Flat extension and phantom homology
Algebra and discrete mathematics, Tome 24 (2017) no. 1, pp. 90-98.

Voir la notice de l'article dans Math-Net.Ru

Phantom homology arises in tight closure theory due to small non-exactness when ‘kernel’ is not equal to ‘image’ but ‘kernel’ is in the tight closure of the ‘image’. In this paper we study a typical flat extension, which we call $*$-flat extension, such that upon tensoring which preserves phantom homology. Along with other properties, we observe that $*$-flat extension preserves ghost regular sequence, which is a typical ‘tight closure’ generalization of regular sequence. We also show that in some situations, under $*$-flat extension, test ideal of the $*$-flat algebra is the expansion of the test ideal of the base ring.
Mots-clés : tight closure, phantom homology. 
@article{ADM_2017_24_1_a5,
     author = {Rajsekhar Bhattacharyya},
     title = {Flat extension and phantom homology},
     journal = {Algebra and discrete mathematics},
     pages = {90--98},
     publisher = {mathdoc},
     volume = {24},
     number = {1},
     year = {2017},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/item/ADM_2017_24_1_a5/}
}
TY  - JOUR
AU  - Rajsekhar Bhattacharyya
TI  - Flat extension and phantom homology
JO  - Algebra and discrete mathematics
PY  - 2017
SP  - 90
EP  - 98
VL  - 24
IS  - 1
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/item/ADM_2017_24_1_a5/
LA  - en
ID  - ADM_2017_24_1_a5
ER  - 
%0 Journal Article
%A Rajsekhar Bhattacharyya
%T Flat extension and phantom homology
%J Algebra and discrete mathematics
%D 2017
%P 90-98
%V 24
%N 1
%I mathdoc
%U https://geodesic-test.mathdoc.fr/item/ADM_2017_24_1_a5/
%G en
%F ADM_2017_24_1_a5
Rajsekhar Bhattacharyya. Flat extension and phantom homology. Algebra and discrete mathematics, Tome 24 (2017) no. 1, pp. 90-98. https://geodesic-test.mathdoc.fr/item/ADM_2017_24_1_a5/

[1] Ana Bravo and Karen E. Smith, “Behavior of test ideals under smooth and etale homomorphism”, Journal of Algebra, 247 (2002), 78–94 | DOI | MR

[2] Winfried Bruns and Jurgen Herzog, Cohen-Macaulay rings, Cambridge University Press, 1997 | MR

[3] Neil Epstein, “Phantom depth and stable phantom exactness”, Trans. Amer. Math. Soc., 359 (2007), 4829–4864, arXiv: 0505235[math.AC] | DOI | MR | Zbl

[4] Melvin Hochster, “Cyclic purity versus purity in escellent Noetherian local rings”, Trans. Amer. Math. Soc., 231 (1977), 463–488 | DOI | MR | Zbl

[5] Melvin Hochster and Craig Huneke, “Tight closure, invariant theory, and the Briançon-Skoda theorem”, J. Amer. Math. Soc., 3:1 (1990), 31–116 | MR | Zbl

[6] Melvin Hochster and Craig Huneke, “$F$-regularity, test elements, and smooth base change”, Trans. Amer. Math. Soc., 346:1 (1994), 1–62 | MR | Zbl

[7] Melvin Hochster and Craig Huneke, “Tight closure of parameter ideals and splitting in module-finite extensions”, J. Algebraic Geom., 3:4 (1994), 599–670 | MR | Zbl

[8] Craig Huneke, Tight closure and its applications, With an appendix by Melvin Hochster, CBMS Reg. Conf. Ser. in Math., 88, Amer. Math. Soc., Providence, RI, 1996 | MR | Zbl

[9] Hideyuki Matsumara, Commutative algebra, Advanced book Program, The Bejamin Cummings Publishing Company inc., Reading, Massachusetts, 1980 | MR | Zbl

[10] Hideyuki Matsumara, Commutative Ring Theory, Cambridge University Press, 1990 | MR