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Algorithms for graded injective resolutions and local cohomology over semigroup rings [article]

David Helm, Ezra Miller
2003 arXiv   pre-print
Let Q be an affine semigroup generating Z^d, and fix a finitely generated Z^d-graded module M over the semigroup algebra k[Q] for a field k.  ...  We provide an algorithm to compute a minimal Z^d-graded injective resolution of M up to any desired cohomological degree.  ...  We wish to thank Gennady Lyubeznik, who motivated us to make our previously abstract methods algorithmic. Both authors were partially supported by the National Science Foundation.  ... 
arXiv:math/0309256v1 fatcat:uhizw2f23bax7py2jccyzpuanu

Algorithms for graded injective resolutions and local cohomology over semigroup rings

David Helm, Ezra Miller
2005 Journal of symbolic computation  
Let Q be an affine semigroup generating Z d , and fix a finitely generated Z d -graded module M over the semigroup algebra k[Q] for a field k.  ...  We provide an algorithm to compute a minimal Z d -graded injective resolution of M up to any desired cohomological degree.  ...  Acknowledgments We wish to thank Gennady Lyubeznik, who motivated us to make our previously abstract methods algorithmic. Both authors were partially supported by the National Science Foundation.  ... 
doi:10.1016/j.jsc.2004.11.009 fatcat:bswrxgbwj5dvzp7j6nxb53oox4

Cohen-Macaulay quotients of normal semigroup rings via irreducible resolutions [article]

Ezra Miller
2001 arXiv   pre-print
Every quotient R/I of a semigroup ring R by a radical monomial ideal I has a unique minimal injective-like resolution by direct sums of quotients of R modulo prime monomial ideals.  ...  The quotient R/I is Cohen-Macaulay if and only if every summand in cohomological degree i has dimension exactly dim(R/I) - i.  ...  over semigroup rings.  ... 
arXiv:math/0110096v1 fatcat:kgnpki6375hprdblhjlgelco7e

Cohen–Macaulay quotients of normal semigroup rings via irreducible resolutions

Ezra Miller
2002 Mathematical Research Letters  
For a radical monomial ideal I in a normal semigroup ring k [Q], there is a unique minimal irreducible resolution 0 sum of quotients of k[Q] by prime ideals.  ...  The proof exploits a graded ring-theoretic analogue of the Zeeman spectral sequence [Zee63], thereby also providing a combinatorial topological version involving no commutative algebra.  ...  The author is indebted to John Greenlees for directing him to Zeeman's work, and Vic Reiner for inspiring discussions. Kohji Yanagawa kindly pointed out Remarks 2.5 and 4.5.  ... 
doi:10.4310/mrl.2002.v9.n1.a9 fatcat:mm53n5j73vb6zbzn4lqlx7g564

Master index to volumes 31–40

1986 Journal of Pure and Applied Algebra  
Heinzer, Finitely generated intermediate rings 37 (1985) Giuli, E., Pro-reflective subcategories 33 (1984) Golan, J.S., Minimal injective resolutions and Bass invariants of modules over left stable left  ...  ) 167-179 Jacobsson, C., Finitely presented graded Lie algebras and homomorphisms of local rings 38 (1985) 243-253 Jacobsson, C., A.R.  ... 
doi:10.1016/0022-4049(86)90052-6 fatcat:y7ythwmtgfg65eaazmihxkgmfe

Contents

2005 Journal of symbolic computation  
, E. and Koiran, P. , Quantum automata and algebraic groups 357 Helm, D. and Miller, E., Algorithms for graded injective resolutions and local cohomology over semigroup rings . . . . . . . . . . . . .  ...  373 Mourrain, B., Bezoutian and quotient ring structure . . . . . . . . 397 Granger, M., Oaku, T. and Takayama Dabbaghian-Abdoly, V., An algorithm for constructing representations of finite groups  ... 
doi:10.1016/s0747-7171(05)00160-4 fatcat:a3hcsoc2orad3fh2g2mvgsxm6q

Notes on C-graded modules over an affine semigroup ring K[C] [article]

Kohji Yanagawa
2006 arXiv   pre-print
Let C ⊂ N^d be an affine semigroup, and R=K[C] its semigroup ring. This paper is a collection of various results on "C-graded" R-modules, especially, monomial ideals.  ...  For example, we show the following: If R is normal and I is a radical monomial ideal (i.e., R/I is a generalization of Stanley-Reisner rings), then the sequentially Cohen-Macaulay property of R/I is a  ...  Acknowledgments The author is grateful to Professor Ngo Viet Trung for telling him the ring given in Example 2.5 (2) .  ... 
arXiv:math/0506457v2 fatcat:4ktd7lu625cqtegmffaxxamsfu

Local cohomology – an invitation [article]

Uli Walther, Wenliang Zhang
2021 arXiv   pre-print
This article is part introduction and part survey to the mathematical area centered around local cohomology.  ...  For homological LOCAL COHOMOLOGY 3 algebra one should know about injective and projective resolutions, Ext and Tor and the principles of  ...  For example, over a ring R of polynomials over C, a local cohomology module HIk pRq is a finite R-module precisely if I “ 0 and k “ 0, or if HIk pRq “ 0.  ... 
arXiv:2106.09796v2 fatcat:uyufd45vobferfub3bhnud644q

Noncommutative Rational Double Points

Daniel Chan
2000 Journal of Algebra  
We prove a version of local duality for such rings, as well as many other classical results from the commutative theory of local cohomology.  ...  The first part is a study of dualising complexes for noncommutative complete local rings.  ...  Let (A, m) be a complete local ring over a field k and (A, m') be its opposite ring.  ... 
doi:10.1006/jabr.2000.8419 fatcat:ught5ekt6raytiah3ie6pv7ndy

Index

1999 Journal of Pure and Applied Algebra  
Lubimcev 133 (1998) 203}208 Vitale, E.M., Localizations of algebraic categories II 133 (1998) 317}326 Walther, U., Algorithmic computation of local cohomology modules and the local cohomological dimension  ...  Dlab, and E. LukaH cs, Well-"ltered algebras 133 (1998) 3} 21 Ajitabh, K., S.P. Smith and J.J. Zhang, Injective resolutions of some regular rings 140 (1999) 1} 21 Ameur, M., S. Benayadi, C.  ...  Martins, Ma.I.R. and J.A. de la Pen a, Comparing the simplicial and the Hochschild cohomologies of a "nite dimensional algebra 138 (1999) (1998)  ... 
doi:10.1016/s0022-4049(99)00087-0 fatcat:jkkwxvnj4zhhvie6yseiuok4im

Master Author Index Volumes 141–160

2001 Journal of Pure and Applied Algebra  
Sharp, On annihilators and associated primes of local cohomology modules 153 (2000) 197}227 Browder, W. and Pakianathan, J., Lifting Lie algebras over the residue "eld of a discrete valuation ring  ...  , pure injectives, and a substitute for projective resolutions 144 (1999) 169}211 Szafraniec, Z., Topological degree and quadratic forms 141 (1999) 299}314 Szemberg, T., see Th.  ... 
doi:10.1016/s0022-4049(01)00057-3 fatcat:ezgzzcilqndjvcyjswky2nfv5u

A survey on the Lyubeznik numbers [article]

Luis Núñez-Betancourt, Emily E. Witt, Wenliang Zhang
2014 arXiv   pre-print
The Lyubeznik numbers are invariants of a local ring containing a field that capture ring-theoretic properties, but also have numerous connections to geometry and topology.  ...  We discuss basic properties of these integer-valued invariants, as well as describe some significant results and recent developments (including certain generalizations) in the area.  ...  We thank the referee for a careful reading of this paper, and for several comments and suggestions that improved it.  ... 
arXiv:1406.7590v1 fatcat:muqprczjjrdwbjexq5im2mse7m

Recent developments around partial actions [article]

Mikhailo Dokuchaev
2018 arXiv   pre-print
We give an overview of publications on partial actions and related concepts, paying main attention to some recent developments.  ...  Acknowledgements The author thanks Fernando Abadie and Mykola Khrypchenko for many useful comments.  ...  Birget [56] applied the partial action definition of Thompson's groups to study algorithmic problems for them.  ... 
arXiv:1801.09105v2 fatcat:v4tcurgvfzaqvljzysb7vz5zem

On liaison, arithmetical Buchsbaum curves and monomial curves in P3

Henrik Bresinsky, Peter Schenzel, Wolfgang Vogel
1984 Journal of Algebra  
ACKNOWLEDGMENTS We would like to thank David Eisenbud for the stimulating discussions that we have had during the preparation of this paper.  ...  We get from Proposition 3.1 of [ 191 that Z(C)/(fi ,f2,f,) differs from the first local cohomology group H:(A) only by grading and duality, where A is the local ring of the (afftne) cone over C at the  ...  Goto [ 111 computes the first local cohomology module H\ of such a Buchsbaum ring in terms of the underlying semigroup. But his methods do not provide a proof for Theorem 3.  ... 
doi:10.1016/0021-8693(84)90034-6 fatcat:mg6cbhkpi5e33ijntgem4d7pya

Tate resolutions on toric varieties [article]

Michael K. Brown, Daniel Erman
2022 arXiv   pre-print
We also develop a Beilinson-type resolution of the diagonal for toric varieties.  ...  We develop an analogue of Eisenbud-Floystad-Schreyer's Tate resolutions for toric varieties.  ...  Many of the ideas in this paper had origins in joint conversations with David Eisenbud and Frank-Olaf Schreyer, and we are tremendously grateful for their generous insights.  ... 
arXiv:2108.03345v2 fatcat:37hjmeutijf5xbjdwoe3cxffcm
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