On the nullstellensätze for stein spaces and C-analytic sets.

In this work we prove the real Nullstellensatz for the ring O(X) of analytic functions on a C-analytic set X ⊂ Rn in terms of the saturation of Łojasiewicz’s radical in O(X): The ideal I(Ƶ(a)) of the zero-set Ƶ(a) of an ideal a of O(X) coincides with the saturation (Formula presented) of Łojasiewicz’s radical (Formula presented). If Ƶ(a) has ‘good properties’ concerning Hilbert’s 17th Problem, then I(Ƶ(a)) = (Formula presented) where (Formula presented) stands for the real radical of a. The same holds if we replace (Formula presented) with the real-analytic radical (Formula presented) of a, which is a natural generalization of the real radical ideal in the C-analytic setting. We revisit the classical results concerning (Hilbert’s) Nullstellensatz in the framework of (complex) Stein spaces. Let a be a saturated ideal of O(Rn) and YRn the germ of the support of the coherent sheaf that extends aORn to a suitable complex open neighborhood of Rn. We study the relationship between a normal primary decomposition of a and the decomposition of YRn as the union of its irreducible components. If a:= p is prime, then I(Ƶ(p)) = p if and only if the (complex) dimension of YRn coincides with the (real) dimension of Ƶ(p).

[C II] and 12^CO(1-0) emission maps in HLSJ091828.6+514223: a strongly lensed interacting system AT z = 5.24

We present Submillimeter Array [C II] 158 μm and Karl G. Jansky Very Large Array 12^CO(1-0) line emission maps for the bright, lensed, submillimeter source at z = 5.2430 behind A 773: HLSJ091828.6+514223 (HLS0918). We combine these measurements with previously reported line profiles, including multiple 12^CO rotational transitions, [C I], water, and [N II], providing some of the best constraints on the properties of the interstellar medium in a galaxy at z > 5. HLS0918 has a total far-infrared (FIR) luminosity L_FIR(8–1000 μm) = (1.6 ± 0.1) × 10^14 L_☉ μ^–1, where the total magnification μ_total = 8.9 ± 1.9, via a new lens model from the [C II] and continuum maps. Despite a HyLIRG luminosity, the FIR continuum shape resembles that of a local LIRG. We simultaneously fit all of the observed spectral line profiles, finding four components that correspond cleanly to discrete spatial structures identified in the maps. The two most redshifted spectral components occupy the nucleus of a massive galaxy, with a source-plane separation <1 kpc. The reddest dominates the continuum map (demagnified L_FIR, component = (1.1 ± 0.2) × 10^13 L_☉) and excites strong water emission in both nuclear components via a powerful FIR radiation field from the intense star formation. A third star-forming component is most likely a region of a merging companion (ΔV ~ 500 km s^–1) exhibiting generally similar gas properties. The bluest component originates from a spatially distinct region and photodissociation region analysis suggests that it is lower density, cooler, and forming stars less vigorously than the other components. Strikingly, it has very strong [N II] emission, which may suggest an ionized, molecular outflow. This comprehensive view of gas properties and morphology in HLS0918 previews the science possible for a large sample of high-redshift galaxies once ALMA attains full sensitivity.