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<head><title>FirstOrderDeformation -- The class of all first order deformations of monomial ideals.</title>
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<div><h1>FirstOrderDeformation -- The class of all first order deformations of monomial ideals.</h1>
<div class="single"><h2>Description</h2>
<div><p>The class of all first order deformations of reduced monomial ideals. Elements represent a big torus (i.e., torus on the variables of a <a href="../../Macaulay2Doc/html/___Polynomial__Ring.html" title="the class of all ordered monoid rings">PolynomialRing</a>) graded part of the vector space of first order deformations. By results of Klaus Altmann and Jan Arthur Christophersen the dimension is either 0 or 1 (for manifolds, though this is not required by the implementation).</p>
<p>First order deformations can be created by <a href="_first__Order__Deformation.html" title="Makes a first order deformation.">firstOrderDeformation</a> by specifying a matrix with generators of a reduced monomial ideal and the exponent vector of a Laurent-monomial (i.e., a big torus degree).</p>
<p></p>
<p><b>Functions producing (sets of) first order deformations:</b></p>
<p><a href="_deform.html" title="Compute the deformations associated to a Stanley-Reisner complex.">deform</a> -- Compute the deformations associated to a Stanley-Reisner complex.</p>
<p><a href="_deformations__Face.html" title="Compute the deformations associated to a face.">deformationsFace</a> -- Compute the deformations associated to a face</p>
<p><a href="_trivial__Deformations.html" title="Compute the trivial deformations.">trivialDeformations</a> -- Compute the trivial deformations</p>
<p><a href="_first__Order__Deformation.html" title="Makes a first order deformation.">firstOrderDeformation</a> -- Makes a first order deformation</p>
<p></p>
<p><b>The data stored in a first order deformation f are</b></p>
<p><i>f.gens</i>, a <a href="../../Macaulay2Doc/html/_matrix.html" title="make a matrix">matrix</a> with generators of <a href="../../Macaulay2Doc/html/_source.html" title="source of a map">source</a> of the homomorphisms represented by f.</p>
<p><i>f.bigTorusDegree</i>, the exponent vector of the Laurent monomial.</p>
<p><i>f.degree</i>, the small torus (i.e., with respect to the grading added to R by <a href="_add__Coker__Grading.html" title="Stores a cokernel grading in a polynomial ring.">addCokerGrading</a>) degree of f.</p>
<p><i>f.isHomogeneous</i>, a <a href="../../Macaulay2Doc/html/___Boolean.html" title="the class of Boolean values">Boolean</a> indicating if f.degree is zero.</p>
<p><i>f.relevantGens</i>, a <a href="../../Macaulay2Doc/html/___Matrix.html" title="the class of all matrices">Matrix</a> with those elements of f.gens which are relevant to the deformation f (i.e., those m which have numerator(m*laurent(f)) not in ideal(f.gens)).</p>
<p><i>f.relationsCoefficients</i>, <a href="../../Macaulay2Doc/html/_matrix.html" title="make a matrix">matrix</a> of relations on coefficients of f. The rows correspond to the generators given in f.relevantGens.</p>
<p><i>f.parameters</i>, a <a href="../../Macaulay2Doc/html/___Matrix.html" title="the class of all matrices">Matrix</a> whose image is the <a href="../../Macaulay2Doc/html/_kernel.html" title="kernel of a ringmap, matrix, or chain complex">kernel</a> of the <a href="../../Macaulay2Doc/html/_transpose.html" title="transpose a table or a matrix">transpose</a> of f.relationsCoefficients extended by zeros for the elements of f.gens not in f.relevantGens. The rows correspond to the generators given in f.gens.</p>
<p></p>
<p><i>f.dim</i>, the dimension of the f-graded part of the deformation space of <a href="../../Macaulay2Doc/html/_ideal.html" title="make an ideal">ideal</a> f.gens.</p>
<p><i>f.isNonzero</i>, a <a href="../../Macaulay2Doc/html/___Boolean.html" title="the class of Boolean values">Boolean</a> indicating whether f is non-zero.</p>
<p><i>f.isTrivial</i>, a <a href="../../Macaulay2Doc/html/___Boolean.html" title="the class of Boolean values">Boolean</a> indicating whether f is trivial, i.e., <a href="_denominator__Monomial.html" title="The denominator monomial of a deformation.">denominatorMonomial</a> f has degree 1.</p>
<p>For an example see <a href="___Example_spfirst_sporder_spdeformation.html" title="Example accessing the data stored in a first order deformation.">Example first order deformation</a>.</p>
<p>This data can also be accessed by the methods listed below.</p>
<p><a href="_laurent.html" title="Converts an exponent vector or a deformation into a Laurent monomial.">laurent</a> represents f as a Laurent monomial, <a href="_to__Hom.html" title="Convert a first order deformation into a homomorphism.">toHom</a> represents f as a homomorphism.</p>
<p><a href="_total__Space.html" title="Total space of a deformation.">totalSpace</a> computes the total space of f.</p>
<div/>
<table class="examples"><tr><td><pre>i1 : R=QQ[x_0..x_4];</pre>
</td></tr>
<tr><td><pre>i2 : addCokerGrading(R)
o2 = | -1 -1 -1 -1 |
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
5 4
o2 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i3 : I=ideal(x_0*x_1,x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_0)
o3 = ideal (x x , x x , x x , x x , x x )
0 1 1 2 2 3 3 4 0 4
o3 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i4 : mg=mingens I;
1 5
o4 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i5 : f=firstOrderDeformation(mg, vector {-1,-1,0,2,0})
2
x
3
o5 = ----
x x
0 1
o5 : first order deformation space of dimension 1</pre>
</td></tr>
<tr><td><pre>i6 : degree f
o6 = 0
o6 : cokernel | -1 -1 -1 -1 |
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |</pre>
</td></tr>
<tr><td><pre>i7 : dim f
o7 = 1</pre>
</td></tr>
<tr><td><pre>i8 : f1=firstOrderDeformation(mg,vector {-1,1,0,0,0})
x
1
o8 = --
x
0
o8 : first order deformation space of dimension 1</pre>
</td></tr>
<tr><td><pre>i9 : isTrivial f1
o9 = true</pre>
</td></tr>
<tr><td><pre>i10 : f2=firstOrderDeformation(mg,vector {0,-1,-1,2,0})
2
x
3
o10 = ----
x x
1 2
o10 : first order deformation space of dimension 0</pre>
</td></tr>
<tr><td><pre>i11 : isNonzero f2
o11 = false</pre>
</td></tr>
</table>
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</div>
<div class="single"><h2>Caveat</h2>
<div><div>If we run into performance issues some of the redundant data will be removed, so for future compatibility access the data by the corresponding <a href="../../Macaulay2Doc/html/_method.html" title="make a new method function">method</a> not via the <a href="../../Macaulay2Doc/html/___Mutable__Hash__Table.html" title="the class of all mutable hash tables">MutableHashTable</a>.</div>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_simplex__Ring.html" title="The underlying polynomial ring of a deformation or face or complex.">simplexRing</a> -- The underlying polynomial ring of a deformation or face or complex.</span></li>
<li><span><a href="_target_lp__First__Order__Deformation_rp.html" title="The target of a deformation.">target(FirstOrderDeformation)</a> -- The target of a deformation.</span></li>
<li><span><a href="_source_lp__First__Order__Deformation_rp.html" title="The source of a deformation.">source(FirstOrderDeformation)</a> -- The source of a deformation.</span></li>
<li><span><a href="_gens__Source.html" title="Generators of the source of a deformation.">gensSource</a> -- Generators of the source of a deformation.</span></li>
<li><span><a href="_big__Torus__Degree.html" title="The big torus degree of a deformation.">bigTorusDegree</a> -- The big torus degree of a deformation.</span></li>
<li><span><a href="_numerator_lp__First__Order__Deformation_rp.html" title="The numerator of a deformation as a vector.">numerator(FirstOrderDeformation)</a> -- The numerator of a deformation as a vector.</span></li>
<li><span><a href="_denominator_lp__First__Order__Deformation_rp.html" title="The denominator of a deformation as a vector.">denominator(FirstOrderDeformation)</a> -- The denominator of a deformation as a vector.</span></li>
<li><span><a href="_numerator__Monomial.html" title="The numerator monomial of a deformation.">numeratorMonomial</a> -- The numerator monomial of a deformation.</span></li>
<li><span><a href="_denominator__Monomial.html" title="The denominator monomial of a deformation.">denominatorMonomial</a> -- The denominator monomial of a deformation.</span></li>
<li><span><a href="_degree_lp__First__Order__Deformation_rp.html" title="The small torus degree of a deformation.">degree(FirstOrderDeformation)</a> -- The small torus degree of a deformation.</span></li>
<li><span><a href="_grading_lp__First__Order__Deformation_rp.html" title="The small torus grading of a deformation.">grading(FirstOrderDeformation)</a> -- The small torus grading of a deformation.</span></li>
<li><span><a href="_is__Homogeneous_lp__First__Order__Deformation_rp.html" title="Check whether a deformation is homogeneous.">isHomogeneous(FirstOrderDeformation)</a> -- Check whether a deformation is homogeneous.</span></li>
<li><span><a href="_relations__Coefficients.html" title="Relations between the coefficients of a deformation.">relationsCoefficients</a> -- Relations between the coefficients of a deformation.</span></li>
<li><span><a href="_parameters.html" title="Parameters of a deformation.">parameters</a> -- Parameters of a deformation.</span></li>
<li><span><a href="_dim_lp__First__Order__Deformation_rp.html" title="Compute the dimension of a deformation.">dim(FirstOrderDeformation)</a> -- Compute the dimension of a deformation.</span></li>
<li><span><a href="_is__Nonzero.html" title="Check whether a deformation is non-zero.">isNonzero</a> -- Check whether a deformation is non-zero.</span></li>
<li><span><a href="_is__Trivial.html" title="Check whether a deformation is trivial.">isTrivial</a> -- Check whether a deformation is trivial.</span></li>
<li><span><a href="_laurent.html" title="Converts an exponent vector or a deformation into a Laurent monomial.">laurent</a> -- Converts an exponent vector or a deformation into a Laurent monomial.</span></li>
<li><span><a href="_to__Hom.html" title="Convert a first order deformation into a homomorphism.">toHom</a> -- Convert a first order deformation into a homomorphism.</span></li>
<li><span><a href="_total__Space.html" title="Total space of a deformation.">totalSpace</a> -- Total space of a deformation.</span></li>
<li><span><a href="_trivial__Deformations.html" title="Compute the trivial deformations.">trivialDeformations</a> -- Compute the trivial deformations.</span></li>
</ul>
</div>
<div class="waystouse"><h2>Methods that use a first order deformation :</h2>
<ul><li><span>bigTorusDegree(FirstOrderDeformation), see <span><a href="_big__Torus__Degree.html" title="The big torus degree of a deformation.">bigTorusDegree</a> -- The big torus degree of a deformation.</span></span></li>
<li><span><a href="_degree_lp__First__Order__Deformation_rp.html" title="The small torus degree of a deformation.">degree(FirstOrderDeformation)</a> -- The small torus degree of a deformation.</span></li>
<li><span><a href="_denominator_lp__First__Order__Deformation_rp.html" title="The denominator of a deformation as a vector.">denominator(FirstOrderDeformation)</a> -- The denominator of a deformation as a vector.</span></li>
<li><span>denominatorMonomial(FirstOrderDeformation), see <span><a href="_denominator__Monomial.html" title="The denominator monomial of a deformation.">denominatorMonomial</a> -- The denominator monomial of a deformation.</span></span></li>
<li><span><a href="_dim_lp__First__Order__Deformation_rp.html" title="Compute the dimension of a deformation.">dim(FirstOrderDeformation)</a> -- Compute the dimension of a deformation.</span></li>
<li><span><a href="___First__Order__Deformation_sp_eq_eq_sp__First__Order__Deformation.html" title="Compare two first order deformations.">FirstOrderDeformation == FirstOrderDeformation</a> -- Compare two first order deformations.</span></li>
<li><span>gensSource(FirstOrderDeformation), see <span><a href="_gens__Source.html" title="Generators of the source of a deformation.">gensSource</a> -- Generators of the source of a deformation.</span></span></li>
<li><span><a href="_grading_lp__First__Order__Deformation_rp.html" title="The small torus grading of a deformation.">grading(FirstOrderDeformation)</a> -- The small torus grading of a deformation.</span></li>
<li><span><a href="_is__Homogeneous_lp__First__Order__Deformation_rp.html" title="Check whether a deformation is homogeneous.">isHomogeneous(FirstOrderDeformation)</a> -- Check whether a deformation is homogeneous.</span></li>
<li><span>isNonzero(FirstOrderDeformation), see <span><a href="_is__Nonzero.html" title="Check whether a deformation is non-zero.">isNonzero</a> -- Check whether a deformation is non-zero.</span></span></li>
<li><span>isTrivial(FirstOrderDeformation), see <span><a href="_is__Trivial.html" title="Check whether a deformation is trivial.">isTrivial</a> -- Check whether a deformation is trivial.</span></span></li>
<li><span>laurent(FirstOrderDeformation), see <span><a href="_laurent.html" title="Converts an exponent vector or a deformation into a Laurent monomial.">laurent</a> -- Converts an exponent vector or a deformation into a Laurent monomial.</span></span></li>
<li><span><a href="_net_lp__First__Order__Deformation_rp.html" title="Pretty print for deformations.">net(FirstOrderDeformation)</a> -- Pretty print for deformations.</span></li>
<li><span><a href="_numerator_lp__First__Order__Deformation_rp.html" title="The numerator of a deformation as a vector.">numerator(FirstOrderDeformation)</a> -- The numerator of a deformation as a vector.</span></li>
<li><span>numeratorMonomial(FirstOrderDeformation), see <span><a href="_numerator__Monomial.html" title="The numerator monomial of a deformation.">numeratorMonomial</a> -- The numerator monomial of a deformation.</span></span></li>
<li><span>parameters(FirstOrderDeformation), see <span><a href="_parameters.html" title="Parameters of a deformation.">parameters</a> -- Parameters of a deformation.</span></span></li>
<li><span>relationsCoefficients(FirstOrderDeformation), see <span><a href="_relations__Coefficients.html" title="Relations between the coefficients of a deformation.">relationsCoefficients</a> -- Relations between the coefficients of a deformation.</span></span></li>
<li><span><a href="_simplex__Ring_lp__First__Order__Deformation_rp.html" title="The underlying polynomial ring of a deformation or face.">simplexRing(FirstOrderDeformation)</a> -- The underlying polynomial ring of a deformation or face.</span></li>
<li><span><a href="_source_lp__First__Order__Deformation_rp.html" title="The source of a deformation.">source(FirstOrderDeformation)</a> -- The source of a deformation.</span></li>
<li><span><a href="_target_lp__First__Order__Deformation_rp.html" title="The target of a deformation.">target(FirstOrderDeformation)</a> -- The target of a deformation.</span></li>
<li><span>toHom(FirstOrderDeformation), see <span><a href="_to__Hom.html" title="Convert a first order deformation into a homomorphism.">toHom</a> -- Convert a first order deformation into a homomorphism.</span></span></li>
<li><span>totalSpace(FirstOrderDeformation,PolynomialRing), see <span><a href="_total__Space.html" title="Total space of a deformation.">totalSpace</a> -- Total space of a deformation.</span></span></li>
</ul>
</div>
<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___First__Order__Deformation.html" title="The class of all first order deformations of monomial ideals.">FirstOrderDeformation</a> is <span>a <a href="../../Macaulay2Doc/html/___Type.html">type</a></span>, with ancestor classes <a href="../../Macaulay2Doc/html/___Mutable__Hash__Table.html" title="the class of all mutable hash tables">MutableHashTable</a> < <a href="../../Macaulay2Doc/html/___Hash__Table.html" title="the class of all hash tables">HashTable</a> < <a href="../../Macaulay2Doc/html/___Thing.html" title="the class of all things">Thing</a>.</p>
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