We here present rudiments of an approach to geometric actions in noncommutative algebraic geometry, based on geometrically admissible actions of monoidal categories. This generalizes the usual (co)module algebras over Hopf algebras which provide affine examples. We introduce a compatibility of monoidal actions and localizations which is a distributive law. There are satisfactory notions of equivariant objects, noncommutative fiber bundles and quotients in this setup.
Deep Dive into Some equivariant constructions in noncommutative algebraic geometry.
We here present rudiments of an approach to geometric actions in noncommutative algebraic geometry, based on geometrically admissible actions of monoidal categories. This generalizes the usual (co)module algebras over Hopf algebras which provide affine examples. We introduce a compatibility of monoidal actions and localizations which is a distributive law. There are satisfactory notions of equivariant objects, noncommutative fiber bundles and quotients in this setup.
arXiv:0811.4770v2 [math.AG] 9 Feb 2009
SOME EQUIVARIANT CONSTRUCTIONS IN
NONCOMMUTATIVE ALGEBRAIC GEOMETRY
ZORAN ˇSKODA
To Mamuka Jibladze on occasion of his 50th birthday
Abstract. We here present rudiments of an approach to geometric actions
in noncommutative algebraic geometry, based on geometrically admissible
actions of monoidal categories. This generalizes the usual (co)module al-
gebras over Hopf algebras which provide affine examples. We introduce a
compatibility of monoidal actions and localizations which is a distributive
law. There are satisfactory notions of equivariant objects, noncommutative
fiber bundles and quotients in this setup.
2000 Mathematics Subject Classification: 14A22, 16W30, 18D10.
Key words and phrases: Hopf module, equivariant sheaf, monoidal cate-
gory, compatible localization.
Modern mathematics is ever creating new kinds of geometries, and again
viewpoints of unification emerge. Somehow category theory seems to be very
effective in making this new order. Grothendieck taught us how important for
geometry the relative method is and the emphasis on general maps rather than
just on the incidence hierarchy of subspaces, intersections and so on. Impor-
tant properties of maps are often just categorical properties of morphisms in a
category (possibly with a structure). Various spectral constructions in algebra
and category theory use valuations, ideals, special kinds of modules, coreflective
subcategories, and so on, to single out genuine “underlying sets” of points, or
of subschemes – to rings, algebras and categories – which appear as objects rep-
resenting ’spaces’. Abstract localization enables us to consider local properties
of objects in categorical setup; sheaf theory and generalizations enable passage
between local and global. It is always enjoyable listening about a rich vision
of categorical and any other geometry from Mamuka, due to his enthusiasm,
width of interests and knowledge.
1. Noncommutative Algebraic Geometry
1.1. Descriptively, a noncommutative space X is a geometric entity which is de-
termined by a structure (algebra AX, category CX . . . ) carried by the collection
of objects (functions, cocycles, modules, sheaves . . . ) which are heuristically, or
in a genuine model, living over X. In this article, our primary interest will be
1
2
Z. ˇSKODA
spaces represented by abelian categories “of quasicoherent sheaves”. Gabriel–
Rosenberg theorem says that every scheme can be reconstructed up to isomor-
phism of schemes from its category of quasicoherent sheaves.
This involves
spectral constructions [26]: from an abelian category, Rosenberg constructs a
genuine set, its spectrum (many different spectra have been defined for various
purposes), which can be equipped with a natural induced topology and a stack
of local categories.
1.2. Noncommutative analogues of group actions, quotients and principal bun-
dles have been abundantly studied earlier, particularly within quantum group
renaissance [12, 22], in the context of study of noncommutative algebras and
graded algebras representing noncommutative affine or projective varieties. As
known from commutative geometry, it is easy to get out of these categories
when performing the most basic constructions, e.g., the quotient spaces. The
Tannakian reconstruction points out the correspondence between group-like ob-
jects and categories of representations, and it is natural to try to extend this
principle not only to symmetry objects but also to actions themselves, consid-
ering thus the actions of monoidal categories of modules over symmetry objects
to some other categories of quasicoherent sheaves. However, not every action
qualifies.
1.3. (Affine morphisms.) Given a ring R, denote by R −Mod the category of
left R-modules. To a morphism of rings f ◦: R →S (which is thought of as a
dual morphism to f : Spec S →Spec R) one associates
• extension of scalars f ∗: R −Mod →S −Mod, M 7→S ⊗R M;
• restriction of scalars (forgetful functor) f∗: S −Mod →R −Mod,
SM 7→RM;
• f ! : R −Mod →S −Mod, M 7→HomR(RS, M).
Denote F ⊣G when a functor F is left adjoint to a functor G. An easy fact:
f ∗⊣f∗⊣f !.
In particular, f ∗is left exact, f ! right exact and f∗exact.
Moreover, f∗is faithful. As maps of commutative rings correspond precisely to
maps of affine schemes, one says that an (additive) functor f ∗is almost affine
if it has a right adjoint f∗which is faithful and that f ∗is affine if, in addition,
f∗has a right adjoint as well (another motivation for this definition: Serre’s
affinity criterion, ´El´ements de g´eom´etrie alg´ebrique, II 5.2.1, IV 1.7.18).
1.4. (Pseudogeometry of functors) Given two abelian categories A, B, (equiv-
alent to small categories) a morphism f : B →A (viewed as a categorical
analogue of a map of spectra or rings) is an isomorphism class of right exact
additive functors from A to B. An inverse image functor f ∗: A →B of f is
a chosen representative of f. If it has a right adjoint, then it will be referred to
the direct image functor of
…(Full text truncated)…
This content is AI-processed based on ArXiv data.