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presentation Ext modules as quotients of free modules
Ext(I: INT, M:TAGGED("Quotient"), Q:TAGGED("Quotient")): TAGGED("Quotient")
Ext(I: LIST, M:TAGGED("Quotient"), Q:TAGGED("Quotient")): TAGGED("$ext.ExtList")
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***** NOT YET IMPLEMENTED *****
In the first form the function computes the I-th Ext module of M and N.
It returns a presentation of
Ext^I_R(M, N) as a quotient of a free module.
IMPORTANT: the only exception to the type of M or N (or even of the
output) is when they are either a zero module or a free module.
In these cases their type is indeed MOD.
It computes Ext via a presentation of the quotient of the two modules
Ker(Phi*_I)
and
Im(Phi*_{I-1}), where
-
Phi_I is the I-th map in the free resolution of M
-
Phi*_I is the map
Hom(Phi_I, N)
in the dual of the free resolution.
Main differences with the previous version include:
- SHIFTS have been removed, consequently only standard homogeneous
modules and quotients are supported
- as a consequence of 1), the type
Tagged("Shifted")
has been
removed. Ext will just be a
Tagged("Quotient")
- The former functions Presentation(), HomPresentation() and
KerPresentation() have been removed
- The algorithm uses Res() to compute the maps needed, and not
SyzOfGens anylonger, believed to cause troubles
- The function
Ext
always has THREE variables, see syntax...
In the second form the variable I is a LIST of nonnegative
integers. In this case the function Ext prints all the Ext modules
corresponding to the integers in I.
The output is of special type
Tagged("$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.
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([[0]])
-------------------------------
Ext(0, R/I, R^1); --- all those things should be isomorphic
Module([[0]])
-------------------------------
Ext(0..4, R/I, R/ideal(0)); -- another way to define the ring R as a quotient
Ext^0 = Module([[0]])
Ext^1 = Module([[0]])
Ext^2 = Module([[0]])
Ext^3 = R^1/Module([[x^5], [y^3], [z^2]])
Ext^4 = Module([[0]])
-------------------------------
N := Module([x^2,y], [x+z,0]);
Ext(0..4, R/I, R^2/N);
Ext^0 = Module([[0]])
Ext^1 = Module([[0]])
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([[0]])
-------------------------------
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Since version 4.7.3 the output modules are presented minimally.