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### 2 Beilinson Monads and $\mathrm{Coh}(\mathbb{P}^m)$

#### 2.1 Operations

##### 2.1-1 TwistedOmegaModule
 ‣ TwistedOmegaModule( A, i ) ( operation )

The input is a graded exterior algebra $$A$$ and an integer $$i$$. The output is the graded $$A$$-lp $$\omega_A(i)$$.

 ‣ TwistedGradedFreeModule( S, i ) ( operation )

The input is a graded polynomial ring $$S$$ and an integer $$i$$. The output is the graded $$S$$-lp $$S(i)$$. The sheafification of $$S(i)$$ is the structure sheaf $$\mathcal{O}_{\mathbb{P}^m}(i)$$.

##### 2.1-3 TwistedCotangentModule
 ‣ TwistedCotangentModule( S, i ) ( operation )

The input is a graded polynomial ring $$S$$ and an integer $$i$$. The output is the graded $$S$$-lp $$\Omega^i(i)$$. The sheafification of $$\Omega^i(i)$$ is the twisted cotangent sheaf $$\Omega^i_{\mathbb{P}^m}(i)$$.

##### 2.1-4 TwistedCotangentModuleAsChain
 ‣ TwistedCotangentModuleAsChain( S, i ) ( operation )

Returns: chain complex

The input is a graded polynomial ring $$S$$ and an integer $$i$$. The output is the chain complex of $$S$$-lp's whose objects are direct sums of twists of $$S$$ and its homology at $$0$$ is $$\Omega^i(i)$$. I.e., a chain complex that is quasi-isomorphic to $$\Omega^i(i)$$.

##### 2.1-5 TwistedCotangentModuleAsCochain
 ‣ TwistedCotangentModuleAsCochain( S, i ) ( operation )

Returns: cochain complex

The input is a graded polynomial ring $$S$$ and an integer $$i$$. The output is the chain complex of $$S$$-lp's whose objects are direct sums of twists of $$S$$ and its homology at $$0$$ is $$\Omega^i(i)$$. I.e., a chain complex that is quasi-isomorphic to $$\Omega^i(i)$$.

##### 2.1-6 BasisBetweenTwistedOmegaModules
 ‣ BasisBetweenTwistedOmegaModules( A, i, j ) ( operation )

Returns: a list

The input is a graded exterior ring A := KoszulDualRing(S) with $$S:=k[x_0,\dots,x_m]$$ and two integers $$i,j$$ with $$i\geq j$$. The output is a basis of the external hom: $$\mathrm{Hom}_A(\omega_A(i), \omega_A(j))$$. If we denote the indeterminates of $$A$$ by $$e_0,\dots, e_m$$ and the $$\ell$$'th entry of the output by $$\sigma_{ij}^{\ell}$$ then we have $$\sigma_{i+1,i}^{\ell}=e_{\ell-1}$$, thus $$\sigma_{i+1,i}^{\ell_1} \sigma_{i,i-1}^{\ell_2}= -\sigma_{i+1,i}^{\ell_2} \sigma_{i,i-1}^{\ell_1}$$.

 ‣ BasisBetweenTwistedGradedFreeModules( S, i, j ) ( operation )

Returns: a list

The input is a graded polynomial ring $$S=k[x_0, \dots, x_m]$$ and two integers $$i,j$$ with $$i\leq j$$. The output is a basis of the external hom: $$\mathrm{Hom}_S(S(i), S(j))$$. If we denote the $$\ell$$'th entry of the output by $$\psi_{ij}^{\ell}$$ then we have $$\psi_{i-1,i}^{\ell}=x_{\ell-1}$$, thus $$\psi_{i-1,i}^{\ell_1} \psi_{i,i+1}^{\ell_2}= \psi_{i-1,i}^{\ell_2} \psi_{i,i+1}^{\ell_1}$$.

##### 2.1-8 BasisBetweenTwistedCotangentModulesAsGLP
 ‣ BasisBetweenTwistedCotangentModulesAsGLP( S, i, j ) ( operation )

Returns: a list

The input is a graded polynomial ring $$S=k[x_0, \dots, x_m]$$ and two integers $$0\leq j \leq i \leq m$$. The output is a basis of the external hom: $$\mathrm{Hom}_S(\Omega^i(i), \Omega^i(j))$$. If we denote the $$\ell$$'th entry of the output by $$\varphi_{ij}^{\ell}$$ then we have $$\varphi_{i+1,i}^{\ell} \varphi_{i,i-1}^{\ell}=0$$ and $$\varphi_{i+1,i}^{\ell_1} \varphi_{i,i-1}^{\ell_2}= -\varphi_{i+1,i}^{\ell_2} \varphi_{i,i-1}^{\ell_1}$$.

##### 2.1-9 BeilinsonReplacement
 ‣ BeilinsonReplacement( M ) ( attribute )

Returns: a chain or cochain complex

The input is graded $$S$$-lp, graded $$A$$-lp, chain complex or cochain complex of $$S$$-lp's. The output is a Beilinson monad of $$M$$.

##### 2.1-10 BeilinsonReplacement
 ‣ BeilinsonReplacement( phi ) ( attribute )

Returns: a chain or cochain complex

The input is graded $$S$$-lp morphism, graded $$A$$-lp morphism, chain morphism or cochain morphism of $$S$$-lp's. The output is a Beilinson monad morphism of $$\phi$$.

##### 2.1-11 MorphismFromGLPToZerothObjectOfBeilinsonReplacement
 ‣ MorphismFromGLPToZerothObjectOfBeilinsonReplacement( M ) ( attribute )

Returns: a morphism

The input is graded $$S$$-lp $$M$$. The output is a morphism from the sheafification of $$M$$ to the sheafification of the $$0$$'th object of its Beilinson replacement. This morphism induces a quasi-isomorphism from sheafification of $$M$$ considered as a chain complex concentrated in degree $$0$$ to the sheafification of the Beilinson replacement of $$M$$.

##### 2.1-12 MorphismFromGLPToZerothHomologyOfBeilinsonReplacement
 ‣ MorphismFromGLPToZerothHomologyOfBeilinsonReplacement( M ) ( attribute )

Returns: a morphism

The input is graded $$S$$-lp $$M$$. The output is an isomorphism from the Sheafification of $$M$$ to the sheafification of the $$0$$'th homology of its Beilinson replacement.

##### 2.1-13 MorphismFromZerothObjectOfBeilinsonReplacementToGLP
 ‣ MorphismFromZerothObjectOfBeilinsonReplacementToGLP( M ) ( attribute )

Returns: a morphism

The input is graded $$S$$-lp $$M$$. The output is a morphism from the sheafification of the $$0$$'th object of the Beilinson replacement of $$M$$ to the sheafification of $$M$$. This morphism induces a quasi-isomorphism from the sheafification of the Beilinson replacement of $$M$$ to sheafification of $$M$$ considered as a chain complex concentrated in degree $$0$$.

##### 2.1-14 MorphismFromZerothHomologyOfBeilinsonReplacementToGLP
 ‣ MorphismFromZerothHomologyOfBeilinsonReplacementToGLP( M ) ( attribute )

Returns: a morphism

The input is graded $$S$$-lp $$M$$. The output is an isomorphism from the the sheafification of the $$0$$'th homology of its Beilinson replacement to Sheafification of $$M$$.

##### 2.1-15 ShowMatrix
 ‣ ShowMatrix( M ) ( function )

Returns: nothing

The inpute is a homalg matrix or anything that has attribute UnderlyingMatrix. It views the entries using the browse package. To quit: q+y.

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