Adelic quotient group for algebraic surfaces

Abstract

We calculate explicitly an adelic quotient group for an excellent Noetherian normal integral two-dimensional separated scheme. An application to an irreducible normal projective algebraic surface over a field is given.Recently there were also found the interesting connections and interactions with many questions between the higher adeles theory and the Langlands program, see [16].As an application, we deduce from our calculations of adelic quotient group (2) the corresponding quotient group when X is a projective irreducible normal surface over a field k , and C is the support of an ample divisor. As in the one-dimensional case, this quotient group will be a linearly compact k -vector space. We note that this quotient group when the surface X is smooth and the sheaf F = O X was calculated in [12, § 14].

Nonterritorial Autonomy in Northern Eurasia: Rooted or Alien?

“…In the immediate aftermath of the monarchy’s downfall in February 1917, most liberal and socialist parties reaffirmed their commitment to “cultural-national autonomy” while the Bolsheviks kept declaring exclusively the unlimited peoples’ right to self-determination, assuming the right to secession (L’vova, Nam, and Naumova 1993; Osipov 2004, 47–48).…”

“…During the “restructuring” and overall liberalization of the USSR in 1987–1991 and the debates about nationalities issues, numerous intellectuals and oppositional activists commenced promoting the ideas of NTA 5 as an alternative to the assumingly flawed Soviet nationalities policy (Osipov 2004, 62–64). The leadership of the Soviet constituent republics, which rapidly started claiming sovereign rights and thus opposing the USSR central authorities, was also resorting to the concept of NTA for gaining the support of their own ethnic minorities (Osipov 2004, 60–62).…”

“…During the “restructuring” and overall liberalization of the USSR in 1987–1991 and the debates about nationalities issues, numerous intellectuals and oppositional activists commenced promoting the ideas of NTA 5 as an alternative to the assumingly flawed Soviet nationalities policy (Osipov 2004, 62–64). The leadership of the Soviet constituent republics, which rapidly started claiming sovereign rights and thus opposing the USSR central authorities, was also resorting to the concept of NTA for gaining the support of their own ethnic minorities (Osipov 2004, 60–62). For example, the 1990 Sovereignty Declaration of the Russian Soviet Federative Socialist Republic (RSFSR) promised “the self-determination of peoples in national-territorial and national-cultural forms”; in 1991, Latvia enacted the law about the cultural autonomy of national minorities.…”

Arithmetic surfaces and adelic quotient groups

“…For a locally free sheaf F on Y , the corresponding adelic quotient group was calculated in [9] and [11, § 14] in the following form:…”

“…We recall that O X (nC) is a reflexive torsion free coherent subsheaf of the constant sheaf of the field of rational functions on X , and this subsheaf consists of elements of j * O U which have discrete valuations given by subschemes C i (where 1 ≤ i ≤ w ) greater or equal to −n , see also the beginning of Section 3.2.1 from[9].…”