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 Ext

presentation Ext modules as quotients of free modules
 Syntax
 ``` Ext(I:INT, M:TAGGED(Quotient), Q:TAGGED(Quotient)): TAGGED(Quotient) Ext(I:LIST, M:TAGGED(Quotient), Q:TAGGED(Quotient)): TAGGED(\$ext.ExtList) ```

 Description
In the first form the function computes the I-th Ext module of M and N. It returns a presentation of Ext^I_R(M, N) as a quotient of a free module.

IMPORTANT: the only exception to the type of M or N (or even of the output) is when they are either a zero module or a free module. In these cases their type is indeed MOD.

It computes Ext via a presentation of the quotient of the two modules Ker(Phi*_I) and Im(Phi*_{I-1}), where

- Phi_I is the I-th map in the free resolution of M

- Phi*_I is the map Hom(Phi_I, N) in the dual of the free resolution.

Main differences with the previous version include:

- SHIFTS have been removed, consequently only standard homogeneous modules and quotients are supported

- as a consequence of 1), the type Tagged("Shifted") has been removed. Ext will just be a Tagged("Quotient")

- The former functions Presentation(), HomPresentation() and KerPresentation() have been removed

- The algorithm uses Res() to compute the maps needed, and not SyzOfGens anylonger, believed to cause troubles

- The function Ext always has THREE variables, see syntax...

In the second form the variable I is a LIST of nonnegative integers. In this case the function Ext prints all the Ext modules corresponding to the integers in I. The output is of special type Tagged("\$ext.ExtList") which is basically just the list of pairs {(J, Ext^J(M, N)) | J in I} in which the first element is an integer of I and the second element is the correpsonding Ext module.

VERY IMPORTANT: CoCoA cannot accept the ring R as one of the inputs, so if you want to calculate the module Ext^I_R(M, R) you need to type something like

Ext(I, M, Ideal(1));

or

Ext(I, M, R^1);

or

Ext(I, M, R/Ideal(0));

NOTE: The input is pretty flexible in terms of what you can use for M and N. For example they can be zero modules or free modules. See some examples below.

 Example
 ``` Use R ::= QQ[x,y,z]; I := Ideal(x^5, y^3, z^2); Ideal(0) : (I); Ideal(0) ------------------------------- \$hom.Hom(R^1/Module(I), R^1); -- from Hom package Module([]) ------------------------------- Ext(0, R/I, R^1); --- all those things should be isomorphic Module([]) ------------------------------- Ext(0..4, R/I, R/Ideal(0)); -- another way to define the ring R as a quotient Ext^0 = Module([]) Ext^1 = Module([]) Ext^2 = Module([]) Ext^3 = R^1/Module([[x^5], [y^3], [z^2]]) Ext^4 = Module([]) ------------------------------- N := Module([x^2,y], [x+z,0]); Ext(0..4, R/I, R^2/N); Ext^0 = Module([]) Ext^1 = Module([]) Ext^2 = R^2/Module([[0, x + z], [y, 0], [0, z^2], [z^2, 0], [0, y^3], [x^5, 0]]) Ext^3 = R^2/Module([[x + z, 0], [0, z^2], [z^2, 0], [y^3, 0], [0, x^5], [0, y]]) Ext^4 = Module([]) ------------------------------- ```
Since version 4.7.3 the output modules are presented minimally.