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undocumented { sheafMap, -- deprecated -- TODO: document some of these (symbol SPACE, SheafMap, ZZ), (symbol +, SheafMap, SheafMap), (symbol *, SheafMap, SheafMap), (symbol *, RingElement, SheafMap), (symbol *, ZZ, SheafMap), (symbol -, SheafMap, SheafMap), (symbol -, SheafMap), (symbol **, SheafMap, SheafMap), (symbol **, SheafMap, SheafOfRings), (symbol **, SheafOfRings, SheafMap), (symbol **, SheafMap, CoherentSheaf), (symbol **, CoherentSheaf, SheafMap), (symbol |, SheafMap, SheafMap), (symbol ||, SheafMap, SheafMap), (symbol ==, SheafMap, SheafMap), (symbol ==, SheafMap, ZZ), (symbol ==, ZZ, SheafMap), } ----------------------------------------------------------------------------- -- Types and basic constructors and methods that return a coherent sheaf map ----------------------------------------------------------------------------- doc /// Node Key SheafMap Headline the class of morphisms of coherent sheaves Description Text The most essential data of the type @TO CoherentSheaf@ in Macaulay2 is the representative module. For morphisms of sheaves, the data type requires a little more care, because even if $\mathcal{F}$ and $\mathcal{G}$ are sheaves represented by modules $M$ and $N$, respectively, a morphism of sheaves $\phi : \mathcal F \to \mathcal G$ is not necessarily the sheaf associated to a module map $\psi : M \to N$. Indeed, the best one can say is that $\phi$ is represented by some map $$\psi : M_{\geq d} \to N,$$ where $d$ is some truncation degree. This means that in Macaulay2, a morphism of sheaves is represented as a morphism from some truncation of the source representative to the target representative. Example Q = QQ[x..z]; --Do example of accessing the truncation degree Text As illustrated in the above example, the source and target are still represented by the sheaves $\mathcal F$ and $\mathcal G$. The key {\tt degree} accesses the truncation degree needed to represent the map as a morphism of modules. To access the actual matrix representing the map, use {\tt matrix}. Subnodes (map, CoherentSheaf, CoherentSheaf, Matrix) (random, CoherentSheaf, CoherentSheaf) (inducedMap, CoherentSheaf, CoherentSheaf) Node Key (map, CoherentSheaf, CoherentSheaf, Matrix) (map, CoherentSheaf, CoherentSheaf, Matrix, InfiniteNumber) (map, CoherentSheaf, CoherentSheaf, Matrix, ZZ) (map, CoherentSheaf, Nothing, Matrix) (map, Nothing, CoherentSheaf, Matrix) Headline the constructor of morphisms of coherent sheaves Node Key (sheaf, Matrix) (sheaf, Matrix, ZZ) (sheaf, Variety, Matrix) (sheaf, Variety, Matrix, ZZ) Headline the sheafification functor for morphisms Node Key (random, CoherentSheaf, CoherentSheaf) Headline generate a random map of coherent sheaves Usage random(F, G) Inputs F:CoherentSheaf G:CoherentSheaf Outputs :SheafMap -- Description -- Text -- Example SeeAlso setRandomSeed (random, Module, Module) (map, CoherentSheaf, CoherentSheaf, Matrix) Node Key (inducedMap, CoherentSheaf, CoherentSheaf) (inducedMap, CoherentSheaf, CoherentSheaf, SheafMap) Headline induced maps on coherent sheaves Node Key cotangentSurjection -- (cotangentSurjection, ProjectiveVariety) Node Key embeddedToAbstract -- (embeddedToAbstract, ProjectiveVariety) ----------------------------------------------------------------------------- -- Basic methods for sheaves ----------------------------------------------------------------------------- Node Key (matrix, SheafMap) Headline the morphism of modules representing a morphisms of coherent sheaves Usage matrix phi phi.map Inputs phi:SheafMap Outputs :Matrix SeeAlso (truncate, List, Module) (truncate, Nothing, List, Matrix) Node Key (isWellDefined, SheafMap) Headline whether a morphism of coherent sheaves is well-defined Usage isWellDefined phi Inputs phi:SheafMap Outputs :Boolean -- Description -- Text -- Example SeeAlso (isWellDefined, CoherentSheaf) (isWellDefined, Variety) (isWellDefined, Matrix) -- Note: this uses F.cache.TorsionFree and F.cache.GlobalSectionLimit, -- which are set by twistedGlobalSectionsModule and called by HH^0(SumOfTwists) and prune(CoherentSheaf) /// ----------------------------------------------------------------------------- -- Arithmetic operations ----------------------------------------------------------------------------- doc /// Node Key (quotient, SheafMap, SheafMap) (quotient', SheafMap, SheafMap) (symbol //, SheafMap, SheafMap) (symbol \\, SheafMap, SheafMap) Headline factoring a morphism of coherent sheaves through another Node Key (symbol ^, SheafMap, ZZ) Headline raises a SheafMap to a power Usage f ^ n Inputs f:SheafMap a sheaf map n:ZZ a non-negative integer Outputs :SheafMap the sheaf map f raised to the power n Description Text This will create a sheaf map g which is the square of f. Example R = QQ[x,y,z] X = variety R f = sheafHom(OO_X, OO_X) g = f ^ 2 Node Key (symbol ^**, SheafMap, ZZ) Headline repeatedly tensors a map Usage f ^** n Inputs f:SheafMap a sheaf map n:ZZ a non-negative integer Outputs :SheafMap the sheaf map f tensor n times Description Text This will create a sheaf map g which is the tensor of f with itself. Example R = QQ[x,y,z] X = variety R f = sheafHom(OO_X, OO_X) g = f ^** 2 Node Key (symbol ++, SheafMap, SheafMap) Headline takes the direct sum of two sheaf maps Usage f ++ g Inputs f:SheafMap a sheaf map g:SheafMap another sheaf map Outputs :SheafMap the direct sum of sheaf maps f and g Description Text This will create a sheaf map h which is the direct sum of f and g. Example R = QQ[x,y,z] /// /// Node Key Headline Usage Inputs Outputs Description Text Example SeeAlso ///
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