Search

CO serves as a tracer for dynamics in the polar winter middle atmosphere. This work presents the retrieval and the characterization of ground-based CO measurements from the winters 2008/2009 and 2009/2010 by the Kiruna microwave radiometer KIMRA, located in northern Sweden (67.8° N, 20.4° E). Furthermore, the dataset is used for an extensive comparison to the recent satellite instruments MLS on Aura, ACE-FTS, and MIPAS on Envisat. The vmr profiles are retrieved using the optimal estimation approach. A detailed analysis of the averaging kernel functions is carried out, showing sensitivity of the measurements between 40 and 80 km altitude, a vertical resolution of 16 to 22 km, as well as a residual influence of the region up to 130 km altitude. An error assessment reveals a total error of the retrieved profile that increases with altitude and is approx. ±0.1 ppmv at 50 km altitude and ±3 ppmv at 80 km altitude. The main contributions to this total error arise from the measurement noise and the uncertainty of the used temperature profiles. The expected dynamical features of the polar winter middle atmosphere are qualitatively identified in the retrieved time series, but are not quantitatively analyzed here. The dense MLS dataset is used to investigate the influence of the collocation criteria on the satellite comparison, showing that relaxing the distance criterion causes a high bias for MLS. The comparison including the other instruments is difficult because of the small number of coincidences. However, it suggests that there is a general agreement between KIMRA and the satellite instruments below 65 km altitude, but a high bias for KIMRA above this altitude. Furthermore, the shape of the KIMRA profile appears to be systematically different from the satellite profiles.

The first long-term measurements of aerosol number and size distributions in South-American tropical free troposphere (FT) were performed from March 2007 until March 2009. The measurements took place at the high altitude Atmospheric Research Station Alexander von Humboldt. The station is located on top of the Sierra Nevada mountain ridge at 4765 m a.s.l. nearby the city of Mérida, Venezuela. Aerosol size distribution and number concentration data was obtained with a custom-built Differential Mobility Particle Sizer (DMPS) system and a Condensational Particle Counter (CPC). The analysis of the annual and diurnal variability of the tropical FT aerosol focused mainly on possible links to the atmospheric general circulation in the tropics. Considerable annual and diurnal cycles of the particle number concentration were observed. Highest total particle number concentrations were measured during the dry season (January–March, 519 ± 613 cm−3), lowest during the wet season (July–September, 318 ± 194 cm−3). The more humid FT (relative humidity (RH) range 50–95 %) contained generally higher aerosol particle number concentrations (573 ± 768 cm−3 during dry season, 320 ± 195 cm−3 during wet season) than the dry FT (RH < 50 %, 454 ± 332 cm−3 during dry season, 275 ± 172 cm−3 during wet season), indicating the importance of convection for aerosol distributions in the tropical FT. The diurnal cycle in the variability of the particle number concentration was mainly driven by local orography.

Since March 2004 the ground-based 268–280 GHz radiometer MIRA 2 has been in quasi-continuous operation at the inner-tropical Merida Atmospheric Research Station (MARS) on Pico Espejo (8.58°N, 71.15°W, 4765 m above sea level) in the Venezuelan Andes. Using the optimal estimation method concentration profiles of ozone have been retrieved in the stratosphere and mesosphere. In the middle stratosphere variations of ozone on a time scale of several months were revealed in the resulting time series that probably result from the turnaround of the Brewer–Dobson circulation. In the mesosphere photochemistry plays the most important role for the ozone concentration. Therefore a diurnal cycle is observed in the upper part of the retrieved ozone profiles. The measurements are used to verify model results of ozone in the mesosphere obtained by a photochemistry model, which reproduces the diurnal variation of ozone very well. Also an inter-annual variation of mesospheric ozone during the month of November in 2004–2007 ...

The signature of five-day planetary waves in ozone and temperature data from the advanced sub-millimeter radiometer aboard the Odin satellite is examined. The period January–March 2005 and heights from 24–56 km are used. We find highest wave amplitudes in both temperature and ozone in the winter hemisphere at 60°N-70°N. The relative phases between ozone and temperature perturbations show the expected antiphase behaviour in the photochemistry-dominated region at about 40 km altitude. We compare the global planetary wave properties from Odin with five-day perturbations in ozone measured by the millimeter wave radiometer in Kiruna (KIMRA, 67°N, 20°E). In the early part of the comparison interval (January–February) at 40 km, we find good correlation between the two in terms of both phase and amplitude of the perturbations. In the latter part of the comparison interval (March) where mean ozone levels are higher, the amplitudes of the ozone five-day perturbations over Kiruna are much higher than the wave amplitudes found using Odin. We conclude that five-day variations in ozone due to planetary waves can be detected by KIMRA in some circumstances, but that other sources of variability dominate at other heights and times. PACS No.: 94.10.Jd

The sub-millimetre radiometer (SMR) on board the Odin satellite measures signatures of ozone in two bands centred at 501.8 and 544.6 GHz. From the measurements, ozone volume mixing ratio profiles in the stratosphere and lower mesosphere are retrieved using the Optimal Estimation Method. In this paper, the ozone profiles measured by Odin–SMR (level-2 data ver. 2.1 and 2.0, respectively) are compared to measurements taken by ground-based millimetre wave radiometers in the Arctic; at Kiruna, Sweden; in the mid-latitudes on the Zugspitze, Germany; and in the tropics at Mérida, Venezuela. The Kiruna Microwave Radiometer (KIMRA) covers the frequency range 195–224 GHz, and the Millimeter Wave Radiometer MIRA 2, which was operated on the Zugspitze and at Mérida, measures in the frequency band 268–281 GHz. From the measurements, ozone profiles in the vertical range between approximately 15–65 km were retrieved using the Optimal Estimation Method. Since the ground-based measurements have a lower vertical resolution than those of Odin the latter were degraded using the averaging kernels of the ground-based retrievals. The comparison of the resulting profiles to the ground-based data enables the identification of biases in the Odin measurements and their possible latitudinal variation. In general, a good agreement between satellite and ground-based measurements for the 501.8 GHz band was found in the stratosphere except for a negative bias in the Odin data of about 10–15% in the tropical measurements. The Odin measurements taken at 544.9 GHz yielded systematically 20–30% lower ozone mixing ratios in the middle stratosphere than the ground-based measurements at all sites. PACS No.: 92.60.hd

The constrained linear inversion technique and the optimal estimation method are applied to joint retrievals of atmospheric constituent profiles from vertical sounding spectra in the millimeterwave range. The algorithms are illustrated on dual profile retrievals of ozone (O 3 ) and chlorine monoxide (ClO) from synthesized spectra and on dual profile retrievals of ozone and chlorine monoxide, of ozone and nitrous oxide (N 2 O), and of ozone and nitric acid (HNO 3 ), respectively, from measurements performed with the ground-based millimeterwave radiometer MIRA II during February and March 1996 at Kiruna. Reconstructed profiles are compared to observations performed by other groups. The joint retrieval concept proves to be a useful tool that provides valuable information about the altitude distribution of atmospheric constituents.

Results of the Karlsruhe Simulation Model of the Middle Atmosphere (KASIMA) are compared with vertical ClO profiles measured by the groundbased Millimeter Wave Radiometer MIRA2 inside the vortex during March 1997 at Ny-Alesund. The influence of the OH + ClO and HO 2 + ClO reaction branching ratio and of the absorption cross section of Cl 2 O 2 on the calculated mixing ratios of ClO and ozone has been investigated. In the upper stratosphere the ClO mixing ratio is reduced by 90% by using a minor channel of the OH + ClO reaction with a branching ratio of 0.07. A temperature dependent minor channel of the HO 2 + ClO reaction reduces the upper stratospheric ClO mixing ratio by 22%. Different absorption spectra of Cl 2 O 2 alter the ClO mixing ratios up to 12% at noon at 20 km. This causes differences of 15% in the ozone loss during winter.

We present middle-atmospheric ozone mixing ratio profiles obtained from measurements around 142 GHz using the ground-based millimeter wave radiometer (MIRA 1) developed at the Forschungszentrum Karlsruhe. Data were recorded during spring season of 1994 from the summit of Hornisgrinde (1165 m, 48.61°N, 8.20°E), Germany. They were severely disturbed by standing waves caused by reflections in the internal transmission path. The profiles are retrieved by using a new extension of the well-known constrained linear inversion and optimal estimation method, respectively. With this new extension both techniques are capable of fitting the amplitudes and phases of standing waves in the inversion algorithm itself rather than during preprocessing. In the present paper, analysis of the retrieval technique is given, including a short analysis of sources of error. The new retrieval technique proves to be a reliable tool for the inversion of measured spectra disturbed by standing waves.

During the CAWSES DFG (German Research Association) priority program measurements of stratospheric and mesospheric O3 using ground based millimeterwave radiometry have been established and analyzed. Instruments have been operated at two different locations, at Merida, Venezuela, a high altitude tropic station and at Ny Alesund, Spitsbergen, an Arctic station. Additionally, data obtained from the millimeterwave radiometer based at Kiruna, Sweden, have been used for an analysis of the 5-day planetary wave.

First long-term observations of South American biomass burning aerosol within the tropical lower free troposphere are presented. The observations were conducted between 2007 and 2009 at a high altitude station (4765 m a.s.l.) on the Pico Espejo, Venezuela. Sub-micron particle volume, number concentrations of primary particles and particle absorption were observed. Orographic lifting and shallow convection leads to a distinct diurnal cycle at the station. It enables measurements within the lower free troposphere during night-time and observations of boundary layer air masses during daytime and at their transitional regions. The seasonal cycle is defined by a wet rainy season and a dry biomass burning season. The particle load of biomass burning aerosol is dominated by fires in the Venezuelan savannah. Increases of aerosol concentrations could not be linked to long-range transport of biomass burning plumes from the Amazon basin or Africa due to effective wet scavenging of particles. Highest particle concentrations were observed within boundary layer air masses during the dry season. Ambient sub-micron particle volume reached 1.4±1.3 μm3 cm−3, refractory particle number concentrations (at 300 °C) 510±420 cm−3 and the absorption coefficient 0.91±1.2 Mm−1. The respective concentrations were lowest within the lower free troposphere during the wet season and averaged at 0.19±0.25 μm3 cm−3, 150±94 cm−3 and 0.15±0.26 Mm−1. A decrease of particle concentrations during the dry seasons from 2007–2009 could be connected to a decrease in fire activity in the wider region of Venezuela using MODIS satellite observations. The variability of biomass burning is most likely linked to the El Niño–Southern Oscillation (ENSO). Low biomass burning activity in the Venezuelan savannah was observed to follow La Niña conditions, high biomass burning activity followed El Niño conditions.

This paper presents the Alpine Radiometer Intercomparison at the Schneefernerhaus (ARIS), which took place in winter 2009 at the high altitude station at the Zugspitze, Germany (47.42° N, 10.98° E, 2650 m). This campaign was the first direct intercomparison between three new ground based 22 GHz water vapor radiometers for middle atmospheric profiling with the following instruments participating: MIRA 5 (Karlsruhe Institute of Technology), cWASPAM3 (Max Planck Institute for Solar System Research, Katlenburg-Lindau) and MIAWARA-C (Institute of Applied Physics, University of Bern). Even though the three radiometers all measure middle atmospheric water vapor using the same rotational transition line and similar fundamental set-ups, there are major differences between the front ends, the back ends, the calibration concepts and the profile retrieval. The spectrum comparison shows that all three radiometers measure spectra without severe baseline artifacts and that the measurements are in good general agreement. The measurement noise shows good agreement to the values theoretically expected from the radiometer noise formula. At the same time the comparison of the noise levels shows that there is room for instrumental and calibration improvement, emphasizing the importance of low elevation angles for the observation, a low receiver noise temperature and an efficient calibration scheme. The comparisons of the retrieved profiles show that the agreement between the profiles of MIAWARA-C and cWASPAM3 with the ones of MLS is better than 0.3 ppmv (6%) at all altitudes. MIRA 5 has a dry bias of approximately 0.5 ppm (8%) below 0.1 hPa with respect to all other instruments. The profiles of cWASPAM3 and MIAWARA-C could not be directly compared because the vertical region of overlap was too small. The comparison of the time series at different altitude levels show a similar evolution of the H2O volume mixing ratio (VMR) for the ground based instruments as well as the space borne sensor MLS.

The first long-term measurements of aerosol number and size distributions in South-American tropical free troposphere were performed from March 2007 until Mai 2009. The measurements took place at the high altitude Atmospheric Research Station Alexander von Humboldt. The station is located on top of the Sierra Nevada mountain ridge at 4765 m a.s.l. nearby the city of Mérida, Venezuela. Aerosol size distribution and number concentration data was obtained with a custom-built Differential Mobility Particle Sizer (DMPS system) and a Condensational Particle Counter (CPC). The analysis of the annual and diurnal variability of the tropical free troposphere (FT) aerosol focused mainly on possible links to the atmospheric general circulation in the tropics. Considerable annual and diurnal cycles of the particle number concentration were observed. Highest total particle number concentrations were measured during the dry season (519±613 cm−3), lowest during the wet season (318±194 cm−3). The more humid FT contained generally higher aerosol particle number concentrations (573±768 cm−3 during dry season, 320±195 cm−3 during wet season) than the dry FT (454±332 cm−3 during dry season, 275±172 cm−3 during wet season), indicating the importance of convection for aerosol distributions in the tropical FT. The diurnal cycle in the variability of the particle number concentration was mainly driven by local orography.

The authors present first results from the new MIRA 5 microwave spectroradiometer for middle-atmospheric observations of water vapour at 22 GHz. The data have been taken at Environmental Research Station Schneefernerhaus, Zugspitze, Germany (47°N, 11°E, 2650 m a.s.l.) in January 2009. Comparison with Aura/MLS yields good agreement in the lower and middle stratosphere and some systematic deviation at higher altitudes, which is attributed to a simplified retrieval set-up used in this study.

The gyrotron is a powerful source of millimetre wave radiation. Fusion applications require more power per tube than is currently available This in turn means that the resonator must be highly overmoded, with a dense mode spectrum, which leads to mode competition. The influence of external parameters, such as the applied magnetic field, electron beam, and resonator geometry, on mode competition in tubes designed to operate at 150 and 140 GHz in the TE03 mode is investigated theoretically and experimentally. It can be shown that even when the mode spectrum is fairly dense, single mode operation of a gyrotron is possible. The influence of startup conditions on output power is investigated in the 150 GHz experiment. In some cases, modes with an axial index of 2 were excited.

This paper characterizes vertical ozone profiles retrieved with the IMK-IAA (Institute for Meteorology and Climate Research, Karlsruhe – Instituto de Astrofisica de Andalucia) science-oriented processor from high spectral resolution data (until March 2004) measured by the Michelson Interferometer for Passive Atmospheric Sounding (MIPAS) aboard the environmental satellite Envisat. Bias determination and precision validation is performed on the basis of correlative measurements by ground-based lidars, Fourier transform infrared spectrometers, and microwave radiometers as well as balloon-borne ozonesondes, the balloon-borne version of MIPAS, and two satellite instruments (Halogen Occultation Experiment and Polar Ozone and Aerosol Measurement III). Percentage mean differences between MIPAS and the comparison instruments for stratospheric ozone are generally within ±10%. The precision in this altitude region is estimated at values between 5 and 10% which gives an accuracy of 15 to 20%. Below 18 km, the spread of the percentage mean differences is larger and the precision degrades to values of more than 20% depending on altitude and latitude. The main reason for the degraded precision at low altitudes is attributed to undetected thin clouds which affect MIPAS retrievals, and to the influence of uncertainties in the water vapor concentration.

International audience; We present ozone measurements from the millimetre wave radiometer installed at the Swedish Institute of Space Physics (Institutet för rymdfysik, IRF) in Kiruna (67.8° N, 20.4° E, 420 m asl). Nearly continuous operation in the winter of 2002/2003 allows us to give an overview of ozone evolution in the stratosphere between 15 and 55 km. In this study we present a detailed analysis of the Arctic winter 2002/2003. By means of a methodology using equivalent latitudes we investigate the meteorological processes in the stratosphere during the entire winter/spring period. During the course of the winter strong mixing into the vortex took place in the middle and upper stratosphere as a result of three minor and one major warming event, but no evidence was found for significant mixing in the lower stratosphere. Ozone depletion in the lower stratosphere during this winter was estimated by measurements on those days when Kiruna was well inside the Arctic polar vortex. The days were carefully chosen using a definition of the vortex edge based on equivalent latitudes. At the 475 K isentropic level a cumulative ozone loss of about 0.5 ppmv was found starting in January and lasting until mid-March. The early ozone loss is probably a result of the very cold temperatures in the lower stratosphere in December and the geographical extension of the vortex to lower latitudes where solar irradiation started photochemical ozone loss in the pre-processed air. In order to correct for dynamic effects of the ozone variation due to diabatic subsidence of air masses inside the vortex, we used N2O measurements from the Odin satellite for the same time period. The derived ozone loss in the lower stratosphere between mid-December and mid-March varies between 1.1±0.1 ppmv on the 150 ppbv N2O isopleth and 1.7±0.1 ppmv on the 50 ppbv N2O isopleth.

International audience; During the Arctic winter of 2000/01, ground-based FTIR and millimetre-wave measurements revealed significant amounts of ClO over Kiruna after the final warming in February 2001. In fact, column amounts of ClO were still increased in March 2001 when temperatures were about 20 K above the PSC (Polar Stratospheric Clouds) threshold. At these temperatures, chlorine activation due to heterogeneous processes on PSCs is not possible even in the presence of strong lee wave effects. In order to discuss possible reasons of this feature, time series of other chemical species will be presented and discussed, too. Measurements of HF and COF2 indicated that vortex air was still observed in mid-March 2001. Since the time series of HNO3 column amounts do not give any evidence of a denitrification later than 11 February, chlorine activation persisting for several weeks after the presence of PSCs due to denitrification is rather unlikely. The photolysis of ClONO2-rich air which had been formed at the end of February and beginning of March 2001 as well as chlorine activation due to the presence of an unusual aerosol layer are discussed as possible causes of the increased ClO column amounts after the final warming.

During two subsequent polar winter measurement campaigns the ground-based mm wave radiometer MIRA2 has been operated from 12 March to 31 March 1997 at Ny-/spl Aring/lesund (78.9/spl deg/N, 11.9/spl deg/E), Svalbard, and from 26 January to 3 April 1998 at Kiruna (67.8/spl deg/N, 20.4/spl deg/E), Sweden. From the measurements at Ny-/spl Aring/lesund a nearly continuous time series of 100 ClO vmr profiles could be retrieved and compared to calculations of the Karlsruhe Simulation Model of the Middle Atmosphere (KASIMA). During the measurement campaign at Kiruna 1998 the weather conditions were not always favorable for ClO measurements. During the periods of bad weather conditions the radiometer was tuned for the detection of ozone resulting in a time series starting from 26 January to 31 March, or for the detection of N/sub 2/O.

This paper describes the new radiometer system operating in the 278 GHz range, which is intended for long term ground-based monitoring of the vertical profiles of chlorine monoxide and ozone in the stratosphere over the Arctic area. To improve the measurement of weak lines, in particular in view of their diurnal variation and to allow operation over a wider range of weather conditions, an advanced balancing technique has been implemented. Results of recent measurements during February and March 1996 at Kiruna, Sweden, are presented.

The 268-280 GHz radiometer MIRA2 was constructed for ground- based monitoring of vertical profiles of minor stratospheric constituents and has been optimized especially for the observation of the extremely weak chlorine monoxide signature at 278 GHz and its diurnal variation. For calibration an adjustable internal load was developed, which provides any required brightness temperature from cryogenic to ambient temperatures, dependent on mechanical adjustment. In comparison to the well established external beam switching method this new technique result in a higher contrast in particular under critical conditions of the troposphere, For inversion of the measured spectra the modified Constrained Linear Inversion and the Optimal Estimation Method were used alternatively. The extensions of these well established methods include the fit of standing waves within the inversion process itself an the joint retrieval of several spectral lines. The advances in calibration and inversion techniques became obvious during the evaluation of the data measured at Kiruna, Sweden, 1996 and 1998, and Ny-Alesund, Svalbard, 1997. Profiles of the trace gases Ozone, N2O, HNO3 and ClO could be retrieved. From the data measured in Ny-Alesund a unique time series of ClO- and Ozone has been achieved which shows diurnal and long term variations of ClO and Ozone, respectively.© (1998) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Ground-based millimeter wave observations of stratospheric trace gases have been carried out by IMK at the NDSC station Ny-/spl Aring/lesund (78.9/spl deg/ N, 11.9/spl deg/ E) during March 1997. This measurement campaign was the authors contribution to the international intercomparison of chlorine monoxide (ClO) radiometers coordinated by IUP, University Bremen, during which 5 individual instruments have been installed at the same site. The radiometer of IMK has been operated continuously from 12 March to 31 March. Within this timespan there were coincident measurements in parallel with 1 or 2 of the other ClO radiometers on a few days, allowing a direct comparison of the measured spectra and the retrieved profiles. For the entire period the location was inside the polar vortex.

A vector network-analyzer is described, which is in use for waveguide component tests over the 130 - 150 GHz frequency range. The set up consists mainly of standard laboratory equipment, only a few components are custom designed for the 2 mm wavelength range. Measurement examples are given. The operation principle may briefly be described as follows: A test signal f1+f2 is generated by mixing a LO signal with fl ? 15 GHz and an auxiliary signal with f2 = 200 MHz. At the mixer output the LO signal and the lower sideband are suppressed. The upper sideband is multiplied by nine (n=9). Having passed the device under test, it is down converted by a harmonic mixer. The resulting IF-level and phase shift are measured at n·200 MHz = 1.8 GHz with the vector-voltmeter. The multiplied test signal and the local oscillator signal of the harmonic mixer are derived from the same signal source. Swept measurements are performed by tuning the LO frequency. The same principle can be used in a wide range of waveguide-bands.

[1] Vertical profiles of stratospheric ozone and ozone-related trace gases were measured in winter 1999/2000 using the ground-based millimeter-wave radiometer MIRA 2 and a Fourier-transform infrared spectrometer (FTIR) located at the Swedish Institute of Space Physics at Kiruna. The MIRA 2 measurements covered all three SOLVE/THESEO 2000 flight phases. An almost complete time series of O3 profiles and complementary profiles of ClO, HNO3, and N2O were achieved. Profiles of O3, HCl, HNO3, and N2O, as well as stratospheric column amounts of NO2, ClONO2, and ClO, were obtained from the continuous ground-based FTIR measurements between January and March 2000. From the measurements of N2O and HF, a diabatic subsidence inside the polar vortex of about 1.2 km between January and March for altitudes of about 20 km was deduced. On 26 and 28 January, an uptake of about 25% of stratospheric HNO3 by polar stratospheric clouds (PSCs) could be measured and between January and March, an significant denitrification of the lower stratosphere was derived from the measurements. Strong chlorine activation was detected by both instruments in January and March, resulting in an ozone loss of more than 1 ppmv in a layer below 23 km. In comparison to recent cold winters, this layer was found to be quite thin. Therefore the total loss measured in column amount of (1.2–1.4) × 1022 molec/m2 was smaller than the losses in previous cold winters.

As a part of a feasibility study the design and the construction of an accelerator section to be tested at the heavy ion beam of an electrostatic tandem Van de Graaff has been started. This test section consists of a cryostat with two independently phased ?/2-helix structures at 108 MHz and of a suitable rf-control system. Design criteria and recent results of measurements on a superconducting test cavity with electric surface peak fields up to 29 MV/m are presented.

At the Kernforschungszentrum Karlsruhe a gyrotron started operation in October 1986. By the end of December 1986 it had delivered 120 kW RF-power during 10 μs pulses in the TE03-mode at 149-5GHz. The efficiency was about 20%.

Introduction. Let C be a field of characteristic p 0, and let K be a finite algebraic extension field of C such that the pth power of every element of K lies in C. Then K/C is called a purely inseparable extension of exponent 1. It was shown by N. Jacobson that there is a Galois theory for such extensions in which the place of the Galois group is taken by the derivation algebra of K/C. In particular, if K is any field of characteristic p, the purely inseparable extensions K/C of exponent 1 are precisely those in which C is the field of constants of a restricted K-Lie ring of derivations of K which is of finite dimension over K. In the classical theory of simple algebras, it is shown that, if K/C is a Galois extension, the Brauer similarity classes of the simple algebras with center C and split by K constitute a group which is canonically isomorphic with the group of equivalence classes of the group extensions of the multiplicative group of K by the Galois group of K/C. The present paper provides the answer to a question put to me by J-P. Serre, of whether one could establish an analogous result, for K/ C purely inseparable of exponent 1, in which restricted Lie algebra extensions [2] of K by the derivation algebra of K/C take the place of the group extensions. Not only is the answer to this question affirmative, but it provides an excellent illustration of the theory of restricted Lie algebra extensions. It turns out, in fact, that the Lie algebra extensions which arise from simple algebras are trivial extensions when regarded as ordinary extensions, so that the essential structural elements are here precisely those which differentiate the restricted extensions from the ordinary ones. ?1 contains the field theoretical background of our problem. In particular, it gives a simple proof of the main theorem of Jacobson's Galois theory [4] which we include here because it gives us the connection, on which many of our subsequent arguments are based, between the structure of the field extension K/C and that of the derivation algebra of K/C. Theorem 2, which is not needed in the sequel, is the analogue for the present situation of a well known result in the classical Galois theory and is significant for the cohomology theory of derivation algebras. In ?2 we give a proof of a theorem of Jacobson's on derivations (in a slightly generalized form) which is fundamental for the crossed product theory that follows, in the same way as the analogous theorem for isomorphisms is the source of the classical theory of crossed products. In ?3 we discuss the special type of restricted Lie algebra

Let R be a ring with identity element 1, and let S be a subring of R containing 1. We consider R-modules on which 1 acts as the identity map, and we shall simultaneously regard such R-modules as S-modules in the natural way. In [4], we have defined the relative analogues of the functors of Cartan-Eilenberg [1], and we have briefly treated the corresponding relative analogues of module dimension and global ring dimension.

Introduction. Let F be a field of characteristic p, and let C be a finite algebraic extension field of F such that CPCF. It has been shown in [4] that there is a one to one correspondence (up to isomorphisms) between the simple finite dimensional algebras with center F and containing C as a maximal commutative subring and the regular restricted Lie algebra extensions of C by the derivation algebra of C/F. This led to a very simple description of the group of similarity classes of these algebras. The main task of the present paper is to investigate the connection between restricted Lie algebras and simple associative algebras of characteristic p quite generally. For this purpose, we generalize the notion of a regular extension of C by the derivation algebra of C/F to that of a regular extension of a simple algebra A with center C by the derivation algebra of C/F. The correspondence mentioned above is then generalized to a one to one correspondence between these extensions and the simple algebras with center F which contain A as the commutator algebra of C (Theorem 3). We shall then show how every simple algebra containing a purely inseparable extension field of its center as a maximal commutative subring can be built up in a series of steps from regular Lie algebra extensions. With a suitable composition of regular extensions, which we define in ?2, this reduces the structure of the group of algebra classes with a fixed purely inseparable splitting field to the structural elements of restricted Lie algebras, at least in principle. However, it does not yield a direct description of this group, as was the case for splitting fields of exponent one. In ?3 we show that the tensor multiplication with a purely inseparable extension field C of F maps the Brauer group of algebra classes over F onto the Brauer group over C (Theorem 5). Although this result seems not to have been stated before, it may well be regarded as one of the culminating points of Albert's theory of p-algebras. In fact, Chapter VII of [1] contains all the essential points of a proof along classical lines, which is the first proof we give in ?3. Here, the main tools are Galois theory and the theory of cyclic p-extensions. Our second proof is entirely different. It proceeds within the framework of the first part of this paper, replacing the field theory with the theory of Lie algebras. In particular, we use the theory of restricted Lie algebra kernels, [3], in order to show that if A is a simple algebra with center C and CPC F then there always exists a regular extension of A by the derivation algebra of C/F, whence (by Theorem 3) A can be imbedded as the

simple (finite-dimensional) algebras. Since the meaning of the designation "Galois theory" has begun to change quite considerably, it may be appropriate to point out that we study the relationships between certain subalgebras of an algebra A on the one hand, and certain groups of automorphisms of A on the other. The ordinary theory of central simple algebras covers the case of subalgebras containing the center of A. Another special case, in which the subalgebras are subfields of the center, has been discussed from a somewhat different point of view by Teichmueller [7] and by Eilenberg and MacLane [4](1). Some of the recent generalizations of Galois theory to far more general rings-in particular J. Dieudonne's [3]-are concerned chiefly with Galois correspondences between certain subrings of a given ring A on the one hand, and certain rings of endomorphisms on the other. These rings of endomorphisms are intimately related to groups of automorphisms of A in the case of division rings, but in the general case this theory is quite far removed from a theory of automorphisms of A. The theory given here resembles the Galois theory for division rings as given by H. Cartan [2] and N. Jacobson [6], and covers the special case of division algebras in the same way. Some of our results are related also to those of Dieudonne [3]. However, our method is independent of the new techniques and consists simply in combining the well known results of the theory of simple algebras with those of the ordinary Galois theory for fields, a knowledge of which we shall assume(2). 1. Auxiliary results.

Let F be a field of characteristic 0, and let V be a finite dimensional vector space over F. Let E denote the algebra of all endomorphisms of V, and let L be any Lie subalgebra of E. Among the algebraic Lie algebras contained in E and containing L, there is one that is contained in all of them, and this is called the algebraic hull of L in E. Here, an algebraic Lie algebra is defined as the Lie algebra of an algebraic group. It is an easy consequence of the definitions that if A and B are algebraic groups of automorphisms of V such that A CB then the Lie algebra of A is contained in the Lie algebra of B. Hence the existence of the algebraic hull of L is an immediate consequence of the following basic result: let G be the intersection of all algebraic groups of automorphisms of V whose Lie algebras contain L. Then the Lie algebra of G contains L. This theorem reduces at once to the case where L is one dimensional. For any xCE, let G. be the intersection of all algebraic groups of automorphisms of V whose Lie algebras contain x. Then we have GZCG, whenever xCEL, and it suffices to show that the Lie algebra of G, contains x. This is part of [1, Theorem 10, p. 165], but it is not clear from the proof given in [1 ] that, although this result is not as obvious as it might seem at first sight, it can be proved quite directly without invoking any special knowledge of algebraic groups. The proof we give here is based on the simple idea of recovering G, from its generic point exp(tx). We use an auxiliary variable t over F and introduce the ring F {t of the integral power series in t with coefficients in F. Let E* denote the dual space of E, and let P be the algebra of all polynomial functions on E. The elements of E* are canonically extended to become F { t } -linear maps of E 0 FF { t } into F { t }, and the elements of P are extended accordingly to become F { t } -valued functions on E 0 FF { t }. With xCE, we interpret the formal power series exp(tx) as the element of E0FF{t} that is determined by the conditions

Introduction. The main purpose of this paper is to draw attention to certain functors, exactly analogous to the functors "Tor" and "Ext" of Cartan-Eilenberg [2], but applicable to a module theory that is relativized with respect to a given subring of the basic ring of operators. In particular, we shall show how certain relative cohomology theories for groups, rings, and Lie algebras can be subsumed under the theory of the relative Ext functor, just as (in [2]) the ordinary cohomology theories have been subsumed under the theory of the ordinary Ext functor. Among the various relative cohomology groups that have been considered so far, some can be expressed in terms of the ordinary Ext functor; these have been studied systematically within the framework of general homological algebra by M. Auslander (to appear). A typical feature of these relative groups is that they appear naturally as terms of exact sequences whose other terms are the ordinary cohomology groups of the algebraic system in question, and of its given subsystem. There is, however, another type of relative cohomology theory whose groups are not so intimately linked to the ordinary cohomology groups and exhibit a more individualized behaviour. Specifically, the relative cohomology groups for Lie algebras, as defined (in [3]) by Chevalley and Eilenberg, and the relative cohomology groups of groups, defined and investigated by I. T. Adamson [1], are of this second type. It is these more genuinely "relative" cohomology theories that fall in our present framework of relative homological algebra. Our plan here is to sketch the general features of the relative Tor and Ext functors (?2) and to illustrate some of their possible uses or interpretations by a selection of unelaborated examples. Thus, ?3 illustrates the use of the relative Ext functor in extending the cohomology theory for algebras. ?4 deals with relative homology and relative cohomology of groups, and involves both the relative Tor functor and the relative Ext functor. ?5 discusses the role played by the relative Ext functor in the cohomology theory for Lie algebras. Since this paper is intended to serve as a preliminary survey, and since the topics dealt with are supplementary to the corresponding topics of the nonrelative theory (contained in [2]), we feel justified in presupposing that the

This method is based on the Cartan-Leray spectral sequence, [3; 1 ], and can be generalized to other algebraic situations, as will be shown in a forthcoming paper of Cartan-Eilenberg [2]. Since the details of the Cartan-Leray technique have not been published (other than in seminar notes of limited circulation), we develop them in Chapter I. The auxiliary theorems we need for this purpose are useful also in other connections. In Chapter II, which is independent of Chapter I, we obtain a spectral sequence quite directly by filtering the group of cochains for G. This filtration leads to the same group E2=H(G/K, H(K)) (although we do not know whether or not the succeeding terms are isomorphic to those of the first spectral sequence) and lends itself more readily to applications, because one can identify the maps which arise from it. This is not always the case with the first filtration, and it is for this reason that we have developed the direct method in spite of the somewhat lengthy computations which are needed for its proofs. Chapter III gives some applications of the spectral sequence of Chapter II. Most of the results could be obtained in the same manner with the spectral sequence of Chapter I. A notable exception is the connection with the theory of simple algebras which we discuss in ?5. Finally, let us remark that the methods and results of this paper can be transferred to Lie Algebras. We intend to take up this subject in a later paper.

Introduction. It has been shown in a previous paper [311 that every algebra A with radical R, such that AIR is separable, is a homomorphic image of a certain maximal algebra which is determined to within an isomorphism by A/R, the A /R-module (two-sided) R/R2, and the index of nilpotency of R. Furthermore, some indication was given of how the structure of maximal algebras can be determined in simple cases. Here, we wish to give a further illustration by describing a rather wide class of maximal and primary algebras whose structure will be shown to resemble that of crossed products, in certain respects. In fact, we shall impose a certain normality condition and then trace the consequences of a few simple facts of the noncommutative Galois theory. An algebra B over the field F, with radical R, is called primary if it has an identity element and if BIR is simple. As is well known,2 B is then isomorphic with a Kronecker product FmXC, where Fm denotes the full matrix algebra of degree m over F, and where C is completely primary, in the sense that it has an identity element and that the quotient of C by its radical is a division algebra over F. We are concerned with primary algebras B for which this division algebra (which is determined to within an isomorphism by B) is normal over F, in the sense of the noncommutative Galois theory.3 This will be the case if and only if the center Z of BIR is a separable normal extension field of F and every automorphism of Z over F is induced by an automorphism of BIR. A completely primary algebra C with radical S will henceforth be called quasinormal if C/S is normal over F. If 4 is an isomorphism of Fm X C onto B then 4 maps the radical Fm XS onto the radical R of B, and B/R is isomorphic with Fm X C/S. Therefore, if C is quasinormal over F, then B/;R is automatically separable over F. By [3], B has then a maximal related extension B* which is evidently primary. Moreover, it is easily seen that B* FmXC*, where C* is the maximal related extension of C, and is quasinormal over F. Finally, the natural extension to B* of a homomorphism of C* onto C is a homomorphism of B* onto B. From these facts it is evident