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**Extra resources for A treatise on the analytical geometry of the point, line, circle, and conical sections**

**Example text**

A, and can forget about sites and topoi, used during its construction. Unary spectrum Specu A always coincides with the prime spectrum Specp A constructed before, so this notion of T ? -spectra indeed generalizes that of prime spectra. e. categories SA? might be distinct from SAu , arising topoi Spec? e. we can write Spec? A = Specu A = Spec A for a classical ring A, regardless of the localization theory chosen. This property is due to the fact that Specu A′ → Specu A is an open map for any open pseudolocalization of classical rings A → A′ , since open pseudolocalizations are flat and finitely presented.

Vector bundles) on a generalized scheme S, thus defining projective bundles PS (F ) = Proj SOS (F ) over S. These projective bundles seem to retain most of their classical properties. For example, sections of PS (F ) over S are in one-to-one correspondence to those strict quotients of F , which are line bundles over S. We can apply [1] [1] this to compute A-valued points of Pn = PnF∅ = Proj F∅[T0 , . . 7. Applications to Arakelov geometry 45 any generalized ring A with Pic(A) = 0: elements of Pn (A) correspond to surjective homomorphisms from free A-module A(n + 1) into |A| = A(1), considered modulo multiplication by elements from |A|× .

They constitute a lattice with respect to inclusion, with inf(a, b) given by the intersection a∩b, and sup(a, b) equal to a+b = (a, b) = Im(a⊕b → |A|). The largest element of this lattice is the unit ideal (1) = (e) = |A|, and the smallest element is the initial ideal ∅A = LA (0) ⊂ LA (1) = |A|. If A admits a zero 0, then ∅A = (0) is called the zero ideal; if not, we say that ∅A is the empty ideal. We have also a notion of principal ideal (a) = a|A| ⊂ |A|, for any a ∈ |A|, and of product of ideals ab, defined for example as Im(a ⊗A b → A); this is the ideal generated by all products ab, a ∈ a, b ∈ b.