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On the canonical map to the local cohomology of a Stanley-Reisner ring. (English) Zbl 0741.55013
This paper contains two parts. In the first one, the author gives an example of a Stanley-Reisner ring \(k[\Delta]\) whose defining ideal is generated by monomials of degree at most 2 and such that the maps \[ \to F_{i+1}\to F_ i\to F_{i-1}\to\cdots\to F_ 0\to k[\Delta]\to 0, \] in a minimal free resolution of \(k[\Delta]\) as a module over a polynomial algebra \(A\), are not represented by matrices with all components of degree at most 2. (A counterexample to a question of Watanabe).
In the second part the author proves that the canonical map \(\hbox{Ext}^ i_ A(k,k[\Delta])\to H^ i_ m(k[\Delta])\) corresponds to the map induced by the inclusion map of certain subcomplexes of \(\Delta\).

MSC:
55U10 Simplicial sets and complexes in algebraic topology
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
14B15 Local cohomology and algebraic geometry
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