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### 1 Operations and attributes

#### 1.1 Attributes

##### 1.1-1 CastelnuovoMumfordRegularity
 ‣ 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 $$M$$.

##### 1.1-2 TateResolution
 ‣ 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.

##### 1.1-3 TateResolution
 ‣ 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.

##### 1.1-4 RCochainFunctor
 ‣ 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).

##### 1.1-5 RChainFunctor
 ‣ 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).

##### 1.1-6 LCochainFunctor
 ‣ 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 $$S$$.

##### 1.1-7 LChainFunctor
 ‣ 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 $$S$$.

##### 1.1-8 TruncationFunctorUsingTateResolution
 ‣ 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 $$f_{\geq n}$$.

##### 1.1-9 NatTransFromTruncationUsingTateResolutionToIdentityFunctor
 ‣ 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.

##### 1.1-10 TruncationFunctorUsingHomalg
 ‣ 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.

##### 1.1-11 NatTransFromTruncationUsingHomalgToIdentityFunctor
 ‣ 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.

##### 1.1-12 NatTransFromTruncationUsingTateResolutionToTruncationFunctorUsingHomalg
 ‣ 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).

##### 1.1-13 GLPGeneratedByHomogeneousPartFunctor
 ‣ 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.

##### 1.1-14 NatTransFromGLPGeneratedByHomogeneousPartToIdentityFunctor
 ‣ 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.

##### 1.1-15 HomogeneousPartOverCoefficientsRingFunctor
 ‣ 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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