870.9     16.9      9.6     0.2     3.6     0.2     90.7     2.6     241.9     14.2    2.67     0.16

0.21     881.2     16.1      9.6     0.2     3.5     0.2     91.8     2.5     251.8     15.1    2.74     0.17

0.27     870.2     15.5      9.6     0.2     3.5     0.2     90.6     2.5     248.6     14.9    2.74     0.17

0.32     859.7     14.2      9.8     0.2     3.7     0.2     87.7     2.3     232.4     13.1    2.65     0.15

$\S$: Keplerian minus systemic velocity () is that of the hf line of HCN at 354507.455 MHz. $\dag$: $W_{12}$, $W_{13}$, and $W_{15}$ are the integrated fluxes (mJy/beam ) derived from Gaussian fitting (see also kgauss). $\ddagger$: integrated line flux ratios $W_{12}/W_{13}$, $W_{12}/W_{15}$, and $W_{13}/W_{15}$. All uncertainties are statistical and at the 1$\sigma$ level.

§.§ Radially dependent line width

Gaussian fits to Keplerian-filtered spectra averaged within concentric annuli of thickness 0.1 were performed in order to measure the dependence of the FWHM of the HCN, H, and HC lines. The results are summarized in fwhm. They are used to infer an upper limit on the kinetic temperature following  and <cit.>.

Radial dependence of the line width (FWHM, ) of HCN, H, and HC.

$r$ ()

0.3     0.954     0.183     0.832     0.101     1.231     0.140

0.4     0.614     0.060     0.603     0.038     0.687     0.043

0.5     0.461     0.020     0.458     0.034     0.479     0.071

0.6     0.389     0.018     0.382     0.020     0.355     0.036

0.7     0.370     0.013     0.369     0.020     0.316     0.048

0.8     0.354     0.014     0.371     0.016     0.334     0.053

0.9     0.345     0.016     0.367     0.030     0.354     0.058

Spectra were averaged in annuli of thickness 0.1. $r$ is the outer radius of each annulus.

§ SINGLE EXCITATION TEMPERATURE ASSUMPTION

The main assumption in our analysis is that of a single excitation temperature characterizing the weak hf lines of HCN and the overlapping hf lines of H and of HC. A thorough investigation of this assumption can only be made with two-dimensional radiative transfer taking into account the overlap of hf transitions in the level population evaluation, which in turn requires a complete physical and kinematic structure of the disk. This type of analysis has been made in the context of pre-stellar starless cores by <cit.>, in spherical geometry, with the 1D ALICO model <cit.>.

In the present context, we were unable to make detailed calculations like this. Instead, we used our 1D ALICO code to investigate the density and kinetic gas temperature at which departures from the single excitation temperature assumption may be obtained. To do so, the level populations were computed while treating the effect of hf line overlap in the excitation of HCN, H, and HC. Our grid of models encompasses a range of  density (6 to 11) and kinetic temperature (from 5 to 30 K) relevant to the TW Hya orbiting disk. The results of these calculations in terms of  are shown in tex. There are significant departures at densities between 6 and 8. At higher density, all lines are thermalized (models at  higher than 9 are not shown). From the generic chemical models of <cit.>, it appears that HCN is concentrated at radii $r=5-50$ au and scale heights $z/r=0.2$ to 0.3, where the gas kinetic temperature is $\approx 200$K and the density in the range 6 to 8, which would indicate that the single excitation temperature assumption fails.

On the other hand, the FWHM of the HCN, H, and HC lines at a radius of 0.6 ($\approx$35 au) and beyond is $\approx 0.35$ (see fwhm), placing a conservative upper limit on the kinetic temperature of <cit.>

\begin{equation}
(FWHM/2.35)^2\times\mu \mh/(2k_B)\approx35 \rm\,K.
\end{equation}

Referring to the tailored TW Hya model of <cit.>, this upper limit, applied to the relevant range of radii, indicates that the density is higher than 9, in a regime where lines should be thermalized. This validates our single excitation temperature assumption. Our upper limit on the kinetic temperature is also well below the gas temperature of $\sim200$ K where the HCN abundance is the highest in the models of <cit.>, although we note that these models are generic and are not tailored to TW Hya.

Excitation temperature of the six hf lines of HCN (see spectro) and the strongest hf lines of H (red) and HC (green). The four overlapping hf lines of HCN are shown as open circles and the two weakest resolved lines with filled dots.


DE RHAM - WITT KZ EQUATIONS

Crystalline bozonization

Vadim Schechtman and Alexander Varchenko

December 5, 2022

Те, кому приходилось столкнуться с векторами Витта

в собственной работе, часто относятся к ним

с теплотой

Д. Каледин111See [K]

Introduction

In this note we propose a de Rham - Witt version of the derived KZ equations,
and of their hypergeometric realizations, cf. [SV2]. In particular we describe
a de Rham - Witt version of Theorems $7.2.5^{\prime}$, $7.2.5^{\prime\prime}$
from [SV1].

We also propose de Rham - Witt versions of some classical theorems related to
arbitrary hyperplane arrangements.

Contents of the paper

In §1 we recall some background material about the de Rham - Witt complex. In
§2 we present a de Rham - Witt version of the first part of [SV1] concerning
arbitrary hyperplane arrangements over $\mathbb{F}_{p}$. In 2.3.3 - 2.3.7 we
discuss also a motivic version as suggested by A.Beilinson. In §3 we present a
de Rham - Witt version of [SV2] which in its turn is a development of [SV1],
Part 2. For simplicity of exposition we discuss only the case
$\mathfrak{g}=\mathfrak{sl}_{2}$ but the generalization to arbitrary Kac -
Moody algebras is straightforward.

Acknowledgements

We are grateful to Alexander Beilinson for an interesting discussion and
consultations; the idea of 2.3.3 - 2.3.7 belongs to him, and the proofs are
obtained jointly with him. We express our gratitude to Hélène Esnault for
important corrections.

Our reflections on the topics of this paper have been triggered by a talk of
Rubén Muñoz-Bertrand; we are thankful to him.

§1. The de Rham - Witt complex

1.1. De Rham - Witt complex of a polynomial ring. Cf. [B], §2.

$p$ \- a prime number. $W(\mathbb{F}_{p})=\mathbb{Z}_{p},\
F=\text{Id},V=p\text{Id}$. Thus

$W_{n}(\mathbb{F}_{p}):=W(\mathbb{F}_{p})/V^{n}W(\mathbb{F}_{p})=\mathbb{Z}/p^{n}\mathbb{Z}.$

(a) Witt ring

Let $A:=\mathbb{F}_{p}[t_{1},\ldots,t_{n}]$.

We consider, together with S.Lubkin et al., cf. [L, I1, I2, B], a
(nonnoetherian) algebra

$C=\cup_{i\geq 0}\mathbb{Q}_{p}[t_{1}^{p^{-i}},\ldots,t_{n}^{p^{-i}}]$

1.1.1. Remark. $C$ is filtered by subalgebras

$C_{m}=\cup_{0\leq i\leq
m}\mathbb{Q}_{p}[t_{1}^{p^{-i}},\ldots,t_{n}^{p^{-i}}],$

and each

$C_{m}=\mathbb{Q}_{p}[t_{1}^{p^{-m}},\ldots,t_{n}^{p^{-m}}].$

We may also consider a bigger ring

$\hat{C}=\\{\sum_{(i_{1},\ldots,i_{n})\in\mathbb{N}[p^{-1}]}a_{i_{1},\ldots,i_{n}}t_{1}^{i_{1}}\ldots
t_{n}^{i_{n}}\\}$

(formal power series).

Both $C$ and $\hat{C}$ are equipped with a $\mathbb{Q}_{p}$-linear
automorphism $F$,

$F(t_{i})=t_{i}^{p},$

and we define $V:=pF^{-1}$.

Notations. For $i\in\mathbb{Z}[p^{-1}]\subset\mathbb{Q}\subset\mathbb{Q}_{p}$,

$d(i):=-v_{p}(i);$

for $I=(i_{1},\ldots i_{n})\in\mathbb{Z}[p^{-1}]^{n}$,

$|I|=\sum i_{k},$ $d(I)=\max\\{d(i_{k})\\}.$

The Witt ring $W(\mathbb{F}_{p}[t_{1},\ldots,t_{n}])$ is a subring of
$\hat{C}$, namely

$W(\mathbb{F}_{p}[t_{1},\ldots,t_{n}])=\\{\sum_{(i_{1},\ldots,i_{n})\in\mathbb{N}[p^{-1}]^{n}}a_{i_{1},\ldots,i_{n}}t_{1}^{i_{1}}\ldots
t_{n}^{i_{n}}|\ \forall I\ v_{p}(a_{I})\geq d(I),\
\lim_{|I|\longrightarrow\infty}a_{I}=0\\},$

cf. [L], VI (a). It is stable with respect to $F$ and $V$.

(b) De Rham complex $\Omega^{\bullet}_{C/\mathbb{Q}_{p}}$

Consider the de Rham complex $\Omega^{\bullet}_{C/\mathbb{Q}_{p}}$.

Each $m$-form $\omega\in\Omega^{m}_{C/\mathbb{Q}_{p}}$ may be uniquely written
as

$\omega=\sum_{1\leq i_{1}<\ldots<i_{m}\leq n}a_{i_{1},\ldots,i_{m}}(t)d\log
t_{i_{1}}\wedge\ldots d\log t_{i_{m}}$

where $a_{I}(t)\in C$.

Define automorphisms $F:\
\Omega^{m}_{C/\mathbb{Q}_{p}}\overset{\sim}{\longrightarrow}\Omega^{m}_{C/\mathbb{Q}_{p}}$
by

$F(\sum_{I}a_{i_{1},\ldots,i_{m}}(t_{1},\ldots,t_{n})d\log
t_{i_{1}}\wedge\ldots d\log t_{i_{m}})=$
$\sum_{I}a_{i_{1},\ldots,i_{m}}(t^{p}_{1},\ldots,t^{p}_{n})d\log
t_{i_{1}}\wedge\ldots d\log t_{i_{m}}.$

Compatibility with $d$:

We have

$dF=pFd$

As before, we put $V:=pF^{-1}$.

Thus we have two morphisms $F,W:\
\Omega^{\bullet}_{C/\mathbb{Q}_{p}}\longrightarrow\Omega^{\bullet}_{C/\mathbb{Q}_{p}}$
which satisfy:

$FV=VF=p,\ FdV=d,\ dF=pFd,\ Vd=pdV,\ $ $V(x\cdot Fy)=Vx\cdot y$

(c) De Rham - Witt complex

Let us say that $\omega\in\Omega^{m}_{C/\mathbb{Q}_{p}}$ is integral if all
coefficients of all $a_{I}(t)$ belong to $\mathbb{Z}_{p}$. Define subspaces

$E^{m}=\\{\omega\in\Omega^{m}_{C/\mathbb{Q}_{p}}|\ \omega,d\omega\text{ are
integral}\\},$

for example

$E^{0}=\cup_{i\geq
0}p^{i}\mathbb{Z}_{p}[t_{1}^{p^{-i}},\ldots,t_{n}^{p^{-i}}].$

They form a subcomplex $E^{\bullet}\subset\Omega^{\bullet}_{C/\mathbb{Q}_{p}}$
stable with respect to $F$ and $V$.

Define

$\operatorname{Fil}^{a}E^{m}=V^{a}E^{m}+d(V^{a}E^{m-1})\subset E^{m};$

these subspaces form a subcomplex $\operatorname{Fil}^{a}E^{\bullet}\subset
E^{\bullet}$, and we may consider the quotient complex

$W_{a}\Omega^{\bullet}_{A}=E^{\bullet}/\operatorname{Fil}^{a}E^{\bullet}.$

When $a\geq 1$ varies, these complexes form a projective system.

$W\Omega^{\bullet}_{A}=\lim_{\longleftarrow}W_{a}\Omega^{\bullet}_{A}.$

1.2. De Rham - Witt complex of Laurent polynomials. The construction is
similar, cf. [I2] II.2.

Consider the case of one variable.

$A=\mathbb{F}_{p}[t,t^{-1}],$ $C=\cup_{r\geq
0}\mathbb{Q}_{p}[t^{1/p^{r}},t^{-1/p^{r}}]$ $F:\
C\overset{\sim}{\longrightarrow}C,\ F(a(t,t^{-1}))=a(t^{p},t^{-p}),\
V=pF^{-1}.$

The de Rham complex $\Omega^{\bullet}_{C/\mathbb{Q}_{p}}$ has length $1$:

$0\longrightarrow
C=\Omega^{0}_{C/\mathbb{Q}_{p}}\longrightarrow\Omega^{1}_{C/\mathbb{Q}_{p}}\longrightarrow
0.$

Each form $\omega\in\Omega^{1}_{C/\mathbb{Q}_{p}}$ can be uniquely written as

$\omega=\sum_{i\in\mathbb{N}[p^{-1}]}a_{i}(t,t^{-1})d\log t,\ a_{i}\in C,$
$F:\
\Omega^{1}_{C/\mathbb{Q}_{p}}\overset{\sim}{\longrightarrow}\Omega^{1}_{C/\mathbb{Q}_{p}},\
$ $F(\sum_{i\in\mathbb{N}[p^{-1}]}a_{i}(t,t^{-1})d\log
t)=\sum_{i\in\mathbb{N}[p^{-1}]}a_{i}(t^{p},t^{-p})d\log t,$ $V=pF^{-1}.$

Let us say that $\omega\in\Omega^{m}_{C/\mathbb{Q}_{p}}$ is integral if all
coefficients of all $a_{I}(t)$ belong to $\mathbb{Z}_{p}$. Define subspaces

$E^{m}=\\{\omega\in\Omega^{m}_{C/\mathbb{Q}_{p}}|\ \omega,d\omega\text{ are
integral}\\},$

for example

$E^{0}=\cup_{i\geq 0}p^{i}\mathbb{Z}_{p}[t^{p^{-i}},t^{-p^{-i}}].$

They form a subcomplex $E^{\bullet}\subset\Omega^{\bullet}_{C/\mathbb{Q}_{p}}$
stable with respect to $F$ and $V$.

Define

$\operatorname{Fil}^{a}E^{m}=V^{a}E^{m}+d(V^{a}E^{m-1})\subset E^{m};$

these subspaces form a subcomplex $\operatorname{Fil}^{a}E^{\bullet}\subset
E^{\bullet}$, and we may consider the quotient complex

$W_{a}\Omega^{\bullet}_{A}=E^{\bullet}/\operatorname{Fil}^{a}E^{\bullet}.$

When $a\geq 1$ varies, these complexes form a projective system.

$W\Omega^{\bullet}_{A}=\lim_{\longleftarrow}W_{a}\Omega^{\bullet}_{A}.$

1.3. Shift. Given $z\in\mathbb{F}_{p}$, let

$A=\mathbb{F}_{p}[t,(t-z)^{-1}]=\mathbb{F}_{p}[t-z,(t-z)^{-1}].$

We introduce an algebra

$C=\cup_{r\geq 0}\mathbb{Q}_{p}[(t-z)^{1/p^{r}},(t-z)^{-1/p^{r}}],$

an isomorphism $F:\ C\overset{\sim}{\longrightarrow}C$,

$\ F(a(t-z,(t-z)^{-1}))=a((t-z)^{p},(t-z)^{-p}),\ $

and an endomorphism $V:C\longrightarrow C$,

$V=pF^{-1}.$

Then we proceed as before: $1$-forms will have the form

$\omega=\sum_{i\in\mathbb{N}[p^{-1}]}a_{i}(t-z,(t-z)^{-1})d\log(t-z),$

etc.

1.4. Laurent polynomials of several variables. Let
$I\subset[n]=\\{1,\ldots,n\\}$, and

$A=\mathbb{F}_{p}[t_{1},\ldots,t_{n},t_{i}^{-1},i\in I]$

We proceed as before, starting with an algebra

$C=\cup_{r\geq
0}\mathbb{Q}_{p}[t_{1}^{1/p^{r}},\ldots,t_{n}^{1/p^{r}},t_{i}^{\pm
1/p^{r}},i\in I],$

etc.

The multiplicative group $A^{*}$ is a direct sum of $\mathbb{F}_{p}^{*}$ and
of a free abelian group generated by $t_{i},\ i\in I$. The Teichmüller
homomorphism

$T:\ A^{*}\longrightarrow W(A)^{*}\subset\hat{C}^{*}$

sends $t_{i}$ to $t_{i}\in\hat{C}^{*}$.

1.5. More generally, we will be interested in the de Rham - Witt complex of
the ring

$A=\mathbb{F}_{p}[t,z_{1},\ldots,z_{n},(t-z_{1})^{-1},\ldots(t-z_{n})^{-1}].$

Similarly to the previous considerations, we introduce a ring

$C=\cup_{r\geq 0}\mathbb{Q}_{p}[t^{1/p^{r}},(t-z_{1})^{\pm
1/p^{r}},\ldots(t-z_{n})^{\pm 1/p^{r}}]$

and its completion $\hat{C}$.

The complex $W\Omega^{\bullet}_{A}$ will be a subcomplex of
$\Omega^{\bullet}_{\hat{C}/\mathbb{Q}_{p}}$.

§2. Du côté de chez Orlik - Solomon

2.1. Orlik - Solomon subalgebra, aka fixed points of Frobenius.

Notation. Given a whatever object $?$ acted upon by a Frobenius $F$, the
superscript $?^{F-1}$ will denote $\\{x\in?|\ Fx=x\\}$.

2.1.1. Let $A=\mathbb{F}_{p}[t,t^{-1}]$. Consider the fixed submodule of
Frobenius

$W\Omega^{\bullet F-1}_{A}\hookrightarrow W\Omega^{\bullet}_{A}.$ $None$

We have

$W\Omega^{0F-1}_{A}=\mathbb{Z}_{p},\ W\Omega^{1F-1}_{A}=\mathbb{Z}_{p}d\log
t,$

the differential in $\Omega^{\bullet F-1}_{A}$ being zero.

The embedding (2.1.1) is a quasiisomorphism.

2.1.2. A more general case of 1.4 is similar: for

$A=\mathbb{F}_{p}[t_{1},\ldots,t_{n},t_{i}^{-1},i\in I]$

the subcomplex $W\Omega^{\bullet F-1}_{A}$ is a free exterior
$\mathbb{Z}_{p}$-algebra with generators $d\log t_{i},i\in I$ and zero
differential,

$W\Omega^{\bullet F-1}_{A}=\Lambda^{\bullet}_{\mathbb{Z}_{p}}(\oplus_{i\in
I}\mathbb{Z}_{p}d\log t_{i}).$

It will be called the Orlik - Solomon subalgebra, and denoted also
$W\Omega^{\bullet OS}_{A}$.

The embedding

$W\Omega^{\bullet OS}_{A}\hookrightarrow W\Omega^{\bullet}_{A}.$ $None$

is a quasiisomorphism.

2.2. This is a general phenomenon, valid for any hyperplane arrangement.
Namely, consider an affine space

$\mathbb{A}^{n}_{\mathbb{F}_{p}}=\operatorname{Spec}\mathbb{F}_{p}[t_{1},\ldots,t_{n}]$

and a finite collection $\mathcal{H}=\\{H_{i}\\}_{i\in I}$ of affine
hyperplanes in it, $H_{i}$ being given by an equation $f_{i}(t)=0$.

Let

$U=\mathbb{A}^{n}_{\mathbb{F}_{p}}\setminus\bigcup_{i\in
I}H_{i}=\operatorname{Spec}A$

where

$A=\mathbb{F}_{p}[t_{1},\ldots,t_{n},f_{i}^{-1},\ i\in I].$

We will use both notations $W\Omega^{\bullet}_{A}=W\Omega^{\bullet}(U)$.

2.2.1. Fixed points of Frobenius are closed. If $x\in W\Omega^{i}_{A}$, $x=Fx$
then

$dx=dFx=pFdx,$

i.e.

$(1-pF)dx=0.$

But the operator $1-pF$ is invertible:

$(1-pF)^{-1}=\sum_{i\geq 0}p^{i}F^{i},$

and the series in the RHS converges. Whence $dx=0$.

We’ve got an embedding of complexes

$os:W\Omega^{\bullet OS}_{A}:=W\Omega^{\bullet F-1}_{A}\hookrightarrow
W\Omega^{\bullet}_{A}.$ $None$

where the first one is equipped with the zero differential.

2.2.2. Logarithmic forms.  Let
$H\subset\mathbb{A}=\mathbb{A}^{n}_{\mathbb{F}_{p}}$ be a hyperplane given by
an equation $f(t)=0$. Let us choose new coordinates $u_{1},\ldots,u_{n}$ in
$\mathbb{A}$ such that in new coordinates $H$ will be given by the equation
$u_{n}=0$. Thus we get an isomorphism

$\phi:\ \mathbb{A}\setminus
H\overset{\sim}{\longrightarrow}\operatorname{Spec}\mathbb{F}_{p}[u_{1},\ldots,u_{n-1},u_{n}^{\pm
1}]$

We may apply 1.4 to the target and get closed $1$-forms

$d\log u_{n}\in W\Omega^{1}(\mathbb{F}_{p}[u_{1},\ldots,u_{n-1},u_{n}^{\pm
1}]),$

whence

$d\log f:=\phi^{*}d\log u_{n}\in W\Omega^{1}(\mathbb{A}\setminus H).$

Otherwise one can define $d\log f$ as in [I2] 3.23, as

$d\log f=d\tilde{f}/\tilde{f}$

where $\tilde{f}\in W\mathcal{O}(\mathbb{A}\setminus H)$ is the Teichmüller
representative of $f$.

2.2.3. Returning to our arrangement, we obtain closed $1$-forms $d\log
f_{i}\in W\Omega^{1}_{A}$; by construction they belong to
$W\Omega^{1F-1}_{A}$.

Compare with [I2] CH. I, (3.23).

According to op. cit. I, Thm. 5.7.2

$W\Omega^{\bullet F-1}_{U}\subset W\Omega^{\bullet}_{U}$

is the same as the subsheaf generated by logarithmic forms.

2.3. Orlik - Solomon. Recall that the Orlik - Solomon algebra
$\mathcal{A}(\mathcal{H})$, cf. [OS], §2, is an $\mathbb{N}$-graded skew
commmmutative ring with $\mathcal{A}(\mathcal{H})_{0}=\mathbb{Z}$, the
quotient of the exterior algebra with generators $(H_{i}),i\in I$, of degree
1, modulo the ideal generated by the elements

$(H_{1},\ldots,H_{p}):=H_{1}\wedge\ldots\wedge H_{p}$

if $H_{i}$ are not in general position, and

$\sum_{i=1}^{p+1}(-1)^{i}(H_{1},\ldots,\hat{H}_{i},\ldots,H_{p+1})=0$

if $H_{1},\ldots,H_{p+1}$ are not in general position and $H_{1}\cap\ldots\cap
H_{p+1}\neq\emptyset$, cf. [SV1], 1.2.

Multiplication is given by concatenation.

Recall that all $\mathcal{A}(\mathcal{H})_{p}$ are free abelian groups.

2.3.1. Proposition. There exists a unique algebra map

$\psi:\ \mathcal{A}(\mathcal{H})\longrightarrow W\Omega^{\bullet
OS}_{A}=W\Omega^{\bullet OS}(U)$

such that $\psi(H_{i})=d\log f_{i}$. $\square$

Integer structure

Let

$W\Omega^{\bullet OS}(U)_{\mathbb{Z}}=\operatorname{Im}(\psi)\subset
W\Omega^{\bullet OS}(U)$

denote the $\mathbb{Z}$-subalgebra generated by $d\log f_{i}$.

2.3.2. Proposition. The map $\psi$ induces an isomorphism

$\psi:\
\mathcal{A}(\mathcal{H})\overset{\sim}{\longrightarrow}W\Omega^{\bullet
OS}(U)_{\mathbb{Z}},$

whence an isomorphism

$\psi_{\mathbb{Z}_{p}}:\
\mathcal{A}(\mathcal{H})\otimes\mathbb{Z}_{p}\overset{\sim}{\longrightarrow}W\Omega^{\bullet
OS}(U).$

Proof the same as in [OS], Thm. 5.2. $\square$

Motivic variation: Milnor $K$-theory and Orlik - Solomon

2.3.3.  As was suggested by Beilinson, we may consider a multiplicative
version of the above.

At the beginning we may replace $\mathbb{F}_{p}$ by an arbitrary base field
$k$.

Let

$k(A)=k(U)=k(\mathbb{A}^{n}_{k})=k(t_{1},\ldots,t_{n})$

be the field of rational functions on $\mathbb{A}^{n}_{k}$. Let
$K_{\bullet}^{M}(k(U))$ be the algebra of Milnor $K$-functors. Thus

$K_{0}^{M}(k(U))=\mathbb{Z},\ K_{1}^{M}(k(U))=k(t_{1},\ldots,t_{n})^{*},$

and $K_{\bullet}^{M}(k(U))$ is a an $\mathbb{N}$-graded skew commutative
algebra generated by $K_{1}^{M}(k(U))$ subject to the Steinberg relation

$\\{x,1-x\\}=0,\ x,1-x\in K_{1}^{M}(k(U)),$ $None$

cf. [M].

Let

$\tilde{K}_{1}^{M}(U)\subset K_{1}^{M}(k(U))$

be the subgroup generated by $f_{i}$; it is a free abelian group with a base
$\\{f_{i},i\in I\\}$. Let

$\tilde{K}_{\bullet}^{M}(U)\subset K_{\bullet}^{M}(k(U))$

be the $\mathbb{Z}$-subalgebra generated by $\tilde{K}_{1}^{M}(U)$.

We have a map

$\chi_{\mathbb{Z}1}:\
\tilde{K}_{1}^{M}(U)\longrightarrow\mathcal{A}_{1}(\mathcal{H})$

sending $f_{i}$ to $(H_{i})$.

2.3.3.1. Claim. The map $\chi_{\mathbb{Z}1}$ extends to an algebra map

$\chi_{\mathbb{Z}}:\
\tilde{K}_{\bullet}^{M}(U)\longrightarrow\mathcal{A}(\mathcal{H})$ $None$

2.3.3.2. Example. Identity "chère à Gelfand": magic of the Steinberg relation.
Consider an arrangement $\mathcal{H}$ in
$\mathbb{A}^{2}_{k}=\operatorname{Spec}k[x,y]$ consisting of $3$ lines

$\ell_{1}:\ x=0,\ \ell_{2}:\ y=0,\ \ell_{3}:\ x=y,$

so

$U=\mathbb{A}^{2}_{k}\setminus\bigcup_{i=1}^{3}\ell_{i}.$

We have a familiar relation

$\frac{1}{x(x-y)}-\frac{1}{y(x-y)}=-\frac{1}{xy},$

and its counterpart in $\mathcal{A}(\mathcal{H})$. Let us write down its
counterpart in $K_{2}^{M}(U)$.

We have the following equalities in $K_{2}^{M}(k(U))\otimes\mathbb{Z}[1/2]$:

$\\{x,x-y\\}=\\{x,x\\}+\\{x,1-x^{-1}y\\}=\\{x,1-x^{-1}y\\};$
$\\{y,x-y\\}=\\{y,xy^{-1}-1\\}=\\{y,xy^{-1}\\}+\\{y,1-x^{-1}y\\}=$
$=\\{y,x\\}+\\{y,y^{-1}\\}+\\{y,1-x^{-1}y\\}=\\{y,x\\}+\\{y,1-x^{-1}y\\}.$

Here we have used the equality $\\{y,y\\}=0$; by that reason we had to
tensorise by $\mathbb{Z}[1/2]$.

On the other hand

$\\{x,1-x^{-1}y\\}=\\{x,1-x^{-1}y\\}+\\{x^{-1}y,1-x^{-1}y\\}=\\{xx^{-1}y,1-x^{-1}y\\}=\\{y,1-x^{-1}y\\},$

whence

$\\{x,x-y\\}-\\{y,x-y\\}=-\\{y,x\\}=\\{x,y\\}.$

This is the Gelfand identity in $K_{2}^{M}$. $\square$

2.3.4. Conjecture (A.Beilinson) . The map $\chi_{\mathbb{Z}}$ (2.3.2) is an
isomorphism.

The map $\chi_{\mathbb{Z}}$ composed with $\psi$ (see Prop. 2.3.2) gives rise
to a map

$d\log:\ \tilde{K}^{M}_{\bullet}(U)\longrightarrow W\Omega^{\bullet
OS}(U)_{\mathbb{Z}}$ $None$

sending $f_{i}$ to $d\log f_{i}$.

2.3.5. Motivic Orlik - Solomon. A canonical map

$H^{\bullet}_{Mot}(U;\mathbb{Q})^{\operatorname{diag}}\longrightarrow
H^{\bullet}_{Mot}(U;\mathbb{Q})$

is an isomorphism.

Here $H^{\bullet}_{Mot}(U;\mathbb{Q})$ denotes the motivic cohomology realized
as appropriate eigenspaces of Adams operations in the Quillen $K$-theory, see
[Be], 2.2, and $H^{\bullet}_{Mot}(U;\mathbb{Q})^{\operatorname{diag}}$ is its
part consisting of all elements of type $(i,i)$.

Proof. Our assertion says that all elements of
$H^{\bullet}_{Mot}(U;\mathbb{Q})$ are of type $(i,i)$. This may be proven by
induction on $n$. $\square$

2.3.6. Instead of rational motivic cohomology we could use more refined
version, the Voevodsky’s motivic cohomology, [MVW].

It is also probable that a reduced version from [ES] works for $U$.

Let us return to the de Rham - Witt realization.

2.4. Theorem. The embedding $os$ (2.2.1) is a quasiisomorphism.

Proof the same as in [OS], Thm. 5.2 which uses [Br] Lemme 5. Brieskorn uses
the Lefschetz theorems on hyperplane sections which hold true for the
crystalline cohomology. $\square$

2.5. Turning on a connection, "Aomoto conjecture". Given a collection of
exponents $a_{i}\in\mathbb{Z}_{p},\ i\in I$, we get a closed one-form

$\omega=\sum_{i}a_{i}d\log f_{i}\in W\Omega^{1OS}_{A},$

and a connection

$\nabla=d+\omega\wedge\bullet:W\Omega_{A}^{i}\longrightarrow
W\Omega_{A}^{i+1}.$

which restricted to $W\Omega^{\bullet OS}_{A}$ is simply the multiplication by
$\omega$.

2.6. Theorem (Aomoto conjecture, a weak form). For sufficiently small $a_{i}$
(in the $p$-adic metric) the embedding

$(W\Omega^{\bullet
OS}_{A},\omega)\hookrightarrow(W\Omega^{\bullet}_{A},d+\omega)$

is a quasiisomorphism.

Proof. The same as in [SV1] Thm 4.6. $\square$

2.7. It is probable that a strong form of the Aomoto conjecture similar to
[ESV] is valid as well.

§3. De Rham - Witt Coulomb - KZ - Chevalley cocycle

3.1. Consider a simple example. Suppose that $p>2$. Suppose we are given $n$
numbers $a_{i}\in\mathbb{Z}_{p}$, $1\leq i\leq n$. Consider an affine space

$\mathbb{A}=\operatorname{Spec}\mathbb{F}_{p}[t,z_{1},\ldots,z_{n}]$

and a hyperplane arrangement in it consisting of hyperplanes $H_{i}:t=z_{i},\
H_{ij}:z_{i}=z_{j}$.

Consider the complement

$U=\mathbb{A}\setminus((\bigcup_{i}H_{i})\cup(\bigcup_{i,j}H_{ij}))=\operatorname{Spec}\mathbb{F}_{p}[t,z_{1},\ldots,z_{n},(t-z_{i})^{-1},(z_{i}-z_{j})^{-1}]$

We have a Coulomb $1$-form

$\omega=-\sum_{i}2a_{i}d\log(t-z_{i})+\sum_{i,j}a_{i}a_{j}d\log(z_{i}-z_{j})\in
W\Omega^{1}(U)$

Construction of a hypergeometric cocycle

We follow the pattern of [SV2], Section 2.

3.2. Coulomb connection.  Suppose we are given natural $n,N\geq 1$, and an
$n$-tuple

$\mathfrak{m}=(m_{1},\ldots,m_{n})\in\mathbb{Z}_{p}^{n}.$

Let

$U_{n}=\operatorname{Spec}\mathbb{F}_{p}[t_{1},\ldots,t_{N},z_{1},\ldots
z_{n},(z_{b}-z_{c})^{-1}],$ $U_{n,N}=\operatorname{Spec}A,\
A=\mathbb{F}_{p}[t_{1},\ldots,t_{N},z_{1},\ldots
z_{n},(t_{i}-t_{j})^{-1},(t_{i}-z_{b})^{-1},(z_{b}-z_{c})^{-1}].$

We define the de Rham - Witt Coulomb closed $1$-form

$\omega_{\mathfrak{m}}=\sum_{1\leq b<c\leq
n}\frac{m_{b}m_{c}}{2}d\log(z_{b}-z_{c})+\sum_{1\leq i<j\leq
N}2d\log(t_{i}-t_{j})-$ $-\sum_{1\leq i\leq N,1\leq b\leq
n}m_{b}d\log(t_{i}-z_{b})\in W\Omega^{1}(A)=W\Omega^{1}(U_{n,N}).$

Given an invertible $\kappa\in\mathbb{Z}_{p}$, we define a Coulomb connection

$\nabla_{Coul,KZ}=d+\frac{1}{\kappa}\omega_{\mathfrak{m}}:\
W\Omega^{\bullet}(U_{n,N})\longrightarrow W\Omega^{\bullet+1}(U_{n,N}).$

3.3. Chevalley complex. Let $\mathfrak{g}$ be the Lie algebra
$\mathfrak{sl}_{2}$ over $\mathbb{Z}_{p}$ with generators $e,f,g$,
$\mathfrak{n}=\mathbb{Z}_{p}f\subset\mathfrak{g}$.

For $m\in\mathbb{Z}_{p}$ let

$M(m)=\oplus_{i\geq 0}\mathbb{Z}_{p}f^{i}v$

be the Verma module, $ev=0$; let

$M(\mathfrak{m})=M(m_{1})\otimes_{\mathbb{Z}_{p}}\ldots\otimes_{\mathbb{Z}_{p}}M(m_{n})=\oplus_{i\geq
0}M(\mathfrak{m})_{i}$

We consider a Chevalley complex

$C^{\bullet}(M(\mathfrak{m})):\ 0\longrightarrow
M(\mathfrak{m})\overset{d}{\longrightarrow}\mathfrak{n}\otimes_{\mathbb{Z}_{p}}M(\mathfrak{m}),\
d(x)=f\otimes ex$

living in degrees $0,1$,

$C^{\bullet}(M(\mathfrak{m}))=\oplus_{i\geq
0}C^{\bullet}(M(\mathfrak{m}))_{i}$

Let

$\Omega=\frac{1}{2}h\otimes h+e\otimes e+f\otimes
f\in\mathfrak{g}\otimes\mathfrak{g}$

be the Casimir element. For $1\leq i<j\leq n$ we will denote by

$\Omega_{ij}:\ M(\mathfrak{m})\longrightarrow M(\mathfrak{m})$

the action of $\Omega$ through $i$-th and $j$-th factors.

3.4. Coulomb - KZ - Chevalley complex. Let

We define a bicomplex

$C^{\bullet\bullet}_{\mathfrak{m},N}=\\{W\Omega^{p}(U_{n,N})\otimes_{\mathbb{Z}_{p}}C^{q}(M(\mathfrak{m}))_{N}\\}_{p\geq
0,0\leq q\leq 1}.$

The horizontal differential

$d^{\prime}=\nabla_{Coul,KZ}=d+\frac{1}{\kappa}\omega_{\mathfrak{m}}-\frac{1}{\kappa}\omega_{KZ}:$
$\
W\Omega^{\bullet}(U_{n,N})\otimes_{\mathbb{Z}_{p}}C^{\bullet}(M(\mathfrak{m}))_{N}\longrightarrow
W\Omega^{\bullet+1}(U_{n,N})\otimes_{\mathbb{Z}_{p}}C^{\bullet}(M(\mathfrak{m}))_{N}$

acts on index $p$, where

$\omega_{KZ}:=\sum_{1\leq b<c\leq n}\Omega_{bc}d\log(z_{b}-z_{c}):\
C^{pq}_{\mathfrak{m},N}\longrightarrow C^{p+1,q}_{\mathfrak{m},N},$

the Casimirs $\Omega_{bc}$ acting through $M(\mathfrak{m})$ factor.

The vertical differential $d^{\prime\prime}$ acts on index $q$ and is induced
by the differential $d_{Ch}$ in the Chevalley complex
$C^{\bullet}(M(\mathfrak{m}))_{N}$.

3.5. Hypergeometric forms. They are defined by the same formulas as in [SV2]
2.2 but now they are considered as elements of the de Rham - Witt complex.

For $b=(b_{1},\ldots,b_{n})\in\mathbb{N}^{n},|b|:=\sum b_{i}=N$, define

$u_{b}=d\log(t_{1}-z_{1})\wedge\ldots\wedge d\log(t_{b_{1}}-z_{1})+$
$+d\log(t_{b_{1}+1}-z_{2})\wedge\ldots\wedge d\log(t_{b_{1}+b_{2}}-z_{2})+$
$\ldots+d\log(t_{b_{1}+\ldots+b_{n-1}+1}-z_{n})\wedge\ldots\wedge
d\log(t_{N}-z_{n})\in W\Omega^{N}(U_{n,N}),$

and for $c=(c_{1},\ldots,c_{n})\in\mathbb{N}^{n},|c|=N-1$, define

$u_{c}=-\kappa\biggl{(}d\log(t_{2}-z_{1})\wedge\ldots\wedge
d\log(t_{c_{1}+1}-z_{1})+$ $+d\log(t_{c_{1}+2}-z_{2})\wedge\ldots\wedge
d\log(t_{c_{1}+c_{2}+1}-z_{2})+$
$\ldots+d\log(t_{c_{1}+\ldots+c_{n-1}+2}-z_{n})\wedge\ldots\wedge
d\log(t_{1}-z_{n})\biggr{)}\in W\Omega^{N-1}(U_{n,N}),$

Symmetrization

Starting from here we suppose that $p>N$.

The symmetric group acts on $W\Omega^{\bullet}(U_{n,N})$ by permuting $t_{i}$.
For $w(t_{1},\ldots,t_{N})\in W\Omega^{\bullet}(U_{n,N})$ we denote

$\operatorname{Alt}w(t_{1},\ldots,t_{N})=\sum_{\sigma\in\Sigma_{N}}(-1)^{|\sigma|}w(t_{\sigma(1)},\ldots,t_{\sigma(N)}).$

We define

$w_{b}=\frac{1}{b_{1}!\ldots b_{n}!}\operatorname{Alt}u_{b},\
w_{c}=\frac{1}{c_{1}!\ldots c_{n}!}\operatorname{Alt}u_{c}.$

3.6. Out of these forms we define a cochain in our bicomplex of total degree
$N$: set

$\mathcal{I}_{0}=\sum_{|b|=N}w_{b}\otimes f^{b}v\in C^{N,0}_{\mathfrak{m},N}$

where

$f^{b}v=f^{b_{1}}\ldots f^{b_{n}}v,$

and

$\mathcal{I}_{1}=\sum_{|c|=N-1}w_{c}\otimes f\otimes f^{c}v\in
C^{N-1,1}_{\mathfrak{m},N}.$

3.7. Theorem. $\mathcal{I}=(\mathcal{I}_{0},\mathcal{I}_{1})$ is a cocycle,
i.e.

$\nabla_{KZ,Coul}\mathcal{I}_{0}=0,$
$d_{Ch}\mathcal{I}_{0}+\nabla_{KZ,Coul}\mathcal{I}_{1}=0.$

Proof the same as in [SV2], 2.5. $\square$

3.8. Bozonization map. Similarly to [SV2], 2.6 we can interpret our cocycle as
a map of (filtered) complexes.

Namely, our bicomplex may be rewritten as a tensor product

$C^{\bullet\bullet}_{\mathfrak{m},N}=W\Omega^{\bullet}(C^{\bullet}(M(\mathfrak{m}))_{N})\otimes_{W\Omega^{\bullet}(U_{n})}W\Omega_{\mathfrak{m}}^{\bullet}(U_{n,N})$

where

$W\Omega^{\bullet}(C^{\bullet}(M(\mathfrak{m}))_{N})=W\Omega^{\bullet}(U_{n})\otimes_{\mathbb{Z}_{p}}C^{\bullet}(M(\mathfrak{m}))_{N}$

equipped with the KZ connection

$\nabla_{KZ}=d_{z}-\frac{1}{\kappa}\omega_{KZ},$

and

$W\Omega_{\mathfrak{m}}^{\bullet}(U_{n,N})=W\Omega^{\bullet}(U_{n,N})$

equipped with the Coulomb connection

$\nabla_{Coul}=d_{z}+\frac{1}{\kappa}\omega_{Coul}.$

Define the dual Chevalley complex

$C_{\bullet}(M(\mathfrak{m})^{*})_{N}=(C^{\bullet}(M(\mathfrak{m}))_{N})^{*}$

living in degrees $-1,0$. The $N$-cocycle $\mathcal{I}$ gives rise to a map of
complexes

$\eta(\mathcal{I}):\
W\Omega^{\bullet}(C_{\bullet}(M(\mathfrak{m})^{*})_{N})\longrightarrow
W\Omega_{\mathfrak{m}}^{\bullet}(U_{n,N})^{\operatorname{Alt}}[N].$ $None$

Moreover, it is a map of filtered complexes, where

$F^{i}W\Omega^{\bullet}(C_{\bullet}(M(\mathfrak{m})^{*})_{N})\subset
W\Omega^{\bullet}(C_{\bullet}(M(\mathfrak{m})^{*})_{N})$

is the subcomplex of degree $\geq i$ forms, and

$F^{i}W\Omega_{\mathfrak{m}}^{\bullet}(U_{n,N})\subset
W\Omega_{\mathfrak{m}}^{\bullet}(U_{n,N})$

is the subcomplex of forms having $\geq i$ of differentials $dz_{a}$.

Let $\mathcal{L}(\mathfrak{m},N)$ denote the structure sheaf
$\mathcal{O}_{U_{n,N}})$ equipped with the Coulomb de Rham - Witt connection

$\nabla_{\mathfrak{m}}=d+\frac{1}{\kappa}\omega_{\mathfrak{m}}.$

Let $p:U_{n,N}\longrightarrow U_{n}$ be the projection. The direct image
$Rp_{*}\mathcal{L}(\mathfrak{m},N)$ carries the Gauss - Manin connection
$\nabla_{GM}$, the de Rham - Witt version of [KO].

The induced by (3.8.1) map of $E_{1}$-pages of spectral sequences will be

$\eta^{i}:\
(W\Omega^{\bullet}(H_{i}(M(\mathfrak{m})^{*})_{N})(U_{n}),\nabla_{KZ})\longrightarrow$
$\longrightarrow(W\Omega^{\bullet}(Rp_{*}^{N-i}(\mathcal{L}(\mathfrak{m},N)^{\operatorname{Alt}})(U_{n}),\nabla_{GM}),\
i=0,1.$

3.9. Theorem. The maps $\eta^{i}$ are isomorphisms for $1/\kappa$ close to
zero in the $p$-adic metrics.

Proof. This is a corollary of Thm. 2.6. $\square$

3.10. All the above generalizes to the case of an arbitrary symmetrizable Kac
- Moody algebra $\mathfrak{g}$. The details are left to an interested reader.

References

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problems in mathematics 24, 181–238, Itogi Nauki i Tekhniki, VINITI, Moscow,
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(2004), 315 - 326.

[Br] E.Brieskorn, Sur les groupes de tresses (d’après V.I.Arnold), Séminaire
Bourbaki 401, 1973.

[ES] J.N.Eberhardt, J.Scholbach, Integral motivic sheaves and geometric
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[ESV] H.Esnault, V.Schechtman, E.Viehweg, Cohomology of local systems on the
complement of hyperplanes, Inv. Math. 109 (1992), 557 - 561.

[I1] L.Illusie, Complexe de de Rham - Witt, Journées de Géométrie Algébrique
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[I2] L.Illusie, Complexe de de Rham - Witt et cohomologie cristalline, Ann.
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[K] D.Kaledin, Witt vectors, commutative and noncommutative, Russian Math.
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[KO] N.Katz, T.Oda, On the differentiation of De Rham cohomology classes with
respect to parameters, J. Math. Kyoto Univ. 8 (1968), 199 - 213.

[L] S.Lubkin, Generalization of $p$-adic cohomology; bounded Witt vectors. A
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characteristic zero, Compos. Math. 34 (1977), 225 - 277.

[MVW] C.Mazza, V.Voevodsky, Ch.Weibel, Lecture notes on motivic cohomology

[M] J.Milnor, Introduction to algebraic $K$-theory, Annals of Math. Studies
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[OS] P.Orlik, L.Solomon, Combinatorics and topology of complements of
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[SV1] V.Schechtman, A.Varchenko, Arrangements of hyperplanes and Lie algebra
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[SV2] V.Schechtman, A.Varchenko, Derived KZ equations, J. Singul. 25 (2022),
422 – 442.

V.S.: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118
route de Narbonne, 31062 Toulouse, France; Kavli IPMU, 5-1-5 Kashiwa-no-ha,
Kashiwa-shi, Chiba, 277-8583 Japan

A.V.: Department of Mathematics, University of North Carolina at Chapel Hill,
Chapel Hill, NC 27599-3250, USA; Faculty of Mathematics and Mechanics,
Lomonosov Moscow State University, Leninskiye Gory 1, 119991 Moscow GSP-1,
Russia