What is tautological ring
First defined by Mumford,
What is , moduli space of curves of (arithmetic genus ) and marked (smooth) points, all singularities simple nodes; stability condition: every irreducible component of genus zero must have distinguished points should be stability condition gurantees no automorphisms, it is a DM stack smooth and proper of dim 3g-3+n
Intersection theory means studying Chow rings , there is a cycle map from this ring to the cohomology , tautological ring is a subring of the Chow ring, classes naturally arise in geometry. If we restrict this to tautological ring, is it an isomorphism to its image.
Maps between : 1. forget a marked point (need to restablize if necessary by contracting unstable component) 2. glueing map
Tautological ring is simultaneously defined as the smallest subring of the Chow ring closed under pushforward under the above maps starting with fundamantal class in each space
Essentially tautological divisors are boundary divisors of specific shapes.
But we can also get chern class?
drawing dual graph of stable curves convey same info
There is a permutation representation of on tautological
Something called Grothendieck-Teichmuller group
profinite version
Goal of Grothendieck: understand
uncountable group, what are its elements, action on for certain schemes .
LHS is acted on by , so is ;
E.g. , then
the action is the cyclotomic character
Remove one more point: , now . Belyi prove that this is a faithful representation.
Idea: constrain image by way of various maps behave between moduli space
Drinfeld’s subgroup subgroup of the form : Let be the two generators of , , , such that 1. 2. if . 3. where is generator of . Theorem (Drinfeld-Ihara): Image is contained in conjecture is it is exactly that
Connection to topology: configuration space (ordered configuration space), unordered analog. braid group, . is a fibration with fiber over $(0,1) equal to , is isomorphic to , fibration split so
Braid group appear as automorphisms of objects in braided monoidal category
Drinfeld’s definition: mess with all structures of bracat
Notice: little disks operad
Theorem (Horel, based on Drinfeld, Bar-Natan)
The free algerba is graded (i.e. bracket preserves degree); There should exist an adjunction from category of lie algebras to that of associative algebras making the diagram of free functors commute. Indeed, this is the functor of forming universal envelopping algebra.
A -coalgebra is a vector space equipped with a map and a counit such that it is coassociative and satisifies counitality ( is identity and another one for right counitality) Bialgebra: is ring hom
Monoidal cat - , braiding - , symmetric if Example: product of two sets, tensor product of vector spaces, graded vector space with braiding given by Koszul sign convention
An algebra in a monoidal category is a monoidal object (bilinearity is distributive law)
If is braided and are algebras, then so is .
A bialgebra in a braided category is a algebra and coalgebra s.t. comultiplication is a mp of algebras. It is moreover a Hopf algebra if there is an antipode (like inverse) such that is equal to the composite .
Example: group algebra, universal envelopping algebra of Lie algebra (Milnor-Moore: There is an equivalence of category between the category of Lie algebras to that if primitively generated Hopf algebras if is characterstic zero, one direction is given by taking universal envelopping algebra, the other is given by taking primitive elements)
Let be the set of group-like elements, then there is a map of algebras .
(Cartier-Kostant-Milnor-Moore) If is Hopf algebra over algebraically closed field, and characteristic 0 and co-commutative, then (does use algebraically closed)
impplies if is generated by group like elements, then is isomorphic to
natural isomorphism (same universal property)
Completed tensor product: , but completion of vector spaces are not unique, https://en.wikipedia.org/wiki/Complete_topological_vector_space, though there is a unique Hausdorff completion. We want to make the category of TVS with completed tensor product to be a monoidal category
If is filtered by , we can define the completion to be the inverse limit of and the topology is given by the local base ; can be filtered
If is graded vector space , then
If is graded Lie algebra, then is completed Lie algbra (need to use that is positively graded), but it is not graded vector space.
Filter: be the ideal generated by such that such that . Let . then becomes a completed Hopf algebra, and it is isomorphic to
Let be a complete positive graded Lie algebra, over a field of characteristic 0. Define , we define multiplication . A priori this only makes sense in , but the content of BCH is that this has an alternate expression
Example: If is the completion of the free Lie algebra , there is an isomorphism of groups from to the group like element of the completed Hopf algebra.
Sketch: , and is primitive iff is group-like (.
completion is complete?