Derivations

A derivation d of a ring R is a map R \to R that obeys the product rule d(rs) = d(r)s + rd(s) for elements r, s in the field, and which is also distributive over addition in the ring d(r+s) = d(r) + d(s)

The HAPDerivation datatype
The HAPDerivation datatype represents a derivation of a polynomial ring R. Given the product and addition rules above, a derivation can be completely specified in terms of the images of the ring indeterminates. Consequently, a HAPDerivation is specified by providing: a polynomial ring R a list of images, under the derivation, of each of the ring's indeterminates. Optionally, you can also provide a set of relations I (which is assumed to be empty if none is provided). The support for derivations over quotient rings using HAPDerivation is limited. The ring is always assumed to simply be the polynomial ring R, but when calculating the kernel and homology (see and ), any relations in I are also included in the result's relations, which is the equivalent of calculating the kernel of homology of the derivation R/I \to R/I.
Computing the kernel and homology of a derivation

Computing the kernel of a derivation is difficult in the general case, but for polynomial rings over \mathbb{F}_p we have a novel solution which allows derivations to be considered as module homomorphisms.

Consider a ring R over GF(2). For any r \in R, d(r^2) = d(rr) = d(r)r + rd(r) = 2d(r) = 0 In general, for a ring R = \mathbb{F}_p[x_1, \ldots, x_n], the subring S = \mathbb{F}_p[x_1^p, \ldots, x_n^p] is in the kernel of d. For this choice of S, R can always be written as a free S-module with 2^n generators.

Consider now taking the derivation of sr where s \in S and r \in R, d(sr) = sd(r) + d(s)r = sd(r) since s is in the kernel of d. In other words, when R is written as an S-module, the derivation d behaves exactly as as an S-module homomorphism.

The &Singular; computer algebra system can compute the kernels of module homomorphisms over polynomial rings, and the &singular; package provides an interface between &Singular; and &GAP;. The function constructs S, expresses d as an S-module homomorphism (as well as converting any ideal), and asks &Singular; for the kernel. &Singular; returns the result as a set of module generators, which are then reduced to a set of ring generators and a presentation found. This makes further use of &Singular;, and the HAPRingHomomorphism datatype (see Chapter ).

The homology of a derivation is defined to be Hom(d) = \ker (d) / im (d) Given the kernel of a derivation and the image (expressed as the images of the module generators), the homology can be readily computed, as is done by . The natural presentation obtained by concatenating the kernel relations and the image typically contains redundancies and the is used to generate a nicer presentation involving fewer indeterminates and relations.

Construction function
Data access function
Example: Constructing and accessing data of a HAPDerivation

We demonstrate creating a HAPDerivation object and reading back its data by creating a derivation d of the polynomial ring over the integers with two indeterminates, \mathbb{Z}[x_1, x_2, x_3], which has the mapping d(x_1) = d(x_2) = 1, d(x_3) = 0. gap> ring := PolynomialRing(Integers, 3);; gap> d := HAPDerivation(ring, [One(ring), One(ring), Zero(ring)]); <Derivation> gap> gap> DerivationRing(d); Integers[x_1,x_2,x_3] gap> DerivationImages(d); [ 1, 1, 0 ] gap> DerivationRelations(d); [ ]
Image, kernel and homology functions
Example: Homology of a HAPDerivation

In this example, we wish to calculate the homology of a derivation d on the ring \mathbb{F}_2[x,y,z]/(yx^2 + xy^2) where d(x) = d(y) = 0 and d(z) = yx^2 + xy^2. To demonstrate some of the other functions, we also calculate the kernel of this derivation and the image of the element xz. gap> ring := PolynomialRing(GF(2), 3);; gap> x := ring.1;; y := ring.2;; z := ring.3;; gap> d := HAPDerivation(ring, [x^2 + x*y + y^2], > [Zero(ring), Zero(ring), x^2*y + x*y^2]); <Derivation> gap> # gap> ImageOfDerivation(d, x*z); x_1^3*x_2+x_1^2*x_2^2 gap> KernelOfDerivation(d); [ GF(2)[x_7,x_8,x_9], [ x_7^2+x_7*x_8+x_8^2 ], <Ring homomorphism> ] gap> HomologyOfDerivation(d); [ GF(2)[x_4,x_5,x_6], [ x_4^2+x_4*x_5+x_5^2, x_5^3 ], <Ring homomorphism> ]

The result we were after, the homology of d, can be read from the final result as follows. The first two entries in the list give the homology group as a quotient ring: the quotient of the polynomial ring \mathbb{F}_2[x_6,x_7,x_8] with the two relations (x_7^2+x_7x_8+x_8^2, x_8^3). The last entry in the list gives the new indeterminates in terms of the original indeterminates, and from this we know that it would be friendlier to write the result as \frac{\mathbb{F}_2[x,y,e^2]}{(x^2+xy+y^2, y^3)} F_2[x,y,e^2] / (x^2+xy+y^2, y^3) The kernel of the derivation can read from the result in a similar manner.