SEMINAR ON COMMUTATIVE ALGEBRA AND RELATED TOPICS | Year:2012 | دوره:9 |Papers Count:19

LET G BE A GROUP, G’ BE A SUBGROUP OF G OF FINITE INDEX AND R BE AN ARBITRARY RING WITH IDENTITY. LET M BE AN RG-MODULE. IT IS WELL KNOWN (AND EASY TO SEE) THAT IF M IS PROJECTIVE OVER RG, THEN IT IS PROJECTIVE AS A MODULE OVER THE SUBRING RG’ (IN FACT THIS IS TRUE FOR ARBITRARY SUBGROUP G’ OF G). IN ADDITION, IT IS NOT DIFFICULT TO CONSTRUCT EXAMPLES WHICH SHOW THAT THE CONVERSE IS NOT TRUE IN GENERAL. IN THIS DIRECTION, IN 1976 J. MOORE POSED THE FOLLOWING CONJECTURE...

LET I DENOTE AN IDEAL OF A NOETHERIAN RING R. THE PURPOSE OF THIS TALK IS TO INTRODUCE THE CONCEPTS OF QUINTASYMPTOTIC SEQUENCES OVER I AND QUINTASYMPTOTIC COGRADE OF I, AND IT IS SHOWN THAT THEY PLAY A ROLE ANALOGOUS TO QUINTESSENTIAL SEQUENCES OVER I AND QUINTESSENTIAL COGRADE OF I. ALSO, WE SHOW THAT, IF R IS LOCAL, THEN THE QUINTASYMPTOTIC COGRADE OF I IS UNAMBIGUOUSLY DEFINED AND BEHAVES WELL WHEN PASSING TO CERTAIN RELATED LOCAL RINGS. FINALLY, WE USE THIS COGRADE TO CHARACTERIZE TWO CLASSES OF LOCAL RINGS.

E USE THE CORRESPONDING NUMERICAL SEMI GROUP TO AN AFFINE MONOMIAL CURVE TO STUDY ARITHMETIC PROPERTIES AND INVARIANTS OF IT’S COORDINATE RING AND TANGENT CONE. AS A MAIN TOOL IN THIS TALK, WE PRESENT THE APRY TABLE AND SHOW HOW TO READ FROM IT MANY PROPERTIES OF THE TANGENT CONE, IN PARTICULAR THE COHEN-MACAULAY, K-BUCHSBAUM AND GORENSTEIN PROPERTIES, OR THE HILBERT FUNCTION AND IT'S BEHAVIOR...

FOR A GRADED IDEAL I IN THE POLYNOMIAL RING (FORMULA) IN N VARIABLES OVER THE FIELD K THE RING S/I IS CALLED GOLOD IF ALL MASSEY OPERATIONS ON THE KOSZUL COMPLEX OF S/I WITH RESPECT TO (FORMULA)...

LET A BE AN IDEAL OF A LOCAL RING (R, M) WITH D=DIM R: THIS TALK IS TWO FOLDED: OF PARTICULAR INTEREST ARE THE SO-CALLED TOP LOCAL COHOMOLOGY MODULES, THAT IS...

IN THIS TALK WE REVIEW SOME RECENT RESULTS ABOUT CLEAN AND PRETTY CLEAN MODULES AND THEIR APPLICATIONS...

LET (R, N) BE A REGULAR LOCAL RING AND I BE AN IDEAL OF R.SUPPOSING THAT (A, M)=(R/I, N/I) IS AN ARTINIAN LOCAL RING, WE SAY THAT A IS AN EXTENDED STRETCHED ARTINIAN LOCAL RING IF (FORMULA)...

IN THIS TALK, WE INTRODUCE SOME NEW CONCEPTS SUCH AS S-GROBNER BASES AND PRIMTIVE S-IRREDUCIBLE POLYNOMIALS AND WE PRESENT A NEW VERSION OF THE F5 CRITERION AND DEVELOP A NEW ALGORITHM. THE NEW CONCEPT ALSO ENABLE US TO REMOVE VARIOUS RESTRICTIONS. AT THE END, WE GIVE SOME EXAMPLES TO COMPARE THE NEW VERSION OF F5 ALGORITHM TO THE OLD ONE.

LET (FORMULA), WHERE K IS A FIELD. FIX AN INTEGER T SUCH THAT (FORMULA).ASSOCIATED TO A DIRECTED GRAPH G IS A MONOMIAL IDEAL, CALLED THE PATH IDEAL (OF LENGTH T), AND DENOTED BY IT (G), WHOSE GENERATORS CORRESPOND TO THE DIRECTED PATHS OF LENGTH TIN G. LET G BE A DIRECTED ROOTED TREE...

LET (FORMULA) BE A STANDARD GRADED RING, (FORMULA) BE THE IRRELEVANT IDEAL OF RAND M BE A NITELY GENERATED GRADED R-MODULE...

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SOLVING SYSTEMS OF MULTIVARIABLE POLYNOMIAL EQUATIONS IS A BASIC PROBLEM IN COMPUTER ALGEBRA WHICH CAN BE SOLVED USING GROBNER BASIS EASILY. ONE OF THE FAMOUS ALGORITHMS FOR COMPUTING GROBNER BASIS IS F5 ALGORITHM WHICH IS A BIT DIFFICULT TO UNDERSTAND AND IMPLEMENT...

WE SHALL EXAMINE THE AVAILABILITY OF A NOTION OF DIMENSION IN THE SPACE OF ALGEBRAIC CYCLES ON AN ALGEBRAIC VARIETY (AFTER SEVERI, MUMFORD, ...). THIS WILL LEAD US TO RETHINK THE NOTION (AFTER KIMURA, O' SULLIVAN, ...) WHEN WORKING WITH THE MORE COMPLICATED SPACES OF MOTIVES ASSOCIATED TO ALGEBRAIC VARIETIES.

LET (FORMULA) BE A REDUCED POLYNOMIAL OF TOTAL DEGREE AT LEAST TWO AND THE IDEAL JF STANDS FOR JACOBIAN IDEAL OF F WHICH DE NES THE SINGULAR SUBSCHEME OF THE HYPERSURFACE DE NED BY F...

LET R BE A COMMUTATIVE NOETHERIAN RING OF PRIME CHARACTERISTIC AND M BE AN X-DIVISIBLE RIGHT R[X, F]-MODULE THAT IS NOETHERIAN AS R-MODULE...

LET V BE A HENSELIAN VALUATION OF ANY RANK OF AFIELD K, AND LET`V BE THE UNIQUE EXTENSION OF V TO A FIXED ALGEBRAIC CLOSURE`K OF K WITH VALUE GROUP`G...

A PROPER SUBRING S OF A RING R IS CALLED MAXIMAL SUBRING IF THERE IS NO SUBRINGS PROPERLY BETWEEN S AND R. IN THIS NOTE WE STUDY THE EXISTENCE OF MAXIMAL SUBRINGS IN COMMUTATIVE RINGS.

LET (R, M) BE A COMMUTATIVE NOETHERIAN LOCAL RING WITH IDENTITY AND I BE AN IDEAL OF R. ASSUME THAT M IS A NONZERO MINIMAX R-MODULE SUCH THAT DIM SUPP HII (M)£1 FOR ALL I...

IN THIS TALK WE EMPLOY THE NOTION OF A QUASI-PRIME SUB MODULE OF AN R-MODULE TO INTRODUCE A NEW CLASS OF MODULES AS STRONGLY TOP MODULES. WE PRESENT SOME ALGEBRAIC PROPERTIES OF THIS MODULES AND PUT A TOPOLOGY ON THE SET OF ALL ITS QUASI-PRIME SUB MODULES, AS QUASI-ZARISKI TOPOLOGY. WE ALSO INVESTIGATE THE RELATIONSHIP BETWEEN ALGEBRAIC PROPERTIES OF STRONGLY TOP MODULES AND TOPOLOGICAL PROPERTIES OF QUASI-ZARISKI TOPOLOGY.