‣ TwistedOmegaModule( A, i ) | ( operation ) |
Returns: graded lp
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 ) |
Returns: graded lp
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 \(\mathcal{O}_{\mathbb{P}^m}(i)\).
‣ TwistedCotangentModule( S, i ) | ( operation ) |
Returns: graded lp
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 \(\Omega^i_{\mathbb{P}^m}(i)\).
‣ 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 \(\Omega^i(i)\).
‣ 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 \(\Omega^i(i)\).
‣ 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 \(\psi_{i-1,i}^{\ell_1} \psi_{i,i+1}^{\ell_2}= \psi_{i-1,i}^{\ell_2} \psi_{i,i+1}^{\ell_1}\).
‣ 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 \(\varphi_{i+1,i}^{\ell_1} \varphi_{i,i-1}^{\ell_2}= -\varphi_{i+1,i}^{\ell_2} \varphi_{i,i-1}^{\ell_1}\).
‣ 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 \(M\).
‣ 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 \(\phi\).
‣ 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 \(M\).
‣ 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.
‣ 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 \(0\).
‣ 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 \(M\).
‣ 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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