‣ 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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