A theorem in representation theory for DG algebras, with an - PowerPoint PPT Presentation

A theorem in representation theory for DG algebras, with an application to a question of Vasconcelos Saeed Nasseh Sean Sather-Wagstaff Department of Mathematics North Dakota State University 16 October 2011 2011 Fall Central Section Meeting University of Nebraska-Lincoln Special Session on Algebraic Geometry and Graded Commutative Algebra Saeed Nasseh, Sean Sather-Wagstaff Representation theory for DG algebras

Much of Algebra Reduces to Linear Algebra Assumption R is a d -dimensional commutative algebra over a field F = F . Facts Every R -module has a canonical F -vector space structure 1 by restriction of scalars. Every non-zero F -vector space has many distinct 2 R -module structures. Example Let R = F [ x ] / ( x 2 ) and M = ( R / xR ) 2 and N = R . Then M and N are isomorphic over F , but not over R . Slogan To study R -modules, fix V = F n and study all the ways to make V into an R -module. Saeed Nasseh, Sean Sather-Wagstaff Representation theory for DG algebras

Algebraic Information Can Be Encoded Geometrically Facts An R -module structure on V is a bilinear map R × V → V 1 satisfying certain axioms (associative and unital). The bilinear maps R × V → V are in bijection with the 2 linear maps R ⊗ F V → V , i.e., the n × dn matrices over F . An R -module structure on V is a matrix in M n × dn ( F ) 3 satisfying certain axioms (associative and unital). Notation Mod R ( V ) ⊆ M n × dn ( F ) is the set of R -module structures on V . Fact Given variables x ij to represent the entries of a matrix in M n × dn ( F ) , the R-module axioms are characterized by polynomial equations in the x ij , so Mod R ( V ) ⊆ M n × dn ( F ) is Zariski closed. Saeed Nasseh, Sean Sather-Wagstaff Representation theory for DG algebras

When are two modules in Mod R ( V ) isomorphic? Facts GL n ( F ) acts on Mod R ( V ) by conjugation: 1 Given φ ∈ GL n ( F ) and µ ∈ Mod R ( V ) , set φ · µ = φ ◦ µ ◦ ( R ⊗ F φ − 1 ) . Two module structures µ, λ ∈ Mod R ( V ) are isomorphic 2 over R if and only if λ = φ · µ for some φ ∈ GL n ( F ) . The isomorphism classes in Mod R ( V ) are precisely the 3 orbits under the action of GL n ( F ) . Each orbit in Mod R ( V ) is locally closed. 4 For M ∈ Mod R ( V ) , there is an inclusion of tangent spaces 5 T GL n ( F ) · M ⊆ T Mod R ( V ) . M M Saeed Nasseh, Sean Sather-Wagstaff Representation theory for DG algebras

Geometric Information Can Be Encoded Algebraically Theorem Given M ∈ Mod R ( V ) , there is an isomorphism T Mod R ( V ) / T GL n ( F ) · M ∼ = Ext 1 R ( M , M ) . M M Corollary Given M ∈ Mod R ( V ) , the orbit GL n ( F ) · M is open in Mod R ( V ) if and only if Ext 1 R ( M , M ) = 0 . Corollary The set of isomorphism classes of R-modules M such that Hom R ( M , M ) ∼ = R and Ext 1 R ( M , M ) = 0 is finite. Question How to prove the second corollary for rings that are not finite dimensional algebras over a field? Saeed Nasseh, Sean Sather-Wagstaff Representation theory for DG algebras

An Extension Answer When R is local, replace R with an appropriate finite dimensional differential graded (DG) F -algebra U : U is a a graded commutative F -algebra U = ⊕ e i = 0 U i , 1 U has a differential, i.e., a sequence of R -linear maps 2 ∂ U i : U i → U i − 1 such that ∂ U i ∂ U i + 1 = 0 for all i , and ∂ U satisfies the Leibniz Rule: for all a i ∈ U i and a j ∈ U j 3 ∂ U i + j ( a i a j ) = ∂ U i ( a i ) a j + ( − 1 ) i a i ∂ U j ( a j ) . Note The starting point for this replacement is to take the Koszul complex on a minimal generating sequence for the maximal ideal m ⊂ R . Saeed Nasseh, Sean Sather-Wagstaff Representation theory for DG algebras

An Extension, cont. Solution One needs to work with DG U -modules: U -modules with 1 extra data (a differential that satisfies the Leibniz Rule), and one has to encode the extra data into the geometric object DGMod U ( V ) . One has to consider a product of GL’s for the group action. 2 The quotient of tangent spaces is still isomorphic to an 3 Ext-module, but it is in general the wrong Ext-module. There are two distinct kinds of Ext over U ! 4 DG-Ext corresponds to Ext 1 R ( M , M ) under passage to U . Yoneda-Ext parametrizes extensions. They are not generally the same. By using truncations of semiprojective DG U -modules we 5 can reduce to the case where DG Ext and Yoneda Ext are the same, and the rest of the proof goes through. Saeed Nasseh, Sean Sather-Wagstaff Representation theory for DG algebras

Conclusion Remarks Commutative algebra does not exist in an algebraic 1 vacuum. Much of algebra reduces to linear algebra. 2 Geometry can encode algebraic information. 3 Group actions are not only useful for the Algebra prelim. 4 Algebra can encode geometric information. 5 Sometime to prove a theorem about rings, you have to be 6 flexible about your definition of “ring”. Saeed Nasseh, Sean Sather-Wagstaff Representation theory for DG algebras