Introduction to Algebraic Geometry and Commutative Algebra

10.1142/7725 ◽  
2010 ◽  
Author(s):  
Dilip P Patil ◽  
Uwe Storch
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra

On Grothendieck's local duality theorem

1979 ◽  
Vol 85 (3) ◽  
pp. 431-437 ◽  
Author(s):  
M. H. Bijan-Zadeh ◽  
R. Y. Sharp
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Noetherian Ring ◽  
Duality Theorem ◽  
Great Emphasis ◽  
Triangulated Category ◽  
Elementary Approach ◽  
Commutative Noetherian Ring ◽  
Local Duality ◽  
Dualizing Complexes

In (11) and (12), a comparatively elementary approach to the use of dualizing complexes in commutative algebra has been developed. Dualizing complexes were introduced by Grothendieck and Hartshorne in (2) for use in algebraic geometry; the approach to dualizing complexes in (11) and (12) differs from that of Grothendieck and Hartshorne in that it avoids use of the concepts of triangulated category, derived category, and localization of categories, and instead places great emphasis on the concept of quasi-isomorphism of complexes of modules over a commutative Noetherian ring.

Cellular reductions of the Pommaret-Seiler resolution for Quasi-stable ideals

2021 ◽  
Vol 55 (3) ◽  
pp. 102-106
Author(s):  
Rodrigo Iglesias ◽  
Eduardo Sáenz de Cabezón
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Combinatorial Properties ◽  
Pommaret Bases ◽  
Involutive Bases

Involutive bases were introduced in [6] as a type of Gröbner bases with additional combinatorial properties. Pommaret bases are a particular kind of involutive bases with strong relations to commutative algebra and algebraic geometry[11, 12].

scholarly journals TROPICAL ALGEBRAIC SETS, IDEALS AND AN ALGEBRAIC NULLSTELLENSATZ

2008 ◽  
Vol 18 (06) ◽  
pp. 1067-1098 ◽  
Author(s):  
ZUR IZHAKIAN
Keyword(s):  
Algebraic Geometry ◽  
Classical Theory ◽  
Commutative Algebra ◽  
Fundamental Theorem ◽  
Polynomial Algebra ◽  
Algebraic Sets ◽  
Tropical Semiring ◽  
Fundamental Theorem Of Algebra

This paper introduces the foundations of the polynomial algebra and basic structures for algebraic geometry over the extended tropical semiring. Our development, which includes the tropical version for the fundamental theorem of algebra, leads to the reduced polynomial semiring — a structure that provides a basis for developing a tropical analogue to the classical theory of commutative algebra. The use of the new notion of tropical algebraic com-sets, built upon the complements of tropical algebraic sets, eventually yields the tropical algebraic Nullstellensatz.

scholarly journals On the cominimaxness of generalized local cohomology modules

2020 ◽  
Vol 23 (1) ◽  
pp. 479-483
Author(s):  
Cam Thi Hong Bui ◽  
Tri Minh Nguyen
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Local Cohomology ◽  
Cohomology Theory ◽  
Negative Integer ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules

The local cohomology theory plays an important role in commutative algebra and algebraic geometry. The I-cofiniteness of local cohomology modules is one of interesting properties which has been studied by many mathematicians. The I-cominimax modules is an extension of I-cofinite modules which was introduced by Hartshorne. An R-module M is I-cominimax if Supp_R(M)\subseteq V(I) and Ext^i_R(R/I,M) is minimax for all  i\ge 0. In this paper, we show some conditions such that the generalized local cohomology module H^i_I(M,N) is I-cominimax for all i\ge 0. We show that if H^i_I(M,K) is I-cofinite for all i<t and all finitely generated R-module K, then  H^i_I(M,N) is I-cominimax for all i<t  and all minimax R-module N.  If M is a finitely generated R-module, N is a minimax R-module and t is a non-negative integer such that  dim Supp_R(H^i_I(M,N))\le 1 for all i<t then H^i_I(M,N) is I-cominimax for all  i<t. When  dim R/I\le 1 and H^i_I(N) is I-cominimax for all  i\ge 0 then H^i_I(M,N) is I-cominimax for all i\ge 0.

Pullback of the Normal Module of Ideals with Low Codimension

2021 ◽  
Author(s):  
Cleto B Miranda-Neto
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Difficult Problem ◽  
Natural Projection ◽  
Analytic Spread ◽  
Reduction Number ◽  
Numerical Invariants ◽  
Special Classes

Abstract The normal module (or sheaf) of an ideal is a celebrated object in commutative algebra and algebraic geometry. In this paper, we prove results about its pullback under the natural projection, focusing on subtle numerical invariants such as, for instance, the reduction number. For certain codimension 2 perfect ideals, we show that the pullback has reduction number two. This is of interest since the determination of this invariant in the context of modules (even for special classes) is a mostly open, difficult problem. The analytic spread is also computed. Finally, for codimension 3 Gorenstein ideals, we determine the depth of the pullback, and we also consider a broader class of ideals provided that the Auslander transpose of the conormal module is almost Cohen–Macaulay.

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