In what
way is a formally smooth algebra a _machine_ producing families of
manifolds? Consider the special case of the path algebra
quiver and recall that an
map
determined by the rule
in
a direct sum decomposition
representation does determine a _dimension vector_
for every arrow
defines a linear map
whole point of writing paths in the quiver from right to left so that a
representation is determined by its _vertex spaces_
linear maps between them as there are arrows). Fixing vectorspace bases
in the vertex-spaces one observes that the space of all
space
base-change in the vertex-spaces does determine the action of the
_base-change group_
a bases in
components are no longer just affine spaces with a groupaction but
essentially equal to them as there is a natural one-to-one
correspondence between
affine space
determines vertex-spaces
determine linear maps between these spaces
vertex-spaces, we can represent these maps by matrices
acts on the representation space
(embedded as block-matrices in
simultaneous conjugation. More generally, if
algebra, then all its representation varieties
affine smooth varieties equipped with a
or less immediately from the definition and [Grothendieck][2]\’s
characterization of commutative regular algebras. For the record, an
algebra
because for every map
vertex-idempotents form an orthogonal set of idempotents which is known
to lift modulo nilpotent ideals and call this lift
every arrow lifts as we can send it to an arbitrary element of
Grothendieck\’s criterium
also clarifies why so few commutative regular algebras are formally
smooth : being formally smooth is a vastly more restrictive property as
the lifting property extends to all algebras
dimension of the commutative variety is at least two one can think of
maps from its coordinate ring to the commutative quotient of a
non-commutative ring by a nilpotent ideal which do not lift (for an
example, see for example [this preprint][3]). The aim of
non-commutative algebraic geometry is to study _families_ of manifolds
https://lievenlb.local/wp-trackback.php/10 [2]:
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Grothendieck.
html [3]: http://www.arxiv.org/abs/math.AG/9904171
Tag: geometry
The
previous part of this sequence was [quiver representations][1]. When
is a formally smooth algebra, we have an infinite family of smooth
affine varieties
representations. On
representations, that is, orbits under this action. Mind you, an orbit
space does not always exist due to the erxistence of non-closed orbits
so one often has to restrict to suitable representations of
which it _is_ possible to construct an orbit-space. But first, let us
give a motivating example to illustrate the fact that many interesting
classification problems can be translated into the setting of this
non-commutative algebraic geometry. Let
curve of genus
classical object of study is
of semi-stable vectorbundles on
space has an open subset (corresponding to the _stable_ vectorbundles)
which classify the isomorphism classes of unitary simple representations
fundamental group of
projective curve
orthogonal idempotents, its representation varieties decompose into
connected components according to dimension vectors
varieties. As mentioned before it is not possible to construct a
variety classifying the orbits in one of these components, but there are
two methods to approximate the orbit space. The first one is the
_algebraic quotient variety_ of which the coordinate ring is the ring of
invariant functions. In this case one merely recovers for this quotient
of
representations_ which is an algebraic quotient of the open subset of
all representations having no subrepresentation of dimension vector
OK! I asked to get side-tracked by comments so now that there is one I’d better deal with it at once. So, is there any relation between the non-commutative (algebraic) geometry based on formally smooth algebras and the non-commutative _differential_ geometry advocated by Alain Connes?
Short answers to this question might be (a) None whatsoever! (b) Morally they are the same! and (c) Their objectives are quite different!
As this only adds to the confusion, let me try to explain each point separately after issuing a _disclaimer_ that I am by no means an expert in Connes’ NOG neither in
(a) _None whatsoever!_ : Connes’ approach via spectral triples is modelled such that one gets (suitable) ordinary (that is, commutative) manifolds into this framework. The obvious algebraic counterpart for this would be a statement to the effect that the affine coordinate ring
(b) _Morally they are the same_ : If you ever want to get some differential geometry done, you’d better have a connection on the tangent bundle! Now, Alain Connes extended the notion of a connection to the non-commutative world (see for example _the_ book) and if you take the algebraic equivalent of it and ask for which algebras possess such a connection, you get _precisely_ the formally smooth algebras (see section 8 of the Cuntz-Quillen paper “Algebra extensions and nonsingularity” Journal AMS Vol 8 (1991). Besides there is a class of
(c) _Their objectives are quite different!_ : Connes’ formalism aims to define a length function on a non-commutative manifold associated to a