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undocumented { (truncate, InfiniteNumber, Thing), [(truncate, InfiniteNumber, Thing), MinimalGenerators], (truncate, InfiniteNumber, InfiniteNumber, Matrix), [(truncate, InfiniteNumber, InfiniteNumber, Matrix), MinimalGenerators], (truncate, Nothing, InfiniteNumber, Matrix), [(truncate, Nothing, InfiniteNumber, Matrix), MinimalGenerators], -- TODO: (truncate, Nothing, List, Matrix), [(truncate, Nothing, List, Matrix), MinimalGenerators], (truncate, Nothing, ZZ, Matrix), [(truncate, Nothing, ZZ, Matrix), MinimalGenerators], (truncate, List, List, Matrix), [(truncate, List, List, Matrix), MinimalGenerators], (truncate, ZZ, ZZ, Matrix), [(truncate, ZZ, ZZ, Matrix), MinimalGenerators], effCone, effGenerators, (effCone, Ring), (effGenerators, Ring), nefCone, nefGenerators, (nefCone, Ring), (nefGenerators, Ring), } doc /// Node Key Truncations Headline truncations of graded ring, ideals and modules Description Text This package provides methods for truncation of a graded ring, or a graded module or ideal over a graded ring (see @TO (truncate, List, Module)@). Truncation is functorial: it can be applied to maps of modules as well, and the truncation of a composition of maps is the composition of the truncations (see @TO (truncate, List, Matrix)@). Let $S$ be a $\ZZ^r$-graded ring whose variables have non-negative degrees and $M$ be a graded $S$-module. Then for a finite subset of degrees $L\subset\ZZ^r$ the method {\tt truncate(L, M)} computes $$M_{\ge L} = \bigoplus_{m\in L+\NN^r} M_m,$$ where the sum is taken over all degrees $m \in \ZZ^r$ which are component-wise greater than or equal to some degree $d\in L$. In this case the truncation is a submodule of $M$. Example R = ZZ/101[a..d, Degrees => {1,2,3,4}]; truncate(4, ideal"a3,b3") Text More generally, let $S$ be the total coordinate ring of a simplicial toric variety $X$ with Picard group $\operatorname{Pic} X$. Then for a finite subset $L\subset\operatorname{Pic} X$, the truncation $F_{\ge L}$ of a free module $F$ may be defined as the submodule generated by $$F_{\ge L} = \bigoplus_{m\in L+\operatorname{Nef} X} F_m,$$ where $\operatorname{Nef} X$ is the semigroup of nef line bundles in $\operatorname{Pic} X$ (c.f. Definition 5.1 in [MS04]). Then for a graded $S$-module $M$ with @TO presentation@ $0 \gets M \gets G \gets H,$ where $G$ and $H$ are free modules, the truncation $M_{\ge L}$ is the $S$-module with presentation $$0 \gets M_{\ge L} \gets G_{\ge L} \gets H_{\ge L}.$$ Note that $M_{\ge L}$ is not a submodule of $M$ in general, but exact sequences are preserved and since $M/M_{\ge L}$ is annihilated by a power of the irrelevant ideal of $S$, a module and its truncation define the same sheaf. Example needsPackage "NormalToricVarieties"; dP6 = smoothFanoToricVariety(2, 4); S = ring dP6; M = S^{-{0,0,1,0}}; N = truncate({0,0,0,0}, M) N == coker truncate({0,0,0,0}, presentation M) Text For the most general case, if $S$ is a $\ZZ^r$-graded ring where the degree components of variables may be negative, the result is the same as the above but we fix the nef cone to be the positive orthant $\NN^r$. Example R = ZZ/101[a..d, Degrees => {2:{1,0},{0,1},{-3,1}}]; M = R^1/ideal d N = truncate({0,0}, M) isSubset(M, N) Text The polyhedral algorithms implemented in this package correctly handle many cases. The behavior of @TO "truncate"@ changed as of Macaulay2 version 1.13 to support @TO "Macaulay2Doc :: exterior algebras"@, and again in Macaulay2 version 1.19 to support Cox rings of simplicial @TO2 {"NormalToricVarieties::NormalToricVarieties", "normal toric varieties"}@. References @UL { {"[MS04]: Maclagan and Smith, Multigraded Castelnuovo-Mumford Regularity (see ", arXiv "math/0305214", ")."} }@ Contributors Lauren Cranton Heller contributed to the code and documentation for this package. Subnodes (truncate, List, Module) (truncate, List, Matrix) SeeAlso basis Node Key (truncate, List, Module) (truncate, List, Ideal) (truncate, List, Ring) (truncate, ZZ, Module) (truncate, ZZ, Ideal) (truncate, ZZ, Ring) [(truncate, List, Module), MinimalGenerators] [(truncate, List, Ideal), MinimalGenerators] [(truncate, List, Ring), MinimalGenerators] [(truncate, ZZ, Module), MinimalGenerators] [(truncate, ZZ, Ideal), MinimalGenerators] [(truncate, ZZ, Ring), MinimalGenerators] Headline truncation of the graded ring, ideal or module at a specified degree or set of degrees Usage truncate(degs, M) Inputs degs:{ZZ,List} a single degree, a multidegree, or a list of degrees or multidegrees M:{Module,Ideal,Ring} MinimalGenerators=>Boolean indicates whether the result should be @TO2 {trim, "trimmed"}@ Outputs :{Module,Ideal} the truncation submodule $M_{\ge degs}$ Description Text The truncation to degree $d$ in the singly graded case of a module (or ring or ideal) is generated by all homogeneous elements of degree at least $d$ in $M$. The resulting truncation is minimally generated (assuming that $M$ is graded). Example R = ZZ/101[a..c]; truncate(2, R) truncate(2, R^1) truncate(2, R^{0,-3}) Text The coefficient ring of $R$ may be $\ZZ$ or another polynomial ring. Over $\ZZ$, the generators may not be minimal, but they do generate. Example A = ZZ[x,y,z]; truncate(2, ideal(3*x, 5*y, 15)) Text If a multi-degree $d$ is given, then the result is the submodule generated by elements of degree $d+\NN\mathcal C$ where $\mathcal C$ is either a generating set for the degree semigroup of $R$ or the Nef cone of the toric variety. The following example finds the 11 generators needed to obtain all graded elements whose degrees are at least $\{7,24\}$. Example S = ZZ/101[x,y,z, Degrees => {{1,3},{1,4},{1,0}}]; trunc = truncate({7,24}, S^1 ++ S^{{-8,-20}}) degrees trunc Text Given a list of multi-degrees $D$, then the result is the submodule generated by elements of degree $d+\NN\mathcal C$ for any $d\in D$. The following example finds the generators needed to obtain all graded elements whose degrees at least $\{3,0\}$ or at least $\{0,1\}$. The resulting module is also minimally generated. Example S = ZZ/101[x,y,z, Degrees => {{1,3},{1,4},{1,0}}]; trunc = truncate({{3,0}, {0,1}}, S^1 ++ S^{{-8,-20}}) degrees trunc Text The coefficient ring may also be a polynomial ring. In this example, the coefficient variables also have degree one. The given generators will generate the truncation over the coefficient ring. Example B = R[x,y,z, Join => false]; degrees B truncate(2, B^1) truncate(4, ideal(b^2*y,x^3)) Text If the coefficient variables have degree 0: Example A1 = ZZ/101[a,b,c, Degrees => {3:{}}]; degrees A1 B1 = A1[x,y]; degrees B1 truncate(2, B1^1) truncate(2, ideal(a^3*x, b*y^2)) Caveat The behavior of this function has changed as of Macaulay2 version 1.13. This is a (potentially) breaking change. Before, it used a less useful notion of truncation, involving the heft vector, and was often not what one wanted in the multi-graded case. Additionally, in the tower ring case, when the coefficient ring had variables of nonzero degree, sometimes incorrect answers resulted. Also, the function expects a graded module, ring, or ideal, but this is not checked, and some answer is returned. SeeAlso basis module comodule Node Key (truncate, List, Matrix) (truncate, ZZ, Matrix) [(truncate, List, Matrix), MinimalGenerators] [(truncate, ZZ, Matrix), MinimalGenerators] Headline truncation of a map of free modules Usage truncate(degs, f) Inputs degs:{ZZ,List} a single degree, a multidegree, or a list of degrees or multidegrees f:Matrix a graded map between graded modules (not necessarily free modules) MinimalGenerators=>Boolean indicates whether the source and target of the result should be @TO2 {trim, "trimmed"}@ Outputs :Matrix a graded map between the truncations of the source and target of $f$ Description Text This function truncates the source and target of a module map, and returns the induced map between them. Example R = ZZ/101[a..d, Degrees => {{1,3},{1,0},{1,3},{1,2}}]; I = ideal "a,b2,c3,d4" C = res I g1 = truncate({1,1}, C.dd_1) g2 = truncate({1,1}, C.dd_2) g3 = truncate({1,1}, C.dd_3) g4 = truncate({1,1}, C.dd_4) needsPackage "Complexes" D = complex {g1, g2, g3, g4} D' = truncate({1,1}, freeResolution I) assert(D == D') Text This functor is exact. Example prune HH D HH_0 D == truncate({1,1}, comodule ideal"a,b2,c3,d4") SeeAlso (truncate, List, Module) /// end-- doc /// Node Key Headline Usage Inputs Outputs Description Text Example Caveat SeeAlso ///