The Injective Spectrum of a Noncommutative Space

Abstract

For a noncommutative space X, we study Inj X , the set of isomorphism classes of indecomposable injective X-modules. In particular, we look at how this set, suitably topologized, can be viewed as an underlying "spectrum" for X. As applications we discuss noncommutative notions of irreducibility and integrality, and a way of associating an integral subspace of X to each element of Inj X which behaves like a "weak point."  2002 Elsevier Science (USA)

Subspaces of non-commutative spaces

“…Our definition of a prime module is modelled on the analogous notion for modules over rings-see, for example, [17,Section 4.3.4]. Our definition of prime module in terms of support emerged in discussions with C. Pappacena and agrees with his definition in [18].…”

The BGQ spectral sequence for noncommutative spaces

“…It will turn out that the appropriate noncommutative analogue of the underlying point set of X will be the set of (isomorphism classes of) indecomposable injective X-modules Inj(X). This gives further evidence that Inj(X) (called the injective spectrum in [7]) is an appropriate spectrum to study in noncommutative algebraic geometry.…”

“…Proof. Part (a) follows immediately from the fact that dim is exact, and (b) is a well-known consequence of the fact that dim is also finitely partitive (see for instance [7,Proposition 3.9]). …”

“…(The proof is a modification of [5,Theorem 6.7] to the graded situation.) Thus E is the sum of its critical submodules, and so E is critical [7,Lemma 3.6]. Thus any nonzero homomorphism E → E(j − i) must be injective, hence an isomorphism.…”

Untitled

“…Key to this approach is the notion of a closed subspace, investigated in detail, for example, in [1], [3], [4], [5], [8]. This brief note records an elementary-but apparently previously unnoticedobservation about closed subspaces, working in the slightly more general setting of an abelian category equipped with a generator and having arbitrary products and coproducts: We observe that there is a duality between the collection of closed subspaces and a suitably defined small poset of ideals within the generator.…”