Maps between non-commutative spaces

Abstract

Let J be a graded ideal in a not necessarily commutative graded k-algebra A = A 0 ⊕ A 1 ⊕• • • in which dim k A i < ∞ for all i. We show that the map A → A/J induces a closed immersion i : Proj nc A/J → Proj nc A between the non-commutative projective spaces with homogeneous coordinate rings A and A/J. We also examine two other kinds of maps between non-commutative spaces. First, a homomorphism φ : A → B between not necessarily commutative N-graded rings induces an affine map Proj nc B ⊃ U → Proj nc A from a non-empty open subspace U ⊂ Proj nc B. Second, if A is a right noetherian connected graded algebra (not necessarily generated in degree one), and A (n) is a Veronese subalgebra of A, there is a map Proj nc A → Proj nc A (n) ; we identify open subspaces on which this map is an isomorphism. Applying these general results when A is (a quotient of) a weighted polynomial ring produces a non-commutative resolution of (a closed subscheme of) a weighted projective space.

On Continuous and Adjoint Morphisms Between Non-Commutative Prime Spectra

“…One of the most elementary and well-known properties of non-commutative rings is the non-functoriality of their prime spectra: there is apparently no natural way of assigning, to an arbitrary ring homomorphism R → S, a function from the prime spectrum of S into the prime spectrum of R. Nevertheless, there is an extensive and deep literature presenting, among many other things, topological and geometric contexts for both non-commutative ring homomorphisms and their generalizations to certain functors between module-like categories. These contexts appear, for example, in the earlier publications [1], [5], [18], [20] and [19], and the more recent ones [2], [10], [12], [11], [14], [13] and [15]. In the present paper we continue a discussion begun in [1, § 4].…”

“…While a few of the definitions and preliminary results in this paper are presented within this broader context, we leave a more complete generalization to the interested reader. Recent studies on non-commutative ring homomorphisms (and generalizations) from this point of view include [13][14][15].…”

Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

“…The sheaf L / is -ample (see §2.3.2 and Theorem 3.2(6)), so a result of Artin and Van den Bergh [AVdB90] (see §2.3.3) tells us that QGr( ( , , L / )) is equivalent to Qcoh( ). Combining this with the main result in [Smi04] (see [Smi16,Theorem 1.2]) implies that there are functors…”

“…between the quotient categories such that * is a fully faithful embedding whose essential image is closed under subobjects and quotients, * is left adjoint to * and ! is right adjoint to * (see [Smi16,VdB01]). The functors * and * behave like the inverse and direct image functors associated to a closed immersion of one scheme in another.…”

Untitled

“…We prefer to say that the closed subspaces are the same things as the closed subschemes. Further evidence is provided by [24,Theorem 3.2], which shows that a two-sided ideal in an N-graded k-algebra A cuts out a closed subspace of ProjA. There is further evidence.…”