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A coequalizer of a pair of maps f,g:X->Y in a category is a map c:Y->C such that 1. c degreesf=c degreesg, where degrees denotes composition. 2. For any other map c^':Y->C^' ...

A discriminant is a quantity (usually invariant under certain classes of transformations) which characterizes certain properties of a quantity's roots. The concept of the ...

Let C be a smooth geometrically connected projective curve over F_q with q=p^s a prime power. Let infty be a fixed closed point of X but not necessarily F_q-rational. A ...

An equalizer of a pair of maps f,g:X->Y in a category is a map e:E->X such that 1. f degreese=g degreese, where degrees denotes composition. 2. For any other map e^':E^'->X ...

A functor between categories of groups or modules is called exact if it preserves the exactness of sequences, or equivalently, if it transforms short exact sequences into ...

An exact sequence is a sequence of maps alpha_i:A_i->A_(i+1) (1) between a sequence of spaces A_i, which satisfies Im(alpha_i)=Ker(alpha_(i+1)), (2) where Im denotes the ...

A forgetful functor (also called underlying functor) is defined from a category of algebraic gadgets (groups, Abelian groups, modules, rings, vector spaces, etc.) to the ...

A free Abelian group is a group G with a subset which generates the group G with the only relation being ab=ba. That is, it has no group torsion. All such groups are a direct ...

An abstract algebra concerned with results valid for many different kinds of spaces. Modules are the basic tools used in homological algebra.

Given a short exact sequence of modules 0->A->B->C->0, (1) let ...->P_2->^(d_2)P_1->^(d_1)P_0->^(d_0)A->0 (2) ...->Q_2->^(f_2)Q_1->^(f_1)Q_0->^(f_0)C->0 (3) be projective ...