type,tub_enc.html

type tub_enc - encoding an indecomposable sheaf with respect to its position within the tubular families

SYNOPSIS:

For X being of tubular representation type, each indecomposable sheaf F lies in an AR-component which is a homogeneous tube in the tubular family C_q (where q = slope(F)). Those sheaves lying in one of the exceptional tubes of rank p_i can be encoded by a discrete set of data. The list of these data is defined to be the type "tub_enc"; namely:

[q,i,s,l]

where:

q - the "slope" (a rational, or infinity)
i - number of an exceptional tube (i=1,..,t)
s - quasi-socle index (an integer, which is taken modulo p_i )
l - the quasi-length (height in corresponding tube)

One of the central tasks of the tubular-routines (and the main application of the telescoping functors) is to relate this encoding to the presentation of a sheaf given by its class in the Grothendieck-group K₀X. (Cf. to get_class and get_tub .)

EXAMPLES:

Assume p = (2,3,6).

[-4/3,2,1,3]

encodes the sheaf lying in C_{-4/3}, lying in the 2nd tube of rank p₂ = 3, having quasi-socle indexed by 1, and quasi-length 3, so its corresponding class in K₀X lies in the radical.

[infinity,3,0,1]

encodes the sheaf in C_{infinity}, in the 3rd tube of rank p₃ = 6, having quasi-socle indexed by 0 and quasi-length 1, so being quasi-simple.

BACK TO: start page of the crep online manual, tubular