# ToDoList

This is a quick list of things we want to implement. This is by no means a complete list of which computations one can do in QPA (hopefully), but an indication of what is possible in not so distant future. If you have suggestions of computations that you want to see implemented and it is not on the list, please contact us.

### Functors

 Compute Hom(M,N) as a vector space Not Done Compute Hom(M,N) using tensor product method Not Done Compute M\otimes N as a vector space Not Done Compute the functor -\otimes B, when B is a bimodule Not Done Compute Hom(M,N) as a End(M)-End(N)-bimodule Not Done Compute Ext^n(M,N) as a vectorspace, modules over End(M) and End(N), and bimodule over these Not Done Compute Tor^n(M,N) as a vectorspace, modules over End(M) and End(N), and bimodule over these Not Done Compute Hom-underline and Hom-overline Not Done Compute the adjuntion isomorphism Not Done Compute the Auslander-Reiten formula Not Done Compute the Hom(T,-) and Ext(T,-) for a tilting module T Not Done Coxeter like operations, functors Not Done

### Algebras

 Is an algebra a (quotient of a) poset-algebra Not Done Tests for finite type of an algebra Partially Done Is an algebra Gorenstein? Not Done Compute the representation dimension of an algebra Not Done Find the Auslander algebra of an algebra of finite type Not Done Compute trivial extension, one-point extensions Soon Done Find matrix constructions of new rings Not Done Compute the ring structure of Ext^*(M,M) for a module M Partially Done Find non-nilpotent elements of an algebra Not Done Find the (graded) center of an algebra Not Done Check if an graded algebra is Koszul, D-Koszul? Not Done Predefined classes of quivers and path algebras, Dynkin, extended Dynkin quivers, preprojective algebras, special biserial algebras Not Done Quiver of an algebra, algebra given by structure constants, find quiver Not Done Hilbert series, Poincare series Not Done Compute the Hochschild cohomology ring of an algebra Not Done Coverings of an algebra, fundamental group of a quiver Not Done

### Modules

 Predefined classes of modules for the predefined classes of algebras Not Done Given a ring homomorphism between two quotients of path algebras and a module over the target of this homomorphism, compute the induced module over the domain of the ring homomorphism. Not Done Decomposition of modules Partially Done Find the multiplicity of an indecomposable module in a given module Not Done Is a module preprojective? Not Done Is a module preinjective? Not Done Is a module regular? Not Done Is a module DTr-periodic, TrD-periodic, eventually periodic? Not Done Given a A-B-bimodule M, view it as a module over A or B. Not Done Given a A-A-bimodule M such that as a right and a left A-module M is isomorphic to A, find the automorphism of A such that M is isomorphic to A twisted on the left with this automorphism. Not Done Compute the almost split sequence starting in an indecomposable module Not Done Compute the support variety of a module Not Done Degenerations of modules? Not Done

### Homomorphisms

 Given n homomorphism from n modules to one module, or vica versa, construct the sum of these maps. Not Done

### Resolutions/homological algebra

 Projective resolutions of a module Done using linear algebra, only partially using Groebner basis Injective resolutions of a module Not Done Minimalization of non-minimal projective and injective resolutions Not Done Injective resolution of the algebra Not Done Projective resolution of dual of the algebra (all injectives) Not Done Compute Ext^i(M,N) and Tor_i(M,N) Not Done Graded resolutions for graded modules over graded algebras Not Done Quasi-isomorphism test for morphisms of complexes Not Done Hom-spaces of complexes Not Done Endomorphism rings of complexes Not Done Approximations/mutations of complexes Not Done AR-triangles of complexes Not Done Tilting complex test Not Done

### Invariants

 Euler form Not Done Knitting in the Auslander-Reiten quiver Not Done Include the combinatorial structures and functionality from CREP Started, not done

### Miscellaneous

 Latex output for every object Not Done

Oyvind.Solberg (at) math ntnu no