‣ CastelnuovoMumfordRegularity( M ) | ( attribute ) |
Returns: an integer
The input is a graded lp M over a graded polynomial ring S. The output is the Castelnuovo-Mumford regularity of M.
‣ TateResolution( M ) | ( attribute ) |
Returns: a chain complex
The input is a graded lp M over a graded polynomial ring S or a chain complex of graded lp's over S or a graded lp over KoszulDualRing(S). The output is the Tate resolution given as a chain complex. To convert this chain complex to a cochain complex we can use the command AsCochainComplex.
‣ TateResolution( phi ) | ( attribute ) |
Returns: a chain morphism
The input is a morphism \phi of graded lp's over a graded polynomial ring S or a chain morphism of graded lp's over S or a graded lp morphism over KoszulDualRing(S). The output is the Tate resolution of \phi given as a chain morphism. To convert this chain morphism to a cochain morphism we can use the command AsCochainMorphism.
‣ RCochainFunctor( S ) | ( attribute ) |
Returns: a CAP functor
The input is a graded polynomial ring with deg(x_i)=1 for all indeterminates x_i's. The output is the R functor from the category of graded lp's over S to the cochains category of the graded lp's over KoszulDualRing(S).
‣ RChainFunctor( S ) | ( attribute ) |
Returns: a CAP functor
The input is a graded polynomial ring with deg(x_i)=1 for all indeterminates x_i's. The output is the R functor from the category of graded lp's over S to the chains category of the graded lp's over KoszulDualRing(S).
‣ LCochainFunctor( S ) | ( attribute ) |
Returns: a CAP functor
The input is a graded polynomial ring with deg(x_i)=1 for all indeterminates x_i's. The output is the L functor from the category of graded lp's over KoszulDualRing(S) to the cochains category of the graded lp's over S.
‣ LChainFunctor( S ) | ( attribute ) |
Returns: a CAP functor
The input is a graded polynomial ring with deg(x_i)=1 for all indeterminates x_i's. The output is the L functor from the category of graded lp's over KoszulDualRing(S) to the chains category of the graded lp's over S.
‣ TruncationFunctorUsingTateResolution( S, n ) | ( operation ) |
Returns: a CAP functor
The input is a graded polynomial ring with deg(x_i)=1 for all indeterminates x_i's and an integer n. The output is an endofunctor on the category of lp's over S that send each M with reg(M)\leq n to M_{\geq n} and each morphism f to f_{\geq n}.
‣ NatTransFromTruncationUsingTateResolutionToIdentityFunctor( S, n ) | ( operation ) |
Returns: a CAP natural transformation
The input is a graded polynomial ring with deg(x_i)=1 for all indeterminates x_i's and an integer n. The output is a natural transformation from the truncation functor (using Tate resolution) to identity functor.
‣ TruncationFunctorUsingHomalg( S, n ) | ( operation ) |
Returns: a CAP functor
The input is a graded polynomial ring with deg(x_i)=1 for all indeterminates x_i's and an integer n. The output is an endofunctor on the category of lp's over S that send each M with reg(M)\leq n to M_{\geq n} and each morphism f to f_{\geq n}. This is much more faster than the truncation using Tate resolition.
‣ NatTransFromTruncationUsingHomalgToIdentityFunctor( S, n ) | ( operation ) |
Returns: a CAP natural transformation
The input is a graded polynomial ring with deg(x_i)=1 for all indeterminates x_i's and an integer n. The output is a natural transformation from the truncation functor (using homalg) to identity functor.
‣ NatTransFromTruncationUsingTateResolutionToTruncationFunctorUsingHomalg( S, n ) | ( operation ) |
Returns: a CAP natural transformation
The input is a graded polynomial ring with deg(x_i)=1 for all indeterminates x_i's and an integer n. The output is a natural transformation from the truncation functor (using Tate resolution) to the truncation functor (using homalg).
‣ GLPGeneratedByHomogeneousPartFunctor( S, n ) | ( operation ) |
Returns: a CAP functor
The input is a graded ring S and an integer n. The output is an endofunctor on the category of lp's over S that send each M to the submodule of M generated by all homogeneous elements of degree n. GLP stands for Graded Left Presentation.
‣ NatTransFromGLPGeneratedByHomogeneousPartToIdentityFunctor( S, n ) | ( operation ) |
Returns: a CAP natural transformation
The input is a graded ring and an integer n. The output is a natural transformation from the GLPGeneratedByHomogeneousPartFunctor(S,n) to the identity functor.
‣ HomogeneousPartOverCoefficientsRingFunctor( S, n ) | ( operation ) |
Returns: a CAP functor
The input is a graded ring an integer n. The output is an functor from the category of lp's over S to the category of vector spaces over the Coefficient field of S that sends each M to the vector space of M_n.
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