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<head><title>AbstractSheaf -- the class of sheaves given by their Chern classes</title>
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<div><h1>AbstractSheaf -- the class of sheaves given by their Chern classes</h1>
<div class="single"><h2>Description</h2>
<div><div>This is the class of a sheaf as defined in the package <a href="index.html" title="A package for computations in Intersection Theory">Schubert2</a>. An abstract sheaf is really the data of its base variety (see <a href="___Abstract__Variety.html" title="The Schubert2 data type of an abstract variety">AbstractVariety</a>), <tt>Rank</tt> (missing documentation<!-- tag: Rank -->), <tt>ChernClass</tt> (missing documentation<!-- tag: ChernClass -->)...</div>
<p>For example, the Horrocks-Mumford bundle on projective 4-space can be represented by the following text. We first compute a “base variety” that is a point <tt>pt</tt>, and a variable integer named <tt>n</tt>, in terms of which we can compute the Hilbert polynomial.</p>
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<table class="examples"><tr><td><pre>i1 : pt = base(n)
o1 = pt
o1 : an abstract variety of dimension 0</pre>
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<tr><td><pre>i2 : X = projectiveSpace(4,pt,VariableName => h)
o2 = X
o2 : a flag bundle with ranks {4, 1}</pre>
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<tr><td><pre>i3 : F = abstractSheaf(X, Rank => 2, ChernClass => 1 + 5*h + 10*h^2)
o3 = F
o3 : an abstract sheaf of rank 2 on X</pre>
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<tr><td><pre>i4 : chi(F**OO_X(n*h))
1 4 5 3 125 2 125
o4 = --n + -n + ---n + ---n + 2
12 3 12 6
o4 : QQ[n]</pre>
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<div>Here in the description of X the number 4 is the rank of the trivial bundle, pt is the base variety of the projective space (in general, we allow projective spaces over an arbitrary base AbstractVariety), and the variable name specifies the variable to use to represent the first Chern Class of the tautological quotient line bundle on the projective space.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Abstract__Variety.html" title="The Schubert2 data type of an abstract variety">AbstractVariety</a> -- The Schubert2 data type of an abstract variety</span></li>
</ul>
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<div class="waystouse"><h2>Functions and methods returning an abstract sheaf :</h2>
<ul><li><span><tt>AbstractSheaf * AbstractSheaf</tt> (missing documentation<!-- tag: (*,AbstractSheaf,AbstractSheaf) -->)</span></li>
<li><span><tt>AbstractSheaf ** AbstractSheaf</tt> (missing documentation<!-- tag: (**,AbstractSheaf,AbstractSheaf) -->)</span></li>
</ul>
<h2>Methods that use an abstract sheaf :</h2>
<ul><li><span><tt>- AbstractSheaf</tt> (missing documentation<!-- tag: (-,AbstractSheaf) -->)</span></li>
<li><span><tt>AbstractSheaf + AbstractSheaf</tt> (missing documentation<!-- tag: (+,AbstractSheaf,AbstractSheaf) -->)</span></li>
<li><span><tt>AbstractSheaf + ZZ</tt> (missing documentation<!-- tag: (+,AbstractSheaf,ZZ) -->)</span></li>
<li><span><tt>AbstractSheaf ++ AbstractSheaf</tt> (missing documentation<!-- tag: (++,AbstractSheaf,AbstractSheaf) -->)</span></li>
<li><span><tt>AbstractSheaf ++ ZZ</tt> (missing documentation<!-- tag: (++,AbstractSheaf,ZZ) -->)</span></li>
<li><span><tt>AbstractSheaf - AbstractSheaf</tt> (missing documentation<!-- tag: (-,AbstractSheaf,AbstractSheaf) -->)</span></li>
<li><span><tt>AbstractSheaf ^ ZZ</tt> (missing documentation<!-- tag: (^,AbstractSheaf,ZZ) -->)</span></li>
<li><span><tt>AbstractSheaf ^** QQ</tt> (missing documentation<!-- tag: (^**,AbstractSheaf,QQ) -->)</span></li>
<li><span><tt>AbstractSheaf ^** RingElement</tt> (missing documentation<!-- tag: (^**,AbstractSheaf,RingElement) -->)</span></li>
<li><span><tt>AbstractSheaf ^** ZZ</tt> (missing documentation<!-- tag: (^**,AbstractSheaf,ZZ) -->)</span></li>
<li><span><tt>AbstractSheaf QQ</tt> (missing documentation<!-- tag: (SPACE,AbstractSheaf,QQ) -->)</span></li>
<li><span><tt>AbstractSheaf RingElement</tt> (missing documentation<!-- tag: (SPACE,AbstractSheaf,RingElement) -->)</span></li>
<li><span><tt>AbstractSheaf ZZ</tt> (missing documentation<!-- tag: (SPACE,AbstractSheaf,ZZ) -->)</span></li>
<li><span><tt>adams(ZZ,AbstractSheaf)</tt> (missing documentation<!-- tag: (adams,ZZ,AbstractSheaf) -->)</span></li>
<li><span><tt>ch(AbstractSheaf)</tt> (missing documentation<!-- tag: (ch,AbstractSheaf) -->)</span></li>
<li><span><tt>ch(ZZ,AbstractSheaf)</tt> (missing documentation<!-- tag: (ch,ZZ,AbstractSheaf) -->)</span></li>
<li><span><tt>chern(AbstractSheaf)</tt> (missing documentation<!-- tag: (chern,AbstractSheaf) -->)</span></li>
<li><span><tt>chern(ZZ,AbstractSheaf)</tt> (missing documentation<!-- tag: (chern,ZZ,AbstractSheaf) -->)</span></li>
<li><span><a href="_chern_lp__Z__Z_cm__Z__Z_cm__Abstract__Sheaf_rp.html" title="Get the Chern class of an abstract sheaf">chern(ZZ,ZZ,AbstractSheaf)</a> -- Get the Chern class of an abstract sheaf</span></li>
<li><span><tt>chi(AbstractSheaf)</tt> (missing documentation<!-- tag: (chi,AbstractSheaf) -->)</span></li>
<li><span><tt>ctop(AbstractSheaf)</tt> (missing documentation<!-- tag: (ctop,AbstractSheaf) -->)</span></li>
<li><span><tt>determinant(AbstractSheaf)</tt> (missing documentation<!-- tag: (determinant,AbstractSheaf) -->)</span></li>
<li><span><tt>dual(AbstractSheaf)</tt> (missing documentation<!-- tag: (dual,AbstractSheaf) -->)</span></li>
<li><span><tt>End(AbstractSheaf)</tt> (missing documentation<!-- tag: (End,AbstractSheaf) -->)</span></li>
<li><span><tt>exteriorPower(ZZ,AbstractSheaf)</tt> (missing documentation<!-- tag: (exteriorPower,ZZ,AbstractSheaf) -->)</span></li>
<li><span><tt>flagBundle(List,AbstractSheaf)</tt> (missing documentation<!-- tag: (flagBundle,List,AbstractSheaf) -->)</span></li>
<li><span><tt>Hom(AbstractSheaf,AbstractSheaf)</tt> (missing documentation<!-- tag: (Hom,AbstractSheaf,AbstractSheaf) -->)</span></li>
<li><span><tt>net(AbstractSheaf)</tt> (missing documentation<!-- tag: (net,AbstractSheaf) -->)</span></li>
<li><span><tt>projectiveBundle(AbstractSheaf)</tt> (missing documentation<!-- tag: (projectiveBundle,AbstractSheaf) -->)</span></li>
<li><span><tt>rank(AbstractSheaf)</tt> (missing documentation<!-- tag: (rank,AbstractSheaf) -->)</span></li>
<li><span><tt>schur(List,AbstractSheaf)</tt> (missing documentation<!-- tag: (schur,List,AbstractSheaf) -->)</span></li>
<li><span><tt>segre(AbstractSheaf)</tt> (missing documentation<!-- tag: (segre,AbstractSheaf) -->)</span></li>
<li><span><tt>segre(ZZ,AbstractSheaf)</tt> (missing documentation<!-- tag: (segre,ZZ,AbstractSheaf) -->)</span></li>
<li><span><tt>symmetricPower(QQ,AbstractSheaf)</tt> (missing documentation<!-- tag: (symmetricPower,QQ,AbstractSheaf) -->)</span></li>
<li><span><tt>symmetricPower(RingElement,AbstractSheaf)</tt> (missing documentation<!-- tag: (symmetricPower,RingElement,AbstractSheaf) -->)</span></li>
<li><span><tt>symmetricPower(ZZ,AbstractSheaf)</tt> (missing documentation<!-- tag: (symmetricPower,ZZ,AbstractSheaf) -->)</span></li>
<li><span><tt>todd(AbstractSheaf)</tt> (missing documentation<!-- tag: (todd,AbstractSheaf) -->)</span></li>
<li><span><tt>variety(AbstractSheaf)</tt> (missing documentation<!-- tag: (variety,AbstractSheaf) -->)</span></li>
<li><span><tt>ZZ + AbstractSheaf</tt> (missing documentation<!-- tag: (+,ZZ,AbstractSheaf) -->)</span></li>
<li><span><tt>ZZ ++ AbstractSheaf</tt> (missing documentation<!-- tag: (++,ZZ,AbstractSheaf) -->)</span></li>
</ul>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Abstract__Sheaf.html" title="the class of sheaves given by their Chern classes">AbstractSheaf</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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